Research Article Heat and Mass Transfer on Squeezing ...
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Research Article Heat and Mass Transfer on Squeezing Unsteady MHD Nanofluid Flow between Parallel Plates with Slip Velocity Effect Khilap Singh, Sawan K. Rawat, and Manoj Kumar Department of Mathematics, Statistics and Computer Science, G. B. Pant University of Agriculture and Technology, Pantnagar, Uttarakhand, India Correspondence should be addressed to Khilap Singh; [email protected] Received 30 June 2016; Revised 17 October 2016; Accepted 24 October 2016 Academic Editor: In Cheol Bang Copyright © 2016 Khilap Singh et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Heat and mass transfer behavior of unsteady flow of squeezing nanofluids between two parallel plates in the sight of uniform magnetic field with slip velocity effect is investigated. e governing equations representing fluid flow have been transformed into nonlinear ordinary differential equations using similarity transformation. e equations thus obtained have been solved numerically using Runge-Kutta-Fehlberg method with shooting technique. Effects on the behavior of velocity, temperature, and concentration for various values of relevant parameters are illustrated graphically. e skin-friction coefficient and heat and mass transfer rate are also tabulated for various governing parameters. e results indicate that, for nanofluid flow, the rates of heat and mass transfer are inversely proportional to nanoparticle volume fraction and magnetic parameter. e rate of mass transfer increases with increasing values of Schmidt number and squeeze number. 1. Introduction e study of heat and mass transfer in unsteady squeezing viscous nanofluid flow between two parallel plates is a stim- ulating topic of exploration because of its industrial use and intense biological situations, some of which include process- ing of polymer, compression, power transmitting, lubricant system, transient loading of mechanical components and the squeezed films in power transmission, food processing, and cooling water, modeling of synthetics transportation inside living bodies, hydromechanical machinery, chemical processing equipment, and crop destruction due to freezing. e first work on the squeezing flow under lubrication approximation was studied by Stefan [1]. e flow analysis between two parallel plates of Cu-water squeezing nanofluid was investigated by Domairry and Hatami [2]. Pourmehran et al. [3] studied the unsteady flow of squeezing nanofluid between parallel plates. Gupta and Ray [4] proposed a problem of unsteady flow of a squeezing nanofluid between two parallel plates. e squeezing flow of Cu-water (or kerosene) nanofluid between two parallel plates under the effects of viscous dissipation and velocity slip was investigated by Khan et al. [5]. Dib et al. [6] obtained an approximate analytic solution of squeezing unsteady nanofluid flow. e word nanofluid represents the fluid in which particles of size with order of nanometer (diameter < 100 nm) are mixed in the base fluid. e nanoparticles used in nanofluids are generally made of metals (Al, Cu), oxides (Al 2 O 3 , CuO, TiO 2 , and SiO 2 ), carbides (SiC), nitrides (AlN, SiN), and nonmetal (graphite, carbon nanotubes) and the base fluid is usually a conductive fluid, such as water or ethylene glycol. Other base fluids are toluene, oil, other lubricants, biofluids, and polymer solution. Nanoparticles are present up to 5% volume fraction in nanofluids. e conventional heat transfer fluids are poor conductors of heat. Nanofluids make an edge over them because they have high heat transfer capability. Since these heating/cooling fluids play a vital role in the development of energy efficient heat transfer equipment for energy supply, to raise the thermal conductivity of these flu- ids, nanosized conducting metal particles are added to them. erefore, their proper understanding is a must to use them efficiently in modern industry. Applications of nanofluids include microelectronics, fuel cells, and pharmaceutical pro- cesses. Choi and Eastman [7] were the first to propose the Hindawi Publishing Corporation Journal of Nanoscience Volume 2016, Article ID 9708562, 11 pages http://dx.doi.org/10.1155/2016/9708562
Transcript of Research Article Heat and Mass Transfer on Squeezing ...
Research ArticleHeat and Mass Transfer on Squeezing Unsteady MHD NanofluidFlow between Parallel Plates with Slip Velocity Effect
Khilap Singh Sawan K Rawat and Manoj Kumar
Department of Mathematics Statistics and Computer Science G B Pant University of Agriculture and TechnologyPantnagar Uttarakhand India
Correspondence should be addressed to Khilap Singh singhkhilap8gmailcom
Received 30 June 2016 Revised 17 October 2016 Accepted 24 October 2016
Academic Editor In Cheol Bang
Copyright copy 2016 Khilap Singh et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Heat and mass transfer behavior of unsteady flow of squeezing nanofluids between two parallel plates in the sight of uniformmagnetic field with slip velocity effect is investigated The governing equations representing fluid flow have been transformedinto nonlinear ordinary differential equations using similarity transformation The equations thus obtained have been solvednumerically using Runge-Kutta-Fehlberg method with shooting technique Effects on the behavior of velocity temperature andconcentration for various values of relevant parameters are illustrated graphically The skin-friction coefficient and heat and masstransfer rate are also tabulated for various governing parameters The results indicate that for nanofluid flow the rates of heatand mass transfer are inversely proportional to nanoparticle volume fraction and magnetic parameter The rate of mass transferincreases with increasing values of Schmidt number and squeeze number
1 Introduction
The study of heat and mass transfer in unsteady squeezingviscous nanofluid flow between two parallel plates is a stim-ulating topic of exploration because of its industrial use andintense biological situations some of which include process-ing of polymer compression power transmitting lubricantsystem transient loading of mechanical components andthe squeezed films in power transmission food processingand cooling water modeling of synthetics transportationinside living bodies hydromechanical machinery chemicalprocessing equipment and crop destruction due to freezingThe first work on the squeezing flow under lubricationapproximation was studied by Stefan [1] The flow analysisbetween two parallel plates of Cu-water squeezing nanofluidwas investigated by Domairry and Hatami [2] Pourmehranet al [3] studied the unsteady flow of squeezing nanofluidbetween parallel plates Gupta and Ray [4] proposed aproblem of unsteady flow of a squeezing nanofluid betweentwo parallel plates The squeezing flow of Cu-water (orkerosene) nanofluid between two parallel plates under theeffects of viscous dissipation and velocity slipwas investigated
by Khan et al [5] Dib et al [6] obtained an approximateanalytic solution of squeezing unsteady nanofluid flow
Theword nanofluid represents the fluid inwhich particlesof size with order of nanometer (diameter lt 100 nm) aremixed in the base fluidThe nanoparticles used in nanofluidsare generally made of metals (Al Cu) oxides (Al2O3 CuOTiO2 and SiO2) carbides (SiC) nitrides (AlN SiN) andnonmetal (graphite carbon nanotubes) and the base fluid isusually a conductive fluid such as water or ethylene glycolOther base fluids are toluene oil other lubricants biofluidsand polymer solution Nanoparticles are present up to 5volume fraction in nanofluidsThe conventional heat transferfluids are poor conductors of heat Nanofluids make an edgeover them because they have high heat transfer capabilitySince these heatingcooling fluids play a vital role in thedevelopment of energy efficient heat transfer equipment forenergy supply to raise the thermal conductivity of these flu-ids nanosized conducting metal particles are added to themTherefore their proper understanding is a must to use themefficiently in modern industry Applications of nanofluidsinclude microelectronics fuel cells and pharmaceutical pro-cesses Choi and Eastman [7] were the first to propose the
Hindawi Publishing CorporationJournal of NanoscienceVolume 2016 Article ID 9708562 11 pageshttpdxdoiorg10115520169708562
2 Journal of Nanoscience
Table 1 Thermophysical properties of pure water and nanoparticles
120588 (Kgm3) 119862119901 (JkgK) 119896 (WmK) 120573 times 105 (Kminus1)Pure water 9971 4179 0613 21Copper (Cu) 8933 385 401 167Silver (Ag) 10500 235 429 189Alumina (Al2O3) 3970 765 40 085Titanium oxide (TiO2) 4250 6862 89538 09
term nanofluid that represents the fluid in which nanoscaleparticles are suspended in the base fluid with low thermalconductivity such as water ethylene glycol and oil In recentyears many researchers have studied and reported nanofluidtechnology experimentally or numerically in the presence ofheat transfer [8ndash23]
Ibrahim and Shankar [24] studied MHD nanofluid flowand heat transfer over a stretching sheet in the presence ofthermal radiation and slip conditions Malvandi and Ganji[25] investigated effect of magnetic field on heat transfer ofaluminawater nanofluid inside a circular microchannel UlHaq et al [26] obtained the influence of thermal radiationand slip on MHD nanofluid flow passed over a stretchingsheet Govindaraju et al [27] solved the problem of magneto-hydrodynamic nanofluid flow on entropy generation in astretching sheet with slip velocity The problem based oneffects of Stefan blowing and the slips on bioconvectionnanofluid flow over a horizontal plate in motion was numer-ically investigated by Uddin et al [28] Hsiao [29] studied theMHD mixed-convection stagnation flow of a nanofluid overstretching sheet in the presence of slip Kameswaran et al[30] investigated chemical reaction and viscous dissipationeffects on nanofluid flow through stretching or shrinkingsheet Matin and Pop [31] studied heat and mass transferflow of a nanofluid with chemical reaction in porous channelPal and Mandal [32] observed mixed-convection heat andmass transfer stagnation-point flow in nanofluids throughstretchingshrinking sheet in a porous medium with thermalradiation The nanofluid flow and heat transfer in porousmedium in the presence of magnetic field and radiationwere made by Zhang et al [33] Elshehabey and Ahmed [34]analyzed effect of mixed convection in nanofluid flow withsinusoidal distribution of temperature on the both verticalwalls using Buongiornorsquos nanofluid model
The novelty of the present study is to account for theslip velocity magnetic field and mass transfer on squeezingunsteady nanofluid flow and heat transfer between two par-allel plates In this study authors have applied Runge-Kutta-Fehlberg fourth-fifth-order method with shooting techniqueto find the solution of nonlinear differential equations Theeffects of governing parameters such as slip magnetic andsqueeze number Schmidt number and nanoparticle volumefraction on velocity temperature and concentration aswell ason skin-friction coefficient Nusselt and Sherwood numberare investigated To the best of our best knowledge suchinvestigation is not studied in the scientific literature Some
x
y
D
Nanoparticales
2lradic1 minus at
Figure 1 Flow configuration and coordinate system
analyticalmethods for squeezing unsteady nanofluid flow canbe found in [4 6]
2 Mathematical Formulation
We consider an unsteady two-dimensional flow to observeheat andmass transfer of a squeezing nanofluid in themiddleof two parallel plates extended infinitely and implanted ina system occupied with nanofluid (water as a base fluid)containing different types of nanoparticles that is copper(Cu) silver (Ag) alumina (Al2O3) and titaniumoxide (TiO2)with slip velocity effectThe thermophysical properties of thenanofluids are given in Table 1 A transverse magnetic fieldof variable strength is imposed in direction perpendicularto both the plates The distance between two plates is 119911 =plusmn119897(1 minus 120572119905)12 = plusmnℎ(119905) where 119897 is the initial position (at time119905 = 0) Flow is incompressible with no chemical reaction insystem Further viscous dissipation effects are retained Thegraphical model support to the present study has been givenin Figure 1 The governing equations representing flow are asfollows
120597119906120597119909 + 120597V
120597119910 = 0 (1)
120588nf (120597119906120597119905 + 119906120597119906120597119909 + V120597119906120597119910)
= minus120597119901120597119909 + 120583nf (12059721199061205971199092 +
12059721199061205971199102) minus 120590nf11986102119906
(2)
120588nf (120597V120597119905 + 119906 120597V120597119909 + V120597V120597119910) = minus120597119901120597119910 + 120583nf (1205972V
1205971199092 +1205972V1205971199102) (3)
Journal of Nanoscience 3
120597119879120597119905 + 119906120597119879120597119909 + V
120597119879120597119910
= 119896nf(120588119862119901)nf
(12059721198791205971199092 +
12059721198791205971199102 )
+ 120583nf(120588119862119901)nf
(4(120597119906120597119909)2 + (120597119906120597119910 + 120597119906
120597119909)2)
(4)
120597119862120597119905 + 119906120597119862120597119909 + V
120597119862120597119910 = 119863(1205972119862
1205971199092 +12059721198621205971199102 ) (5)
The associated boundary conditions are given as
119906 = minus119871120597119906120597119910
V = V119908 = 119889ℎ119889119905
119879 = 119879ℎ119862 = 119862ℎ at 119910 = ℎ (119905) V = 120597119906
120597119910 = 120597119879120597119910 = 0
119862 = 1198620 at 119910 = 0
(6)
where
120588nf = (1 minus 120593) 120588119891 + 120593120588119904(120588119862119901)nf = (1 minus 120593) (120588119862119901)119891 + 120593 (120588119862119901)119904 120583nf = 120583119891
(1 minus 120593)25 (Brinkman)
119896nf119896119891 = 119896119904 + 2119896119891 minus 2120593 (119896119891 minus 119896119904)119896119904 + 2119896119891 + 2120593 (119896119891 minus 119896119904) (Maxwell-Garnetts)
120590nf = (1 minus 120593) 120590119891 + 120593120590119904
(7)
Equations (2)ndash(5) can be converted to a system of nonlin-ear ordinary differential equations via the following similarityvariables
120578 = 119910119897 (1 minus 120572119905)12
119906 = 1205721199092 (1 minus 120572119905)1198911015840 (120578)
V = minus 1205721198972 (1 minus 120572119905)12119891 (120578)
120579 = 119879 minus 1198790119879ℎ minus 1198790
120601 = 119862 minus 1198620119862ℎ minus 1198620
(8)
The transformed equations are
119891119894V minus 1198781198601 (1 minus 120593)25 (120578119891101584010158401015840 + 311989110158401015840 + 119891101584011989110158401015840 minus 119891119891101584010158401015840)minus11987211989110158401015840 = 0
12057910158401015840 + Pr 119878 (11986021198603) (1205791015840119891 minus 1205781205791015840)
+ Pr Ec1198603 (1 minus 120593)25 [119891
101584010158402 + 4119895211989110158402] = 012060110158401015840 + 119878119888119878 (1198911206011015840 minus 1205781206011015840) = 0
(9)
where11986011198602 and1198603 are dimensionless constants defined asfollows
1198601 = (1 minus 120593) + 120593 120588119904120588119891
1198602 = (1 minus 120593) + 120593 (120588119862119901)119904(120588119862119901)119891
1198603 = 119896nf119896119891
(10)
The boundary conditions (6) in the terms of similarityvariables (8) become
119891 (0) = 011989110158401015840 (0) = 01205791015840 (0) = 0120601 (0) = 0119891 (1) = 11198911015840 (1) = minus12057511989110158401015840 (1) 120579 (1) = 1120601 (1) = 1
(11)
where 119878 = 12057211989722]119891 is the squeeze number Pr = 120583119891(120588119862119901)119891120588119891119896119891 is the Prandtl number Sc = 120592119891119863nf is Schmidt number119872 = 2120590nf11986120(ℎ(119905))2120588119891120583nf is the magnetic parameter 119895 = 119897119909is the reference length and 120575 = 119871119897(1 minus 120572119905)12 is the velocityslip parameter and Ec = (120588119891(120588119862119901)119891)(1205721199092(1 minus 120572119905))2 is theEckert number
The physical quantities of interest are the skin-frictioncoefficient 119862119891 the Nusselt number Nu119909 and the Sherwoodnumber Sh119909 defined as
119862119891 = 120591119908120588nfV2119908
Nu119909 = 119897119902119908119896119891 (119879ℎ minus 1198790)
Sh119909 = 119897119898119908119863nf (119862ℎ minus 1198620)
(12)
4 Journal of Nanoscience
where
120591119908 = 120583nf (120597119906120597119910)119910=ℎ(119905)
119902119908 = minus119896nf (120597119879120597119910 )119910=ℎ(119905)
119898119908 = minus119863nf (120597119862120597119910 )119910=ℎ(119905)
(13)
Using (8) and (13) in (12) we get
119862119891lowast = 1199092 (1 minus 120572119905)Re1199091198621198911198972 = 11989110158401015840 (1)
1198601 (1 minus 120601)25 Nu119909lowast = radic1 minus 120572119905Nu119909 = minus11986031205791015840 (1)
Sh119909lowast = radic1 minus 120572119905Sh119909 = minus1206011015840 (1)
(14)
where Re119909 = 120572119897521199093(1 minus 120572119905)12120592119891 is the local Reynoldsnumber
3 Method of Solution
In this present paper Runge-Kutta-Fehlberg fourth-fifth-ordermethod (RKF45) has been employed to solve the systemof nonlinear ordinary differential (9) with the boundaryconditions given by (11) for different values of governingparametersTheRKF45method has a procedure to determineif the suitable step size ℎ is being used The formula offifth-order Runge-Kutta-Fehlberg method can be defined asfollows
119911119899+1 = 119911119899+ ( 16
1351198960 +6656128251198962 +
28561564301198963 minus
9501198964 +
2551198965)
sdot ℎ(15)
where the coefficients 1198960 to 1198965 are defined as follows
1198960 = 119891 (119909119899 119910119899) 1198961 = 119891(119909119899 + 1
4ℎ 119910119899 +14ℎ1198960)
1198962 = 119891(119909119899 + 38ℎ 119910119899 + ( 3
321198960 +9321198961) ℎ)
1198963 = 119891(119909119899 + 1213ℎ 119910119899
+ (193221971198960 minus720021971198961 +
729621971198962) ℎ)
1198964 = 119891(119909119899 + ℎ 119910119899+ (4392161198960 minus 81198961 + 3680
513 1198962 minus 84541041198963) ℎ)
1198965 = 119891(119909119899 + 12ℎ 119910119899
+ (minus 8271198960 + 21198961 minus 3544
25651198962 +185941041198963 minus
11401198964) ℎ)
(16)
The computation of the error can be achieved by subtractingthe fifth-order from the fourth-order method
119910119899+1 = 119910119899 + ( 252161198960 +
140825651198962 +
219741041198963 minus
151198964) ℎ (17)
If the error goes beyond a specified antechamber the resultscan be recalculated using a smaller step size The approach tocomputing the new step size is shown as follows
ℎnew = ℎold ( 120576ℎold2 1003816100381610038161003816119911119899+1 minus 119910119899+11003816100381610038161003816)14
(18)
The variation of the dimensionless velocity temperature andconcentration is ensured to be less than 10minus6 between any twoconsecutive iterations for the convergence criterion
To solve the nonlinear differential equations (9) subjectto the boundary conditions (11) first boundary conditions for120578 = 1 are replaced by 119891(1) = 1 1198911015840(1) = minus12057511989110158401015840(1) 120579(1) = 1and 120601(1) = 1 We consider that 119891 = 1198911 1198911015840 = 1198912 11989110158401015840 =1198913 119891101584010158401015840 = 1198914 120579 = 1198915 1205791015840 = 1198916 120601 = 1198917 and 1206011015840 = 1198918Thenonlinear equations (9) are first converted into first-orderordinary linear differential equations as follows
1198911015840 = 119891211989110158402 = 119891311989110158403 = 119891411989110158404 = 1198781198601 (1 minus 120593)25 (1205781198914 + 31198913 + 11989121198913 minus 11989111198914) + 119872119891311989110158405 = 119891611989110158406 = minusPr 119878 (11986021198603) (11989161198911 minus 1205781198916)
minus Pr Ec1198603 (1 minus 120593)25 [1198913
2 + 4119895211989122] 11989110158407 = 119891811989110158408 = minus119878119888119878 (11989111198918 minus 1205781198918)
(19)
Journal of Nanoscience 5
Table 2 Comparison of 119891(120578) and 120579(120578) for Cu-water nanofluid when 119878 = 1 Pr = 62 Ec = 119895 = 001 120593 = 002 and119872 = Sc = 120575 = 0120578 Gupta and Ray [4] Present result
119891(120578) 120579(120578) 119891(120578) 120579(120578)0 0 103206637 0 10299663701 014135866 103206407 014135879 10299642802 028066605 103203202 028066639 10299353103 041578075 103189263 041578137 10298093204 054437882 103150811 054437979 10294617505 066385692 103066423 066385837 10286989606 077122923 102903983 077123132 10272306607 086301562 102615205 086301853 10246203708 093511971 102125916 093512364 10201976509 098269524 101318608 098270044 10129003110 100000000 100000000 100000000 100000000
subject to the following initial conditions
1198911 (0) = 01198912 (0) = 12057211198913 (0) = 01198914 (0) = 12057221198915 (0) = 12057231198916 (0) = 01198917 (0) = 01198918 (0) = 1205724
(20)
We ran the computer code written in the MATLAB fordifferent values of step lengthnabla120578We found that there is no oronly a negligible change in the physical quantities of interestlike the skin-friction coefficient the couple stress coefficientNusselt number and Sherwood number for various values ofnabla120578 gt 001 Therefore in the present paper we have set stepsize nabla120578 = 001 There are four initial conditions at 120578 = 0 andfour conditions on boundary 120578 = 1 To get the solution of theproblem four more initial conditions at 120578 = 0 that is valuesof 1205721 1205722 1205723 and 1205724 are required which have been obtainedby shooting technique Finally the transformed initial valueproblem is solved by employing the Runge-Kutta-Fehlbergfourth-fifth-order method along with calculated boundaryconditions
4 Results and Discussion
In order to validate the numerical results obtained wecompare our results with those reported by Gupta and Ray[4] as shown in Table 2 and they are found to be in agood agreement The effects of the volume fraction of solidnanoparticles magnetic parameter velocity slip parametersqueeze number and Schmidt number are inspected fordifferent kinds of nanoparticles when the base fluid is waterEc = 001 Pr = 62 and 119895 = 001
00 02 04 06 08 1002
04
06
08
10
12
14
M = 0M = 1M = 2
Cu-water
f998400 (120578)
120578
120593 = 002 120575 = 01 S = 1 Sc = 1
Figure 2 Effects of119872 on velocity 1198911015840(120578)
Figures 2 and 3 illustrate the behavior of the velocity1198911015840(120578)and temperature 120579(120578) of the Cu-water nanofluid for differentvalues of magnetic parameter 119872 Figure 2 shows that withan increase in the values of119872 the velocity decreases near thelower plate surface but after a certain distance it increasesIt is noticed from Figure 3 that the temperature decreasesmonotonically with increasing values of119872
The variations of the velocity temperature and concen-tration for different values of slip parameter 120575 are shownin Figures 4ndash6 It is noted from Figure 4 that near the wallthe value of velocity 1198911015840(120578) increases with rising values of 120575when 0 le 120578 le 06 and for 120578 ge 06 the velocity decreasesas 120575 increases Figure 5 depicts that a rise in the values ofslip parameter 120575 increases temperature monotonically It isevident from Figure 6 that the change in the concentration120601(120578) with rising values of slip parameter 120575 is negligible
6 Journal of Nanoscience
00 02 04 06 08 101000
1002
1004
1006
1008
1010
1012
1014
1016
1018
M = 0M = 1M = 2
Cu-water
120578
120579(120578
)
120593 = 002 120575 = 01 S = 1 Sc = 1
Figure 3 Effects of119872 on temperature 120579(120578)
00 02 04 06 08 1000
02
04
06
08
10
12
14Cu-water
f998400 (120578)
120578
120575 = 01
120575 = 02
120575 = 05
120593 = 002 M = 1 S = 1 Sc = 1
Figure 4 Effects of 120575 on velocity 1198911015840(120578)
The effects of the squeeze number 119878 on velocity temper-ature and concentration profiles are depicted in Figures 7ndash9Physically the squeeze number (119878) describes the movementof the plates (119878 gt 0 corresponds to the plates moving apartwhile 119878 lt 0 corresponds to the plates moving together) Itcan be easily seen from Figure 7 that the value of velocity1198911015840(120578) near the lower plate surface decreases regularly withthe increase in the value of 119878 and as we move away fromlower plate surface this value increases Figure 8 showsthat increasing value of squeeze number 119878 decreases thetemperature 120579(120578) Figure 9 depicts that the value of 120601(120578) goes
00 02 04 06 08 101000
1005
1010
1015
1020
1025
Cu-water
120578
120579(120578
)
120593 = 002 M = 1 S = 1 Sc = 1
120575 = 01
120575 = 02
120575 = 05
Figure 5 Effects of 120575 on temperature 120579(120578)
00 02 04 06 08 1000
02
04
06
08
10
Cu-water
120578
120575 = 01
120575 = 02
120575 = 05
120593 = 002 M = 1 S = 1 Sc = 1
120601(120578
)
Figure 6 Effects of 120575 on concentration 120601(120578)
up with increasing 120578 and high squeeze number implies slightdrop in concentration 120601(120578)
Figures 10 and 11 show the effects of the volume fraction120593 on the velocity and temperature profiles Figure 10 exhibitsinitially an increase in the values of nanoparticles volumefraction 120593 the velocity 1198911015840(120578) decreases and after a fixeddistance from lower plate surface it increases slightly FromFigure 11 it is observed that the temperature decreases withincreasing values of 120593
Figures 12 and 13 illiterate the effect of the differentnanoparticles on velocity and temperature profile whenthe base fluid is water It is observed from Figure 12 that
Journal of Nanoscience 7
00 02 04 06 08 10
02
04
06
08
10
12
14
16Cu-water
f998400 (120578)
120578
S = minus10S = minus05S = 00
S = 05
S = 10
120593 = 002 M = 1 120575 = 01 Sc = 1
Figure 7 Effects of 119878 on velocity 1198911015840(120578)
00 02 04 06 08 10100
101
102
103
104
105
106
107
120578
120579(120578
)
Cu-water
S = minus10S = minus05S = 00
S = 05
S = 10
120593 = 002 M = 1 120575 = 01 Sc = 1
Figure 8 Effects of 119878 on temperature 120579(120578)
the different nanofluids have different velocities and alsonoted that titanium oxide (TiO2) has higher velocity whencompared to other nanoparticles such as Al2O3 Cu andAg for 0 le 120578 le 05 while for 120578 gt 05 the results getreversed It is noted from Figure 13 that the order of nanofluidfor decreasing value of 120579(120578) when 120578 change from 0 to 1 isTiO2-water Al2O3-water Cu-water and Ag-water nanofluidFigure 14 depicts the effect of the Schmidt number Sc onconcentration profile The values of the temperature increasewith the increase in the value of Sc
The effects of the squeeze number 119878 on the skin-frictioncoefficient119862119891lowast the Nusselt number Nu119909
lowast and the Sherwood
00 02 04 06 08 1000
02
04
06
08
10
120578
Cu-water
S = minus10S = minus05S = 00
S = 05
S = 10
120593 = 002 M = 1 120575 = 01 Sc = 1
120601(120578
)
Figure 9 Effects of 119878 on concentration 120601(120578)
00 02 04 06 08 1002
04
06
08
10
12
14
120578
Cu-water
f998400 (120578)
M = 1 120575 = 01 Sc = 1
120593 = 000
120593 = 002120593 = 005
Figure 10 Effects of 120593 on velocity 1198911015840(120578)
number Sh119909lowast are given in Table 3 From Table 3 it is obvious
that the skin-friction coefficient and the Nusselt number areinversely proportional to 119878 whereas the Sherwood number isdirectly proportional to 119878
Table 4 displays the effects of the skin-friction coefficientthe Nusselt number and Sherwood number for differentvalues of the magnetic parameter119872 and slip parameter 120575 Itis noticed from the table that the effect of increasing valuesof119872 is to increase the skin-friction coefficient 119862119891lowast the heattransfer rate Nu119909
lowast andmass transfer rate Sh119909lowast Further from
Table 4 it is concluded that the increasing value of 120575 decreases
8 Journal of Nanoscience
00 02 04 06 08 101000
1002
1004
1006
1008
1010
1012
1014
1016
1018
120578
Cu-water
M = 1 120575 = 01 Sc = 1
120593 = 000
120593 = 002120593 = 005
120579(120578
)
Figure 11 Effects of 120593 on temperature 120579(120578)
00 02 04 06 08 1002
04
06
08
10
12
14
120578
f998400 (120578)
AgCu
Al2O3
TiO2
120593 = 002 M = 1 120575 = 01 S = 1 Sc = 1
Figure 12 Velocity of different nanofluids
the skin-friction coefficient and increases the heat and masstransfer rate
The effects of the nanoparticle volume fraction 120593 andSchmidt number Sc on the skin-friction coefficient 119862119891lowastNusselt number (the heat transfer rate) Nu119909
lowast and Sherwoodnumber (the mass transfer rate) Sh119909
lowast are given in Table 5From this table it is concluded that the increasing value of 120593increases the skin-friction coefficient and decreases the heatand mass transfer rate In addition the rate of mass transferincreases with the increase in Sc but there is no effect of Scon the skin-friction coefficient and the heat transfer rate
00 02 04 06 08 101000
1002
1004
1006
1008
1010
1012
1014
1016
120578
120579(120578
)
AgCu
Al2O3
TiO2
120593 = 002 M = 1 120575 = 01 S = 1 Sc = 1
Figure 13 Temperature of different nanofluids
00 02 04 06 08 1000
02
04
06
08
10
Sc = 0Sc = 1Sc = 10
Sc = 50Sc = 100
120578
Cu-water
120593 = 002 M = 1 120575 = 01 S = 1
120601(120578
)
Figure 14 Effects of Sc on concentration 120601(120578)
Table 3 Variation of 119862119891lowast Nu119909lowast and Sh119909lowast with different values
of Squeeze number 119878 for Cu-water nanofluid when 119872 = 1 Sc =1 120575 = 01 and 120593 = 002119878 119862119891lowast Nu119909
lowast Sh119909lowast
minus10 minus134295 0187977 minus10694254minus05 minus177787 0127657 minus1029323300 minus209419 0104693 minus1000000005 minus234148 0094096 minus0977084110 minus254426 0088577 minus09583107
The comparison for metallic nanoparticles (Ag Cu)and nonmetallic nanoparticles (Al2O3 TiO2) is done in
Journal of Nanoscience 9
Table 4 Variation of119862119891lowast Nu119909lowast and Sh119909lowast with different values of magnetic parameter119872 and slip parameter 120575 for Cu-water nanofluid when119878 = 1 Sc = 1 and 120593 = 002119872 120575 119862119891lowast Nu119909
lowast Sh119909lowast
0 01 minus246084 0089198 minus0956761 01 minus254426 0088577 minus0958312 01 minus26223 0088075 minus0959741 02 minus295257 0118598 minus0951581 05 minus326534 0144432 minus094633Table 5 Variation of 119862119891lowast Nu119909lowast and Sh119909lowast with different values of volume fraction 120593 and Schmidt number Sc for Cu-water nanofluid when119872 = 1 119878 = 1 and 120575 = 01120593 Sc 119862119891lowast Nu119909
lowast Sh119909lowast
000 1 minus276550 0092885 minus095773002 1 minus254426 0088577 minus095831005 1 minus231844 0082295 minus095911002 0 minus254426 0088577 minus100000002 10 minus254426 0088577 minus063318002 50 minus254426 0088577 minus005952002 100 minus254426 0088577 minus000182
Table 6 Variation of 119862119891lowast Nu119909lowast and Sh119909lowast for different nanoparti-
cles when119872 = 1 119878 = 1 Sc = 1 120575 = 01 and 120593 = 002Nanoparticle 119862119891lowast Nu119909
lowast Sh119909lowast
Ag minus248678 0088535 minus095851Cu minus254426 0088577 minus095831TiO2 minus273607 0088742 minus095777Al2O3 minus274807 0088794 minus095772
Table 6 It is observed form Table 6 that value of the skin-friction coefficient is more for the metallic nanoparticlesthan nonmetallic nanoparticles whereas the nonmetallicnanoparticles have higher heat and mass transfer rate incomparison with metallic nanoparticles
5 Conclusions
The present paper deals with the numerical solution ofcombined heat andmass transfer effects of an unsteadyMHDlaminar two-dimensional flow of incompressible viscousnanofluids inmiddle of two parallel plates extended infinitelywith slip velocity effect The relevant nonlinear partial dif-ferential equations were transformed to a set of ordinarydifferential equations and then are solved numerically usingthe Runge-Kutta-Fehlberg fourth-fifth-order method alongwith shooting technique From the above discussion the eye-catching results are as follows
(i) Temperature drops in Cu-water nanofluid withincrease in the magnetic field strength
(ii) The velocity of the nanofluid with nonmetallicnanoparticles Al2O3 and TiO2 is greater thanmetallic
nanoparticles Cu and Ag initially when water is thebase fluid but nature gets reversed after 120578 = 05The temperature of themetallic nanoparticles is lowerthan the nonmetallic nanoparticles
(iii) When plates start moving apart then temperaturestarts decreasing
(iv) Initially with the increasing values of magnetic fieldstrength and squeeze number the velocity of Cu-water slows down but the nature in both the case getsreversed after 120578 = 05
(v) The Nusselt number and skin-friction coefficientdecrease as plates move apart for Cu-water nanofluid
(vi) Nusselt number and Sherwood number for non-metallic nanoparticles are higher than the metallicnanoparticles
(vii) Varying value of slip parameter has negligible effecton concentration and as values of squeeze numberincrease the value of concentration profile also getsslightly higher
(viii) Considerable amount of enhancement in concentra-tion can be seen as Schmidt number rises
Nomenclature
1198601 1198602 1198603 Dimensionless constant1198610 Strength of magnetic field119862 Fluid concentration119862119891 Skin-friction coefficient119862119901 Specific heat at constant pressure119901 [Jkgminus1 Kminus1]119863 Mass diffusivity
10 Journal of Nanoscience
Ec Eckert number119891 Dimensionless velocity of the fluid119895 Reference length119896nf Nanofluid effective thermalconductivity119897 Distance of plate [m]119871 Slip length119872 Magnetic parameter119898119908 Surface mass flux
Nu119909 Nusselt number119901 Pressure [Pa]Pr Prandtl number119902119908 Surface heat fluxRe119909 Local Reynolds number119878 Squeeze numberSc Schmidt numberSh119909 Sherwood number119905 Time [s]119879 Temperature [K]119906 V Velocities in 119909119910-directions [m sminus1]119909 119910 Axial and perpendicular coordinates
[m]
Greek Symbols
120583119891 Dynamic viscosity of the fluid120593 Nanoparticle volume fraction120575 Velocity slip parameter120592119891 Kinematic viscosity of the fluid [m2 sminus1]120578 Similarity variable120588119891 Density of base fluid [kgmminus3]119896119904 Thermal conductivity of the solidnanoparticle [mminus1 Kminus1]119896119891 Thermal conductivity of the base fluid[Wmminus1 Kminus1]120579 Nondimensional temperature120601 Nondimensional concentration120588nf Nanofluid density [kgmminus3]120583nf Effective dynamic viscosity of nanofluid[Pa s]120572 Squeeze parameter120590nf Electrical conductivity of nanofluid120588119904 Density of solid particle [kgmminus3]120591119908 Surface shear stress
Subscripts
119891 Base fluidℎ Condition at the wallnf Nanofluid119904 Solid
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] M J Stefan ldquoVersuchuber die scheinbare adhesion sitzungs-bersachsrdquo Akademie der Wissenschaften in Wien Math-ematisch-Naturwissenschaftliche Klasse vol 69 pp 713ndash7211874
[2] G Domairry and M Hatami ldquoSqueezing Cundashwater nanofluidflow analysis between parallel plates by DTM-Pade MethodrdquoJournal of Molecular Liquids vol 193 pp 37ndash44 2014
[3] O Pourmehran M Rahimi-Gorji M Gorji-Bandpy and D DGanji ldquoAnalytical investigation of squeezing unsteady nanofluidflow between parallel plates by LSM and CMrdquo AlexandriaEngineering Journal vol 54 no 1 pp 17ndash26 2015
[4] A K Gupta and S S Ray ldquoNumerical treatment for investiga-tion of squeezing unsteady nanofluid flow between two parallelplatesrdquo Powder Technology vol 279 pp 282ndash289 2015
[5] U Khan N Ahmed M Asadullah and S T Mohyud-dinldquoEffects of viscous dissipation and slip velocity on two-dim-ensional and axisymmetric squeezing flow of Cu-water and Cu-kerosene nanofluidsrdquo Propulsion and Power Research vol 4 no1 pp 40ndash49 2015
[6] A Dib A Haiahem and B Bou-said ldquoApproximate analyticalsolution of squeezing unsteady nanofluid flowrdquo Powder Technol-ogy vol 269 pp 193ndash199 2015
[7] S U S Choi and J A Eastman ldquoEnhancing thermal con-ductivity of fluids with nanoparticlerdquo in Proceedings of theInternational Mechanical Engineering Congress and Exhibitionvol 231 pp 99ndash105 San Francisco Calif USA November 1995
[8] M Azimi and R Riazi ldquoHeat transfer analysis of GO-waternanofluid flow between two parallel disksrdquo Propulsion andPower Research vol 4 no 1 pp 23ndash30 2015
[9] A Aziz W A Khan and I Pop ldquoFree convection boundarylayer flow past a horizontal flat plate embedded in porousmedium filled by nanofluid containing gyrotactic microorgan-ismsrdquo International Journal of Thermal Sciences vol 56 pp 48ndash57 2012
[10] M Fakour D D Ganji and M Abbasi ldquoScrutiny of underde-veloped nanofluidMHD flow and heat conduction in a channelwith porous wallsrdquo Case Studies in Thermal Engineering vol 4pp 202ndash214 2014
[11] T Grosan C Revnic I Pop andD B Ingham ldquoFree convectionheat transfer in a square cavity filled with a porous mediumsaturated by a nano-fluidrdquo International Journal of Heat andMass Transfer vol 87 pp 36ndash41 2015
[12] A V Kuznetsov and D A Nield ldquoThe Cheng-Minkowyczproblem for natural convective boundary layer flow in a porousmedium saturated by a nanofluid a revised modelrdquo Inter-national Journal of Heat andMass Transfer vol 65 pp 682ndash6852013
[13] P Rana R Bhargava and O A Begb ldquoNumerical solution formixed convection boundary layerflow of ananofluid along aninclined plate embedded in a porous mediumrdquo Computers andMathematics with Applications vol 64 no 9 pp 2816ndash28322012
[14] A M Rashad A J Chamkha and M Modather ldquoMixed con-vection boundary-layer flow past a horizontal circular cylinderembedded in a porous medium filled with a nanofluid underconvective boundary conditionrdquo Computers amp Fluids vol 86pp 380ndash388 2013
[15] M Sheikholeslami and D D Ganji ldquoNanofluid flow and heattransfer between parallel plates considering Brownian motion
Journal of Nanoscience 11
using DTMrdquo Computer Methods in Applied Mechanics andEngineering vol 283 pp 651ndash663 2015
[16] N V Ganesh B Ganga and A K Abdul Hakeem ldquoLiesymmetry group analysis of magnetic field effects on free con-vective flow of a nanofluid over a semi-infinite stretching sheetrdquoJournal of the Egyptian Mathematical Society vol 22 no 2pp 304ndash310 2014
[17] Y Lin L Zheng and G Chen ldquoUnsteady flow and heattransfer of pseudo-plastic nanoliquid in a finite thin film ona stretching surface with variable thermal conductivity andviscous dissipationrdquo Powder Technology vol 274 pp 324ndash3322015
[18] M Sheikholeslami MM Rashidi and D D Ganji ldquoNumericalinvestigation of magnetic nanofluid forced convective heattransfer in existence of variable magnetic field using two phasemodelrdquo Journal of Molecular Liquids vol 212 pp 117ndash126 2015
[19] T Hayat M Waqas S A Shehzad and A Alsaedi ldquoMixedconvection flow of a Burgers nanofluid in the presence ofstratifications and heat generationabsorptionrdquo The EuropeanPhysical Journal Plus vol 131 no 8 p 253 2016
[20] F M Abbasi T Hayat S A Shehzad F Alsaadi and NAltoaibi ldquoHydromagnetic peristaltic transport of copper-waternanofluid with temperature-dependent effective viscosityrdquo Par-ticuology vol 27 pp 133ndash140 2016
[21] M Sheikholeslami D D Ganji and M M Rashidi ldquoMagneticfield effect on unsteady nanofluid flow and heat transferusing Buongiorno modelrdquo Journal of Magnetism and MagneticMaterials vol 416 pp 164ndash173 2016
[22] O Pourmehran M Rahimi-Gorji and D D Ganji ldquoHeattransfer and flow analysis of nanofluid flow induced by astretching sheet in the presence of an external magnetic fieldrdquoJournal of the Taiwan Institute of Chemical Engineers vol 65 pp162ndash171 2016
[23] M Rahimi-Gorji O Pourmehran M Hatami and D D GanjildquoStatistical optimization of microchannel heat sink (MCHS)geometry cooled by different nanofluids using RSM analysisrdquoEuropean Physical Journal Plus vol 130 no 2 article 22 pp 1ndash21 2015
[24] W Ibrahim and B Shankar ldquoMHD boundary layer flow andheat transfer of a nanofluid past a permeable stretching sheetwith velocity thermal and solutal slip boundary conditionsrdquoComputers amp Fluids vol 75 pp 1ndash10 2013
[25] A Malvandi and D D Ganji ldquoBrownian motion and ther-mophoresis effects on slip flow of aluminawater nanofluidinside a circular microchannel in the presence of a magneticfieldrdquo International Journal ofThermal Sciences vol 84 pp 196ndash206 2014
[26] R Ul Haq S Nadeem Z H Khan and N S Akbar ldquoThermalradiation and slip effects on MHD stagnation point flow ofnanofluid over a stretching sheetrdquo Physica E vol 65 pp 17ndash232015
[27] M Govindaraju S Saranya A A Hakeem R Jayaprakash andB Ganga ldquoAnalysis of slip MHD nanofluid flow on entropygeneration in a stretching sheetrdquo Procedia Engineering vol 127pp 501ndash507 2015
[28] M J Uddin M N Kabir and O A Beg ldquoComputationalinvestigation of Stefan blowing and multiple-slip effects onbuoyancy-driven bioconvection nanofluid flow with microor-ganismsrdquo International Journal of Heat and Mass Transfer vol95 pp 116ndash130 2016
[29] K-L Hsiao ldquoStagnation electrical MHD nanofluid mixed con-vection with slip boundary on a stretching sheetrdquo AppliedThermal Engineering vol 98 pp 850ndash861 2016
[30] P K Kameswaran M Narayana P Sibanda and P V S NMurthy ldquoHydromagnetic nanofluid flow due to a stretching orshrinking sheet with viscous dissipation and chemical reactioneffectsrdquo International Journal of Heat andMass Transfer vol 55no 25-26 pp 7587ndash7595 2012
[31] M H Matin and I Pop ldquoForced convection heat and masstransfer flow of a nanofluid through a porous channel with afirst order chemical reaction on thewallrdquo International Commu-nications in Heat and Mass Transfer vol 46 pp 134ndash141 2013
[32] D Pal andGMandal ldquoInfluence of thermal radiation onmixedconvection heat and mass transfer stagnation-point flow innanofluids over stretchingshrinking sheet in a porous mediumwith chemical reactionrdquo Nuclear Engineering and Design vol273 pp 644ndash652 2014
[33] C Zhang L Zheng X Zhang and G Chen ldquoMHD flow andradiation heat transfer of nanofluids in porous media withvariable surface heat flux and chemical reactionrdquoAppliedMath-ematical Modelling vol 39 no 1 pp 165ndash181 2015
[34] HM Elshehabey and S E Ahmed ldquoMHDmixed convection ina lid-driven cavity filled by a nanofluid with sinusoidal temper-ature distribution on the both vertical walls using Buongiornorsquosnanofluid modelrdquo International Journal of Heat and MassTransfer vol 88 pp 181ndash202 2015
Submit your manuscripts athttpwwwhindawicom
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Biomaterials
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Advances in
Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Smart Materials Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MetallurgyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
MaterialsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nano
materials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofNanomaterials
2 Journal of Nanoscience
Table 1 Thermophysical properties of pure water and nanoparticles
120588 (Kgm3) 119862119901 (JkgK) 119896 (WmK) 120573 times 105 (Kminus1)Pure water 9971 4179 0613 21Copper (Cu) 8933 385 401 167Silver (Ag) 10500 235 429 189Alumina (Al2O3) 3970 765 40 085Titanium oxide (TiO2) 4250 6862 89538 09
term nanofluid that represents the fluid in which nanoscaleparticles are suspended in the base fluid with low thermalconductivity such as water ethylene glycol and oil In recentyears many researchers have studied and reported nanofluidtechnology experimentally or numerically in the presence ofheat transfer [8ndash23]
Ibrahim and Shankar [24] studied MHD nanofluid flowand heat transfer over a stretching sheet in the presence ofthermal radiation and slip conditions Malvandi and Ganji[25] investigated effect of magnetic field on heat transfer ofaluminawater nanofluid inside a circular microchannel UlHaq et al [26] obtained the influence of thermal radiationand slip on MHD nanofluid flow passed over a stretchingsheet Govindaraju et al [27] solved the problem of magneto-hydrodynamic nanofluid flow on entropy generation in astretching sheet with slip velocity The problem based oneffects of Stefan blowing and the slips on bioconvectionnanofluid flow over a horizontal plate in motion was numer-ically investigated by Uddin et al [28] Hsiao [29] studied theMHD mixed-convection stagnation flow of a nanofluid overstretching sheet in the presence of slip Kameswaran et al[30] investigated chemical reaction and viscous dissipationeffects on nanofluid flow through stretching or shrinkingsheet Matin and Pop [31] studied heat and mass transferflow of a nanofluid with chemical reaction in porous channelPal and Mandal [32] observed mixed-convection heat andmass transfer stagnation-point flow in nanofluids throughstretchingshrinking sheet in a porous medium with thermalradiation The nanofluid flow and heat transfer in porousmedium in the presence of magnetic field and radiationwere made by Zhang et al [33] Elshehabey and Ahmed [34]analyzed effect of mixed convection in nanofluid flow withsinusoidal distribution of temperature on the both verticalwalls using Buongiornorsquos nanofluid model
The novelty of the present study is to account for theslip velocity magnetic field and mass transfer on squeezingunsteady nanofluid flow and heat transfer between two par-allel plates In this study authors have applied Runge-Kutta-Fehlberg fourth-fifth-order method with shooting techniqueto find the solution of nonlinear differential equations Theeffects of governing parameters such as slip magnetic andsqueeze number Schmidt number and nanoparticle volumefraction on velocity temperature and concentration aswell ason skin-friction coefficient Nusselt and Sherwood numberare investigated To the best of our best knowledge suchinvestigation is not studied in the scientific literature Some
x
y
D
Nanoparticales
2lradic1 minus at
Figure 1 Flow configuration and coordinate system
analyticalmethods for squeezing unsteady nanofluid flow canbe found in [4 6]
2 Mathematical Formulation
We consider an unsteady two-dimensional flow to observeheat andmass transfer of a squeezing nanofluid in themiddleof two parallel plates extended infinitely and implanted ina system occupied with nanofluid (water as a base fluid)containing different types of nanoparticles that is copper(Cu) silver (Ag) alumina (Al2O3) and titaniumoxide (TiO2)with slip velocity effectThe thermophysical properties of thenanofluids are given in Table 1 A transverse magnetic fieldof variable strength is imposed in direction perpendicularto both the plates The distance between two plates is 119911 =plusmn119897(1 minus 120572119905)12 = plusmnℎ(119905) where 119897 is the initial position (at time119905 = 0) Flow is incompressible with no chemical reaction insystem Further viscous dissipation effects are retained Thegraphical model support to the present study has been givenin Figure 1 The governing equations representing flow are asfollows
120597119906120597119909 + 120597V
120597119910 = 0 (1)
120588nf (120597119906120597119905 + 119906120597119906120597119909 + V120597119906120597119910)
= minus120597119901120597119909 + 120583nf (12059721199061205971199092 +
12059721199061205971199102) minus 120590nf11986102119906
(2)
120588nf (120597V120597119905 + 119906 120597V120597119909 + V120597V120597119910) = minus120597119901120597119910 + 120583nf (1205972V
1205971199092 +1205972V1205971199102) (3)
Journal of Nanoscience 3
120597119879120597119905 + 119906120597119879120597119909 + V
120597119879120597119910
= 119896nf(120588119862119901)nf
(12059721198791205971199092 +
12059721198791205971199102 )
+ 120583nf(120588119862119901)nf
(4(120597119906120597119909)2 + (120597119906120597119910 + 120597119906
120597119909)2)
(4)
120597119862120597119905 + 119906120597119862120597119909 + V
120597119862120597119910 = 119863(1205972119862
1205971199092 +12059721198621205971199102 ) (5)
The associated boundary conditions are given as
119906 = minus119871120597119906120597119910
V = V119908 = 119889ℎ119889119905
119879 = 119879ℎ119862 = 119862ℎ at 119910 = ℎ (119905) V = 120597119906
120597119910 = 120597119879120597119910 = 0
119862 = 1198620 at 119910 = 0
(6)
where
120588nf = (1 minus 120593) 120588119891 + 120593120588119904(120588119862119901)nf = (1 minus 120593) (120588119862119901)119891 + 120593 (120588119862119901)119904 120583nf = 120583119891
(1 minus 120593)25 (Brinkman)
119896nf119896119891 = 119896119904 + 2119896119891 minus 2120593 (119896119891 minus 119896119904)119896119904 + 2119896119891 + 2120593 (119896119891 minus 119896119904) (Maxwell-Garnetts)
120590nf = (1 minus 120593) 120590119891 + 120593120590119904
(7)
Equations (2)ndash(5) can be converted to a system of nonlin-ear ordinary differential equations via the following similarityvariables
120578 = 119910119897 (1 minus 120572119905)12
119906 = 1205721199092 (1 minus 120572119905)1198911015840 (120578)
V = minus 1205721198972 (1 minus 120572119905)12119891 (120578)
120579 = 119879 minus 1198790119879ℎ minus 1198790
120601 = 119862 minus 1198620119862ℎ minus 1198620
(8)
The transformed equations are
119891119894V minus 1198781198601 (1 minus 120593)25 (120578119891101584010158401015840 + 311989110158401015840 + 119891101584011989110158401015840 minus 119891119891101584010158401015840)minus11987211989110158401015840 = 0
12057910158401015840 + Pr 119878 (11986021198603) (1205791015840119891 minus 1205781205791015840)
+ Pr Ec1198603 (1 minus 120593)25 [119891
101584010158402 + 4119895211989110158402] = 012060110158401015840 + 119878119888119878 (1198911206011015840 minus 1205781206011015840) = 0
(9)
where11986011198602 and1198603 are dimensionless constants defined asfollows
1198601 = (1 minus 120593) + 120593 120588119904120588119891
1198602 = (1 minus 120593) + 120593 (120588119862119901)119904(120588119862119901)119891
1198603 = 119896nf119896119891
(10)
The boundary conditions (6) in the terms of similarityvariables (8) become
119891 (0) = 011989110158401015840 (0) = 01205791015840 (0) = 0120601 (0) = 0119891 (1) = 11198911015840 (1) = minus12057511989110158401015840 (1) 120579 (1) = 1120601 (1) = 1
(11)
where 119878 = 12057211989722]119891 is the squeeze number Pr = 120583119891(120588119862119901)119891120588119891119896119891 is the Prandtl number Sc = 120592119891119863nf is Schmidt number119872 = 2120590nf11986120(ℎ(119905))2120588119891120583nf is the magnetic parameter 119895 = 119897119909is the reference length and 120575 = 119871119897(1 minus 120572119905)12 is the velocityslip parameter and Ec = (120588119891(120588119862119901)119891)(1205721199092(1 minus 120572119905))2 is theEckert number
The physical quantities of interest are the skin-frictioncoefficient 119862119891 the Nusselt number Nu119909 and the Sherwoodnumber Sh119909 defined as
119862119891 = 120591119908120588nfV2119908
Nu119909 = 119897119902119908119896119891 (119879ℎ minus 1198790)
Sh119909 = 119897119898119908119863nf (119862ℎ minus 1198620)
(12)
4 Journal of Nanoscience
where
120591119908 = 120583nf (120597119906120597119910)119910=ℎ(119905)
119902119908 = minus119896nf (120597119879120597119910 )119910=ℎ(119905)
119898119908 = minus119863nf (120597119862120597119910 )119910=ℎ(119905)
(13)
Using (8) and (13) in (12) we get
119862119891lowast = 1199092 (1 minus 120572119905)Re1199091198621198911198972 = 11989110158401015840 (1)
1198601 (1 minus 120601)25 Nu119909lowast = radic1 minus 120572119905Nu119909 = minus11986031205791015840 (1)
Sh119909lowast = radic1 minus 120572119905Sh119909 = minus1206011015840 (1)
(14)
where Re119909 = 120572119897521199093(1 minus 120572119905)12120592119891 is the local Reynoldsnumber
3 Method of Solution
In this present paper Runge-Kutta-Fehlberg fourth-fifth-ordermethod (RKF45) has been employed to solve the systemof nonlinear ordinary differential (9) with the boundaryconditions given by (11) for different values of governingparametersTheRKF45method has a procedure to determineif the suitable step size ℎ is being used The formula offifth-order Runge-Kutta-Fehlberg method can be defined asfollows
119911119899+1 = 119911119899+ ( 16
1351198960 +6656128251198962 +
28561564301198963 minus
9501198964 +
2551198965)
sdot ℎ(15)
where the coefficients 1198960 to 1198965 are defined as follows
1198960 = 119891 (119909119899 119910119899) 1198961 = 119891(119909119899 + 1
4ℎ 119910119899 +14ℎ1198960)
1198962 = 119891(119909119899 + 38ℎ 119910119899 + ( 3
321198960 +9321198961) ℎ)
1198963 = 119891(119909119899 + 1213ℎ 119910119899
+ (193221971198960 minus720021971198961 +
729621971198962) ℎ)
1198964 = 119891(119909119899 + ℎ 119910119899+ (4392161198960 minus 81198961 + 3680
513 1198962 minus 84541041198963) ℎ)
1198965 = 119891(119909119899 + 12ℎ 119910119899
+ (minus 8271198960 + 21198961 minus 3544
25651198962 +185941041198963 minus
11401198964) ℎ)
(16)
The computation of the error can be achieved by subtractingthe fifth-order from the fourth-order method
119910119899+1 = 119910119899 + ( 252161198960 +
140825651198962 +
219741041198963 minus
151198964) ℎ (17)
If the error goes beyond a specified antechamber the resultscan be recalculated using a smaller step size The approach tocomputing the new step size is shown as follows
ℎnew = ℎold ( 120576ℎold2 1003816100381610038161003816119911119899+1 minus 119910119899+11003816100381610038161003816)14
(18)
The variation of the dimensionless velocity temperature andconcentration is ensured to be less than 10minus6 between any twoconsecutive iterations for the convergence criterion
To solve the nonlinear differential equations (9) subjectto the boundary conditions (11) first boundary conditions for120578 = 1 are replaced by 119891(1) = 1 1198911015840(1) = minus12057511989110158401015840(1) 120579(1) = 1and 120601(1) = 1 We consider that 119891 = 1198911 1198911015840 = 1198912 11989110158401015840 =1198913 119891101584010158401015840 = 1198914 120579 = 1198915 1205791015840 = 1198916 120601 = 1198917 and 1206011015840 = 1198918Thenonlinear equations (9) are first converted into first-orderordinary linear differential equations as follows
1198911015840 = 119891211989110158402 = 119891311989110158403 = 119891411989110158404 = 1198781198601 (1 minus 120593)25 (1205781198914 + 31198913 + 11989121198913 minus 11989111198914) + 119872119891311989110158405 = 119891611989110158406 = minusPr 119878 (11986021198603) (11989161198911 minus 1205781198916)
minus Pr Ec1198603 (1 minus 120593)25 [1198913
2 + 4119895211989122] 11989110158407 = 119891811989110158408 = minus119878119888119878 (11989111198918 minus 1205781198918)
(19)
Journal of Nanoscience 5
Table 2 Comparison of 119891(120578) and 120579(120578) for Cu-water nanofluid when 119878 = 1 Pr = 62 Ec = 119895 = 001 120593 = 002 and119872 = Sc = 120575 = 0120578 Gupta and Ray [4] Present result
119891(120578) 120579(120578) 119891(120578) 120579(120578)0 0 103206637 0 10299663701 014135866 103206407 014135879 10299642802 028066605 103203202 028066639 10299353103 041578075 103189263 041578137 10298093204 054437882 103150811 054437979 10294617505 066385692 103066423 066385837 10286989606 077122923 102903983 077123132 10272306607 086301562 102615205 086301853 10246203708 093511971 102125916 093512364 10201976509 098269524 101318608 098270044 10129003110 100000000 100000000 100000000 100000000
subject to the following initial conditions
1198911 (0) = 01198912 (0) = 12057211198913 (0) = 01198914 (0) = 12057221198915 (0) = 12057231198916 (0) = 01198917 (0) = 01198918 (0) = 1205724
(20)
We ran the computer code written in the MATLAB fordifferent values of step lengthnabla120578We found that there is no oronly a negligible change in the physical quantities of interestlike the skin-friction coefficient the couple stress coefficientNusselt number and Sherwood number for various values ofnabla120578 gt 001 Therefore in the present paper we have set stepsize nabla120578 = 001 There are four initial conditions at 120578 = 0 andfour conditions on boundary 120578 = 1 To get the solution of theproblem four more initial conditions at 120578 = 0 that is valuesof 1205721 1205722 1205723 and 1205724 are required which have been obtainedby shooting technique Finally the transformed initial valueproblem is solved by employing the Runge-Kutta-Fehlbergfourth-fifth-order method along with calculated boundaryconditions
4 Results and Discussion
In order to validate the numerical results obtained wecompare our results with those reported by Gupta and Ray[4] as shown in Table 2 and they are found to be in agood agreement The effects of the volume fraction of solidnanoparticles magnetic parameter velocity slip parametersqueeze number and Schmidt number are inspected fordifferent kinds of nanoparticles when the base fluid is waterEc = 001 Pr = 62 and 119895 = 001
00 02 04 06 08 1002
04
06
08
10
12
14
M = 0M = 1M = 2
Cu-water
f998400 (120578)
120578
120593 = 002 120575 = 01 S = 1 Sc = 1
Figure 2 Effects of119872 on velocity 1198911015840(120578)
Figures 2 and 3 illustrate the behavior of the velocity1198911015840(120578)and temperature 120579(120578) of the Cu-water nanofluid for differentvalues of magnetic parameter 119872 Figure 2 shows that withan increase in the values of119872 the velocity decreases near thelower plate surface but after a certain distance it increasesIt is noticed from Figure 3 that the temperature decreasesmonotonically with increasing values of119872
The variations of the velocity temperature and concen-tration for different values of slip parameter 120575 are shownin Figures 4ndash6 It is noted from Figure 4 that near the wallthe value of velocity 1198911015840(120578) increases with rising values of 120575when 0 le 120578 le 06 and for 120578 ge 06 the velocity decreasesas 120575 increases Figure 5 depicts that a rise in the values ofslip parameter 120575 increases temperature monotonically It isevident from Figure 6 that the change in the concentration120601(120578) with rising values of slip parameter 120575 is negligible
6 Journal of Nanoscience
00 02 04 06 08 101000
1002
1004
1006
1008
1010
1012
1014
1016
1018
M = 0M = 1M = 2
Cu-water
120578
120579(120578
)
120593 = 002 120575 = 01 S = 1 Sc = 1
Figure 3 Effects of119872 on temperature 120579(120578)
00 02 04 06 08 1000
02
04
06
08
10
12
14Cu-water
f998400 (120578)
120578
120575 = 01
120575 = 02
120575 = 05
120593 = 002 M = 1 S = 1 Sc = 1
Figure 4 Effects of 120575 on velocity 1198911015840(120578)
The effects of the squeeze number 119878 on velocity temper-ature and concentration profiles are depicted in Figures 7ndash9Physically the squeeze number (119878) describes the movementof the plates (119878 gt 0 corresponds to the plates moving apartwhile 119878 lt 0 corresponds to the plates moving together) Itcan be easily seen from Figure 7 that the value of velocity1198911015840(120578) near the lower plate surface decreases regularly withthe increase in the value of 119878 and as we move away fromlower plate surface this value increases Figure 8 showsthat increasing value of squeeze number 119878 decreases thetemperature 120579(120578) Figure 9 depicts that the value of 120601(120578) goes
00 02 04 06 08 101000
1005
1010
1015
1020
1025
Cu-water
120578
120579(120578
)
120593 = 002 M = 1 S = 1 Sc = 1
120575 = 01
120575 = 02
120575 = 05
Figure 5 Effects of 120575 on temperature 120579(120578)
00 02 04 06 08 1000
02
04
06
08
10
Cu-water
120578
120575 = 01
120575 = 02
120575 = 05
120593 = 002 M = 1 S = 1 Sc = 1
120601(120578
)
Figure 6 Effects of 120575 on concentration 120601(120578)
up with increasing 120578 and high squeeze number implies slightdrop in concentration 120601(120578)
Figures 10 and 11 show the effects of the volume fraction120593 on the velocity and temperature profiles Figure 10 exhibitsinitially an increase in the values of nanoparticles volumefraction 120593 the velocity 1198911015840(120578) decreases and after a fixeddistance from lower plate surface it increases slightly FromFigure 11 it is observed that the temperature decreases withincreasing values of 120593
Figures 12 and 13 illiterate the effect of the differentnanoparticles on velocity and temperature profile whenthe base fluid is water It is observed from Figure 12 that
Journal of Nanoscience 7
00 02 04 06 08 10
02
04
06
08
10
12
14
16Cu-water
f998400 (120578)
120578
S = minus10S = minus05S = 00
S = 05
S = 10
120593 = 002 M = 1 120575 = 01 Sc = 1
Figure 7 Effects of 119878 on velocity 1198911015840(120578)
00 02 04 06 08 10100
101
102
103
104
105
106
107
120578
120579(120578
)
Cu-water
S = minus10S = minus05S = 00
S = 05
S = 10
120593 = 002 M = 1 120575 = 01 Sc = 1
Figure 8 Effects of 119878 on temperature 120579(120578)
the different nanofluids have different velocities and alsonoted that titanium oxide (TiO2) has higher velocity whencompared to other nanoparticles such as Al2O3 Cu andAg for 0 le 120578 le 05 while for 120578 gt 05 the results getreversed It is noted from Figure 13 that the order of nanofluidfor decreasing value of 120579(120578) when 120578 change from 0 to 1 isTiO2-water Al2O3-water Cu-water and Ag-water nanofluidFigure 14 depicts the effect of the Schmidt number Sc onconcentration profile The values of the temperature increasewith the increase in the value of Sc
The effects of the squeeze number 119878 on the skin-frictioncoefficient119862119891lowast the Nusselt number Nu119909
lowast and the Sherwood
00 02 04 06 08 1000
02
04
06
08
10
120578
Cu-water
S = minus10S = minus05S = 00
S = 05
S = 10
120593 = 002 M = 1 120575 = 01 Sc = 1
120601(120578
)
Figure 9 Effects of 119878 on concentration 120601(120578)
00 02 04 06 08 1002
04
06
08
10
12
14
120578
Cu-water
f998400 (120578)
M = 1 120575 = 01 Sc = 1
120593 = 000
120593 = 002120593 = 005
Figure 10 Effects of 120593 on velocity 1198911015840(120578)
number Sh119909lowast are given in Table 3 From Table 3 it is obvious
that the skin-friction coefficient and the Nusselt number areinversely proportional to 119878 whereas the Sherwood number isdirectly proportional to 119878
Table 4 displays the effects of the skin-friction coefficientthe Nusselt number and Sherwood number for differentvalues of the magnetic parameter119872 and slip parameter 120575 Itis noticed from the table that the effect of increasing valuesof119872 is to increase the skin-friction coefficient 119862119891lowast the heattransfer rate Nu119909
lowast andmass transfer rate Sh119909lowast Further from
Table 4 it is concluded that the increasing value of 120575 decreases
8 Journal of Nanoscience
00 02 04 06 08 101000
1002
1004
1006
1008
1010
1012
1014
1016
1018
120578
Cu-water
M = 1 120575 = 01 Sc = 1
120593 = 000
120593 = 002120593 = 005
120579(120578
)
Figure 11 Effects of 120593 on temperature 120579(120578)
00 02 04 06 08 1002
04
06
08
10
12
14
120578
f998400 (120578)
AgCu
Al2O3
TiO2
120593 = 002 M = 1 120575 = 01 S = 1 Sc = 1
Figure 12 Velocity of different nanofluids
the skin-friction coefficient and increases the heat and masstransfer rate
The effects of the nanoparticle volume fraction 120593 andSchmidt number Sc on the skin-friction coefficient 119862119891lowastNusselt number (the heat transfer rate) Nu119909
lowast and Sherwoodnumber (the mass transfer rate) Sh119909
lowast are given in Table 5From this table it is concluded that the increasing value of 120593increases the skin-friction coefficient and decreases the heatand mass transfer rate In addition the rate of mass transferincreases with the increase in Sc but there is no effect of Scon the skin-friction coefficient and the heat transfer rate
00 02 04 06 08 101000
1002
1004
1006
1008
1010
1012
1014
1016
120578
120579(120578
)
AgCu
Al2O3
TiO2
120593 = 002 M = 1 120575 = 01 S = 1 Sc = 1
Figure 13 Temperature of different nanofluids
00 02 04 06 08 1000
02
04
06
08
10
Sc = 0Sc = 1Sc = 10
Sc = 50Sc = 100
120578
Cu-water
120593 = 002 M = 1 120575 = 01 S = 1
120601(120578
)
Figure 14 Effects of Sc on concentration 120601(120578)
Table 3 Variation of 119862119891lowast Nu119909lowast and Sh119909lowast with different values
of Squeeze number 119878 for Cu-water nanofluid when 119872 = 1 Sc =1 120575 = 01 and 120593 = 002119878 119862119891lowast Nu119909
lowast Sh119909lowast
minus10 minus134295 0187977 minus10694254minus05 minus177787 0127657 minus1029323300 minus209419 0104693 minus1000000005 minus234148 0094096 minus0977084110 minus254426 0088577 minus09583107
The comparison for metallic nanoparticles (Ag Cu)and nonmetallic nanoparticles (Al2O3 TiO2) is done in
Journal of Nanoscience 9
Table 4 Variation of119862119891lowast Nu119909lowast and Sh119909lowast with different values of magnetic parameter119872 and slip parameter 120575 for Cu-water nanofluid when119878 = 1 Sc = 1 and 120593 = 002119872 120575 119862119891lowast Nu119909
lowast Sh119909lowast
0 01 minus246084 0089198 minus0956761 01 minus254426 0088577 minus0958312 01 minus26223 0088075 minus0959741 02 minus295257 0118598 minus0951581 05 minus326534 0144432 minus094633Table 5 Variation of 119862119891lowast Nu119909lowast and Sh119909lowast with different values of volume fraction 120593 and Schmidt number Sc for Cu-water nanofluid when119872 = 1 119878 = 1 and 120575 = 01120593 Sc 119862119891lowast Nu119909
lowast Sh119909lowast
000 1 minus276550 0092885 minus095773002 1 minus254426 0088577 minus095831005 1 minus231844 0082295 minus095911002 0 minus254426 0088577 minus100000002 10 minus254426 0088577 minus063318002 50 minus254426 0088577 minus005952002 100 minus254426 0088577 minus000182
Table 6 Variation of 119862119891lowast Nu119909lowast and Sh119909lowast for different nanoparti-
cles when119872 = 1 119878 = 1 Sc = 1 120575 = 01 and 120593 = 002Nanoparticle 119862119891lowast Nu119909
lowast Sh119909lowast
Ag minus248678 0088535 minus095851Cu minus254426 0088577 minus095831TiO2 minus273607 0088742 minus095777Al2O3 minus274807 0088794 minus095772
Table 6 It is observed form Table 6 that value of the skin-friction coefficient is more for the metallic nanoparticlesthan nonmetallic nanoparticles whereas the nonmetallicnanoparticles have higher heat and mass transfer rate incomparison with metallic nanoparticles
5 Conclusions
The present paper deals with the numerical solution ofcombined heat andmass transfer effects of an unsteadyMHDlaminar two-dimensional flow of incompressible viscousnanofluids inmiddle of two parallel plates extended infinitelywith slip velocity effect The relevant nonlinear partial dif-ferential equations were transformed to a set of ordinarydifferential equations and then are solved numerically usingthe Runge-Kutta-Fehlberg fourth-fifth-order method alongwith shooting technique From the above discussion the eye-catching results are as follows
(i) Temperature drops in Cu-water nanofluid withincrease in the magnetic field strength
(ii) The velocity of the nanofluid with nonmetallicnanoparticles Al2O3 and TiO2 is greater thanmetallic
nanoparticles Cu and Ag initially when water is thebase fluid but nature gets reversed after 120578 = 05The temperature of themetallic nanoparticles is lowerthan the nonmetallic nanoparticles
(iii) When plates start moving apart then temperaturestarts decreasing
(iv) Initially with the increasing values of magnetic fieldstrength and squeeze number the velocity of Cu-water slows down but the nature in both the case getsreversed after 120578 = 05
(v) The Nusselt number and skin-friction coefficientdecrease as plates move apart for Cu-water nanofluid
(vi) Nusselt number and Sherwood number for non-metallic nanoparticles are higher than the metallicnanoparticles
(vii) Varying value of slip parameter has negligible effecton concentration and as values of squeeze numberincrease the value of concentration profile also getsslightly higher
(viii) Considerable amount of enhancement in concentra-tion can be seen as Schmidt number rises
Nomenclature
1198601 1198602 1198603 Dimensionless constant1198610 Strength of magnetic field119862 Fluid concentration119862119891 Skin-friction coefficient119862119901 Specific heat at constant pressure119901 [Jkgminus1 Kminus1]119863 Mass diffusivity
10 Journal of Nanoscience
Ec Eckert number119891 Dimensionless velocity of the fluid119895 Reference length119896nf Nanofluid effective thermalconductivity119897 Distance of plate [m]119871 Slip length119872 Magnetic parameter119898119908 Surface mass flux
Nu119909 Nusselt number119901 Pressure [Pa]Pr Prandtl number119902119908 Surface heat fluxRe119909 Local Reynolds number119878 Squeeze numberSc Schmidt numberSh119909 Sherwood number119905 Time [s]119879 Temperature [K]119906 V Velocities in 119909119910-directions [m sminus1]119909 119910 Axial and perpendicular coordinates
[m]
Greek Symbols
120583119891 Dynamic viscosity of the fluid120593 Nanoparticle volume fraction120575 Velocity slip parameter120592119891 Kinematic viscosity of the fluid [m2 sminus1]120578 Similarity variable120588119891 Density of base fluid [kgmminus3]119896119904 Thermal conductivity of the solidnanoparticle [mminus1 Kminus1]119896119891 Thermal conductivity of the base fluid[Wmminus1 Kminus1]120579 Nondimensional temperature120601 Nondimensional concentration120588nf Nanofluid density [kgmminus3]120583nf Effective dynamic viscosity of nanofluid[Pa s]120572 Squeeze parameter120590nf Electrical conductivity of nanofluid120588119904 Density of solid particle [kgmminus3]120591119908 Surface shear stress
Subscripts
119891 Base fluidℎ Condition at the wallnf Nanofluid119904 Solid
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] M J Stefan ldquoVersuchuber die scheinbare adhesion sitzungs-bersachsrdquo Akademie der Wissenschaften in Wien Math-ematisch-Naturwissenschaftliche Klasse vol 69 pp 713ndash7211874
[2] G Domairry and M Hatami ldquoSqueezing Cundashwater nanofluidflow analysis between parallel plates by DTM-Pade MethodrdquoJournal of Molecular Liquids vol 193 pp 37ndash44 2014
[3] O Pourmehran M Rahimi-Gorji M Gorji-Bandpy and D DGanji ldquoAnalytical investigation of squeezing unsteady nanofluidflow between parallel plates by LSM and CMrdquo AlexandriaEngineering Journal vol 54 no 1 pp 17ndash26 2015
[4] A K Gupta and S S Ray ldquoNumerical treatment for investiga-tion of squeezing unsteady nanofluid flow between two parallelplatesrdquo Powder Technology vol 279 pp 282ndash289 2015
[5] U Khan N Ahmed M Asadullah and S T Mohyud-dinldquoEffects of viscous dissipation and slip velocity on two-dim-ensional and axisymmetric squeezing flow of Cu-water and Cu-kerosene nanofluidsrdquo Propulsion and Power Research vol 4 no1 pp 40ndash49 2015
[6] A Dib A Haiahem and B Bou-said ldquoApproximate analyticalsolution of squeezing unsteady nanofluid flowrdquo Powder Technol-ogy vol 269 pp 193ndash199 2015
[7] S U S Choi and J A Eastman ldquoEnhancing thermal con-ductivity of fluids with nanoparticlerdquo in Proceedings of theInternational Mechanical Engineering Congress and Exhibitionvol 231 pp 99ndash105 San Francisco Calif USA November 1995
[8] M Azimi and R Riazi ldquoHeat transfer analysis of GO-waternanofluid flow between two parallel disksrdquo Propulsion andPower Research vol 4 no 1 pp 23ndash30 2015
[9] A Aziz W A Khan and I Pop ldquoFree convection boundarylayer flow past a horizontal flat plate embedded in porousmedium filled by nanofluid containing gyrotactic microorgan-ismsrdquo International Journal of Thermal Sciences vol 56 pp 48ndash57 2012
[10] M Fakour D D Ganji and M Abbasi ldquoScrutiny of underde-veloped nanofluidMHD flow and heat conduction in a channelwith porous wallsrdquo Case Studies in Thermal Engineering vol 4pp 202ndash214 2014
[11] T Grosan C Revnic I Pop andD B Ingham ldquoFree convectionheat transfer in a square cavity filled with a porous mediumsaturated by a nano-fluidrdquo International Journal of Heat andMass Transfer vol 87 pp 36ndash41 2015
[12] A V Kuznetsov and D A Nield ldquoThe Cheng-Minkowyczproblem for natural convective boundary layer flow in a porousmedium saturated by a nanofluid a revised modelrdquo Inter-national Journal of Heat andMass Transfer vol 65 pp 682ndash6852013
[13] P Rana R Bhargava and O A Begb ldquoNumerical solution formixed convection boundary layerflow of ananofluid along aninclined plate embedded in a porous mediumrdquo Computers andMathematics with Applications vol 64 no 9 pp 2816ndash28322012
[14] A M Rashad A J Chamkha and M Modather ldquoMixed con-vection boundary-layer flow past a horizontal circular cylinderembedded in a porous medium filled with a nanofluid underconvective boundary conditionrdquo Computers amp Fluids vol 86pp 380ndash388 2013
[15] M Sheikholeslami and D D Ganji ldquoNanofluid flow and heattransfer between parallel plates considering Brownian motion
Journal of Nanoscience 11
using DTMrdquo Computer Methods in Applied Mechanics andEngineering vol 283 pp 651ndash663 2015
[16] N V Ganesh B Ganga and A K Abdul Hakeem ldquoLiesymmetry group analysis of magnetic field effects on free con-vective flow of a nanofluid over a semi-infinite stretching sheetrdquoJournal of the Egyptian Mathematical Society vol 22 no 2pp 304ndash310 2014
[17] Y Lin L Zheng and G Chen ldquoUnsteady flow and heattransfer of pseudo-plastic nanoliquid in a finite thin film ona stretching surface with variable thermal conductivity andviscous dissipationrdquo Powder Technology vol 274 pp 324ndash3322015
[18] M Sheikholeslami MM Rashidi and D D Ganji ldquoNumericalinvestigation of magnetic nanofluid forced convective heattransfer in existence of variable magnetic field using two phasemodelrdquo Journal of Molecular Liquids vol 212 pp 117ndash126 2015
[19] T Hayat M Waqas S A Shehzad and A Alsaedi ldquoMixedconvection flow of a Burgers nanofluid in the presence ofstratifications and heat generationabsorptionrdquo The EuropeanPhysical Journal Plus vol 131 no 8 p 253 2016
[20] F M Abbasi T Hayat S A Shehzad F Alsaadi and NAltoaibi ldquoHydromagnetic peristaltic transport of copper-waternanofluid with temperature-dependent effective viscosityrdquo Par-ticuology vol 27 pp 133ndash140 2016
[21] M Sheikholeslami D D Ganji and M M Rashidi ldquoMagneticfield effect on unsteady nanofluid flow and heat transferusing Buongiorno modelrdquo Journal of Magnetism and MagneticMaterials vol 416 pp 164ndash173 2016
[22] O Pourmehran M Rahimi-Gorji and D D Ganji ldquoHeattransfer and flow analysis of nanofluid flow induced by astretching sheet in the presence of an external magnetic fieldrdquoJournal of the Taiwan Institute of Chemical Engineers vol 65 pp162ndash171 2016
[23] M Rahimi-Gorji O Pourmehran M Hatami and D D GanjildquoStatistical optimization of microchannel heat sink (MCHS)geometry cooled by different nanofluids using RSM analysisrdquoEuropean Physical Journal Plus vol 130 no 2 article 22 pp 1ndash21 2015
[24] W Ibrahim and B Shankar ldquoMHD boundary layer flow andheat transfer of a nanofluid past a permeable stretching sheetwith velocity thermal and solutal slip boundary conditionsrdquoComputers amp Fluids vol 75 pp 1ndash10 2013
[25] A Malvandi and D D Ganji ldquoBrownian motion and ther-mophoresis effects on slip flow of aluminawater nanofluidinside a circular microchannel in the presence of a magneticfieldrdquo International Journal ofThermal Sciences vol 84 pp 196ndash206 2014
[26] R Ul Haq S Nadeem Z H Khan and N S Akbar ldquoThermalradiation and slip effects on MHD stagnation point flow ofnanofluid over a stretching sheetrdquo Physica E vol 65 pp 17ndash232015
[27] M Govindaraju S Saranya A A Hakeem R Jayaprakash andB Ganga ldquoAnalysis of slip MHD nanofluid flow on entropygeneration in a stretching sheetrdquo Procedia Engineering vol 127pp 501ndash507 2015
[28] M J Uddin M N Kabir and O A Beg ldquoComputationalinvestigation of Stefan blowing and multiple-slip effects onbuoyancy-driven bioconvection nanofluid flow with microor-ganismsrdquo International Journal of Heat and Mass Transfer vol95 pp 116ndash130 2016
[29] K-L Hsiao ldquoStagnation electrical MHD nanofluid mixed con-vection with slip boundary on a stretching sheetrdquo AppliedThermal Engineering vol 98 pp 850ndash861 2016
[30] P K Kameswaran M Narayana P Sibanda and P V S NMurthy ldquoHydromagnetic nanofluid flow due to a stretching orshrinking sheet with viscous dissipation and chemical reactioneffectsrdquo International Journal of Heat andMass Transfer vol 55no 25-26 pp 7587ndash7595 2012
[31] M H Matin and I Pop ldquoForced convection heat and masstransfer flow of a nanofluid through a porous channel with afirst order chemical reaction on thewallrdquo International Commu-nications in Heat and Mass Transfer vol 46 pp 134ndash141 2013
[32] D Pal andGMandal ldquoInfluence of thermal radiation onmixedconvection heat and mass transfer stagnation-point flow innanofluids over stretchingshrinking sheet in a porous mediumwith chemical reactionrdquo Nuclear Engineering and Design vol273 pp 644ndash652 2014
[33] C Zhang L Zheng X Zhang and G Chen ldquoMHD flow andradiation heat transfer of nanofluids in porous media withvariable surface heat flux and chemical reactionrdquoAppliedMath-ematical Modelling vol 39 no 1 pp 165ndash181 2015
[34] HM Elshehabey and S E Ahmed ldquoMHDmixed convection ina lid-driven cavity filled by a nanofluid with sinusoidal temper-ature distribution on the both vertical walls using Buongiornorsquosnanofluid modelrdquo International Journal of Heat and MassTransfer vol 88 pp 181ndash202 2015
Submit your manuscripts athttpwwwhindawicom
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Nano
materials
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Journal ofNanomaterials
Journal of Nanoscience 3
120597119879120597119905 + 119906120597119879120597119909 + V
120597119879120597119910
= 119896nf(120588119862119901)nf
(12059721198791205971199092 +
12059721198791205971199102 )
+ 120583nf(120588119862119901)nf
(4(120597119906120597119909)2 + (120597119906120597119910 + 120597119906
120597119909)2)
(4)
120597119862120597119905 + 119906120597119862120597119909 + V
120597119862120597119910 = 119863(1205972119862
1205971199092 +12059721198621205971199102 ) (5)
The associated boundary conditions are given as
119906 = minus119871120597119906120597119910
V = V119908 = 119889ℎ119889119905
119879 = 119879ℎ119862 = 119862ℎ at 119910 = ℎ (119905) V = 120597119906
120597119910 = 120597119879120597119910 = 0
119862 = 1198620 at 119910 = 0
(6)
where
120588nf = (1 minus 120593) 120588119891 + 120593120588119904(120588119862119901)nf = (1 minus 120593) (120588119862119901)119891 + 120593 (120588119862119901)119904 120583nf = 120583119891
(1 minus 120593)25 (Brinkman)
119896nf119896119891 = 119896119904 + 2119896119891 minus 2120593 (119896119891 minus 119896119904)119896119904 + 2119896119891 + 2120593 (119896119891 minus 119896119904) (Maxwell-Garnetts)
120590nf = (1 minus 120593) 120590119891 + 120593120590119904
(7)
Equations (2)ndash(5) can be converted to a system of nonlin-ear ordinary differential equations via the following similarityvariables
120578 = 119910119897 (1 minus 120572119905)12
119906 = 1205721199092 (1 minus 120572119905)1198911015840 (120578)
V = minus 1205721198972 (1 minus 120572119905)12119891 (120578)
120579 = 119879 minus 1198790119879ℎ minus 1198790
120601 = 119862 minus 1198620119862ℎ minus 1198620
(8)
The transformed equations are
119891119894V minus 1198781198601 (1 minus 120593)25 (120578119891101584010158401015840 + 311989110158401015840 + 119891101584011989110158401015840 minus 119891119891101584010158401015840)minus11987211989110158401015840 = 0
12057910158401015840 + Pr 119878 (11986021198603) (1205791015840119891 minus 1205781205791015840)
+ Pr Ec1198603 (1 minus 120593)25 [119891
101584010158402 + 4119895211989110158402] = 012060110158401015840 + 119878119888119878 (1198911206011015840 minus 1205781206011015840) = 0
(9)
where11986011198602 and1198603 are dimensionless constants defined asfollows
1198601 = (1 minus 120593) + 120593 120588119904120588119891
1198602 = (1 minus 120593) + 120593 (120588119862119901)119904(120588119862119901)119891
1198603 = 119896nf119896119891
(10)
The boundary conditions (6) in the terms of similarityvariables (8) become
119891 (0) = 011989110158401015840 (0) = 01205791015840 (0) = 0120601 (0) = 0119891 (1) = 11198911015840 (1) = minus12057511989110158401015840 (1) 120579 (1) = 1120601 (1) = 1
(11)
where 119878 = 12057211989722]119891 is the squeeze number Pr = 120583119891(120588119862119901)119891120588119891119896119891 is the Prandtl number Sc = 120592119891119863nf is Schmidt number119872 = 2120590nf11986120(ℎ(119905))2120588119891120583nf is the magnetic parameter 119895 = 119897119909is the reference length and 120575 = 119871119897(1 minus 120572119905)12 is the velocityslip parameter and Ec = (120588119891(120588119862119901)119891)(1205721199092(1 minus 120572119905))2 is theEckert number
The physical quantities of interest are the skin-frictioncoefficient 119862119891 the Nusselt number Nu119909 and the Sherwoodnumber Sh119909 defined as
119862119891 = 120591119908120588nfV2119908
Nu119909 = 119897119902119908119896119891 (119879ℎ minus 1198790)
Sh119909 = 119897119898119908119863nf (119862ℎ minus 1198620)
(12)
4 Journal of Nanoscience
where
120591119908 = 120583nf (120597119906120597119910)119910=ℎ(119905)
119902119908 = minus119896nf (120597119879120597119910 )119910=ℎ(119905)
119898119908 = minus119863nf (120597119862120597119910 )119910=ℎ(119905)
(13)
Using (8) and (13) in (12) we get
119862119891lowast = 1199092 (1 minus 120572119905)Re1199091198621198911198972 = 11989110158401015840 (1)
1198601 (1 minus 120601)25 Nu119909lowast = radic1 minus 120572119905Nu119909 = minus11986031205791015840 (1)
Sh119909lowast = radic1 minus 120572119905Sh119909 = minus1206011015840 (1)
(14)
where Re119909 = 120572119897521199093(1 minus 120572119905)12120592119891 is the local Reynoldsnumber
3 Method of Solution
In this present paper Runge-Kutta-Fehlberg fourth-fifth-ordermethod (RKF45) has been employed to solve the systemof nonlinear ordinary differential (9) with the boundaryconditions given by (11) for different values of governingparametersTheRKF45method has a procedure to determineif the suitable step size ℎ is being used The formula offifth-order Runge-Kutta-Fehlberg method can be defined asfollows
119911119899+1 = 119911119899+ ( 16
1351198960 +6656128251198962 +
28561564301198963 minus
9501198964 +
2551198965)
sdot ℎ(15)
where the coefficients 1198960 to 1198965 are defined as follows
1198960 = 119891 (119909119899 119910119899) 1198961 = 119891(119909119899 + 1
4ℎ 119910119899 +14ℎ1198960)
1198962 = 119891(119909119899 + 38ℎ 119910119899 + ( 3
321198960 +9321198961) ℎ)
1198963 = 119891(119909119899 + 1213ℎ 119910119899
+ (193221971198960 minus720021971198961 +
729621971198962) ℎ)
1198964 = 119891(119909119899 + ℎ 119910119899+ (4392161198960 minus 81198961 + 3680
513 1198962 minus 84541041198963) ℎ)
1198965 = 119891(119909119899 + 12ℎ 119910119899
+ (minus 8271198960 + 21198961 minus 3544
25651198962 +185941041198963 minus
11401198964) ℎ)
(16)
The computation of the error can be achieved by subtractingthe fifth-order from the fourth-order method
119910119899+1 = 119910119899 + ( 252161198960 +
140825651198962 +
219741041198963 minus
151198964) ℎ (17)
If the error goes beyond a specified antechamber the resultscan be recalculated using a smaller step size The approach tocomputing the new step size is shown as follows
ℎnew = ℎold ( 120576ℎold2 1003816100381610038161003816119911119899+1 minus 119910119899+11003816100381610038161003816)14
(18)
The variation of the dimensionless velocity temperature andconcentration is ensured to be less than 10minus6 between any twoconsecutive iterations for the convergence criterion
To solve the nonlinear differential equations (9) subjectto the boundary conditions (11) first boundary conditions for120578 = 1 are replaced by 119891(1) = 1 1198911015840(1) = minus12057511989110158401015840(1) 120579(1) = 1and 120601(1) = 1 We consider that 119891 = 1198911 1198911015840 = 1198912 11989110158401015840 =1198913 119891101584010158401015840 = 1198914 120579 = 1198915 1205791015840 = 1198916 120601 = 1198917 and 1206011015840 = 1198918Thenonlinear equations (9) are first converted into first-orderordinary linear differential equations as follows
1198911015840 = 119891211989110158402 = 119891311989110158403 = 119891411989110158404 = 1198781198601 (1 minus 120593)25 (1205781198914 + 31198913 + 11989121198913 minus 11989111198914) + 119872119891311989110158405 = 119891611989110158406 = minusPr 119878 (11986021198603) (11989161198911 minus 1205781198916)
minus Pr Ec1198603 (1 minus 120593)25 [1198913
2 + 4119895211989122] 11989110158407 = 119891811989110158408 = minus119878119888119878 (11989111198918 minus 1205781198918)
(19)
Journal of Nanoscience 5
Table 2 Comparison of 119891(120578) and 120579(120578) for Cu-water nanofluid when 119878 = 1 Pr = 62 Ec = 119895 = 001 120593 = 002 and119872 = Sc = 120575 = 0120578 Gupta and Ray [4] Present result
119891(120578) 120579(120578) 119891(120578) 120579(120578)0 0 103206637 0 10299663701 014135866 103206407 014135879 10299642802 028066605 103203202 028066639 10299353103 041578075 103189263 041578137 10298093204 054437882 103150811 054437979 10294617505 066385692 103066423 066385837 10286989606 077122923 102903983 077123132 10272306607 086301562 102615205 086301853 10246203708 093511971 102125916 093512364 10201976509 098269524 101318608 098270044 10129003110 100000000 100000000 100000000 100000000
subject to the following initial conditions
1198911 (0) = 01198912 (0) = 12057211198913 (0) = 01198914 (0) = 12057221198915 (0) = 12057231198916 (0) = 01198917 (0) = 01198918 (0) = 1205724
(20)
We ran the computer code written in the MATLAB fordifferent values of step lengthnabla120578We found that there is no oronly a negligible change in the physical quantities of interestlike the skin-friction coefficient the couple stress coefficientNusselt number and Sherwood number for various values ofnabla120578 gt 001 Therefore in the present paper we have set stepsize nabla120578 = 001 There are four initial conditions at 120578 = 0 andfour conditions on boundary 120578 = 1 To get the solution of theproblem four more initial conditions at 120578 = 0 that is valuesof 1205721 1205722 1205723 and 1205724 are required which have been obtainedby shooting technique Finally the transformed initial valueproblem is solved by employing the Runge-Kutta-Fehlbergfourth-fifth-order method along with calculated boundaryconditions
4 Results and Discussion
In order to validate the numerical results obtained wecompare our results with those reported by Gupta and Ray[4] as shown in Table 2 and they are found to be in agood agreement The effects of the volume fraction of solidnanoparticles magnetic parameter velocity slip parametersqueeze number and Schmidt number are inspected fordifferent kinds of nanoparticles when the base fluid is waterEc = 001 Pr = 62 and 119895 = 001
00 02 04 06 08 1002
04
06
08
10
12
14
M = 0M = 1M = 2
Cu-water
f998400 (120578)
120578
120593 = 002 120575 = 01 S = 1 Sc = 1
Figure 2 Effects of119872 on velocity 1198911015840(120578)
Figures 2 and 3 illustrate the behavior of the velocity1198911015840(120578)and temperature 120579(120578) of the Cu-water nanofluid for differentvalues of magnetic parameter 119872 Figure 2 shows that withan increase in the values of119872 the velocity decreases near thelower plate surface but after a certain distance it increasesIt is noticed from Figure 3 that the temperature decreasesmonotonically with increasing values of119872
The variations of the velocity temperature and concen-tration for different values of slip parameter 120575 are shownin Figures 4ndash6 It is noted from Figure 4 that near the wallthe value of velocity 1198911015840(120578) increases with rising values of 120575when 0 le 120578 le 06 and for 120578 ge 06 the velocity decreasesas 120575 increases Figure 5 depicts that a rise in the values ofslip parameter 120575 increases temperature monotonically It isevident from Figure 6 that the change in the concentration120601(120578) with rising values of slip parameter 120575 is negligible
6 Journal of Nanoscience
00 02 04 06 08 101000
1002
1004
1006
1008
1010
1012
1014
1016
1018
M = 0M = 1M = 2
Cu-water
120578
120579(120578
)
120593 = 002 120575 = 01 S = 1 Sc = 1
Figure 3 Effects of119872 on temperature 120579(120578)
00 02 04 06 08 1000
02
04
06
08
10
12
14Cu-water
f998400 (120578)
120578
120575 = 01
120575 = 02
120575 = 05
120593 = 002 M = 1 S = 1 Sc = 1
Figure 4 Effects of 120575 on velocity 1198911015840(120578)
The effects of the squeeze number 119878 on velocity temper-ature and concentration profiles are depicted in Figures 7ndash9Physically the squeeze number (119878) describes the movementof the plates (119878 gt 0 corresponds to the plates moving apartwhile 119878 lt 0 corresponds to the plates moving together) Itcan be easily seen from Figure 7 that the value of velocity1198911015840(120578) near the lower plate surface decreases regularly withthe increase in the value of 119878 and as we move away fromlower plate surface this value increases Figure 8 showsthat increasing value of squeeze number 119878 decreases thetemperature 120579(120578) Figure 9 depicts that the value of 120601(120578) goes
00 02 04 06 08 101000
1005
1010
1015
1020
1025
Cu-water
120578
120579(120578
)
120593 = 002 M = 1 S = 1 Sc = 1
120575 = 01
120575 = 02
120575 = 05
Figure 5 Effects of 120575 on temperature 120579(120578)
00 02 04 06 08 1000
02
04
06
08
10
Cu-water
120578
120575 = 01
120575 = 02
120575 = 05
120593 = 002 M = 1 S = 1 Sc = 1
120601(120578
)
Figure 6 Effects of 120575 on concentration 120601(120578)
up with increasing 120578 and high squeeze number implies slightdrop in concentration 120601(120578)
Figures 10 and 11 show the effects of the volume fraction120593 on the velocity and temperature profiles Figure 10 exhibitsinitially an increase in the values of nanoparticles volumefraction 120593 the velocity 1198911015840(120578) decreases and after a fixeddistance from lower plate surface it increases slightly FromFigure 11 it is observed that the temperature decreases withincreasing values of 120593
Figures 12 and 13 illiterate the effect of the differentnanoparticles on velocity and temperature profile whenthe base fluid is water It is observed from Figure 12 that
Journal of Nanoscience 7
00 02 04 06 08 10
02
04
06
08
10
12
14
16Cu-water
f998400 (120578)
120578
S = minus10S = minus05S = 00
S = 05
S = 10
120593 = 002 M = 1 120575 = 01 Sc = 1
Figure 7 Effects of 119878 on velocity 1198911015840(120578)
00 02 04 06 08 10100
101
102
103
104
105
106
107
120578
120579(120578
)
Cu-water
S = minus10S = minus05S = 00
S = 05
S = 10
120593 = 002 M = 1 120575 = 01 Sc = 1
Figure 8 Effects of 119878 on temperature 120579(120578)
the different nanofluids have different velocities and alsonoted that titanium oxide (TiO2) has higher velocity whencompared to other nanoparticles such as Al2O3 Cu andAg for 0 le 120578 le 05 while for 120578 gt 05 the results getreversed It is noted from Figure 13 that the order of nanofluidfor decreasing value of 120579(120578) when 120578 change from 0 to 1 isTiO2-water Al2O3-water Cu-water and Ag-water nanofluidFigure 14 depicts the effect of the Schmidt number Sc onconcentration profile The values of the temperature increasewith the increase in the value of Sc
The effects of the squeeze number 119878 on the skin-frictioncoefficient119862119891lowast the Nusselt number Nu119909
lowast and the Sherwood
00 02 04 06 08 1000
02
04
06
08
10
120578
Cu-water
S = minus10S = minus05S = 00
S = 05
S = 10
120593 = 002 M = 1 120575 = 01 Sc = 1
120601(120578
)
Figure 9 Effects of 119878 on concentration 120601(120578)
00 02 04 06 08 1002
04
06
08
10
12
14
120578
Cu-water
f998400 (120578)
M = 1 120575 = 01 Sc = 1
120593 = 000
120593 = 002120593 = 005
Figure 10 Effects of 120593 on velocity 1198911015840(120578)
number Sh119909lowast are given in Table 3 From Table 3 it is obvious
that the skin-friction coefficient and the Nusselt number areinversely proportional to 119878 whereas the Sherwood number isdirectly proportional to 119878
Table 4 displays the effects of the skin-friction coefficientthe Nusselt number and Sherwood number for differentvalues of the magnetic parameter119872 and slip parameter 120575 Itis noticed from the table that the effect of increasing valuesof119872 is to increase the skin-friction coefficient 119862119891lowast the heattransfer rate Nu119909
lowast andmass transfer rate Sh119909lowast Further from
Table 4 it is concluded that the increasing value of 120575 decreases
8 Journal of Nanoscience
00 02 04 06 08 101000
1002
1004
1006
1008
1010
1012
1014
1016
1018
120578
Cu-water
M = 1 120575 = 01 Sc = 1
120593 = 000
120593 = 002120593 = 005
120579(120578
)
Figure 11 Effects of 120593 on temperature 120579(120578)
00 02 04 06 08 1002
04
06
08
10
12
14
120578
f998400 (120578)
AgCu
Al2O3
TiO2
120593 = 002 M = 1 120575 = 01 S = 1 Sc = 1
Figure 12 Velocity of different nanofluids
the skin-friction coefficient and increases the heat and masstransfer rate
The effects of the nanoparticle volume fraction 120593 andSchmidt number Sc on the skin-friction coefficient 119862119891lowastNusselt number (the heat transfer rate) Nu119909
lowast and Sherwoodnumber (the mass transfer rate) Sh119909
lowast are given in Table 5From this table it is concluded that the increasing value of 120593increases the skin-friction coefficient and decreases the heatand mass transfer rate In addition the rate of mass transferincreases with the increase in Sc but there is no effect of Scon the skin-friction coefficient and the heat transfer rate
00 02 04 06 08 101000
1002
1004
1006
1008
1010
1012
1014
1016
120578
120579(120578
)
AgCu
Al2O3
TiO2
120593 = 002 M = 1 120575 = 01 S = 1 Sc = 1
Figure 13 Temperature of different nanofluids
00 02 04 06 08 1000
02
04
06
08
10
Sc = 0Sc = 1Sc = 10
Sc = 50Sc = 100
120578
Cu-water
120593 = 002 M = 1 120575 = 01 S = 1
120601(120578
)
Figure 14 Effects of Sc on concentration 120601(120578)
Table 3 Variation of 119862119891lowast Nu119909lowast and Sh119909lowast with different values
of Squeeze number 119878 for Cu-water nanofluid when 119872 = 1 Sc =1 120575 = 01 and 120593 = 002119878 119862119891lowast Nu119909
lowast Sh119909lowast
minus10 minus134295 0187977 minus10694254minus05 minus177787 0127657 minus1029323300 minus209419 0104693 minus1000000005 minus234148 0094096 minus0977084110 minus254426 0088577 minus09583107
The comparison for metallic nanoparticles (Ag Cu)and nonmetallic nanoparticles (Al2O3 TiO2) is done in
Journal of Nanoscience 9
Table 4 Variation of119862119891lowast Nu119909lowast and Sh119909lowast with different values of magnetic parameter119872 and slip parameter 120575 for Cu-water nanofluid when119878 = 1 Sc = 1 and 120593 = 002119872 120575 119862119891lowast Nu119909
lowast Sh119909lowast
0 01 minus246084 0089198 minus0956761 01 minus254426 0088577 minus0958312 01 minus26223 0088075 minus0959741 02 minus295257 0118598 minus0951581 05 minus326534 0144432 minus094633Table 5 Variation of 119862119891lowast Nu119909lowast and Sh119909lowast with different values of volume fraction 120593 and Schmidt number Sc for Cu-water nanofluid when119872 = 1 119878 = 1 and 120575 = 01120593 Sc 119862119891lowast Nu119909
lowast Sh119909lowast
000 1 minus276550 0092885 minus095773002 1 minus254426 0088577 minus095831005 1 minus231844 0082295 minus095911002 0 minus254426 0088577 minus100000002 10 minus254426 0088577 minus063318002 50 minus254426 0088577 minus005952002 100 minus254426 0088577 minus000182
Table 6 Variation of 119862119891lowast Nu119909lowast and Sh119909lowast for different nanoparti-
cles when119872 = 1 119878 = 1 Sc = 1 120575 = 01 and 120593 = 002Nanoparticle 119862119891lowast Nu119909
lowast Sh119909lowast
Ag minus248678 0088535 minus095851Cu minus254426 0088577 minus095831TiO2 minus273607 0088742 minus095777Al2O3 minus274807 0088794 minus095772
Table 6 It is observed form Table 6 that value of the skin-friction coefficient is more for the metallic nanoparticlesthan nonmetallic nanoparticles whereas the nonmetallicnanoparticles have higher heat and mass transfer rate incomparison with metallic nanoparticles
5 Conclusions
The present paper deals with the numerical solution ofcombined heat andmass transfer effects of an unsteadyMHDlaminar two-dimensional flow of incompressible viscousnanofluids inmiddle of two parallel plates extended infinitelywith slip velocity effect The relevant nonlinear partial dif-ferential equations were transformed to a set of ordinarydifferential equations and then are solved numerically usingthe Runge-Kutta-Fehlberg fourth-fifth-order method alongwith shooting technique From the above discussion the eye-catching results are as follows
(i) Temperature drops in Cu-water nanofluid withincrease in the magnetic field strength
(ii) The velocity of the nanofluid with nonmetallicnanoparticles Al2O3 and TiO2 is greater thanmetallic
nanoparticles Cu and Ag initially when water is thebase fluid but nature gets reversed after 120578 = 05The temperature of themetallic nanoparticles is lowerthan the nonmetallic nanoparticles
(iii) When plates start moving apart then temperaturestarts decreasing
(iv) Initially with the increasing values of magnetic fieldstrength and squeeze number the velocity of Cu-water slows down but the nature in both the case getsreversed after 120578 = 05
(v) The Nusselt number and skin-friction coefficientdecrease as plates move apart for Cu-water nanofluid
(vi) Nusselt number and Sherwood number for non-metallic nanoparticles are higher than the metallicnanoparticles
(vii) Varying value of slip parameter has negligible effecton concentration and as values of squeeze numberincrease the value of concentration profile also getsslightly higher
(viii) Considerable amount of enhancement in concentra-tion can be seen as Schmidt number rises
Nomenclature
1198601 1198602 1198603 Dimensionless constant1198610 Strength of magnetic field119862 Fluid concentration119862119891 Skin-friction coefficient119862119901 Specific heat at constant pressure119901 [Jkgminus1 Kminus1]119863 Mass diffusivity
10 Journal of Nanoscience
Ec Eckert number119891 Dimensionless velocity of the fluid119895 Reference length119896nf Nanofluid effective thermalconductivity119897 Distance of plate [m]119871 Slip length119872 Magnetic parameter119898119908 Surface mass flux
Nu119909 Nusselt number119901 Pressure [Pa]Pr Prandtl number119902119908 Surface heat fluxRe119909 Local Reynolds number119878 Squeeze numberSc Schmidt numberSh119909 Sherwood number119905 Time [s]119879 Temperature [K]119906 V Velocities in 119909119910-directions [m sminus1]119909 119910 Axial and perpendicular coordinates
[m]
Greek Symbols
120583119891 Dynamic viscosity of the fluid120593 Nanoparticle volume fraction120575 Velocity slip parameter120592119891 Kinematic viscosity of the fluid [m2 sminus1]120578 Similarity variable120588119891 Density of base fluid [kgmminus3]119896119904 Thermal conductivity of the solidnanoparticle [mminus1 Kminus1]119896119891 Thermal conductivity of the base fluid[Wmminus1 Kminus1]120579 Nondimensional temperature120601 Nondimensional concentration120588nf Nanofluid density [kgmminus3]120583nf Effective dynamic viscosity of nanofluid[Pa s]120572 Squeeze parameter120590nf Electrical conductivity of nanofluid120588119904 Density of solid particle [kgmminus3]120591119908 Surface shear stress
Subscripts
119891 Base fluidℎ Condition at the wallnf Nanofluid119904 Solid
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] M J Stefan ldquoVersuchuber die scheinbare adhesion sitzungs-bersachsrdquo Akademie der Wissenschaften in Wien Math-ematisch-Naturwissenschaftliche Klasse vol 69 pp 713ndash7211874
[2] G Domairry and M Hatami ldquoSqueezing Cundashwater nanofluidflow analysis between parallel plates by DTM-Pade MethodrdquoJournal of Molecular Liquids vol 193 pp 37ndash44 2014
[3] O Pourmehran M Rahimi-Gorji M Gorji-Bandpy and D DGanji ldquoAnalytical investigation of squeezing unsteady nanofluidflow between parallel plates by LSM and CMrdquo AlexandriaEngineering Journal vol 54 no 1 pp 17ndash26 2015
[4] A K Gupta and S S Ray ldquoNumerical treatment for investiga-tion of squeezing unsteady nanofluid flow between two parallelplatesrdquo Powder Technology vol 279 pp 282ndash289 2015
[5] U Khan N Ahmed M Asadullah and S T Mohyud-dinldquoEffects of viscous dissipation and slip velocity on two-dim-ensional and axisymmetric squeezing flow of Cu-water and Cu-kerosene nanofluidsrdquo Propulsion and Power Research vol 4 no1 pp 40ndash49 2015
[6] A Dib A Haiahem and B Bou-said ldquoApproximate analyticalsolution of squeezing unsteady nanofluid flowrdquo Powder Technol-ogy vol 269 pp 193ndash199 2015
[7] S U S Choi and J A Eastman ldquoEnhancing thermal con-ductivity of fluids with nanoparticlerdquo in Proceedings of theInternational Mechanical Engineering Congress and Exhibitionvol 231 pp 99ndash105 San Francisco Calif USA November 1995
[8] M Azimi and R Riazi ldquoHeat transfer analysis of GO-waternanofluid flow between two parallel disksrdquo Propulsion andPower Research vol 4 no 1 pp 23ndash30 2015
[9] A Aziz W A Khan and I Pop ldquoFree convection boundarylayer flow past a horizontal flat plate embedded in porousmedium filled by nanofluid containing gyrotactic microorgan-ismsrdquo International Journal of Thermal Sciences vol 56 pp 48ndash57 2012
[10] M Fakour D D Ganji and M Abbasi ldquoScrutiny of underde-veloped nanofluidMHD flow and heat conduction in a channelwith porous wallsrdquo Case Studies in Thermal Engineering vol 4pp 202ndash214 2014
[11] T Grosan C Revnic I Pop andD B Ingham ldquoFree convectionheat transfer in a square cavity filled with a porous mediumsaturated by a nano-fluidrdquo International Journal of Heat andMass Transfer vol 87 pp 36ndash41 2015
[12] A V Kuznetsov and D A Nield ldquoThe Cheng-Minkowyczproblem for natural convective boundary layer flow in a porousmedium saturated by a nanofluid a revised modelrdquo Inter-national Journal of Heat andMass Transfer vol 65 pp 682ndash6852013
[13] P Rana R Bhargava and O A Begb ldquoNumerical solution formixed convection boundary layerflow of ananofluid along aninclined plate embedded in a porous mediumrdquo Computers andMathematics with Applications vol 64 no 9 pp 2816ndash28322012
[14] A M Rashad A J Chamkha and M Modather ldquoMixed con-vection boundary-layer flow past a horizontal circular cylinderembedded in a porous medium filled with a nanofluid underconvective boundary conditionrdquo Computers amp Fluids vol 86pp 380ndash388 2013
[15] M Sheikholeslami and D D Ganji ldquoNanofluid flow and heattransfer between parallel plates considering Brownian motion
Journal of Nanoscience 11
using DTMrdquo Computer Methods in Applied Mechanics andEngineering vol 283 pp 651ndash663 2015
[16] N V Ganesh B Ganga and A K Abdul Hakeem ldquoLiesymmetry group analysis of magnetic field effects on free con-vective flow of a nanofluid over a semi-infinite stretching sheetrdquoJournal of the Egyptian Mathematical Society vol 22 no 2pp 304ndash310 2014
[17] Y Lin L Zheng and G Chen ldquoUnsteady flow and heattransfer of pseudo-plastic nanoliquid in a finite thin film ona stretching surface with variable thermal conductivity andviscous dissipationrdquo Powder Technology vol 274 pp 324ndash3322015
[18] M Sheikholeslami MM Rashidi and D D Ganji ldquoNumericalinvestigation of magnetic nanofluid forced convective heattransfer in existence of variable magnetic field using two phasemodelrdquo Journal of Molecular Liquids vol 212 pp 117ndash126 2015
[19] T Hayat M Waqas S A Shehzad and A Alsaedi ldquoMixedconvection flow of a Burgers nanofluid in the presence ofstratifications and heat generationabsorptionrdquo The EuropeanPhysical Journal Plus vol 131 no 8 p 253 2016
[20] F M Abbasi T Hayat S A Shehzad F Alsaadi and NAltoaibi ldquoHydromagnetic peristaltic transport of copper-waternanofluid with temperature-dependent effective viscosityrdquo Par-ticuology vol 27 pp 133ndash140 2016
[21] M Sheikholeslami D D Ganji and M M Rashidi ldquoMagneticfield effect on unsteady nanofluid flow and heat transferusing Buongiorno modelrdquo Journal of Magnetism and MagneticMaterials vol 416 pp 164ndash173 2016
[22] O Pourmehran M Rahimi-Gorji and D D Ganji ldquoHeattransfer and flow analysis of nanofluid flow induced by astretching sheet in the presence of an external magnetic fieldrdquoJournal of the Taiwan Institute of Chemical Engineers vol 65 pp162ndash171 2016
[23] M Rahimi-Gorji O Pourmehran M Hatami and D D GanjildquoStatistical optimization of microchannel heat sink (MCHS)geometry cooled by different nanofluids using RSM analysisrdquoEuropean Physical Journal Plus vol 130 no 2 article 22 pp 1ndash21 2015
[24] W Ibrahim and B Shankar ldquoMHD boundary layer flow andheat transfer of a nanofluid past a permeable stretching sheetwith velocity thermal and solutal slip boundary conditionsrdquoComputers amp Fluids vol 75 pp 1ndash10 2013
[25] A Malvandi and D D Ganji ldquoBrownian motion and ther-mophoresis effects on slip flow of aluminawater nanofluidinside a circular microchannel in the presence of a magneticfieldrdquo International Journal ofThermal Sciences vol 84 pp 196ndash206 2014
[26] R Ul Haq S Nadeem Z H Khan and N S Akbar ldquoThermalradiation and slip effects on MHD stagnation point flow ofnanofluid over a stretching sheetrdquo Physica E vol 65 pp 17ndash232015
[27] M Govindaraju S Saranya A A Hakeem R Jayaprakash andB Ganga ldquoAnalysis of slip MHD nanofluid flow on entropygeneration in a stretching sheetrdquo Procedia Engineering vol 127pp 501ndash507 2015
[28] M J Uddin M N Kabir and O A Beg ldquoComputationalinvestigation of Stefan blowing and multiple-slip effects onbuoyancy-driven bioconvection nanofluid flow with microor-ganismsrdquo International Journal of Heat and Mass Transfer vol95 pp 116ndash130 2016
[29] K-L Hsiao ldquoStagnation electrical MHD nanofluid mixed con-vection with slip boundary on a stretching sheetrdquo AppliedThermal Engineering vol 98 pp 850ndash861 2016
[30] P K Kameswaran M Narayana P Sibanda and P V S NMurthy ldquoHydromagnetic nanofluid flow due to a stretching orshrinking sheet with viscous dissipation and chemical reactioneffectsrdquo International Journal of Heat andMass Transfer vol 55no 25-26 pp 7587ndash7595 2012
[31] M H Matin and I Pop ldquoForced convection heat and masstransfer flow of a nanofluid through a porous channel with afirst order chemical reaction on thewallrdquo International Commu-nications in Heat and Mass Transfer vol 46 pp 134ndash141 2013
[32] D Pal andGMandal ldquoInfluence of thermal radiation onmixedconvection heat and mass transfer stagnation-point flow innanofluids over stretchingshrinking sheet in a porous mediumwith chemical reactionrdquo Nuclear Engineering and Design vol273 pp 644ndash652 2014
[33] C Zhang L Zheng X Zhang and G Chen ldquoMHD flow andradiation heat transfer of nanofluids in porous media withvariable surface heat flux and chemical reactionrdquoAppliedMath-ematical Modelling vol 39 no 1 pp 165ndash181 2015
[34] HM Elshehabey and S E Ahmed ldquoMHDmixed convection ina lid-driven cavity filled by a nanofluid with sinusoidal temper-ature distribution on the both vertical walls using Buongiornorsquosnanofluid modelrdquo International Journal of Heat and MassTransfer vol 88 pp 181ndash202 2015
Submit your manuscripts athttpwwwhindawicom
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Nano
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Journal ofNanomaterials
4 Journal of Nanoscience
where
120591119908 = 120583nf (120597119906120597119910)119910=ℎ(119905)
119902119908 = minus119896nf (120597119879120597119910 )119910=ℎ(119905)
119898119908 = minus119863nf (120597119862120597119910 )119910=ℎ(119905)
(13)
Using (8) and (13) in (12) we get
119862119891lowast = 1199092 (1 minus 120572119905)Re1199091198621198911198972 = 11989110158401015840 (1)
1198601 (1 minus 120601)25 Nu119909lowast = radic1 minus 120572119905Nu119909 = minus11986031205791015840 (1)
Sh119909lowast = radic1 minus 120572119905Sh119909 = minus1206011015840 (1)
(14)
where Re119909 = 120572119897521199093(1 minus 120572119905)12120592119891 is the local Reynoldsnumber
3 Method of Solution
In this present paper Runge-Kutta-Fehlberg fourth-fifth-ordermethod (RKF45) has been employed to solve the systemof nonlinear ordinary differential (9) with the boundaryconditions given by (11) for different values of governingparametersTheRKF45method has a procedure to determineif the suitable step size ℎ is being used The formula offifth-order Runge-Kutta-Fehlberg method can be defined asfollows
119911119899+1 = 119911119899+ ( 16
1351198960 +6656128251198962 +
28561564301198963 minus
9501198964 +
2551198965)
sdot ℎ(15)
where the coefficients 1198960 to 1198965 are defined as follows
1198960 = 119891 (119909119899 119910119899) 1198961 = 119891(119909119899 + 1
4ℎ 119910119899 +14ℎ1198960)
1198962 = 119891(119909119899 + 38ℎ 119910119899 + ( 3
321198960 +9321198961) ℎ)
1198963 = 119891(119909119899 + 1213ℎ 119910119899
+ (193221971198960 minus720021971198961 +
729621971198962) ℎ)
1198964 = 119891(119909119899 + ℎ 119910119899+ (4392161198960 minus 81198961 + 3680
513 1198962 minus 84541041198963) ℎ)
1198965 = 119891(119909119899 + 12ℎ 119910119899
+ (minus 8271198960 + 21198961 minus 3544
25651198962 +185941041198963 minus
11401198964) ℎ)
(16)
The computation of the error can be achieved by subtractingthe fifth-order from the fourth-order method
119910119899+1 = 119910119899 + ( 252161198960 +
140825651198962 +
219741041198963 minus
151198964) ℎ (17)
If the error goes beyond a specified antechamber the resultscan be recalculated using a smaller step size The approach tocomputing the new step size is shown as follows
ℎnew = ℎold ( 120576ℎold2 1003816100381610038161003816119911119899+1 minus 119910119899+11003816100381610038161003816)14
(18)
The variation of the dimensionless velocity temperature andconcentration is ensured to be less than 10minus6 between any twoconsecutive iterations for the convergence criterion
To solve the nonlinear differential equations (9) subjectto the boundary conditions (11) first boundary conditions for120578 = 1 are replaced by 119891(1) = 1 1198911015840(1) = minus12057511989110158401015840(1) 120579(1) = 1and 120601(1) = 1 We consider that 119891 = 1198911 1198911015840 = 1198912 11989110158401015840 =1198913 119891101584010158401015840 = 1198914 120579 = 1198915 1205791015840 = 1198916 120601 = 1198917 and 1206011015840 = 1198918Thenonlinear equations (9) are first converted into first-orderordinary linear differential equations as follows
1198911015840 = 119891211989110158402 = 119891311989110158403 = 119891411989110158404 = 1198781198601 (1 minus 120593)25 (1205781198914 + 31198913 + 11989121198913 minus 11989111198914) + 119872119891311989110158405 = 119891611989110158406 = minusPr 119878 (11986021198603) (11989161198911 minus 1205781198916)
minus Pr Ec1198603 (1 minus 120593)25 [1198913
2 + 4119895211989122] 11989110158407 = 119891811989110158408 = minus119878119888119878 (11989111198918 minus 1205781198918)
(19)
Journal of Nanoscience 5
Table 2 Comparison of 119891(120578) and 120579(120578) for Cu-water nanofluid when 119878 = 1 Pr = 62 Ec = 119895 = 001 120593 = 002 and119872 = Sc = 120575 = 0120578 Gupta and Ray [4] Present result
119891(120578) 120579(120578) 119891(120578) 120579(120578)0 0 103206637 0 10299663701 014135866 103206407 014135879 10299642802 028066605 103203202 028066639 10299353103 041578075 103189263 041578137 10298093204 054437882 103150811 054437979 10294617505 066385692 103066423 066385837 10286989606 077122923 102903983 077123132 10272306607 086301562 102615205 086301853 10246203708 093511971 102125916 093512364 10201976509 098269524 101318608 098270044 10129003110 100000000 100000000 100000000 100000000
subject to the following initial conditions
1198911 (0) = 01198912 (0) = 12057211198913 (0) = 01198914 (0) = 12057221198915 (0) = 12057231198916 (0) = 01198917 (0) = 01198918 (0) = 1205724
(20)
We ran the computer code written in the MATLAB fordifferent values of step lengthnabla120578We found that there is no oronly a negligible change in the physical quantities of interestlike the skin-friction coefficient the couple stress coefficientNusselt number and Sherwood number for various values ofnabla120578 gt 001 Therefore in the present paper we have set stepsize nabla120578 = 001 There are four initial conditions at 120578 = 0 andfour conditions on boundary 120578 = 1 To get the solution of theproblem four more initial conditions at 120578 = 0 that is valuesof 1205721 1205722 1205723 and 1205724 are required which have been obtainedby shooting technique Finally the transformed initial valueproblem is solved by employing the Runge-Kutta-Fehlbergfourth-fifth-order method along with calculated boundaryconditions
4 Results and Discussion
In order to validate the numerical results obtained wecompare our results with those reported by Gupta and Ray[4] as shown in Table 2 and they are found to be in agood agreement The effects of the volume fraction of solidnanoparticles magnetic parameter velocity slip parametersqueeze number and Schmidt number are inspected fordifferent kinds of nanoparticles when the base fluid is waterEc = 001 Pr = 62 and 119895 = 001
00 02 04 06 08 1002
04
06
08
10
12
14
M = 0M = 1M = 2
Cu-water
f998400 (120578)
120578
120593 = 002 120575 = 01 S = 1 Sc = 1
Figure 2 Effects of119872 on velocity 1198911015840(120578)
Figures 2 and 3 illustrate the behavior of the velocity1198911015840(120578)and temperature 120579(120578) of the Cu-water nanofluid for differentvalues of magnetic parameter 119872 Figure 2 shows that withan increase in the values of119872 the velocity decreases near thelower plate surface but after a certain distance it increasesIt is noticed from Figure 3 that the temperature decreasesmonotonically with increasing values of119872
The variations of the velocity temperature and concen-tration for different values of slip parameter 120575 are shownin Figures 4ndash6 It is noted from Figure 4 that near the wallthe value of velocity 1198911015840(120578) increases with rising values of 120575when 0 le 120578 le 06 and for 120578 ge 06 the velocity decreasesas 120575 increases Figure 5 depicts that a rise in the values ofslip parameter 120575 increases temperature monotonically It isevident from Figure 6 that the change in the concentration120601(120578) with rising values of slip parameter 120575 is negligible
6 Journal of Nanoscience
00 02 04 06 08 101000
1002
1004
1006
1008
1010
1012
1014
1016
1018
M = 0M = 1M = 2
Cu-water
120578
120579(120578
)
120593 = 002 120575 = 01 S = 1 Sc = 1
Figure 3 Effects of119872 on temperature 120579(120578)
00 02 04 06 08 1000
02
04
06
08
10
12
14Cu-water
f998400 (120578)
120578
120575 = 01
120575 = 02
120575 = 05
120593 = 002 M = 1 S = 1 Sc = 1
Figure 4 Effects of 120575 on velocity 1198911015840(120578)
The effects of the squeeze number 119878 on velocity temper-ature and concentration profiles are depicted in Figures 7ndash9Physically the squeeze number (119878) describes the movementof the plates (119878 gt 0 corresponds to the plates moving apartwhile 119878 lt 0 corresponds to the plates moving together) Itcan be easily seen from Figure 7 that the value of velocity1198911015840(120578) near the lower plate surface decreases regularly withthe increase in the value of 119878 and as we move away fromlower plate surface this value increases Figure 8 showsthat increasing value of squeeze number 119878 decreases thetemperature 120579(120578) Figure 9 depicts that the value of 120601(120578) goes
00 02 04 06 08 101000
1005
1010
1015
1020
1025
Cu-water
120578
120579(120578
)
120593 = 002 M = 1 S = 1 Sc = 1
120575 = 01
120575 = 02
120575 = 05
Figure 5 Effects of 120575 on temperature 120579(120578)
00 02 04 06 08 1000
02
04
06
08
10
Cu-water
120578
120575 = 01
120575 = 02
120575 = 05
120593 = 002 M = 1 S = 1 Sc = 1
120601(120578
)
Figure 6 Effects of 120575 on concentration 120601(120578)
up with increasing 120578 and high squeeze number implies slightdrop in concentration 120601(120578)
Figures 10 and 11 show the effects of the volume fraction120593 on the velocity and temperature profiles Figure 10 exhibitsinitially an increase in the values of nanoparticles volumefraction 120593 the velocity 1198911015840(120578) decreases and after a fixeddistance from lower plate surface it increases slightly FromFigure 11 it is observed that the temperature decreases withincreasing values of 120593
Figures 12 and 13 illiterate the effect of the differentnanoparticles on velocity and temperature profile whenthe base fluid is water It is observed from Figure 12 that
Journal of Nanoscience 7
00 02 04 06 08 10
02
04
06
08
10
12
14
16Cu-water
f998400 (120578)
120578
S = minus10S = minus05S = 00
S = 05
S = 10
120593 = 002 M = 1 120575 = 01 Sc = 1
Figure 7 Effects of 119878 on velocity 1198911015840(120578)
00 02 04 06 08 10100
101
102
103
104
105
106
107
120578
120579(120578
)
Cu-water
S = minus10S = minus05S = 00
S = 05
S = 10
120593 = 002 M = 1 120575 = 01 Sc = 1
Figure 8 Effects of 119878 on temperature 120579(120578)
the different nanofluids have different velocities and alsonoted that titanium oxide (TiO2) has higher velocity whencompared to other nanoparticles such as Al2O3 Cu andAg for 0 le 120578 le 05 while for 120578 gt 05 the results getreversed It is noted from Figure 13 that the order of nanofluidfor decreasing value of 120579(120578) when 120578 change from 0 to 1 isTiO2-water Al2O3-water Cu-water and Ag-water nanofluidFigure 14 depicts the effect of the Schmidt number Sc onconcentration profile The values of the temperature increasewith the increase in the value of Sc
The effects of the squeeze number 119878 on the skin-frictioncoefficient119862119891lowast the Nusselt number Nu119909
lowast and the Sherwood
00 02 04 06 08 1000
02
04
06
08
10
120578
Cu-water
S = minus10S = minus05S = 00
S = 05
S = 10
120593 = 002 M = 1 120575 = 01 Sc = 1
120601(120578
)
Figure 9 Effects of 119878 on concentration 120601(120578)
00 02 04 06 08 1002
04
06
08
10
12
14
120578
Cu-water
f998400 (120578)
M = 1 120575 = 01 Sc = 1
120593 = 000
120593 = 002120593 = 005
Figure 10 Effects of 120593 on velocity 1198911015840(120578)
number Sh119909lowast are given in Table 3 From Table 3 it is obvious
that the skin-friction coefficient and the Nusselt number areinversely proportional to 119878 whereas the Sherwood number isdirectly proportional to 119878
Table 4 displays the effects of the skin-friction coefficientthe Nusselt number and Sherwood number for differentvalues of the magnetic parameter119872 and slip parameter 120575 Itis noticed from the table that the effect of increasing valuesof119872 is to increase the skin-friction coefficient 119862119891lowast the heattransfer rate Nu119909
lowast andmass transfer rate Sh119909lowast Further from
Table 4 it is concluded that the increasing value of 120575 decreases
8 Journal of Nanoscience
00 02 04 06 08 101000
1002
1004
1006
1008
1010
1012
1014
1016
1018
120578
Cu-water
M = 1 120575 = 01 Sc = 1
120593 = 000
120593 = 002120593 = 005
120579(120578
)
Figure 11 Effects of 120593 on temperature 120579(120578)
00 02 04 06 08 1002
04
06
08
10
12
14
120578
f998400 (120578)
AgCu
Al2O3
TiO2
120593 = 002 M = 1 120575 = 01 S = 1 Sc = 1
Figure 12 Velocity of different nanofluids
the skin-friction coefficient and increases the heat and masstransfer rate
The effects of the nanoparticle volume fraction 120593 andSchmidt number Sc on the skin-friction coefficient 119862119891lowastNusselt number (the heat transfer rate) Nu119909
lowast and Sherwoodnumber (the mass transfer rate) Sh119909
lowast are given in Table 5From this table it is concluded that the increasing value of 120593increases the skin-friction coefficient and decreases the heatand mass transfer rate In addition the rate of mass transferincreases with the increase in Sc but there is no effect of Scon the skin-friction coefficient and the heat transfer rate
00 02 04 06 08 101000
1002
1004
1006
1008
1010
1012
1014
1016
120578
120579(120578
)
AgCu
Al2O3
TiO2
120593 = 002 M = 1 120575 = 01 S = 1 Sc = 1
Figure 13 Temperature of different nanofluids
00 02 04 06 08 1000
02
04
06
08
10
Sc = 0Sc = 1Sc = 10
Sc = 50Sc = 100
120578
Cu-water
120593 = 002 M = 1 120575 = 01 S = 1
120601(120578
)
Figure 14 Effects of Sc on concentration 120601(120578)
Table 3 Variation of 119862119891lowast Nu119909lowast and Sh119909lowast with different values
of Squeeze number 119878 for Cu-water nanofluid when 119872 = 1 Sc =1 120575 = 01 and 120593 = 002119878 119862119891lowast Nu119909
lowast Sh119909lowast
minus10 minus134295 0187977 minus10694254minus05 minus177787 0127657 minus1029323300 minus209419 0104693 minus1000000005 minus234148 0094096 minus0977084110 minus254426 0088577 minus09583107
The comparison for metallic nanoparticles (Ag Cu)and nonmetallic nanoparticles (Al2O3 TiO2) is done in
Journal of Nanoscience 9
Table 4 Variation of119862119891lowast Nu119909lowast and Sh119909lowast with different values of magnetic parameter119872 and slip parameter 120575 for Cu-water nanofluid when119878 = 1 Sc = 1 and 120593 = 002119872 120575 119862119891lowast Nu119909
lowast Sh119909lowast
0 01 minus246084 0089198 minus0956761 01 minus254426 0088577 minus0958312 01 minus26223 0088075 minus0959741 02 minus295257 0118598 minus0951581 05 minus326534 0144432 minus094633Table 5 Variation of 119862119891lowast Nu119909lowast and Sh119909lowast with different values of volume fraction 120593 and Schmidt number Sc for Cu-water nanofluid when119872 = 1 119878 = 1 and 120575 = 01120593 Sc 119862119891lowast Nu119909
lowast Sh119909lowast
000 1 minus276550 0092885 minus095773002 1 minus254426 0088577 minus095831005 1 minus231844 0082295 minus095911002 0 minus254426 0088577 minus100000002 10 minus254426 0088577 minus063318002 50 minus254426 0088577 minus005952002 100 minus254426 0088577 minus000182
Table 6 Variation of 119862119891lowast Nu119909lowast and Sh119909lowast for different nanoparti-
cles when119872 = 1 119878 = 1 Sc = 1 120575 = 01 and 120593 = 002Nanoparticle 119862119891lowast Nu119909
lowast Sh119909lowast
Ag minus248678 0088535 minus095851Cu minus254426 0088577 minus095831TiO2 minus273607 0088742 minus095777Al2O3 minus274807 0088794 minus095772
Table 6 It is observed form Table 6 that value of the skin-friction coefficient is more for the metallic nanoparticlesthan nonmetallic nanoparticles whereas the nonmetallicnanoparticles have higher heat and mass transfer rate incomparison with metallic nanoparticles
5 Conclusions
The present paper deals with the numerical solution ofcombined heat andmass transfer effects of an unsteadyMHDlaminar two-dimensional flow of incompressible viscousnanofluids inmiddle of two parallel plates extended infinitelywith slip velocity effect The relevant nonlinear partial dif-ferential equations were transformed to a set of ordinarydifferential equations and then are solved numerically usingthe Runge-Kutta-Fehlberg fourth-fifth-order method alongwith shooting technique From the above discussion the eye-catching results are as follows
(i) Temperature drops in Cu-water nanofluid withincrease in the magnetic field strength
(ii) The velocity of the nanofluid with nonmetallicnanoparticles Al2O3 and TiO2 is greater thanmetallic
nanoparticles Cu and Ag initially when water is thebase fluid but nature gets reversed after 120578 = 05The temperature of themetallic nanoparticles is lowerthan the nonmetallic nanoparticles
(iii) When plates start moving apart then temperaturestarts decreasing
(iv) Initially with the increasing values of magnetic fieldstrength and squeeze number the velocity of Cu-water slows down but the nature in both the case getsreversed after 120578 = 05
(v) The Nusselt number and skin-friction coefficientdecrease as plates move apart for Cu-water nanofluid
(vi) Nusselt number and Sherwood number for non-metallic nanoparticles are higher than the metallicnanoparticles
(vii) Varying value of slip parameter has negligible effecton concentration and as values of squeeze numberincrease the value of concentration profile also getsslightly higher
(viii) Considerable amount of enhancement in concentra-tion can be seen as Schmidt number rises
Nomenclature
1198601 1198602 1198603 Dimensionless constant1198610 Strength of magnetic field119862 Fluid concentration119862119891 Skin-friction coefficient119862119901 Specific heat at constant pressure119901 [Jkgminus1 Kminus1]119863 Mass diffusivity
10 Journal of Nanoscience
Ec Eckert number119891 Dimensionless velocity of the fluid119895 Reference length119896nf Nanofluid effective thermalconductivity119897 Distance of plate [m]119871 Slip length119872 Magnetic parameter119898119908 Surface mass flux
Nu119909 Nusselt number119901 Pressure [Pa]Pr Prandtl number119902119908 Surface heat fluxRe119909 Local Reynolds number119878 Squeeze numberSc Schmidt numberSh119909 Sherwood number119905 Time [s]119879 Temperature [K]119906 V Velocities in 119909119910-directions [m sminus1]119909 119910 Axial and perpendicular coordinates
[m]
Greek Symbols
120583119891 Dynamic viscosity of the fluid120593 Nanoparticle volume fraction120575 Velocity slip parameter120592119891 Kinematic viscosity of the fluid [m2 sminus1]120578 Similarity variable120588119891 Density of base fluid [kgmminus3]119896119904 Thermal conductivity of the solidnanoparticle [mminus1 Kminus1]119896119891 Thermal conductivity of the base fluid[Wmminus1 Kminus1]120579 Nondimensional temperature120601 Nondimensional concentration120588nf Nanofluid density [kgmminus3]120583nf Effective dynamic viscosity of nanofluid[Pa s]120572 Squeeze parameter120590nf Electrical conductivity of nanofluid120588119904 Density of solid particle [kgmminus3]120591119908 Surface shear stress
Subscripts
119891 Base fluidℎ Condition at the wallnf Nanofluid119904 Solid
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] M J Stefan ldquoVersuchuber die scheinbare adhesion sitzungs-bersachsrdquo Akademie der Wissenschaften in Wien Math-ematisch-Naturwissenschaftliche Klasse vol 69 pp 713ndash7211874
[2] G Domairry and M Hatami ldquoSqueezing Cundashwater nanofluidflow analysis between parallel plates by DTM-Pade MethodrdquoJournal of Molecular Liquids vol 193 pp 37ndash44 2014
[3] O Pourmehran M Rahimi-Gorji M Gorji-Bandpy and D DGanji ldquoAnalytical investigation of squeezing unsteady nanofluidflow between parallel plates by LSM and CMrdquo AlexandriaEngineering Journal vol 54 no 1 pp 17ndash26 2015
[4] A K Gupta and S S Ray ldquoNumerical treatment for investiga-tion of squeezing unsteady nanofluid flow between two parallelplatesrdquo Powder Technology vol 279 pp 282ndash289 2015
[5] U Khan N Ahmed M Asadullah and S T Mohyud-dinldquoEffects of viscous dissipation and slip velocity on two-dim-ensional and axisymmetric squeezing flow of Cu-water and Cu-kerosene nanofluidsrdquo Propulsion and Power Research vol 4 no1 pp 40ndash49 2015
[6] A Dib A Haiahem and B Bou-said ldquoApproximate analyticalsolution of squeezing unsteady nanofluid flowrdquo Powder Technol-ogy vol 269 pp 193ndash199 2015
[7] S U S Choi and J A Eastman ldquoEnhancing thermal con-ductivity of fluids with nanoparticlerdquo in Proceedings of theInternational Mechanical Engineering Congress and Exhibitionvol 231 pp 99ndash105 San Francisco Calif USA November 1995
[8] M Azimi and R Riazi ldquoHeat transfer analysis of GO-waternanofluid flow between two parallel disksrdquo Propulsion andPower Research vol 4 no 1 pp 23ndash30 2015
[9] A Aziz W A Khan and I Pop ldquoFree convection boundarylayer flow past a horizontal flat plate embedded in porousmedium filled by nanofluid containing gyrotactic microorgan-ismsrdquo International Journal of Thermal Sciences vol 56 pp 48ndash57 2012
[10] M Fakour D D Ganji and M Abbasi ldquoScrutiny of underde-veloped nanofluidMHD flow and heat conduction in a channelwith porous wallsrdquo Case Studies in Thermal Engineering vol 4pp 202ndash214 2014
[11] T Grosan C Revnic I Pop andD B Ingham ldquoFree convectionheat transfer in a square cavity filled with a porous mediumsaturated by a nano-fluidrdquo International Journal of Heat andMass Transfer vol 87 pp 36ndash41 2015
[12] A V Kuznetsov and D A Nield ldquoThe Cheng-Minkowyczproblem for natural convective boundary layer flow in a porousmedium saturated by a nanofluid a revised modelrdquo Inter-national Journal of Heat andMass Transfer vol 65 pp 682ndash6852013
[13] P Rana R Bhargava and O A Begb ldquoNumerical solution formixed convection boundary layerflow of ananofluid along aninclined plate embedded in a porous mediumrdquo Computers andMathematics with Applications vol 64 no 9 pp 2816ndash28322012
[14] A M Rashad A J Chamkha and M Modather ldquoMixed con-vection boundary-layer flow past a horizontal circular cylinderembedded in a porous medium filled with a nanofluid underconvective boundary conditionrdquo Computers amp Fluids vol 86pp 380ndash388 2013
[15] M Sheikholeslami and D D Ganji ldquoNanofluid flow and heattransfer between parallel plates considering Brownian motion
Journal of Nanoscience 11
using DTMrdquo Computer Methods in Applied Mechanics andEngineering vol 283 pp 651ndash663 2015
[16] N V Ganesh B Ganga and A K Abdul Hakeem ldquoLiesymmetry group analysis of magnetic field effects on free con-vective flow of a nanofluid over a semi-infinite stretching sheetrdquoJournal of the Egyptian Mathematical Society vol 22 no 2pp 304ndash310 2014
[17] Y Lin L Zheng and G Chen ldquoUnsteady flow and heattransfer of pseudo-plastic nanoliquid in a finite thin film ona stretching surface with variable thermal conductivity andviscous dissipationrdquo Powder Technology vol 274 pp 324ndash3322015
[18] M Sheikholeslami MM Rashidi and D D Ganji ldquoNumericalinvestigation of magnetic nanofluid forced convective heattransfer in existence of variable magnetic field using two phasemodelrdquo Journal of Molecular Liquids vol 212 pp 117ndash126 2015
[19] T Hayat M Waqas S A Shehzad and A Alsaedi ldquoMixedconvection flow of a Burgers nanofluid in the presence ofstratifications and heat generationabsorptionrdquo The EuropeanPhysical Journal Plus vol 131 no 8 p 253 2016
[20] F M Abbasi T Hayat S A Shehzad F Alsaadi and NAltoaibi ldquoHydromagnetic peristaltic transport of copper-waternanofluid with temperature-dependent effective viscosityrdquo Par-ticuology vol 27 pp 133ndash140 2016
[21] M Sheikholeslami D D Ganji and M M Rashidi ldquoMagneticfield effect on unsteady nanofluid flow and heat transferusing Buongiorno modelrdquo Journal of Magnetism and MagneticMaterials vol 416 pp 164ndash173 2016
[22] O Pourmehran M Rahimi-Gorji and D D Ganji ldquoHeattransfer and flow analysis of nanofluid flow induced by astretching sheet in the presence of an external magnetic fieldrdquoJournal of the Taiwan Institute of Chemical Engineers vol 65 pp162ndash171 2016
[23] M Rahimi-Gorji O Pourmehran M Hatami and D D GanjildquoStatistical optimization of microchannel heat sink (MCHS)geometry cooled by different nanofluids using RSM analysisrdquoEuropean Physical Journal Plus vol 130 no 2 article 22 pp 1ndash21 2015
[24] W Ibrahim and B Shankar ldquoMHD boundary layer flow andheat transfer of a nanofluid past a permeable stretching sheetwith velocity thermal and solutal slip boundary conditionsrdquoComputers amp Fluids vol 75 pp 1ndash10 2013
[25] A Malvandi and D D Ganji ldquoBrownian motion and ther-mophoresis effects on slip flow of aluminawater nanofluidinside a circular microchannel in the presence of a magneticfieldrdquo International Journal ofThermal Sciences vol 84 pp 196ndash206 2014
[26] R Ul Haq S Nadeem Z H Khan and N S Akbar ldquoThermalradiation and slip effects on MHD stagnation point flow ofnanofluid over a stretching sheetrdquo Physica E vol 65 pp 17ndash232015
[27] M Govindaraju S Saranya A A Hakeem R Jayaprakash andB Ganga ldquoAnalysis of slip MHD nanofluid flow on entropygeneration in a stretching sheetrdquo Procedia Engineering vol 127pp 501ndash507 2015
[28] M J Uddin M N Kabir and O A Beg ldquoComputationalinvestigation of Stefan blowing and multiple-slip effects onbuoyancy-driven bioconvection nanofluid flow with microor-ganismsrdquo International Journal of Heat and Mass Transfer vol95 pp 116ndash130 2016
[29] K-L Hsiao ldquoStagnation electrical MHD nanofluid mixed con-vection with slip boundary on a stretching sheetrdquo AppliedThermal Engineering vol 98 pp 850ndash861 2016
[30] P K Kameswaran M Narayana P Sibanda and P V S NMurthy ldquoHydromagnetic nanofluid flow due to a stretching orshrinking sheet with viscous dissipation and chemical reactioneffectsrdquo International Journal of Heat andMass Transfer vol 55no 25-26 pp 7587ndash7595 2012
[31] M H Matin and I Pop ldquoForced convection heat and masstransfer flow of a nanofluid through a porous channel with afirst order chemical reaction on thewallrdquo International Commu-nications in Heat and Mass Transfer vol 46 pp 134ndash141 2013
[32] D Pal andGMandal ldquoInfluence of thermal radiation onmixedconvection heat and mass transfer stagnation-point flow innanofluids over stretchingshrinking sheet in a porous mediumwith chemical reactionrdquo Nuclear Engineering and Design vol273 pp 644ndash652 2014
[33] C Zhang L Zheng X Zhang and G Chen ldquoMHD flow andradiation heat transfer of nanofluids in porous media withvariable surface heat flux and chemical reactionrdquoAppliedMath-ematical Modelling vol 39 no 1 pp 165ndash181 2015
[34] HM Elshehabey and S E Ahmed ldquoMHDmixed convection ina lid-driven cavity filled by a nanofluid with sinusoidal temper-ature distribution on the both vertical walls using Buongiornorsquosnanofluid modelrdquo International Journal of Heat and MassTransfer vol 88 pp 181ndash202 2015
Submit your manuscripts athttpwwwhindawicom
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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CeramicsJournal of
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International Journal of
Biomaterials
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Journal of
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CrystallographyJournal of
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Advances in
Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Smart Materials Research
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MetallurgyJournal of
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BioMed Research International
MaterialsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nano
materials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofNanomaterials
Journal of Nanoscience 5
Table 2 Comparison of 119891(120578) and 120579(120578) for Cu-water nanofluid when 119878 = 1 Pr = 62 Ec = 119895 = 001 120593 = 002 and119872 = Sc = 120575 = 0120578 Gupta and Ray [4] Present result
119891(120578) 120579(120578) 119891(120578) 120579(120578)0 0 103206637 0 10299663701 014135866 103206407 014135879 10299642802 028066605 103203202 028066639 10299353103 041578075 103189263 041578137 10298093204 054437882 103150811 054437979 10294617505 066385692 103066423 066385837 10286989606 077122923 102903983 077123132 10272306607 086301562 102615205 086301853 10246203708 093511971 102125916 093512364 10201976509 098269524 101318608 098270044 10129003110 100000000 100000000 100000000 100000000
subject to the following initial conditions
1198911 (0) = 01198912 (0) = 12057211198913 (0) = 01198914 (0) = 12057221198915 (0) = 12057231198916 (0) = 01198917 (0) = 01198918 (0) = 1205724
(20)
We ran the computer code written in the MATLAB fordifferent values of step lengthnabla120578We found that there is no oronly a negligible change in the physical quantities of interestlike the skin-friction coefficient the couple stress coefficientNusselt number and Sherwood number for various values ofnabla120578 gt 001 Therefore in the present paper we have set stepsize nabla120578 = 001 There are four initial conditions at 120578 = 0 andfour conditions on boundary 120578 = 1 To get the solution of theproblem four more initial conditions at 120578 = 0 that is valuesof 1205721 1205722 1205723 and 1205724 are required which have been obtainedby shooting technique Finally the transformed initial valueproblem is solved by employing the Runge-Kutta-Fehlbergfourth-fifth-order method along with calculated boundaryconditions
4 Results and Discussion
In order to validate the numerical results obtained wecompare our results with those reported by Gupta and Ray[4] as shown in Table 2 and they are found to be in agood agreement The effects of the volume fraction of solidnanoparticles magnetic parameter velocity slip parametersqueeze number and Schmidt number are inspected fordifferent kinds of nanoparticles when the base fluid is waterEc = 001 Pr = 62 and 119895 = 001
00 02 04 06 08 1002
04
06
08
10
12
14
M = 0M = 1M = 2
Cu-water
f998400 (120578)
120578
120593 = 002 120575 = 01 S = 1 Sc = 1
Figure 2 Effects of119872 on velocity 1198911015840(120578)
Figures 2 and 3 illustrate the behavior of the velocity1198911015840(120578)and temperature 120579(120578) of the Cu-water nanofluid for differentvalues of magnetic parameter 119872 Figure 2 shows that withan increase in the values of119872 the velocity decreases near thelower plate surface but after a certain distance it increasesIt is noticed from Figure 3 that the temperature decreasesmonotonically with increasing values of119872
The variations of the velocity temperature and concen-tration for different values of slip parameter 120575 are shownin Figures 4ndash6 It is noted from Figure 4 that near the wallthe value of velocity 1198911015840(120578) increases with rising values of 120575when 0 le 120578 le 06 and for 120578 ge 06 the velocity decreasesas 120575 increases Figure 5 depicts that a rise in the values ofslip parameter 120575 increases temperature monotonically It isevident from Figure 6 that the change in the concentration120601(120578) with rising values of slip parameter 120575 is negligible
6 Journal of Nanoscience
00 02 04 06 08 101000
1002
1004
1006
1008
1010
1012
1014
1016
1018
M = 0M = 1M = 2
Cu-water
120578
120579(120578
)
120593 = 002 120575 = 01 S = 1 Sc = 1
Figure 3 Effects of119872 on temperature 120579(120578)
00 02 04 06 08 1000
02
04
06
08
10
12
14Cu-water
f998400 (120578)
120578
120575 = 01
120575 = 02
120575 = 05
120593 = 002 M = 1 S = 1 Sc = 1
Figure 4 Effects of 120575 on velocity 1198911015840(120578)
The effects of the squeeze number 119878 on velocity temper-ature and concentration profiles are depicted in Figures 7ndash9Physically the squeeze number (119878) describes the movementof the plates (119878 gt 0 corresponds to the plates moving apartwhile 119878 lt 0 corresponds to the plates moving together) Itcan be easily seen from Figure 7 that the value of velocity1198911015840(120578) near the lower plate surface decreases regularly withthe increase in the value of 119878 and as we move away fromlower plate surface this value increases Figure 8 showsthat increasing value of squeeze number 119878 decreases thetemperature 120579(120578) Figure 9 depicts that the value of 120601(120578) goes
00 02 04 06 08 101000
1005
1010
1015
1020
1025
Cu-water
120578
120579(120578
)
120593 = 002 M = 1 S = 1 Sc = 1
120575 = 01
120575 = 02
120575 = 05
Figure 5 Effects of 120575 on temperature 120579(120578)
00 02 04 06 08 1000
02
04
06
08
10
Cu-water
120578
120575 = 01
120575 = 02
120575 = 05
120593 = 002 M = 1 S = 1 Sc = 1
120601(120578
)
Figure 6 Effects of 120575 on concentration 120601(120578)
up with increasing 120578 and high squeeze number implies slightdrop in concentration 120601(120578)
Figures 10 and 11 show the effects of the volume fraction120593 on the velocity and temperature profiles Figure 10 exhibitsinitially an increase in the values of nanoparticles volumefraction 120593 the velocity 1198911015840(120578) decreases and after a fixeddistance from lower plate surface it increases slightly FromFigure 11 it is observed that the temperature decreases withincreasing values of 120593
Figures 12 and 13 illiterate the effect of the differentnanoparticles on velocity and temperature profile whenthe base fluid is water It is observed from Figure 12 that
Journal of Nanoscience 7
00 02 04 06 08 10
02
04
06
08
10
12
14
16Cu-water
f998400 (120578)
120578
S = minus10S = minus05S = 00
S = 05
S = 10
120593 = 002 M = 1 120575 = 01 Sc = 1
Figure 7 Effects of 119878 on velocity 1198911015840(120578)
00 02 04 06 08 10100
101
102
103
104
105
106
107
120578
120579(120578
)
Cu-water
S = minus10S = minus05S = 00
S = 05
S = 10
120593 = 002 M = 1 120575 = 01 Sc = 1
Figure 8 Effects of 119878 on temperature 120579(120578)
the different nanofluids have different velocities and alsonoted that titanium oxide (TiO2) has higher velocity whencompared to other nanoparticles such as Al2O3 Cu andAg for 0 le 120578 le 05 while for 120578 gt 05 the results getreversed It is noted from Figure 13 that the order of nanofluidfor decreasing value of 120579(120578) when 120578 change from 0 to 1 isTiO2-water Al2O3-water Cu-water and Ag-water nanofluidFigure 14 depicts the effect of the Schmidt number Sc onconcentration profile The values of the temperature increasewith the increase in the value of Sc
The effects of the squeeze number 119878 on the skin-frictioncoefficient119862119891lowast the Nusselt number Nu119909
lowast and the Sherwood
00 02 04 06 08 1000
02
04
06
08
10
120578
Cu-water
S = minus10S = minus05S = 00
S = 05
S = 10
120593 = 002 M = 1 120575 = 01 Sc = 1
120601(120578
)
Figure 9 Effects of 119878 on concentration 120601(120578)
00 02 04 06 08 1002
04
06
08
10
12
14
120578
Cu-water
f998400 (120578)
M = 1 120575 = 01 Sc = 1
120593 = 000
120593 = 002120593 = 005
Figure 10 Effects of 120593 on velocity 1198911015840(120578)
number Sh119909lowast are given in Table 3 From Table 3 it is obvious
that the skin-friction coefficient and the Nusselt number areinversely proportional to 119878 whereas the Sherwood number isdirectly proportional to 119878
Table 4 displays the effects of the skin-friction coefficientthe Nusselt number and Sherwood number for differentvalues of the magnetic parameter119872 and slip parameter 120575 Itis noticed from the table that the effect of increasing valuesof119872 is to increase the skin-friction coefficient 119862119891lowast the heattransfer rate Nu119909
lowast andmass transfer rate Sh119909lowast Further from
Table 4 it is concluded that the increasing value of 120575 decreases
8 Journal of Nanoscience
00 02 04 06 08 101000
1002
1004
1006
1008
1010
1012
1014
1016
1018
120578
Cu-water
M = 1 120575 = 01 Sc = 1
120593 = 000
120593 = 002120593 = 005
120579(120578
)
Figure 11 Effects of 120593 on temperature 120579(120578)
00 02 04 06 08 1002
04
06
08
10
12
14
120578
f998400 (120578)
AgCu
Al2O3
TiO2
120593 = 002 M = 1 120575 = 01 S = 1 Sc = 1
Figure 12 Velocity of different nanofluids
the skin-friction coefficient and increases the heat and masstransfer rate
The effects of the nanoparticle volume fraction 120593 andSchmidt number Sc on the skin-friction coefficient 119862119891lowastNusselt number (the heat transfer rate) Nu119909
lowast and Sherwoodnumber (the mass transfer rate) Sh119909
lowast are given in Table 5From this table it is concluded that the increasing value of 120593increases the skin-friction coefficient and decreases the heatand mass transfer rate In addition the rate of mass transferincreases with the increase in Sc but there is no effect of Scon the skin-friction coefficient and the heat transfer rate
00 02 04 06 08 101000
1002
1004
1006
1008
1010
1012
1014
1016
120578
120579(120578
)
AgCu
Al2O3
TiO2
120593 = 002 M = 1 120575 = 01 S = 1 Sc = 1
Figure 13 Temperature of different nanofluids
00 02 04 06 08 1000
02
04
06
08
10
Sc = 0Sc = 1Sc = 10
Sc = 50Sc = 100
120578
Cu-water
120593 = 002 M = 1 120575 = 01 S = 1
120601(120578
)
Figure 14 Effects of Sc on concentration 120601(120578)
Table 3 Variation of 119862119891lowast Nu119909lowast and Sh119909lowast with different values
of Squeeze number 119878 for Cu-water nanofluid when 119872 = 1 Sc =1 120575 = 01 and 120593 = 002119878 119862119891lowast Nu119909
lowast Sh119909lowast
minus10 minus134295 0187977 minus10694254minus05 minus177787 0127657 minus1029323300 minus209419 0104693 minus1000000005 minus234148 0094096 minus0977084110 minus254426 0088577 minus09583107
The comparison for metallic nanoparticles (Ag Cu)and nonmetallic nanoparticles (Al2O3 TiO2) is done in
Journal of Nanoscience 9
Table 4 Variation of119862119891lowast Nu119909lowast and Sh119909lowast with different values of magnetic parameter119872 and slip parameter 120575 for Cu-water nanofluid when119878 = 1 Sc = 1 and 120593 = 002119872 120575 119862119891lowast Nu119909
lowast Sh119909lowast
0 01 minus246084 0089198 minus0956761 01 minus254426 0088577 minus0958312 01 minus26223 0088075 minus0959741 02 minus295257 0118598 minus0951581 05 minus326534 0144432 minus094633Table 5 Variation of 119862119891lowast Nu119909lowast and Sh119909lowast with different values of volume fraction 120593 and Schmidt number Sc for Cu-water nanofluid when119872 = 1 119878 = 1 and 120575 = 01120593 Sc 119862119891lowast Nu119909
lowast Sh119909lowast
000 1 minus276550 0092885 minus095773002 1 minus254426 0088577 minus095831005 1 minus231844 0082295 minus095911002 0 minus254426 0088577 minus100000002 10 minus254426 0088577 minus063318002 50 minus254426 0088577 minus005952002 100 minus254426 0088577 minus000182
Table 6 Variation of 119862119891lowast Nu119909lowast and Sh119909lowast for different nanoparti-
cles when119872 = 1 119878 = 1 Sc = 1 120575 = 01 and 120593 = 002Nanoparticle 119862119891lowast Nu119909
lowast Sh119909lowast
Ag minus248678 0088535 minus095851Cu minus254426 0088577 minus095831TiO2 minus273607 0088742 minus095777Al2O3 minus274807 0088794 minus095772
Table 6 It is observed form Table 6 that value of the skin-friction coefficient is more for the metallic nanoparticlesthan nonmetallic nanoparticles whereas the nonmetallicnanoparticles have higher heat and mass transfer rate incomparison with metallic nanoparticles
5 Conclusions
The present paper deals with the numerical solution ofcombined heat andmass transfer effects of an unsteadyMHDlaminar two-dimensional flow of incompressible viscousnanofluids inmiddle of two parallel plates extended infinitelywith slip velocity effect The relevant nonlinear partial dif-ferential equations were transformed to a set of ordinarydifferential equations and then are solved numerically usingthe Runge-Kutta-Fehlberg fourth-fifth-order method alongwith shooting technique From the above discussion the eye-catching results are as follows
(i) Temperature drops in Cu-water nanofluid withincrease in the magnetic field strength
(ii) The velocity of the nanofluid with nonmetallicnanoparticles Al2O3 and TiO2 is greater thanmetallic
nanoparticles Cu and Ag initially when water is thebase fluid but nature gets reversed after 120578 = 05The temperature of themetallic nanoparticles is lowerthan the nonmetallic nanoparticles
(iii) When plates start moving apart then temperaturestarts decreasing
(iv) Initially with the increasing values of magnetic fieldstrength and squeeze number the velocity of Cu-water slows down but the nature in both the case getsreversed after 120578 = 05
(v) The Nusselt number and skin-friction coefficientdecrease as plates move apart for Cu-water nanofluid
(vi) Nusselt number and Sherwood number for non-metallic nanoparticles are higher than the metallicnanoparticles
(vii) Varying value of slip parameter has negligible effecton concentration and as values of squeeze numberincrease the value of concentration profile also getsslightly higher
(viii) Considerable amount of enhancement in concentra-tion can be seen as Schmidt number rises
Nomenclature
1198601 1198602 1198603 Dimensionless constant1198610 Strength of magnetic field119862 Fluid concentration119862119891 Skin-friction coefficient119862119901 Specific heat at constant pressure119901 [Jkgminus1 Kminus1]119863 Mass diffusivity
10 Journal of Nanoscience
Ec Eckert number119891 Dimensionless velocity of the fluid119895 Reference length119896nf Nanofluid effective thermalconductivity119897 Distance of plate [m]119871 Slip length119872 Magnetic parameter119898119908 Surface mass flux
Nu119909 Nusselt number119901 Pressure [Pa]Pr Prandtl number119902119908 Surface heat fluxRe119909 Local Reynolds number119878 Squeeze numberSc Schmidt numberSh119909 Sherwood number119905 Time [s]119879 Temperature [K]119906 V Velocities in 119909119910-directions [m sminus1]119909 119910 Axial and perpendicular coordinates
[m]
Greek Symbols
120583119891 Dynamic viscosity of the fluid120593 Nanoparticle volume fraction120575 Velocity slip parameter120592119891 Kinematic viscosity of the fluid [m2 sminus1]120578 Similarity variable120588119891 Density of base fluid [kgmminus3]119896119904 Thermal conductivity of the solidnanoparticle [mminus1 Kminus1]119896119891 Thermal conductivity of the base fluid[Wmminus1 Kminus1]120579 Nondimensional temperature120601 Nondimensional concentration120588nf Nanofluid density [kgmminus3]120583nf Effective dynamic viscosity of nanofluid[Pa s]120572 Squeeze parameter120590nf Electrical conductivity of nanofluid120588119904 Density of solid particle [kgmminus3]120591119908 Surface shear stress
Subscripts
119891 Base fluidℎ Condition at the wallnf Nanofluid119904 Solid
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] M J Stefan ldquoVersuchuber die scheinbare adhesion sitzungs-bersachsrdquo Akademie der Wissenschaften in Wien Math-ematisch-Naturwissenschaftliche Klasse vol 69 pp 713ndash7211874
[2] G Domairry and M Hatami ldquoSqueezing Cundashwater nanofluidflow analysis between parallel plates by DTM-Pade MethodrdquoJournal of Molecular Liquids vol 193 pp 37ndash44 2014
[3] O Pourmehran M Rahimi-Gorji M Gorji-Bandpy and D DGanji ldquoAnalytical investigation of squeezing unsteady nanofluidflow between parallel plates by LSM and CMrdquo AlexandriaEngineering Journal vol 54 no 1 pp 17ndash26 2015
[4] A K Gupta and S S Ray ldquoNumerical treatment for investiga-tion of squeezing unsteady nanofluid flow between two parallelplatesrdquo Powder Technology vol 279 pp 282ndash289 2015
[5] U Khan N Ahmed M Asadullah and S T Mohyud-dinldquoEffects of viscous dissipation and slip velocity on two-dim-ensional and axisymmetric squeezing flow of Cu-water and Cu-kerosene nanofluidsrdquo Propulsion and Power Research vol 4 no1 pp 40ndash49 2015
[6] A Dib A Haiahem and B Bou-said ldquoApproximate analyticalsolution of squeezing unsteady nanofluid flowrdquo Powder Technol-ogy vol 269 pp 193ndash199 2015
[7] S U S Choi and J A Eastman ldquoEnhancing thermal con-ductivity of fluids with nanoparticlerdquo in Proceedings of theInternational Mechanical Engineering Congress and Exhibitionvol 231 pp 99ndash105 San Francisco Calif USA November 1995
[8] M Azimi and R Riazi ldquoHeat transfer analysis of GO-waternanofluid flow between two parallel disksrdquo Propulsion andPower Research vol 4 no 1 pp 23ndash30 2015
[9] A Aziz W A Khan and I Pop ldquoFree convection boundarylayer flow past a horizontal flat plate embedded in porousmedium filled by nanofluid containing gyrotactic microorgan-ismsrdquo International Journal of Thermal Sciences vol 56 pp 48ndash57 2012
[10] M Fakour D D Ganji and M Abbasi ldquoScrutiny of underde-veloped nanofluidMHD flow and heat conduction in a channelwith porous wallsrdquo Case Studies in Thermal Engineering vol 4pp 202ndash214 2014
[11] T Grosan C Revnic I Pop andD B Ingham ldquoFree convectionheat transfer in a square cavity filled with a porous mediumsaturated by a nano-fluidrdquo International Journal of Heat andMass Transfer vol 87 pp 36ndash41 2015
[12] A V Kuznetsov and D A Nield ldquoThe Cheng-Minkowyczproblem for natural convective boundary layer flow in a porousmedium saturated by a nanofluid a revised modelrdquo Inter-national Journal of Heat andMass Transfer vol 65 pp 682ndash6852013
[13] P Rana R Bhargava and O A Begb ldquoNumerical solution formixed convection boundary layerflow of ananofluid along aninclined plate embedded in a porous mediumrdquo Computers andMathematics with Applications vol 64 no 9 pp 2816ndash28322012
[14] A M Rashad A J Chamkha and M Modather ldquoMixed con-vection boundary-layer flow past a horizontal circular cylinderembedded in a porous medium filled with a nanofluid underconvective boundary conditionrdquo Computers amp Fluids vol 86pp 380ndash388 2013
[15] M Sheikholeslami and D D Ganji ldquoNanofluid flow and heattransfer between parallel plates considering Brownian motion
Journal of Nanoscience 11
using DTMrdquo Computer Methods in Applied Mechanics andEngineering vol 283 pp 651ndash663 2015
[16] N V Ganesh B Ganga and A K Abdul Hakeem ldquoLiesymmetry group analysis of magnetic field effects on free con-vective flow of a nanofluid over a semi-infinite stretching sheetrdquoJournal of the Egyptian Mathematical Society vol 22 no 2pp 304ndash310 2014
[17] Y Lin L Zheng and G Chen ldquoUnsteady flow and heattransfer of pseudo-plastic nanoliquid in a finite thin film ona stretching surface with variable thermal conductivity andviscous dissipationrdquo Powder Technology vol 274 pp 324ndash3322015
[18] M Sheikholeslami MM Rashidi and D D Ganji ldquoNumericalinvestigation of magnetic nanofluid forced convective heattransfer in existence of variable magnetic field using two phasemodelrdquo Journal of Molecular Liquids vol 212 pp 117ndash126 2015
[19] T Hayat M Waqas S A Shehzad and A Alsaedi ldquoMixedconvection flow of a Burgers nanofluid in the presence ofstratifications and heat generationabsorptionrdquo The EuropeanPhysical Journal Plus vol 131 no 8 p 253 2016
[20] F M Abbasi T Hayat S A Shehzad F Alsaadi and NAltoaibi ldquoHydromagnetic peristaltic transport of copper-waternanofluid with temperature-dependent effective viscosityrdquo Par-ticuology vol 27 pp 133ndash140 2016
[21] M Sheikholeslami D D Ganji and M M Rashidi ldquoMagneticfield effect on unsteady nanofluid flow and heat transferusing Buongiorno modelrdquo Journal of Magnetism and MagneticMaterials vol 416 pp 164ndash173 2016
[22] O Pourmehran M Rahimi-Gorji and D D Ganji ldquoHeattransfer and flow analysis of nanofluid flow induced by astretching sheet in the presence of an external magnetic fieldrdquoJournal of the Taiwan Institute of Chemical Engineers vol 65 pp162ndash171 2016
[23] M Rahimi-Gorji O Pourmehran M Hatami and D D GanjildquoStatistical optimization of microchannel heat sink (MCHS)geometry cooled by different nanofluids using RSM analysisrdquoEuropean Physical Journal Plus vol 130 no 2 article 22 pp 1ndash21 2015
[24] W Ibrahim and B Shankar ldquoMHD boundary layer flow andheat transfer of a nanofluid past a permeable stretching sheetwith velocity thermal and solutal slip boundary conditionsrdquoComputers amp Fluids vol 75 pp 1ndash10 2013
[25] A Malvandi and D D Ganji ldquoBrownian motion and ther-mophoresis effects on slip flow of aluminawater nanofluidinside a circular microchannel in the presence of a magneticfieldrdquo International Journal ofThermal Sciences vol 84 pp 196ndash206 2014
[26] R Ul Haq S Nadeem Z H Khan and N S Akbar ldquoThermalradiation and slip effects on MHD stagnation point flow ofnanofluid over a stretching sheetrdquo Physica E vol 65 pp 17ndash232015
[27] M Govindaraju S Saranya A A Hakeem R Jayaprakash andB Ganga ldquoAnalysis of slip MHD nanofluid flow on entropygeneration in a stretching sheetrdquo Procedia Engineering vol 127pp 501ndash507 2015
[28] M J Uddin M N Kabir and O A Beg ldquoComputationalinvestigation of Stefan blowing and multiple-slip effects onbuoyancy-driven bioconvection nanofluid flow with microor-ganismsrdquo International Journal of Heat and Mass Transfer vol95 pp 116ndash130 2016
[29] K-L Hsiao ldquoStagnation electrical MHD nanofluid mixed con-vection with slip boundary on a stretching sheetrdquo AppliedThermal Engineering vol 98 pp 850ndash861 2016
[30] P K Kameswaran M Narayana P Sibanda and P V S NMurthy ldquoHydromagnetic nanofluid flow due to a stretching orshrinking sheet with viscous dissipation and chemical reactioneffectsrdquo International Journal of Heat andMass Transfer vol 55no 25-26 pp 7587ndash7595 2012
[31] M H Matin and I Pop ldquoForced convection heat and masstransfer flow of a nanofluid through a porous channel with afirst order chemical reaction on thewallrdquo International Commu-nications in Heat and Mass Transfer vol 46 pp 134ndash141 2013
[32] D Pal andGMandal ldquoInfluence of thermal radiation onmixedconvection heat and mass transfer stagnation-point flow innanofluids over stretchingshrinking sheet in a porous mediumwith chemical reactionrdquo Nuclear Engineering and Design vol273 pp 644ndash652 2014
[33] C Zhang L Zheng X Zhang and G Chen ldquoMHD flow andradiation heat transfer of nanofluids in porous media withvariable surface heat flux and chemical reactionrdquoAppliedMath-ematical Modelling vol 39 no 1 pp 165ndash181 2015
[34] HM Elshehabey and S E Ahmed ldquoMHDmixed convection ina lid-driven cavity filled by a nanofluid with sinusoidal temper-ature distribution on the both vertical walls using Buongiornorsquosnanofluid modelrdquo International Journal of Heat and MassTransfer vol 88 pp 181ndash202 2015
Submit your manuscripts athttpwwwhindawicom
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CorrosionInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Polymer ScienceInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CeramicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CompositesJournal of
NanoparticlesJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Biomaterials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
NanoscienceJournal of
TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Journal of
NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
CrystallographyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CoatingsJournal of
Advances in
Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Smart Materials Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MetallurgyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
MaterialsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nano
materials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofNanomaterials
6 Journal of Nanoscience
00 02 04 06 08 101000
1002
1004
1006
1008
1010
1012
1014
1016
1018
M = 0M = 1M = 2
Cu-water
120578
120579(120578
)
120593 = 002 120575 = 01 S = 1 Sc = 1
Figure 3 Effects of119872 on temperature 120579(120578)
00 02 04 06 08 1000
02
04
06
08
10
12
14Cu-water
f998400 (120578)
120578
120575 = 01
120575 = 02
120575 = 05
120593 = 002 M = 1 S = 1 Sc = 1
Figure 4 Effects of 120575 on velocity 1198911015840(120578)
The effects of the squeeze number 119878 on velocity temper-ature and concentration profiles are depicted in Figures 7ndash9Physically the squeeze number (119878) describes the movementof the plates (119878 gt 0 corresponds to the plates moving apartwhile 119878 lt 0 corresponds to the plates moving together) Itcan be easily seen from Figure 7 that the value of velocity1198911015840(120578) near the lower plate surface decreases regularly withthe increase in the value of 119878 and as we move away fromlower plate surface this value increases Figure 8 showsthat increasing value of squeeze number 119878 decreases thetemperature 120579(120578) Figure 9 depicts that the value of 120601(120578) goes
00 02 04 06 08 101000
1005
1010
1015
1020
1025
Cu-water
120578
120579(120578
)
120593 = 002 M = 1 S = 1 Sc = 1
120575 = 01
120575 = 02
120575 = 05
Figure 5 Effects of 120575 on temperature 120579(120578)
00 02 04 06 08 1000
02
04
06
08
10
Cu-water
120578
120575 = 01
120575 = 02
120575 = 05
120593 = 002 M = 1 S = 1 Sc = 1
120601(120578
)
Figure 6 Effects of 120575 on concentration 120601(120578)
up with increasing 120578 and high squeeze number implies slightdrop in concentration 120601(120578)
Figures 10 and 11 show the effects of the volume fraction120593 on the velocity and temperature profiles Figure 10 exhibitsinitially an increase in the values of nanoparticles volumefraction 120593 the velocity 1198911015840(120578) decreases and after a fixeddistance from lower plate surface it increases slightly FromFigure 11 it is observed that the temperature decreases withincreasing values of 120593
Figures 12 and 13 illiterate the effect of the differentnanoparticles on velocity and temperature profile whenthe base fluid is water It is observed from Figure 12 that
Journal of Nanoscience 7
00 02 04 06 08 10
02
04
06
08
10
12
14
16Cu-water
f998400 (120578)
120578
S = minus10S = minus05S = 00
S = 05
S = 10
120593 = 002 M = 1 120575 = 01 Sc = 1
Figure 7 Effects of 119878 on velocity 1198911015840(120578)
00 02 04 06 08 10100
101
102
103
104
105
106
107
120578
120579(120578
)
Cu-water
S = minus10S = minus05S = 00
S = 05
S = 10
120593 = 002 M = 1 120575 = 01 Sc = 1
Figure 8 Effects of 119878 on temperature 120579(120578)
the different nanofluids have different velocities and alsonoted that titanium oxide (TiO2) has higher velocity whencompared to other nanoparticles such as Al2O3 Cu andAg for 0 le 120578 le 05 while for 120578 gt 05 the results getreversed It is noted from Figure 13 that the order of nanofluidfor decreasing value of 120579(120578) when 120578 change from 0 to 1 isTiO2-water Al2O3-water Cu-water and Ag-water nanofluidFigure 14 depicts the effect of the Schmidt number Sc onconcentration profile The values of the temperature increasewith the increase in the value of Sc
The effects of the squeeze number 119878 on the skin-frictioncoefficient119862119891lowast the Nusselt number Nu119909
lowast and the Sherwood
00 02 04 06 08 1000
02
04
06
08
10
120578
Cu-water
S = minus10S = minus05S = 00
S = 05
S = 10
120593 = 002 M = 1 120575 = 01 Sc = 1
120601(120578
)
Figure 9 Effects of 119878 on concentration 120601(120578)
00 02 04 06 08 1002
04
06
08
10
12
14
120578
Cu-water
f998400 (120578)
M = 1 120575 = 01 Sc = 1
120593 = 000
120593 = 002120593 = 005
Figure 10 Effects of 120593 on velocity 1198911015840(120578)
number Sh119909lowast are given in Table 3 From Table 3 it is obvious
that the skin-friction coefficient and the Nusselt number areinversely proportional to 119878 whereas the Sherwood number isdirectly proportional to 119878
Table 4 displays the effects of the skin-friction coefficientthe Nusselt number and Sherwood number for differentvalues of the magnetic parameter119872 and slip parameter 120575 Itis noticed from the table that the effect of increasing valuesof119872 is to increase the skin-friction coefficient 119862119891lowast the heattransfer rate Nu119909
lowast andmass transfer rate Sh119909lowast Further from
Table 4 it is concluded that the increasing value of 120575 decreases
8 Journal of Nanoscience
00 02 04 06 08 101000
1002
1004
1006
1008
1010
1012
1014
1016
1018
120578
Cu-water
M = 1 120575 = 01 Sc = 1
120593 = 000
120593 = 002120593 = 005
120579(120578
)
Figure 11 Effects of 120593 on temperature 120579(120578)
00 02 04 06 08 1002
04
06
08
10
12
14
120578
f998400 (120578)
AgCu
Al2O3
TiO2
120593 = 002 M = 1 120575 = 01 S = 1 Sc = 1
Figure 12 Velocity of different nanofluids
the skin-friction coefficient and increases the heat and masstransfer rate
The effects of the nanoparticle volume fraction 120593 andSchmidt number Sc on the skin-friction coefficient 119862119891lowastNusselt number (the heat transfer rate) Nu119909
lowast and Sherwoodnumber (the mass transfer rate) Sh119909
lowast are given in Table 5From this table it is concluded that the increasing value of 120593increases the skin-friction coefficient and decreases the heatand mass transfer rate In addition the rate of mass transferincreases with the increase in Sc but there is no effect of Scon the skin-friction coefficient and the heat transfer rate
00 02 04 06 08 101000
1002
1004
1006
1008
1010
1012
1014
1016
120578
120579(120578
)
AgCu
Al2O3
TiO2
120593 = 002 M = 1 120575 = 01 S = 1 Sc = 1
Figure 13 Temperature of different nanofluids
00 02 04 06 08 1000
02
04
06
08
10
Sc = 0Sc = 1Sc = 10
Sc = 50Sc = 100
120578
Cu-water
120593 = 002 M = 1 120575 = 01 S = 1
120601(120578
)
Figure 14 Effects of Sc on concentration 120601(120578)
Table 3 Variation of 119862119891lowast Nu119909lowast and Sh119909lowast with different values
of Squeeze number 119878 for Cu-water nanofluid when 119872 = 1 Sc =1 120575 = 01 and 120593 = 002119878 119862119891lowast Nu119909
lowast Sh119909lowast
minus10 minus134295 0187977 minus10694254minus05 minus177787 0127657 minus1029323300 minus209419 0104693 minus1000000005 minus234148 0094096 minus0977084110 minus254426 0088577 minus09583107
The comparison for metallic nanoparticles (Ag Cu)and nonmetallic nanoparticles (Al2O3 TiO2) is done in
Journal of Nanoscience 9
Table 4 Variation of119862119891lowast Nu119909lowast and Sh119909lowast with different values of magnetic parameter119872 and slip parameter 120575 for Cu-water nanofluid when119878 = 1 Sc = 1 and 120593 = 002119872 120575 119862119891lowast Nu119909
lowast Sh119909lowast
0 01 minus246084 0089198 minus0956761 01 minus254426 0088577 minus0958312 01 minus26223 0088075 minus0959741 02 minus295257 0118598 minus0951581 05 minus326534 0144432 minus094633Table 5 Variation of 119862119891lowast Nu119909lowast and Sh119909lowast with different values of volume fraction 120593 and Schmidt number Sc for Cu-water nanofluid when119872 = 1 119878 = 1 and 120575 = 01120593 Sc 119862119891lowast Nu119909
lowast Sh119909lowast
000 1 minus276550 0092885 minus095773002 1 minus254426 0088577 minus095831005 1 minus231844 0082295 minus095911002 0 minus254426 0088577 minus100000002 10 minus254426 0088577 minus063318002 50 minus254426 0088577 minus005952002 100 minus254426 0088577 minus000182
Table 6 Variation of 119862119891lowast Nu119909lowast and Sh119909lowast for different nanoparti-
cles when119872 = 1 119878 = 1 Sc = 1 120575 = 01 and 120593 = 002Nanoparticle 119862119891lowast Nu119909
lowast Sh119909lowast
Ag minus248678 0088535 minus095851Cu minus254426 0088577 minus095831TiO2 minus273607 0088742 minus095777Al2O3 minus274807 0088794 minus095772
Table 6 It is observed form Table 6 that value of the skin-friction coefficient is more for the metallic nanoparticlesthan nonmetallic nanoparticles whereas the nonmetallicnanoparticles have higher heat and mass transfer rate incomparison with metallic nanoparticles
5 Conclusions
The present paper deals with the numerical solution ofcombined heat andmass transfer effects of an unsteadyMHDlaminar two-dimensional flow of incompressible viscousnanofluids inmiddle of two parallel plates extended infinitelywith slip velocity effect The relevant nonlinear partial dif-ferential equations were transformed to a set of ordinarydifferential equations and then are solved numerically usingthe Runge-Kutta-Fehlberg fourth-fifth-order method alongwith shooting technique From the above discussion the eye-catching results are as follows
(i) Temperature drops in Cu-water nanofluid withincrease in the magnetic field strength
(ii) The velocity of the nanofluid with nonmetallicnanoparticles Al2O3 and TiO2 is greater thanmetallic
nanoparticles Cu and Ag initially when water is thebase fluid but nature gets reversed after 120578 = 05The temperature of themetallic nanoparticles is lowerthan the nonmetallic nanoparticles
(iii) When plates start moving apart then temperaturestarts decreasing
(iv) Initially with the increasing values of magnetic fieldstrength and squeeze number the velocity of Cu-water slows down but the nature in both the case getsreversed after 120578 = 05
(v) The Nusselt number and skin-friction coefficientdecrease as plates move apart for Cu-water nanofluid
(vi) Nusselt number and Sherwood number for non-metallic nanoparticles are higher than the metallicnanoparticles
(vii) Varying value of slip parameter has negligible effecton concentration and as values of squeeze numberincrease the value of concentration profile also getsslightly higher
(viii) Considerable amount of enhancement in concentra-tion can be seen as Schmidt number rises
Nomenclature
1198601 1198602 1198603 Dimensionless constant1198610 Strength of magnetic field119862 Fluid concentration119862119891 Skin-friction coefficient119862119901 Specific heat at constant pressure119901 [Jkgminus1 Kminus1]119863 Mass diffusivity
10 Journal of Nanoscience
Ec Eckert number119891 Dimensionless velocity of the fluid119895 Reference length119896nf Nanofluid effective thermalconductivity119897 Distance of plate [m]119871 Slip length119872 Magnetic parameter119898119908 Surface mass flux
Nu119909 Nusselt number119901 Pressure [Pa]Pr Prandtl number119902119908 Surface heat fluxRe119909 Local Reynolds number119878 Squeeze numberSc Schmidt numberSh119909 Sherwood number119905 Time [s]119879 Temperature [K]119906 V Velocities in 119909119910-directions [m sminus1]119909 119910 Axial and perpendicular coordinates
[m]
Greek Symbols
120583119891 Dynamic viscosity of the fluid120593 Nanoparticle volume fraction120575 Velocity slip parameter120592119891 Kinematic viscosity of the fluid [m2 sminus1]120578 Similarity variable120588119891 Density of base fluid [kgmminus3]119896119904 Thermal conductivity of the solidnanoparticle [mminus1 Kminus1]119896119891 Thermal conductivity of the base fluid[Wmminus1 Kminus1]120579 Nondimensional temperature120601 Nondimensional concentration120588nf Nanofluid density [kgmminus3]120583nf Effective dynamic viscosity of nanofluid[Pa s]120572 Squeeze parameter120590nf Electrical conductivity of nanofluid120588119904 Density of solid particle [kgmminus3]120591119908 Surface shear stress
Subscripts
119891 Base fluidℎ Condition at the wallnf Nanofluid119904 Solid
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] M J Stefan ldquoVersuchuber die scheinbare adhesion sitzungs-bersachsrdquo Akademie der Wissenschaften in Wien Math-ematisch-Naturwissenschaftliche Klasse vol 69 pp 713ndash7211874
[2] G Domairry and M Hatami ldquoSqueezing Cundashwater nanofluidflow analysis between parallel plates by DTM-Pade MethodrdquoJournal of Molecular Liquids vol 193 pp 37ndash44 2014
[3] O Pourmehran M Rahimi-Gorji M Gorji-Bandpy and D DGanji ldquoAnalytical investigation of squeezing unsteady nanofluidflow between parallel plates by LSM and CMrdquo AlexandriaEngineering Journal vol 54 no 1 pp 17ndash26 2015
[4] A K Gupta and S S Ray ldquoNumerical treatment for investiga-tion of squeezing unsteady nanofluid flow between two parallelplatesrdquo Powder Technology vol 279 pp 282ndash289 2015
[5] U Khan N Ahmed M Asadullah and S T Mohyud-dinldquoEffects of viscous dissipation and slip velocity on two-dim-ensional and axisymmetric squeezing flow of Cu-water and Cu-kerosene nanofluidsrdquo Propulsion and Power Research vol 4 no1 pp 40ndash49 2015
[6] A Dib A Haiahem and B Bou-said ldquoApproximate analyticalsolution of squeezing unsteady nanofluid flowrdquo Powder Technol-ogy vol 269 pp 193ndash199 2015
[7] S U S Choi and J A Eastman ldquoEnhancing thermal con-ductivity of fluids with nanoparticlerdquo in Proceedings of theInternational Mechanical Engineering Congress and Exhibitionvol 231 pp 99ndash105 San Francisco Calif USA November 1995
[8] M Azimi and R Riazi ldquoHeat transfer analysis of GO-waternanofluid flow between two parallel disksrdquo Propulsion andPower Research vol 4 no 1 pp 23ndash30 2015
[9] A Aziz W A Khan and I Pop ldquoFree convection boundarylayer flow past a horizontal flat plate embedded in porousmedium filled by nanofluid containing gyrotactic microorgan-ismsrdquo International Journal of Thermal Sciences vol 56 pp 48ndash57 2012
[10] M Fakour D D Ganji and M Abbasi ldquoScrutiny of underde-veloped nanofluidMHD flow and heat conduction in a channelwith porous wallsrdquo Case Studies in Thermal Engineering vol 4pp 202ndash214 2014
[11] T Grosan C Revnic I Pop andD B Ingham ldquoFree convectionheat transfer in a square cavity filled with a porous mediumsaturated by a nano-fluidrdquo International Journal of Heat andMass Transfer vol 87 pp 36ndash41 2015
[12] A V Kuznetsov and D A Nield ldquoThe Cheng-Minkowyczproblem for natural convective boundary layer flow in a porousmedium saturated by a nanofluid a revised modelrdquo Inter-national Journal of Heat andMass Transfer vol 65 pp 682ndash6852013
[13] P Rana R Bhargava and O A Begb ldquoNumerical solution formixed convection boundary layerflow of ananofluid along aninclined plate embedded in a porous mediumrdquo Computers andMathematics with Applications vol 64 no 9 pp 2816ndash28322012
[14] A M Rashad A J Chamkha and M Modather ldquoMixed con-vection boundary-layer flow past a horizontal circular cylinderembedded in a porous medium filled with a nanofluid underconvective boundary conditionrdquo Computers amp Fluids vol 86pp 380ndash388 2013
[15] M Sheikholeslami and D D Ganji ldquoNanofluid flow and heattransfer between parallel plates considering Brownian motion
Journal of Nanoscience 11
using DTMrdquo Computer Methods in Applied Mechanics andEngineering vol 283 pp 651ndash663 2015
[16] N V Ganesh B Ganga and A K Abdul Hakeem ldquoLiesymmetry group analysis of magnetic field effects on free con-vective flow of a nanofluid over a semi-infinite stretching sheetrdquoJournal of the Egyptian Mathematical Society vol 22 no 2pp 304ndash310 2014
[17] Y Lin L Zheng and G Chen ldquoUnsteady flow and heattransfer of pseudo-plastic nanoliquid in a finite thin film ona stretching surface with variable thermal conductivity andviscous dissipationrdquo Powder Technology vol 274 pp 324ndash3322015
[18] M Sheikholeslami MM Rashidi and D D Ganji ldquoNumericalinvestigation of magnetic nanofluid forced convective heattransfer in existence of variable magnetic field using two phasemodelrdquo Journal of Molecular Liquids vol 212 pp 117ndash126 2015
[19] T Hayat M Waqas S A Shehzad and A Alsaedi ldquoMixedconvection flow of a Burgers nanofluid in the presence ofstratifications and heat generationabsorptionrdquo The EuropeanPhysical Journal Plus vol 131 no 8 p 253 2016
[20] F M Abbasi T Hayat S A Shehzad F Alsaadi and NAltoaibi ldquoHydromagnetic peristaltic transport of copper-waternanofluid with temperature-dependent effective viscosityrdquo Par-ticuology vol 27 pp 133ndash140 2016
[21] M Sheikholeslami D D Ganji and M M Rashidi ldquoMagneticfield effect on unsteady nanofluid flow and heat transferusing Buongiorno modelrdquo Journal of Magnetism and MagneticMaterials vol 416 pp 164ndash173 2016
[22] O Pourmehran M Rahimi-Gorji and D D Ganji ldquoHeattransfer and flow analysis of nanofluid flow induced by astretching sheet in the presence of an external magnetic fieldrdquoJournal of the Taiwan Institute of Chemical Engineers vol 65 pp162ndash171 2016
[23] M Rahimi-Gorji O Pourmehran M Hatami and D D GanjildquoStatistical optimization of microchannel heat sink (MCHS)geometry cooled by different nanofluids using RSM analysisrdquoEuropean Physical Journal Plus vol 130 no 2 article 22 pp 1ndash21 2015
[24] W Ibrahim and B Shankar ldquoMHD boundary layer flow andheat transfer of a nanofluid past a permeable stretching sheetwith velocity thermal and solutal slip boundary conditionsrdquoComputers amp Fluids vol 75 pp 1ndash10 2013
[25] A Malvandi and D D Ganji ldquoBrownian motion and ther-mophoresis effects on slip flow of aluminawater nanofluidinside a circular microchannel in the presence of a magneticfieldrdquo International Journal ofThermal Sciences vol 84 pp 196ndash206 2014
[26] R Ul Haq S Nadeem Z H Khan and N S Akbar ldquoThermalradiation and slip effects on MHD stagnation point flow ofnanofluid over a stretching sheetrdquo Physica E vol 65 pp 17ndash232015
[27] M Govindaraju S Saranya A A Hakeem R Jayaprakash andB Ganga ldquoAnalysis of slip MHD nanofluid flow on entropygeneration in a stretching sheetrdquo Procedia Engineering vol 127pp 501ndash507 2015
[28] M J Uddin M N Kabir and O A Beg ldquoComputationalinvestigation of Stefan blowing and multiple-slip effects onbuoyancy-driven bioconvection nanofluid flow with microor-ganismsrdquo International Journal of Heat and Mass Transfer vol95 pp 116ndash130 2016
[29] K-L Hsiao ldquoStagnation electrical MHD nanofluid mixed con-vection with slip boundary on a stretching sheetrdquo AppliedThermal Engineering vol 98 pp 850ndash861 2016
[30] P K Kameswaran M Narayana P Sibanda and P V S NMurthy ldquoHydromagnetic nanofluid flow due to a stretching orshrinking sheet with viscous dissipation and chemical reactioneffectsrdquo International Journal of Heat andMass Transfer vol 55no 25-26 pp 7587ndash7595 2012
[31] M H Matin and I Pop ldquoForced convection heat and masstransfer flow of a nanofluid through a porous channel with afirst order chemical reaction on thewallrdquo International Commu-nications in Heat and Mass Transfer vol 46 pp 134ndash141 2013
[32] D Pal andGMandal ldquoInfluence of thermal radiation onmixedconvection heat and mass transfer stagnation-point flow innanofluids over stretchingshrinking sheet in a porous mediumwith chemical reactionrdquo Nuclear Engineering and Design vol273 pp 644ndash652 2014
[33] C Zhang L Zheng X Zhang and G Chen ldquoMHD flow andradiation heat transfer of nanofluids in porous media withvariable surface heat flux and chemical reactionrdquoAppliedMath-ematical Modelling vol 39 no 1 pp 165ndash181 2015
[34] HM Elshehabey and S E Ahmed ldquoMHDmixed convection ina lid-driven cavity filled by a nanofluid with sinusoidal temper-ature distribution on the both vertical walls using Buongiornorsquosnanofluid modelrdquo International Journal of Heat and MassTransfer vol 88 pp 181ndash202 2015
Submit your manuscripts athttpwwwhindawicom
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CorrosionInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Polymer ScienceInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CeramicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CompositesJournal of
NanoparticlesJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Biomaterials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
NanoscienceJournal of
TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Journal of
NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
CrystallographyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CoatingsJournal of
Advances in
Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Smart Materials Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MetallurgyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
MaterialsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nano
materials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofNanomaterials
Journal of Nanoscience 7
00 02 04 06 08 10
02
04
06
08
10
12
14
16Cu-water
f998400 (120578)
120578
S = minus10S = minus05S = 00
S = 05
S = 10
120593 = 002 M = 1 120575 = 01 Sc = 1
Figure 7 Effects of 119878 on velocity 1198911015840(120578)
00 02 04 06 08 10100
101
102
103
104
105
106
107
120578
120579(120578
)
Cu-water
S = minus10S = minus05S = 00
S = 05
S = 10
120593 = 002 M = 1 120575 = 01 Sc = 1
Figure 8 Effects of 119878 on temperature 120579(120578)
the different nanofluids have different velocities and alsonoted that titanium oxide (TiO2) has higher velocity whencompared to other nanoparticles such as Al2O3 Cu andAg for 0 le 120578 le 05 while for 120578 gt 05 the results getreversed It is noted from Figure 13 that the order of nanofluidfor decreasing value of 120579(120578) when 120578 change from 0 to 1 isTiO2-water Al2O3-water Cu-water and Ag-water nanofluidFigure 14 depicts the effect of the Schmidt number Sc onconcentration profile The values of the temperature increasewith the increase in the value of Sc
The effects of the squeeze number 119878 on the skin-frictioncoefficient119862119891lowast the Nusselt number Nu119909
lowast and the Sherwood
00 02 04 06 08 1000
02
04
06
08
10
120578
Cu-water
S = minus10S = minus05S = 00
S = 05
S = 10
120593 = 002 M = 1 120575 = 01 Sc = 1
120601(120578
)
Figure 9 Effects of 119878 on concentration 120601(120578)
00 02 04 06 08 1002
04
06
08
10
12
14
120578
Cu-water
f998400 (120578)
M = 1 120575 = 01 Sc = 1
120593 = 000
120593 = 002120593 = 005
Figure 10 Effects of 120593 on velocity 1198911015840(120578)
number Sh119909lowast are given in Table 3 From Table 3 it is obvious
that the skin-friction coefficient and the Nusselt number areinversely proportional to 119878 whereas the Sherwood number isdirectly proportional to 119878
Table 4 displays the effects of the skin-friction coefficientthe Nusselt number and Sherwood number for differentvalues of the magnetic parameter119872 and slip parameter 120575 Itis noticed from the table that the effect of increasing valuesof119872 is to increase the skin-friction coefficient 119862119891lowast the heattransfer rate Nu119909
lowast andmass transfer rate Sh119909lowast Further from
Table 4 it is concluded that the increasing value of 120575 decreases
8 Journal of Nanoscience
00 02 04 06 08 101000
1002
1004
1006
1008
1010
1012
1014
1016
1018
120578
Cu-water
M = 1 120575 = 01 Sc = 1
120593 = 000
120593 = 002120593 = 005
120579(120578
)
Figure 11 Effects of 120593 on temperature 120579(120578)
00 02 04 06 08 1002
04
06
08
10
12
14
120578
f998400 (120578)
AgCu
Al2O3
TiO2
120593 = 002 M = 1 120575 = 01 S = 1 Sc = 1
Figure 12 Velocity of different nanofluids
the skin-friction coefficient and increases the heat and masstransfer rate
The effects of the nanoparticle volume fraction 120593 andSchmidt number Sc on the skin-friction coefficient 119862119891lowastNusselt number (the heat transfer rate) Nu119909
lowast and Sherwoodnumber (the mass transfer rate) Sh119909
lowast are given in Table 5From this table it is concluded that the increasing value of 120593increases the skin-friction coefficient and decreases the heatand mass transfer rate In addition the rate of mass transferincreases with the increase in Sc but there is no effect of Scon the skin-friction coefficient and the heat transfer rate
00 02 04 06 08 101000
1002
1004
1006
1008
1010
1012
1014
1016
120578
120579(120578
)
AgCu
Al2O3
TiO2
120593 = 002 M = 1 120575 = 01 S = 1 Sc = 1
Figure 13 Temperature of different nanofluids
00 02 04 06 08 1000
02
04
06
08
10
Sc = 0Sc = 1Sc = 10
Sc = 50Sc = 100
120578
Cu-water
120593 = 002 M = 1 120575 = 01 S = 1
120601(120578
)
Figure 14 Effects of Sc on concentration 120601(120578)
Table 3 Variation of 119862119891lowast Nu119909lowast and Sh119909lowast with different values
of Squeeze number 119878 for Cu-water nanofluid when 119872 = 1 Sc =1 120575 = 01 and 120593 = 002119878 119862119891lowast Nu119909
lowast Sh119909lowast
minus10 minus134295 0187977 minus10694254minus05 minus177787 0127657 minus1029323300 minus209419 0104693 minus1000000005 minus234148 0094096 minus0977084110 minus254426 0088577 minus09583107
The comparison for metallic nanoparticles (Ag Cu)and nonmetallic nanoparticles (Al2O3 TiO2) is done in
Journal of Nanoscience 9
Table 4 Variation of119862119891lowast Nu119909lowast and Sh119909lowast with different values of magnetic parameter119872 and slip parameter 120575 for Cu-water nanofluid when119878 = 1 Sc = 1 and 120593 = 002119872 120575 119862119891lowast Nu119909
lowast Sh119909lowast
0 01 minus246084 0089198 minus0956761 01 minus254426 0088577 minus0958312 01 minus26223 0088075 minus0959741 02 minus295257 0118598 minus0951581 05 minus326534 0144432 minus094633Table 5 Variation of 119862119891lowast Nu119909lowast and Sh119909lowast with different values of volume fraction 120593 and Schmidt number Sc for Cu-water nanofluid when119872 = 1 119878 = 1 and 120575 = 01120593 Sc 119862119891lowast Nu119909
lowast Sh119909lowast
000 1 minus276550 0092885 minus095773002 1 minus254426 0088577 minus095831005 1 minus231844 0082295 minus095911002 0 minus254426 0088577 minus100000002 10 minus254426 0088577 minus063318002 50 minus254426 0088577 minus005952002 100 minus254426 0088577 minus000182
Table 6 Variation of 119862119891lowast Nu119909lowast and Sh119909lowast for different nanoparti-
cles when119872 = 1 119878 = 1 Sc = 1 120575 = 01 and 120593 = 002Nanoparticle 119862119891lowast Nu119909
lowast Sh119909lowast
Ag minus248678 0088535 minus095851Cu minus254426 0088577 minus095831TiO2 minus273607 0088742 minus095777Al2O3 minus274807 0088794 minus095772
Table 6 It is observed form Table 6 that value of the skin-friction coefficient is more for the metallic nanoparticlesthan nonmetallic nanoparticles whereas the nonmetallicnanoparticles have higher heat and mass transfer rate incomparison with metallic nanoparticles
5 Conclusions
The present paper deals with the numerical solution ofcombined heat andmass transfer effects of an unsteadyMHDlaminar two-dimensional flow of incompressible viscousnanofluids inmiddle of two parallel plates extended infinitelywith slip velocity effect The relevant nonlinear partial dif-ferential equations were transformed to a set of ordinarydifferential equations and then are solved numerically usingthe Runge-Kutta-Fehlberg fourth-fifth-order method alongwith shooting technique From the above discussion the eye-catching results are as follows
(i) Temperature drops in Cu-water nanofluid withincrease in the magnetic field strength
(ii) The velocity of the nanofluid with nonmetallicnanoparticles Al2O3 and TiO2 is greater thanmetallic
nanoparticles Cu and Ag initially when water is thebase fluid but nature gets reversed after 120578 = 05The temperature of themetallic nanoparticles is lowerthan the nonmetallic nanoparticles
(iii) When plates start moving apart then temperaturestarts decreasing
(iv) Initially with the increasing values of magnetic fieldstrength and squeeze number the velocity of Cu-water slows down but the nature in both the case getsreversed after 120578 = 05
(v) The Nusselt number and skin-friction coefficientdecrease as plates move apart for Cu-water nanofluid
(vi) Nusselt number and Sherwood number for non-metallic nanoparticles are higher than the metallicnanoparticles
(vii) Varying value of slip parameter has negligible effecton concentration and as values of squeeze numberincrease the value of concentration profile also getsslightly higher
(viii) Considerable amount of enhancement in concentra-tion can be seen as Schmidt number rises
Nomenclature
1198601 1198602 1198603 Dimensionless constant1198610 Strength of magnetic field119862 Fluid concentration119862119891 Skin-friction coefficient119862119901 Specific heat at constant pressure119901 [Jkgminus1 Kminus1]119863 Mass diffusivity
10 Journal of Nanoscience
Ec Eckert number119891 Dimensionless velocity of the fluid119895 Reference length119896nf Nanofluid effective thermalconductivity119897 Distance of plate [m]119871 Slip length119872 Magnetic parameter119898119908 Surface mass flux
Nu119909 Nusselt number119901 Pressure [Pa]Pr Prandtl number119902119908 Surface heat fluxRe119909 Local Reynolds number119878 Squeeze numberSc Schmidt numberSh119909 Sherwood number119905 Time [s]119879 Temperature [K]119906 V Velocities in 119909119910-directions [m sminus1]119909 119910 Axial and perpendicular coordinates
[m]
Greek Symbols
120583119891 Dynamic viscosity of the fluid120593 Nanoparticle volume fraction120575 Velocity slip parameter120592119891 Kinematic viscosity of the fluid [m2 sminus1]120578 Similarity variable120588119891 Density of base fluid [kgmminus3]119896119904 Thermal conductivity of the solidnanoparticle [mminus1 Kminus1]119896119891 Thermal conductivity of the base fluid[Wmminus1 Kminus1]120579 Nondimensional temperature120601 Nondimensional concentration120588nf Nanofluid density [kgmminus3]120583nf Effective dynamic viscosity of nanofluid[Pa s]120572 Squeeze parameter120590nf Electrical conductivity of nanofluid120588119904 Density of solid particle [kgmminus3]120591119908 Surface shear stress
Subscripts
119891 Base fluidℎ Condition at the wallnf Nanofluid119904 Solid
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] M J Stefan ldquoVersuchuber die scheinbare adhesion sitzungs-bersachsrdquo Akademie der Wissenschaften in Wien Math-ematisch-Naturwissenschaftliche Klasse vol 69 pp 713ndash7211874
[2] G Domairry and M Hatami ldquoSqueezing Cundashwater nanofluidflow analysis between parallel plates by DTM-Pade MethodrdquoJournal of Molecular Liquids vol 193 pp 37ndash44 2014
[3] O Pourmehran M Rahimi-Gorji M Gorji-Bandpy and D DGanji ldquoAnalytical investigation of squeezing unsteady nanofluidflow between parallel plates by LSM and CMrdquo AlexandriaEngineering Journal vol 54 no 1 pp 17ndash26 2015
[4] A K Gupta and S S Ray ldquoNumerical treatment for investiga-tion of squeezing unsteady nanofluid flow between two parallelplatesrdquo Powder Technology vol 279 pp 282ndash289 2015
[5] U Khan N Ahmed M Asadullah and S T Mohyud-dinldquoEffects of viscous dissipation and slip velocity on two-dim-ensional and axisymmetric squeezing flow of Cu-water and Cu-kerosene nanofluidsrdquo Propulsion and Power Research vol 4 no1 pp 40ndash49 2015
[6] A Dib A Haiahem and B Bou-said ldquoApproximate analyticalsolution of squeezing unsteady nanofluid flowrdquo Powder Technol-ogy vol 269 pp 193ndash199 2015
[7] S U S Choi and J A Eastman ldquoEnhancing thermal con-ductivity of fluids with nanoparticlerdquo in Proceedings of theInternational Mechanical Engineering Congress and Exhibitionvol 231 pp 99ndash105 San Francisco Calif USA November 1995
[8] M Azimi and R Riazi ldquoHeat transfer analysis of GO-waternanofluid flow between two parallel disksrdquo Propulsion andPower Research vol 4 no 1 pp 23ndash30 2015
[9] A Aziz W A Khan and I Pop ldquoFree convection boundarylayer flow past a horizontal flat plate embedded in porousmedium filled by nanofluid containing gyrotactic microorgan-ismsrdquo International Journal of Thermal Sciences vol 56 pp 48ndash57 2012
[10] M Fakour D D Ganji and M Abbasi ldquoScrutiny of underde-veloped nanofluidMHD flow and heat conduction in a channelwith porous wallsrdquo Case Studies in Thermal Engineering vol 4pp 202ndash214 2014
[11] T Grosan C Revnic I Pop andD B Ingham ldquoFree convectionheat transfer in a square cavity filled with a porous mediumsaturated by a nano-fluidrdquo International Journal of Heat andMass Transfer vol 87 pp 36ndash41 2015
[12] A V Kuznetsov and D A Nield ldquoThe Cheng-Minkowyczproblem for natural convective boundary layer flow in a porousmedium saturated by a nanofluid a revised modelrdquo Inter-national Journal of Heat andMass Transfer vol 65 pp 682ndash6852013
[13] P Rana R Bhargava and O A Begb ldquoNumerical solution formixed convection boundary layerflow of ananofluid along aninclined plate embedded in a porous mediumrdquo Computers andMathematics with Applications vol 64 no 9 pp 2816ndash28322012
[14] A M Rashad A J Chamkha and M Modather ldquoMixed con-vection boundary-layer flow past a horizontal circular cylinderembedded in a porous medium filled with a nanofluid underconvective boundary conditionrdquo Computers amp Fluids vol 86pp 380ndash388 2013
[15] M Sheikholeslami and D D Ganji ldquoNanofluid flow and heattransfer between parallel plates considering Brownian motion
Journal of Nanoscience 11
using DTMrdquo Computer Methods in Applied Mechanics andEngineering vol 283 pp 651ndash663 2015
[16] N V Ganesh B Ganga and A K Abdul Hakeem ldquoLiesymmetry group analysis of magnetic field effects on free con-vective flow of a nanofluid over a semi-infinite stretching sheetrdquoJournal of the Egyptian Mathematical Society vol 22 no 2pp 304ndash310 2014
[17] Y Lin L Zheng and G Chen ldquoUnsteady flow and heattransfer of pseudo-plastic nanoliquid in a finite thin film ona stretching surface with variable thermal conductivity andviscous dissipationrdquo Powder Technology vol 274 pp 324ndash3322015
[18] M Sheikholeslami MM Rashidi and D D Ganji ldquoNumericalinvestigation of magnetic nanofluid forced convective heattransfer in existence of variable magnetic field using two phasemodelrdquo Journal of Molecular Liquids vol 212 pp 117ndash126 2015
[19] T Hayat M Waqas S A Shehzad and A Alsaedi ldquoMixedconvection flow of a Burgers nanofluid in the presence ofstratifications and heat generationabsorptionrdquo The EuropeanPhysical Journal Plus vol 131 no 8 p 253 2016
[20] F M Abbasi T Hayat S A Shehzad F Alsaadi and NAltoaibi ldquoHydromagnetic peristaltic transport of copper-waternanofluid with temperature-dependent effective viscosityrdquo Par-ticuology vol 27 pp 133ndash140 2016
[21] M Sheikholeslami D D Ganji and M M Rashidi ldquoMagneticfield effect on unsteady nanofluid flow and heat transferusing Buongiorno modelrdquo Journal of Magnetism and MagneticMaterials vol 416 pp 164ndash173 2016
[22] O Pourmehran M Rahimi-Gorji and D D Ganji ldquoHeattransfer and flow analysis of nanofluid flow induced by astretching sheet in the presence of an external magnetic fieldrdquoJournal of the Taiwan Institute of Chemical Engineers vol 65 pp162ndash171 2016
[23] M Rahimi-Gorji O Pourmehran M Hatami and D D GanjildquoStatistical optimization of microchannel heat sink (MCHS)geometry cooled by different nanofluids using RSM analysisrdquoEuropean Physical Journal Plus vol 130 no 2 article 22 pp 1ndash21 2015
[24] W Ibrahim and B Shankar ldquoMHD boundary layer flow andheat transfer of a nanofluid past a permeable stretching sheetwith velocity thermal and solutal slip boundary conditionsrdquoComputers amp Fluids vol 75 pp 1ndash10 2013
[25] A Malvandi and D D Ganji ldquoBrownian motion and ther-mophoresis effects on slip flow of aluminawater nanofluidinside a circular microchannel in the presence of a magneticfieldrdquo International Journal ofThermal Sciences vol 84 pp 196ndash206 2014
[26] R Ul Haq S Nadeem Z H Khan and N S Akbar ldquoThermalradiation and slip effects on MHD stagnation point flow ofnanofluid over a stretching sheetrdquo Physica E vol 65 pp 17ndash232015
[27] M Govindaraju S Saranya A A Hakeem R Jayaprakash andB Ganga ldquoAnalysis of slip MHD nanofluid flow on entropygeneration in a stretching sheetrdquo Procedia Engineering vol 127pp 501ndash507 2015
[28] M J Uddin M N Kabir and O A Beg ldquoComputationalinvestigation of Stefan blowing and multiple-slip effects onbuoyancy-driven bioconvection nanofluid flow with microor-ganismsrdquo International Journal of Heat and Mass Transfer vol95 pp 116ndash130 2016
[29] K-L Hsiao ldquoStagnation electrical MHD nanofluid mixed con-vection with slip boundary on a stretching sheetrdquo AppliedThermal Engineering vol 98 pp 850ndash861 2016
[30] P K Kameswaran M Narayana P Sibanda and P V S NMurthy ldquoHydromagnetic nanofluid flow due to a stretching orshrinking sheet with viscous dissipation and chemical reactioneffectsrdquo International Journal of Heat andMass Transfer vol 55no 25-26 pp 7587ndash7595 2012
[31] M H Matin and I Pop ldquoForced convection heat and masstransfer flow of a nanofluid through a porous channel with afirst order chemical reaction on thewallrdquo International Commu-nications in Heat and Mass Transfer vol 46 pp 134ndash141 2013
[32] D Pal andGMandal ldquoInfluence of thermal radiation onmixedconvection heat and mass transfer stagnation-point flow innanofluids over stretchingshrinking sheet in a porous mediumwith chemical reactionrdquo Nuclear Engineering and Design vol273 pp 644ndash652 2014
[33] C Zhang L Zheng X Zhang and G Chen ldquoMHD flow andradiation heat transfer of nanofluids in porous media withvariable surface heat flux and chemical reactionrdquoAppliedMath-ematical Modelling vol 39 no 1 pp 165ndash181 2015
[34] HM Elshehabey and S E Ahmed ldquoMHDmixed convection ina lid-driven cavity filled by a nanofluid with sinusoidal temper-ature distribution on the both vertical walls using Buongiornorsquosnanofluid modelrdquo International Journal of Heat and MassTransfer vol 88 pp 181ndash202 2015
Submit your manuscripts athttpwwwhindawicom
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CorrosionInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Polymer ScienceInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CeramicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CompositesJournal of
NanoparticlesJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Biomaterials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
NanoscienceJournal of
TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Journal of
NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
CrystallographyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CoatingsJournal of
Advances in
Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Smart Materials Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MetallurgyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
MaterialsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nano
materials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofNanomaterials
8 Journal of Nanoscience
00 02 04 06 08 101000
1002
1004
1006
1008
1010
1012
1014
1016
1018
120578
Cu-water
M = 1 120575 = 01 Sc = 1
120593 = 000
120593 = 002120593 = 005
120579(120578
)
Figure 11 Effects of 120593 on temperature 120579(120578)
00 02 04 06 08 1002
04
06
08
10
12
14
120578
f998400 (120578)
AgCu
Al2O3
TiO2
120593 = 002 M = 1 120575 = 01 S = 1 Sc = 1
Figure 12 Velocity of different nanofluids
the skin-friction coefficient and increases the heat and masstransfer rate
The effects of the nanoparticle volume fraction 120593 andSchmidt number Sc on the skin-friction coefficient 119862119891lowastNusselt number (the heat transfer rate) Nu119909
lowast and Sherwoodnumber (the mass transfer rate) Sh119909
lowast are given in Table 5From this table it is concluded that the increasing value of 120593increases the skin-friction coefficient and decreases the heatand mass transfer rate In addition the rate of mass transferincreases with the increase in Sc but there is no effect of Scon the skin-friction coefficient and the heat transfer rate
00 02 04 06 08 101000
1002
1004
1006
1008
1010
1012
1014
1016
120578
120579(120578
)
AgCu
Al2O3
TiO2
120593 = 002 M = 1 120575 = 01 S = 1 Sc = 1
Figure 13 Temperature of different nanofluids
00 02 04 06 08 1000
02
04
06
08
10
Sc = 0Sc = 1Sc = 10
Sc = 50Sc = 100
120578
Cu-water
120593 = 002 M = 1 120575 = 01 S = 1
120601(120578
)
Figure 14 Effects of Sc on concentration 120601(120578)
Table 3 Variation of 119862119891lowast Nu119909lowast and Sh119909lowast with different values
of Squeeze number 119878 for Cu-water nanofluid when 119872 = 1 Sc =1 120575 = 01 and 120593 = 002119878 119862119891lowast Nu119909
lowast Sh119909lowast
minus10 minus134295 0187977 minus10694254minus05 minus177787 0127657 minus1029323300 minus209419 0104693 minus1000000005 minus234148 0094096 minus0977084110 minus254426 0088577 minus09583107
The comparison for metallic nanoparticles (Ag Cu)and nonmetallic nanoparticles (Al2O3 TiO2) is done in
Journal of Nanoscience 9
Table 4 Variation of119862119891lowast Nu119909lowast and Sh119909lowast with different values of magnetic parameter119872 and slip parameter 120575 for Cu-water nanofluid when119878 = 1 Sc = 1 and 120593 = 002119872 120575 119862119891lowast Nu119909
lowast Sh119909lowast
0 01 minus246084 0089198 minus0956761 01 minus254426 0088577 minus0958312 01 minus26223 0088075 minus0959741 02 minus295257 0118598 minus0951581 05 minus326534 0144432 minus094633Table 5 Variation of 119862119891lowast Nu119909lowast and Sh119909lowast with different values of volume fraction 120593 and Schmidt number Sc for Cu-water nanofluid when119872 = 1 119878 = 1 and 120575 = 01120593 Sc 119862119891lowast Nu119909
lowast Sh119909lowast
000 1 minus276550 0092885 minus095773002 1 minus254426 0088577 minus095831005 1 minus231844 0082295 minus095911002 0 minus254426 0088577 minus100000002 10 minus254426 0088577 minus063318002 50 minus254426 0088577 minus005952002 100 minus254426 0088577 minus000182
Table 6 Variation of 119862119891lowast Nu119909lowast and Sh119909lowast for different nanoparti-
cles when119872 = 1 119878 = 1 Sc = 1 120575 = 01 and 120593 = 002Nanoparticle 119862119891lowast Nu119909
lowast Sh119909lowast
Ag minus248678 0088535 minus095851Cu minus254426 0088577 minus095831TiO2 minus273607 0088742 minus095777Al2O3 minus274807 0088794 minus095772
Table 6 It is observed form Table 6 that value of the skin-friction coefficient is more for the metallic nanoparticlesthan nonmetallic nanoparticles whereas the nonmetallicnanoparticles have higher heat and mass transfer rate incomparison with metallic nanoparticles
5 Conclusions
The present paper deals with the numerical solution ofcombined heat andmass transfer effects of an unsteadyMHDlaminar two-dimensional flow of incompressible viscousnanofluids inmiddle of two parallel plates extended infinitelywith slip velocity effect The relevant nonlinear partial dif-ferential equations were transformed to a set of ordinarydifferential equations and then are solved numerically usingthe Runge-Kutta-Fehlberg fourth-fifth-order method alongwith shooting technique From the above discussion the eye-catching results are as follows
(i) Temperature drops in Cu-water nanofluid withincrease in the magnetic field strength
(ii) The velocity of the nanofluid with nonmetallicnanoparticles Al2O3 and TiO2 is greater thanmetallic
nanoparticles Cu and Ag initially when water is thebase fluid but nature gets reversed after 120578 = 05The temperature of themetallic nanoparticles is lowerthan the nonmetallic nanoparticles
(iii) When plates start moving apart then temperaturestarts decreasing
(iv) Initially with the increasing values of magnetic fieldstrength and squeeze number the velocity of Cu-water slows down but the nature in both the case getsreversed after 120578 = 05
(v) The Nusselt number and skin-friction coefficientdecrease as plates move apart for Cu-water nanofluid
(vi) Nusselt number and Sherwood number for non-metallic nanoparticles are higher than the metallicnanoparticles
(vii) Varying value of slip parameter has negligible effecton concentration and as values of squeeze numberincrease the value of concentration profile also getsslightly higher
(viii) Considerable amount of enhancement in concentra-tion can be seen as Schmidt number rises
Nomenclature
1198601 1198602 1198603 Dimensionless constant1198610 Strength of magnetic field119862 Fluid concentration119862119891 Skin-friction coefficient119862119901 Specific heat at constant pressure119901 [Jkgminus1 Kminus1]119863 Mass diffusivity
10 Journal of Nanoscience
Ec Eckert number119891 Dimensionless velocity of the fluid119895 Reference length119896nf Nanofluid effective thermalconductivity119897 Distance of plate [m]119871 Slip length119872 Magnetic parameter119898119908 Surface mass flux
Nu119909 Nusselt number119901 Pressure [Pa]Pr Prandtl number119902119908 Surface heat fluxRe119909 Local Reynolds number119878 Squeeze numberSc Schmidt numberSh119909 Sherwood number119905 Time [s]119879 Temperature [K]119906 V Velocities in 119909119910-directions [m sminus1]119909 119910 Axial and perpendicular coordinates
[m]
Greek Symbols
120583119891 Dynamic viscosity of the fluid120593 Nanoparticle volume fraction120575 Velocity slip parameter120592119891 Kinematic viscosity of the fluid [m2 sminus1]120578 Similarity variable120588119891 Density of base fluid [kgmminus3]119896119904 Thermal conductivity of the solidnanoparticle [mminus1 Kminus1]119896119891 Thermal conductivity of the base fluid[Wmminus1 Kminus1]120579 Nondimensional temperature120601 Nondimensional concentration120588nf Nanofluid density [kgmminus3]120583nf Effective dynamic viscosity of nanofluid[Pa s]120572 Squeeze parameter120590nf Electrical conductivity of nanofluid120588119904 Density of solid particle [kgmminus3]120591119908 Surface shear stress
Subscripts
119891 Base fluidℎ Condition at the wallnf Nanofluid119904 Solid
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] M J Stefan ldquoVersuchuber die scheinbare adhesion sitzungs-bersachsrdquo Akademie der Wissenschaften in Wien Math-ematisch-Naturwissenschaftliche Klasse vol 69 pp 713ndash7211874
[2] G Domairry and M Hatami ldquoSqueezing Cundashwater nanofluidflow analysis between parallel plates by DTM-Pade MethodrdquoJournal of Molecular Liquids vol 193 pp 37ndash44 2014
[3] O Pourmehran M Rahimi-Gorji M Gorji-Bandpy and D DGanji ldquoAnalytical investigation of squeezing unsteady nanofluidflow between parallel plates by LSM and CMrdquo AlexandriaEngineering Journal vol 54 no 1 pp 17ndash26 2015
[4] A K Gupta and S S Ray ldquoNumerical treatment for investiga-tion of squeezing unsteady nanofluid flow between two parallelplatesrdquo Powder Technology vol 279 pp 282ndash289 2015
[5] U Khan N Ahmed M Asadullah and S T Mohyud-dinldquoEffects of viscous dissipation and slip velocity on two-dim-ensional and axisymmetric squeezing flow of Cu-water and Cu-kerosene nanofluidsrdquo Propulsion and Power Research vol 4 no1 pp 40ndash49 2015
[6] A Dib A Haiahem and B Bou-said ldquoApproximate analyticalsolution of squeezing unsteady nanofluid flowrdquo Powder Technol-ogy vol 269 pp 193ndash199 2015
[7] S U S Choi and J A Eastman ldquoEnhancing thermal con-ductivity of fluids with nanoparticlerdquo in Proceedings of theInternational Mechanical Engineering Congress and Exhibitionvol 231 pp 99ndash105 San Francisco Calif USA November 1995
[8] M Azimi and R Riazi ldquoHeat transfer analysis of GO-waternanofluid flow between two parallel disksrdquo Propulsion andPower Research vol 4 no 1 pp 23ndash30 2015
[9] A Aziz W A Khan and I Pop ldquoFree convection boundarylayer flow past a horizontal flat plate embedded in porousmedium filled by nanofluid containing gyrotactic microorgan-ismsrdquo International Journal of Thermal Sciences vol 56 pp 48ndash57 2012
[10] M Fakour D D Ganji and M Abbasi ldquoScrutiny of underde-veloped nanofluidMHD flow and heat conduction in a channelwith porous wallsrdquo Case Studies in Thermal Engineering vol 4pp 202ndash214 2014
[11] T Grosan C Revnic I Pop andD B Ingham ldquoFree convectionheat transfer in a square cavity filled with a porous mediumsaturated by a nano-fluidrdquo International Journal of Heat andMass Transfer vol 87 pp 36ndash41 2015
[12] A V Kuznetsov and D A Nield ldquoThe Cheng-Minkowyczproblem for natural convective boundary layer flow in a porousmedium saturated by a nanofluid a revised modelrdquo Inter-national Journal of Heat andMass Transfer vol 65 pp 682ndash6852013
[13] P Rana R Bhargava and O A Begb ldquoNumerical solution formixed convection boundary layerflow of ananofluid along aninclined plate embedded in a porous mediumrdquo Computers andMathematics with Applications vol 64 no 9 pp 2816ndash28322012
[14] A M Rashad A J Chamkha and M Modather ldquoMixed con-vection boundary-layer flow past a horizontal circular cylinderembedded in a porous medium filled with a nanofluid underconvective boundary conditionrdquo Computers amp Fluids vol 86pp 380ndash388 2013
[15] M Sheikholeslami and D D Ganji ldquoNanofluid flow and heattransfer between parallel plates considering Brownian motion
Journal of Nanoscience 11
using DTMrdquo Computer Methods in Applied Mechanics andEngineering vol 283 pp 651ndash663 2015
[16] N V Ganesh B Ganga and A K Abdul Hakeem ldquoLiesymmetry group analysis of magnetic field effects on free con-vective flow of a nanofluid over a semi-infinite stretching sheetrdquoJournal of the Egyptian Mathematical Society vol 22 no 2pp 304ndash310 2014
[17] Y Lin L Zheng and G Chen ldquoUnsteady flow and heattransfer of pseudo-plastic nanoliquid in a finite thin film ona stretching surface with variable thermal conductivity andviscous dissipationrdquo Powder Technology vol 274 pp 324ndash3322015
[18] M Sheikholeslami MM Rashidi and D D Ganji ldquoNumericalinvestigation of magnetic nanofluid forced convective heattransfer in existence of variable magnetic field using two phasemodelrdquo Journal of Molecular Liquids vol 212 pp 117ndash126 2015
[19] T Hayat M Waqas S A Shehzad and A Alsaedi ldquoMixedconvection flow of a Burgers nanofluid in the presence ofstratifications and heat generationabsorptionrdquo The EuropeanPhysical Journal Plus vol 131 no 8 p 253 2016
[20] F M Abbasi T Hayat S A Shehzad F Alsaadi and NAltoaibi ldquoHydromagnetic peristaltic transport of copper-waternanofluid with temperature-dependent effective viscosityrdquo Par-ticuology vol 27 pp 133ndash140 2016
[21] M Sheikholeslami D D Ganji and M M Rashidi ldquoMagneticfield effect on unsteady nanofluid flow and heat transferusing Buongiorno modelrdquo Journal of Magnetism and MagneticMaterials vol 416 pp 164ndash173 2016
[22] O Pourmehran M Rahimi-Gorji and D D Ganji ldquoHeattransfer and flow analysis of nanofluid flow induced by astretching sheet in the presence of an external magnetic fieldrdquoJournal of the Taiwan Institute of Chemical Engineers vol 65 pp162ndash171 2016
[23] M Rahimi-Gorji O Pourmehran M Hatami and D D GanjildquoStatistical optimization of microchannel heat sink (MCHS)geometry cooled by different nanofluids using RSM analysisrdquoEuropean Physical Journal Plus vol 130 no 2 article 22 pp 1ndash21 2015
[24] W Ibrahim and B Shankar ldquoMHD boundary layer flow andheat transfer of a nanofluid past a permeable stretching sheetwith velocity thermal and solutal slip boundary conditionsrdquoComputers amp Fluids vol 75 pp 1ndash10 2013
[25] A Malvandi and D D Ganji ldquoBrownian motion and ther-mophoresis effects on slip flow of aluminawater nanofluidinside a circular microchannel in the presence of a magneticfieldrdquo International Journal ofThermal Sciences vol 84 pp 196ndash206 2014
[26] R Ul Haq S Nadeem Z H Khan and N S Akbar ldquoThermalradiation and slip effects on MHD stagnation point flow ofnanofluid over a stretching sheetrdquo Physica E vol 65 pp 17ndash232015
[27] M Govindaraju S Saranya A A Hakeem R Jayaprakash andB Ganga ldquoAnalysis of slip MHD nanofluid flow on entropygeneration in a stretching sheetrdquo Procedia Engineering vol 127pp 501ndash507 2015
[28] M J Uddin M N Kabir and O A Beg ldquoComputationalinvestigation of Stefan blowing and multiple-slip effects onbuoyancy-driven bioconvection nanofluid flow with microor-ganismsrdquo International Journal of Heat and Mass Transfer vol95 pp 116ndash130 2016
[29] K-L Hsiao ldquoStagnation electrical MHD nanofluid mixed con-vection with slip boundary on a stretching sheetrdquo AppliedThermal Engineering vol 98 pp 850ndash861 2016
[30] P K Kameswaran M Narayana P Sibanda and P V S NMurthy ldquoHydromagnetic nanofluid flow due to a stretching orshrinking sheet with viscous dissipation and chemical reactioneffectsrdquo International Journal of Heat andMass Transfer vol 55no 25-26 pp 7587ndash7595 2012
[31] M H Matin and I Pop ldquoForced convection heat and masstransfer flow of a nanofluid through a porous channel with afirst order chemical reaction on thewallrdquo International Commu-nications in Heat and Mass Transfer vol 46 pp 134ndash141 2013
[32] D Pal andGMandal ldquoInfluence of thermal radiation onmixedconvection heat and mass transfer stagnation-point flow innanofluids over stretchingshrinking sheet in a porous mediumwith chemical reactionrdquo Nuclear Engineering and Design vol273 pp 644ndash652 2014
[33] C Zhang L Zheng X Zhang and G Chen ldquoMHD flow andradiation heat transfer of nanofluids in porous media withvariable surface heat flux and chemical reactionrdquoAppliedMath-ematical Modelling vol 39 no 1 pp 165ndash181 2015
[34] HM Elshehabey and S E Ahmed ldquoMHDmixed convection ina lid-driven cavity filled by a nanofluid with sinusoidal temper-ature distribution on the both vertical walls using Buongiornorsquosnanofluid modelrdquo International Journal of Heat and MassTransfer vol 88 pp 181ndash202 2015
Submit your manuscripts athttpwwwhindawicom
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CorrosionInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Polymer ScienceInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CeramicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CompositesJournal of
NanoparticlesJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Biomaterials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
NanoscienceJournal of
TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Journal of
NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
CrystallographyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CoatingsJournal of
Advances in
Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Smart Materials Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MetallurgyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
MaterialsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nano
materials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofNanomaterials
Journal of Nanoscience 9
Table 4 Variation of119862119891lowast Nu119909lowast and Sh119909lowast with different values of magnetic parameter119872 and slip parameter 120575 for Cu-water nanofluid when119878 = 1 Sc = 1 and 120593 = 002119872 120575 119862119891lowast Nu119909
lowast Sh119909lowast
0 01 minus246084 0089198 minus0956761 01 minus254426 0088577 minus0958312 01 minus26223 0088075 minus0959741 02 minus295257 0118598 minus0951581 05 minus326534 0144432 minus094633Table 5 Variation of 119862119891lowast Nu119909lowast and Sh119909lowast with different values of volume fraction 120593 and Schmidt number Sc for Cu-water nanofluid when119872 = 1 119878 = 1 and 120575 = 01120593 Sc 119862119891lowast Nu119909
lowast Sh119909lowast
000 1 minus276550 0092885 minus095773002 1 minus254426 0088577 minus095831005 1 minus231844 0082295 minus095911002 0 minus254426 0088577 minus100000002 10 minus254426 0088577 minus063318002 50 minus254426 0088577 minus005952002 100 minus254426 0088577 minus000182
Table 6 Variation of 119862119891lowast Nu119909lowast and Sh119909lowast for different nanoparti-
cles when119872 = 1 119878 = 1 Sc = 1 120575 = 01 and 120593 = 002Nanoparticle 119862119891lowast Nu119909
lowast Sh119909lowast
Ag minus248678 0088535 minus095851Cu minus254426 0088577 minus095831TiO2 minus273607 0088742 minus095777Al2O3 minus274807 0088794 minus095772
Table 6 It is observed form Table 6 that value of the skin-friction coefficient is more for the metallic nanoparticlesthan nonmetallic nanoparticles whereas the nonmetallicnanoparticles have higher heat and mass transfer rate incomparison with metallic nanoparticles
5 Conclusions
The present paper deals with the numerical solution ofcombined heat andmass transfer effects of an unsteadyMHDlaminar two-dimensional flow of incompressible viscousnanofluids inmiddle of two parallel plates extended infinitelywith slip velocity effect The relevant nonlinear partial dif-ferential equations were transformed to a set of ordinarydifferential equations and then are solved numerically usingthe Runge-Kutta-Fehlberg fourth-fifth-order method alongwith shooting technique From the above discussion the eye-catching results are as follows
(i) Temperature drops in Cu-water nanofluid withincrease in the magnetic field strength
(ii) The velocity of the nanofluid with nonmetallicnanoparticles Al2O3 and TiO2 is greater thanmetallic
nanoparticles Cu and Ag initially when water is thebase fluid but nature gets reversed after 120578 = 05The temperature of themetallic nanoparticles is lowerthan the nonmetallic nanoparticles
(iii) When plates start moving apart then temperaturestarts decreasing
(iv) Initially with the increasing values of magnetic fieldstrength and squeeze number the velocity of Cu-water slows down but the nature in both the case getsreversed after 120578 = 05
(v) The Nusselt number and skin-friction coefficientdecrease as plates move apart for Cu-water nanofluid
(vi) Nusselt number and Sherwood number for non-metallic nanoparticles are higher than the metallicnanoparticles
(vii) Varying value of slip parameter has negligible effecton concentration and as values of squeeze numberincrease the value of concentration profile also getsslightly higher
(viii) Considerable amount of enhancement in concentra-tion can be seen as Schmidt number rises
Nomenclature
1198601 1198602 1198603 Dimensionless constant1198610 Strength of magnetic field119862 Fluid concentration119862119891 Skin-friction coefficient119862119901 Specific heat at constant pressure119901 [Jkgminus1 Kminus1]119863 Mass diffusivity
10 Journal of Nanoscience
Ec Eckert number119891 Dimensionless velocity of the fluid119895 Reference length119896nf Nanofluid effective thermalconductivity119897 Distance of plate [m]119871 Slip length119872 Magnetic parameter119898119908 Surface mass flux
Nu119909 Nusselt number119901 Pressure [Pa]Pr Prandtl number119902119908 Surface heat fluxRe119909 Local Reynolds number119878 Squeeze numberSc Schmidt numberSh119909 Sherwood number119905 Time [s]119879 Temperature [K]119906 V Velocities in 119909119910-directions [m sminus1]119909 119910 Axial and perpendicular coordinates
[m]
Greek Symbols
120583119891 Dynamic viscosity of the fluid120593 Nanoparticle volume fraction120575 Velocity slip parameter120592119891 Kinematic viscosity of the fluid [m2 sminus1]120578 Similarity variable120588119891 Density of base fluid [kgmminus3]119896119904 Thermal conductivity of the solidnanoparticle [mminus1 Kminus1]119896119891 Thermal conductivity of the base fluid[Wmminus1 Kminus1]120579 Nondimensional temperature120601 Nondimensional concentration120588nf Nanofluid density [kgmminus3]120583nf Effective dynamic viscosity of nanofluid[Pa s]120572 Squeeze parameter120590nf Electrical conductivity of nanofluid120588119904 Density of solid particle [kgmminus3]120591119908 Surface shear stress
Subscripts
119891 Base fluidℎ Condition at the wallnf Nanofluid119904 Solid
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] M J Stefan ldquoVersuchuber die scheinbare adhesion sitzungs-bersachsrdquo Akademie der Wissenschaften in Wien Math-ematisch-Naturwissenschaftliche Klasse vol 69 pp 713ndash7211874
[2] G Domairry and M Hatami ldquoSqueezing Cundashwater nanofluidflow analysis between parallel plates by DTM-Pade MethodrdquoJournal of Molecular Liquids vol 193 pp 37ndash44 2014
[3] O Pourmehran M Rahimi-Gorji M Gorji-Bandpy and D DGanji ldquoAnalytical investigation of squeezing unsteady nanofluidflow between parallel plates by LSM and CMrdquo AlexandriaEngineering Journal vol 54 no 1 pp 17ndash26 2015
[4] A K Gupta and S S Ray ldquoNumerical treatment for investiga-tion of squeezing unsteady nanofluid flow between two parallelplatesrdquo Powder Technology vol 279 pp 282ndash289 2015
[5] U Khan N Ahmed M Asadullah and S T Mohyud-dinldquoEffects of viscous dissipation and slip velocity on two-dim-ensional and axisymmetric squeezing flow of Cu-water and Cu-kerosene nanofluidsrdquo Propulsion and Power Research vol 4 no1 pp 40ndash49 2015
[6] A Dib A Haiahem and B Bou-said ldquoApproximate analyticalsolution of squeezing unsteady nanofluid flowrdquo Powder Technol-ogy vol 269 pp 193ndash199 2015
[7] S U S Choi and J A Eastman ldquoEnhancing thermal con-ductivity of fluids with nanoparticlerdquo in Proceedings of theInternational Mechanical Engineering Congress and Exhibitionvol 231 pp 99ndash105 San Francisco Calif USA November 1995
[8] M Azimi and R Riazi ldquoHeat transfer analysis of GO-waternanofluid flow between two parallel disksrdquo Propulsion andPower Research vol 4 no 1 pp 23ndash30 2015
[9] A Aziz W A Khan and I Pop ldquoFree convection boundarylayer flow past a horizontal flat plate embedded in porousmedium filled by nanofluid containing gyrotactic microorgan-ismsrdquo International Journal of Thermal Sciences vol 56 pp 48ndash57 2012
[10] M Fakour D D Ganji and M Abbasi ldquoScrutiny of underde-veloped nanofluidMHD flow and heat conduction in a channelwith porous wallsrdquo Case Studies in Thermal Engineering vol 4pp 202ndash214 2014
[11] T Grosan C Revnic I Pop andD B Ingham ldquoFree convectionheat transfer in a square cavity filled with a porous mediumsaturated by a nano-fluidrdquo International Journal of Heat andMass Transfer vol 87 pp 36ndash41 2015
[12] A V Kuznetsov and D A Nield ldquoThe Cheng-Minkowyczproblem for natural convective boundary layer flow in a porousmedium saturated by a nanofluid a revised modelrdquo Inter-national Journal of Heat andMass Transfer vol 65 pp 682ndash6852013
[13] P Rana R Bhargava and O A Begb ldquoNumerical solution formixed convection boundary layerflow of ananofluid along aninclined plate embedded in a porous mediumrdquo Computers andMathematics with Applications vol 64 no 9 pp 2816ndash28322012
[14] A M Rashad A J Chamkha and M Modather ldquoMixed con-vection boundary-layer flow past a horizontal circular cylinderembedded in a porous medium filled with a nanofluid underconvective boundary conditionrdquo Computers amp Fluids vol 86pp 380ndash388 2013
[15] M Sheikholeslami and D D Ganji ldquoNanofluid flow and heattransfer between parallel plates considering Brownian motion
Journal of Nanoscience 11
using DTMrdquo Computer Methods in Applied Mechanics andEngineering vol 283 pp 651ndash663 2015
[16] N V Ganesh B Ganga and A K Abdul Hakeem ldquoLiesymmetry group analysis of magnetic field effects on free con-vective flow of a nanofluid over a semi-infinite stretching sheetrdquoJournal of the Egyptian Mathematical Society vol 22 no 2pp 304ndash310 2014
[17] Y Lin L Zheng and G Chen ldquoUnsteady flow and heattransfer of pseudo-plastic nanoliquid in a finite thin film ona stretching surface with variable thermal conductivity andviscous dissipationrdquo Powder Technology vol 274 pp 324ndash3322015
[18] M Sheikholeslami MM Rashidi and D D Ganji ldquoNumericalinvestigation of magnetic nanofluid forced convective heattransfer in existence of variable magnetic field using two phasemodelrdquo Journal of Molecular Liquids vol 212 pp 117ndash126 2015
[19] T Hayat M Waqas S A Shehzad and A Alsaedi ldquoMixedconvection flow of a Burgers nanofluid in the presence ofstratifications and heat generationabsorptionrdquo The EuropeanPhysical Journal Plus vol 131 no 8 p 253 2016
[20] F M Abbasi T Hayat S A Shehzad F Alsaadi and NAltoaibi ldquoHydromagnetic peristaltic transport of copper-waternanofluid with temperature-dependent effective viscosityrdquo Par-ticuology vol 27 pp 133ndash140 2016
[21] M Sheikholeslami D D Ganji and M M Rashidi ldquoMagneticfield effect on unsteady nanofluid flow and heat transferusing Buongiorno modelrdquo Journal of Magnetism and MagneticMaterials vol 416 pp 164ndash173 2016
[22] O Pourmehran M Rahimi-Gorji and D D Ganji ldquoHeattransfer and flow analysis of nanofluid flow induced by astretching sheet in the presence of an external magnetic fieldrdquoJournal of the Taiwan Institute of Chemical Engineers vol 65 pp162ndash171 2016
[23] M Rahimi-Gorji O Pourmehran M Hatami and D D GanjildquoStatistical optimization of microchannel heat sink (MCHS)geometry cooled by different nanofluids using RSM analysisrdquoEuropean Physical Journal Plus vol 130 no 2 article 22 pp 1ndash21 2015
[24] W Ibrahim and B Shankar ldquoMHD boundary layer flow andheat transfer of a nanofluid past a permeable stretching sheetwith velocity thermal and solutal slip boundary conditionsrdquoComputers amp Fluids vol 75 pp 1ndash10 2013
[25] A Malvandi and D D Ganji ldquoBrownian motion and ther-mophoresis effects on slip flow of aluminawater nanofluidinside a circular microchannel in the presence of a magneticfieldrdquo International Journal ofThermal Sciences vol 84 pp 196ndash206 2014
[26] R Ul Haq S Nadeem Z H Khan and N S Akbar ldquoThermalradiation and slip effects on MHD stagnation point flow ofnanofluid over a stretching sheetrdquo Physica E vol 65 pp 17ndash232015
[27] M Govindaraju S Saranya A A Hakeem R Jayaprakash andB Ganga ldquoAnalysis of slip MHD nanofluid flow on entropygeneration in a stretching sheetrdquo Procedia Engineering vol 127pp 501ndash507 2015
[28] M J Uddin M N Kabir and O A Beg ldquoComputationalinvestigation of Stefan blowing and multiple-slip effects onbuoyancy-driven bioconvection nanofluid flow with microor-ganismsrdquo International Journal of Heat and Mass Transfer vol95 pp 116ndash130 2016
[29] K-L Hsiao ldquoStagnation electrical MHD nanofluid mixed con-vection with slip boundary on a stretching sheetrdquo AppliedThermal Engineering vol 98 pp 850ndash861 2016
[30] P K Kameswaran M Narayana P Sibanda and P V S NMurthy ldquoHydromagnetic nanofluid flow due to a stretching orshrinking sheet with viscous dissipation and chemical reactioneffectsrdquo International Journal of Heat andMass Transfer vol 55no 25-26 pp 7587ndash7595 2012
[31] M H Matin and I Pop ldquoForced convection heat and masstransfer flow of a nanofluid through a porous channel with afirst order chemical reaction on thewallrdquo International Commu-nications in Heat and Mass Transfer vol 46 pp 134ndash141 2013
[32] D Pal andGMandal ldquoInfluence of thermal radiation onmixedconvection heat and mass transfer stagnation-point flow innanofluids over stretchingshrinking sheet in a porous mediumwith chemical reactionrdquo Nuclear Engineering and Design vol273 pp 644ndash652 2014
[33] C Zhang L Zheng X Zhang and G Chen ldquoMHD flow andradiation heat transfer of nanofluids in porous media withvariable surface heat flux and chemical reactionrdquoAppliedMath-ematical Modelling vol 39 no 1 pp 165ndash181 2015
[34] HM Elshehabey and S E Ahmed ldquoMHDmixed convection ina lid-driven cavity filled by a nanofluid with sinusoidal temper-ature distribution on the both vertical walls using Buongiornorsquosnanofluid modelrdquo International Journal of Heat and MassTransfer vol 88 pp 181ndash202 2015
Submit your manuscripts athttpwwwhindawicom
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CorrosionInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Polymer ScienceInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CeramicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CompositesJournal of
NanoparticlesJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Biomaterials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
NanoscienceJournal of
TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Journal of
NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
CrystallographyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CoatingsJournal of
Advances in
Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Smart Materials Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MetallurgyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
MaterialsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nano
materials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofNanomaterials
10 Journal of Nanoscience
Ec Eckert number119891 Dimensionless velocity of the fluid119895 Reference length119896nf Nanofluid effective thermalconductivity119897 Distance of plate [m]119871 Slip length119872 Magnetic parameter119898119908 Surface mass flux
Nu119909 Nusselt number119901 Pressure [Pa]Pr Prandtl number119902119908 Surface heat fluxRe119909 Local Reynolds number119878 Squeeze numberSc Schmidt numberSh119909 Sherwood number119905 Time [s]119879 Temperature [K]119906 V Velocities in 119909119910-directions [m sminus1]119909 119910 Axial and perpendicular coordinates
[m]
Greek Symbols
120583119891 Dynamic viscosity of the fluid120593 Nanoparticle volume fraction120575 Velocity slip parameter120592119891 Kinematic viscosity of the fluid [m2 sminus1]120578 Similarity variable120588119891 Density of base fluid [kgmminus3]119896119904 Thermal conductivity of the solidnanoparticle [mminus1 Kminus1]119896119891 Thermal conductivity of the base fluid[Wmminus1 Kminus1]120579 Nondimensional temperature120601 Nondimensional concentration120588nf Nanofluid density [kgmminus3]120583nf Effective dynamic viscosity of nanofluid[Pa s]120572 Squeeze parameter120590nf Electrical conductivity of nanofluid120588119904 Density of solid particle [kgmminus3]120591119908 Surface shear stress
Subscripts
119891 Base fluidℎ Condition at the wallnf Nanofluid119904 Solid
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] M J Stefan ldquoVersuchuber die scheinbare adhesion sitzungs-bersachsrdquo Akademie der Wissenschaften in Wien Math-ematisch-Naturwissenschaftliche Klasse vol 69 pp 713ndash7211874
[2] G Domairry and M Hatami ldquoSqueezing Cundashwater nanofluidflow analysis between parallel plates by DTM-Pade MethodrdquoJournal of Molecular Liquids vol 193 pp 37ndash44 2014
[3] O Pourmehran M Rahimi-Gorji M Gorji-Bandpy and D DGanji ldquoAnalytical investigation of squeezing unsteady nanofluidflow between parallel plates by LSM and CMrdquo AlexandriaEngineering Journal vol 54 no 1 pp 17ndash26 2015
[4] A K Gupta and S S Ray ldquoNumerical treatment for investiga-tion of squeezing unsteady nanofluid flow between two parallelplatesrdquo Powder Technology vol 279 pp 282ndash289 2015
[5] U Khan N Ahmed M Asadullah and S T Mohyud-dinldquoEffects of viscous dissipation and slip velocity on two-dim-ensional and axisymmetric squeezing flow of Cu-water and Cu-kerosene nanofluidsrdquo Propulsion and Power Research vol 4 no1 pp 40ndash49 2015
[6] A Dib A Haiahem and B Bou-said ldquoApproximate analyticalsolution of squeezing unsteady nanofluid flowrdquo Powder Technol-ogy vol 269 pp 193ndash199 2015
[7] S U S Choi and J A Eastman ldquoEnhancing thermal con-ductivity of fluids with nanoparticlerdquo in Proceedings of theInternational Mechanical Engineering Congress and Exhibitionvol 231 pp 99ndash105 San Francisco Calif USA November 1995
[8] M Azimi and R Riazi ldquoHeat transfer analysis of GO-waternanofluid flow between two parallel disksrdquo Propulsion andPower Research vol 4 no 1 pp 23ndash30 2015
[9] A Aziz W A Khan and I Pop ldquoFree convection boundarylayer flow past a horizontal flat plate embedded in porousmedium filled by nanofluid containing gyrotactic microorgan-ismsrdquo International Journal of Thermal Sciences vol 56 pp 48ndash57 2012
[10] M Fakour D D Ganji and M Abbasi ldquoScrutiny of underde-veloped nanofluidMHD flow and heat conduction in a channelwith porous wallsrdquo Case Studies in Thermal Engineering vol 4pp 202ndash214 2014
[11] T Grosan C Revnic I Pop andD B Ingham ldquoFree convectionheat transfer in a square cavity filled with a porous mediumsaturated by a nano-fluidrdquo International Journal of Heat andMass Transfer vol 87 pp 36ndash41 2015
[12] A V Kuznetsov and D A Nield ldquoThe Cheng-Minkowyczproblem for natural convective boundary layer flow in a porousmedium saturated by a nanofluid a revised modelrdquo Inter-national Journal of Heat andMass Transfer vol 65 pp 682ndash6852013
[13] P Rana R Bhargava and O A Begb ldquoNumerical solution formixed convection boundary layerflow of ananofluid along aninclined plate embedded in a porous mediumrdquo Computers andMathematics with Applications vol 64 no 9 pp 2816ndash28322012
[14] A M Rashad A J Chamkha and M Modather ldquoMixed con-vection boundary-layer flow past a horizontal circular cylinderembedded in a porous medium filled with a nanofluid underconvective boundary conditionrdquo Computers amp Fluids vol 86pp 380ndash388 2013
[15] M Sheikholeslami and D D Ganji ldquoNanofluid flow and heattransfer between parallel plates considering Brownian motion
Journal of Nanoscience 11
using DTMrdquo Computer Methods in Applied Mechanics andEngineering vol 283 pp 651ndash663 2015
[16] N V Ganesh B Ganga and A K Abdul Hakeem ldquoLiesymmetry group analysis of magnetic field effects on free con-vective flow of a nanofluid over a semi-infinite stretching sheetrdquoJournal of the Egyptian Mathematical Society vol 22 no 2pp 304ndash310 2014
[17] Y Lin L Zheng and G Chen ldquoUnsteady flow and heattransfer of pseudo-plastic nanoliquid in a finite thin film ona stretching surface with variable thermal conductivity andviscous dissipationrdquo Powder Technology vol 274 pp 324ndash3322015
[18] M Sheikholeslami MM Rashidi and D D Ganji ldquoNumericalinvestigation of magnetic nanofluid forced convective heattransfer in existence of variable magnetic field using two phasemodelrdquo Journal of Molecular Liquids vol 212 pp 117ndash126 2015
[19] T Hayat M Waqas S A Shehzad and A Alsaedi ldquoMixedconvection flow of a Burgers nanofluid in the presence ofstratifications and heat generationabsorptionrdquo The EuropeanPhysical Journal Plus vol 131 no 8 p 253 2016
[20] F M Abbasi T Hayat S A Shehzad F Alsaadi and NAltoaibi ldquoHydromagnetic peristaltic transport of copper-waternanofluid with temperature-dependent effective viscosityrdquo Par-ticuology vol 27 pp 133ndash140 2016
[21] M Sheikholeslami D D Ganji and M M Rashidi ldquoMagneticfield effect on unsteady nanofluid flow and heat transferusing Buongiorno modelrdquo Journal of Magnetism and MagneticMaterials vol 416 pp 164ndash173 2016
[22] O Pourmehran M Rahimi-Gorji and D D Ganji ldquoHeattransfer and flow analysis of nanofluid flow induced by astretching sheet in the presence of an external magnetic fieldrdquoJournal of the Taiwan Institute of Chemical Engineers vol 65 pp162ndash171 2016
[23] M Rahimi-Gorji O Pourmehran M Hatami and D D GanjildquoStatistical optimization of microchannel heat sink (MCHS)geometry cooled by different nanofluids using RSM analysisrdquoEuropean Physical Journal Plus vol 130 no 2 article 22 pp 1ndash21 2015
[24] W Ibrahim and B Shankar ldquoMHD boundary layer flow andheat transfer of a nanofluid past a permeable stretching sheetwith velocity thermal and solutal slip boundary conditionsrdquoComputers amp Fluids vol 75 pp 1ndash10 2013
[25] A Malvandi and D D Ganji ldquoBrownian motion and ther-mophoresis effects on slip flow of aluminawater nanofluidinside a circular microchannel in the presence of a magneticfieldrdquo International Journal ofThermal Sciences vol 84 pp 196ndash206 2014
[26] R Ul Haq S Nadeem Z H Khan and N S Akbar ldquoThermalradiation and slip effects on MHD stagnation point flow ofnanofluid over a stretching sheetrdquo Physica E vol 65 pp 17ndash232015
[27] M Govindaraju S Saranya A A Hakeem R Jayaprakash andB Ganga ldquoAnalysis of slip MHD nanofluid flow on entropygeneration in a stretching sheetrdquo Procedia Engineering vol 127pp 501ndash507 2015
[28] M J Uddin M N Kabir and O A Beg ldquoComputationalinvestigation of Stefan blowing and multiple-slip effects onbuoyancy-driven bioconvection nanofluid flow with microor-ganismsrdquo International Journal of Heat and Mass Transfer vol95 pp 116ndash130 2016
[29] K-L Hsiao ldquoStagnation electrical MHD nanofluid mixed con-vection with slip boundary on a stretching sheetrdquo AppliedThermal Engineering vol 98 pp 850ndash861 2016
[30] P K Kameswaran M Narayana P Sibanda and P V S NMurthy ldquoHydromagnetic nanofluid flow due to a stretching orshrinking sheet with viscous dissipation and chemical reactioneffectsrdquo International Journal of Heat andMass Transfer vol 55no 25-26 pp 7587ndash7595 2012
[31] M H Matin and I Pop ldquoForced convection heat and masstransfer flow of a nanofluid through a porous channel with afirst order chemical reaction on thewallrdquo International Commu-nications in Heat and Mass Transfer vol 46 pp 134ndash141 2013
[32] D Pal andGMandal ldquoInfluence of thermal radiation onmixedconvection heat and mass transfer stagnation-point flow innanofluids over stretchingshrinking sheet in a porous mediumwith chemical reactionrdquo Nuclear Engineering and Design vol273 pp 644ndash652 2014
[33] C Zhang L Zheng X Zhang and G Chen ldquoMHD flow andradiation heat transfer of nanofluids in porous media withvariable surface heat flux and chemical reactionrdquoAppliedMath-ematical Modelling vol 39 no 1 pp 165ndash181 2015
[34] HM Elshehabey and S E Ahmed ldquoMHDmixed convection ina lid-driven cavity filled by a nanofluid with sinusoidal temper-ature distribution on the both vertical walls using Buongiornorsquosnanofluid modelrdquo International Journal of Heat and MassTransfer vol 88 pp 181ndash202 2015
Submit your manuscripts athttpwwwhindawicom
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CorrosionInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Polymer ScienceInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CeramicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CompositesJournal of
NanoparticlesJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Biomaterials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
NanoscienceJournal of
TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Journal of
NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
CrystallographyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CoatingsJournal of
Advances in
Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Smart Materials Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MetallurgyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
MaterialsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nano
materials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofNanomaterials
Journal of Nanoscience 11
using DTMrdquo Computer Methods in Applied Mechanics andEngineering vol 283 pp 651ndash663 2015
[16] N V Ganesh B Ganga and A K Abdul Hakeem ldquoLiesymmetry group analysis of magnetic field effects on free con-vective flow of a nanofluid over a semi-infinite stretching sheetrdquoJournal of the Egyptian Mathematical Society vol 22 no 2pp 304ndash310 2014
[17] Y Lin L Zheng and G Chen ldquoUnsteady flow and heattransfer of pseudo-plastic nanoliquid in a finite thin film ona stretching surface with variable thermal conductivity andviscous dissipationrdquo Powder Technology vol 274 pp 324ndash3322015
[18] M Sheikholeslami MM Rashidi and D D Ganji ldquoNumericalinvestigation of magnetic nanofluid forced convective heattransfer in existence of variable magnetic field using two phasemodelrdquo Journal of Molecular Liquids vol 212 pp 117ndash126 2015
[19] T Hayat M Waqas S A Shehzad and A Alsaedi ldquoMixedconvection flow of a Burgers nanofluid in the presence ofstratifications and heat generationabsorptionrdquo The EuropeanPhysical Journal Plus vol 131 no 8 p 253 2016
[20] F M Abbasi T Hayat S A Shehzad F Alsaadi and NAltoaibi ldquoHydromagnetic peristaltic transport of copper-waternanofluid with temperature-dependent effective viscosityrdquo Par-ticuology vol 27 pp 133ndash140 2016
[21] M Sheikholeslami D D Ganji and M M Rashidi ldquoMagneticfield effect on unsteady nanofluid flow and heat transferusing Buongiorno modelrdquo Journal of Magnetism and MagneticMaterials vol 416 pp 164ndash173 2016
[22] O Pourmehran M Rahimi-Gorji and D D Ganji ldquoHeattransfer and flow analysis of nanofluid flow induced by astretching sheet in the presence of an external magnetic fieldrdquoJournal of the Taiwan Institute of Chemical Engineers vol 65 pp162ndash171 2016
[23] M Rahimi-Gorji O Pourmehran M Hatami and D D GanjildquoStatistical optimization of microchannel heat sink (MCHS)geometry cooled by different nanofluids using RSM analysisrdquoEuropean Physical Journal Plus vol 130 no 2 article 22 pp 1ndash21 2015
[24] W Ibrahim and B Shankar ldquoMHD boundary layer flow andheat transfer of a nanofluid past a permeable stretching sheetwith velocity thermal and solutal slip boundary conditionsrdquoComputers amp Fluids vol 75 pp 1ndash10 2013
[25] A Malvandi and D D Ganji ldquoBrownian motion and ther-mophoresis effects on slip flow of aluminawater nanofluidinside a circular microchannel in the presence of a magneticfieldrdquo International Journal ofThermal Sciences vol 84 pp 196ndash206 2014
[26] R Ul Haq S Nadeem Z H Khan and N S Akbar ldquoThermalradiation and slip effects on MHD stagnation point flow ofnanofluid over a stretching sheetrdquo Physica E vol 65 pp 17ndash232015
[27] M Govindaraju S Saranya A A Hakeem R Jayaprakash andB Ganga ldquoAnalysis of slip MHD nanofluid flow on entropygeneration in a stretching sheetrdquo Procedia Engineering vol 127pp 501ndash507 2015
[28] M J Uddin M N Kabir and O A Beg ldquoComputationalinvestigation of Stefan blowing and multiple-slip effects onbuoyancy-driven bioconvection nanofluid flow with microor-ganismsrdquo International Journal of Heat and Mass Transfer vol95 pp 116ndash130 2016
[29] K-L Hsiao ldquoStagnation electrical MHD nanofluid mixed con-vection with slip boundary on a stretching sheetrdquo AppliedThermal Engineering vol 98 pp 850ndash861 2016
[30] P K Kameswaran M Narayana P Sibanda and P V S NMurthy ldquoHydromagnetic nanofluid flow due to a stretching orshrinking sheet with viscous dissipation and chemical reactioneffectsrdquo International Journal of Heat andMass Transfer vol 55no 25-26 pp 7587ndash7595 2012
[31] M H Matin and I Pop ldquoForced convection heat and masstransfer flow of a nanofluid through a porous channel with afirst order chemical reaction on thewallrdquo International Commu-nications in Heat and Mass Transfer vol 46 pp 134ndash141 2013
[32] D Pal andGMandal ldquoInfluence of thermal radiation onmixedconvection heat and mass transfer stagnation-point flow innanofluids over stretchingshrinking sheet in a porous mediumwith chemical reactionrdquo Nuclear Engineering and Design vol273 pp 644ndash652 2014
[33] C Zhang L Zheng X Zhang and G Chen ldquoMHD flow andradiation heat transfer of nanofluids in porous media withvariable surface heat flux and chemical reactionrdquoAppliedMath-ematical Modelling vol 39 no 1 pp 165ndash181 2015
[34] HM Elshehabey and S E Ahmed ldquoMHDmixed convection ina lid-driven cavity filled by a nanofluid with sinusoidal temper-ature distribution on the both vertical walls using Buongiornorsquosnanofluid modelrdquo International Journal of Heat and MassTransfer vol 88 pp 181ndash202 2015
Submit your manuscripts athttpwwwhindawicom
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CorrosionInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Polymer ScienceInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CeramicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CompositesJournal of
NanoparticlesJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Biomaterials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
NanoscienceJournal of
TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Journal of
NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
CrystallographyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CoatingsJournal of
Advances in
Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Smart Materials Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MetallurgyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
MaterialsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nano
materials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofNanomaterials
Submit your manuscripts athttpwwwhindawicom
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CorrosionInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Polymer ScienceInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CeramicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CompositesJournal of
NanoparticlesJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Biomaterials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
NanoscienceJournal of
TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Journal of
NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
CrystallographyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CoatingsJournal of
Advances in
Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Smart Materials Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MetallurgyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
MaterialsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nano
materials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofNanomaterials
