Research Article Free Convection in Heat Transfer Flow...

7
Research Article Free Convection in Heat Transfer Flow over a Moving Sheet in Alumina Water Nanofluid Padam Singh and Manoj Kumar Department of Mathematics, Statistics and Computer Science, G.B. Pant University of Agriculture and Technology, Pantnagar, Uttarakhand 263145, India Correspondence should be addressed to Padam Singh; [email protected] Received 24 May 2014; Accepted 4 October 2014; Published 2 November 2014 Academic Editor: Oronzio Manca Copyright © 2014 P. Singh and M. Kumar. 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. e present paper deals with study of free convection in two-dimensional magnetohydrodynamic (MHD) boundary layer flow of an incompressible, viscous, electrically conducting, and steady nanofluid. e governing equations representing fluid flow are transformed into a set of simultaneous ordinary differential equations by using appropriate similarity transformation. e equations thus obtained have been solved numerically using adaptive Runge-Kutta method with shooting technique. e effects of physical parameters like magnetic parameter, temperature buoyancy parameter on relative velocity and temperature distribution profile, shear stress profile, and temperature gradient profile were depicted graphically and analyzed. Significant changes were observed due to these parameters in velocity and temperature profiles. 1. Introduction Applications of heat transfer through moving material in a moving fluid medium have wide range of applications in real world. Various researchers studied the concepts of moving sheet in a flowing fluid medium. Numerical and analytical solution for momentum and energy in laminar boundary layer flow over continuously moving sheet was discussed by Zheng and Zhang [1]. e authors found that the effect of velocity ratio parameter and other parameters on heat transfer were significant. Al-Sanea [2] studied the flow and thermal characteristic of a moving vertical sheet of extruded material close to and far downstream from the extrusion slot. Regimes of forced, mixed, and natural convection have been delineated, in buoyancy flow, as a function of Reynolds and thermal Grashof numbers for various values of Prandtl number and buoyancy parameter. Seddeek [3] analyzed the effect of magnetic field on the flow of micropolar fluid past a continuously moving plate. Patel et al. [4] suggested a new model for thermal conductivity of nanofluids which is found to agree excellently with a wide range of experimental published data. e momentum and heat transfer flow of an incompressible laminar fluid past a moving sheet based on composite reference velocity was studied by Cortell [5]. e author found that the direction of the wall shear stress changes in such an interval and an increase of the relative velocity parameter yields an increase in temperature. Ishak et al. [6] studied the magnetohydrodynamic boundary layer flow due to a moving extensible surface. It was found that the dual solutions exist for the flow near -axis, where the velocity profiles show a reversed flow. Bachok et al. [7] investigated steady, laminar boundary layer flow of a nanofluid past a moving semi infinite flat plate. e results indicate that dual solutions exist when the plate and the free stream move in the opposite directions. Habib and El-Zahar [8] proposed a model for determining the heat transfer between a moving sheet and flowing fluid. e authors found that the heat transfer depends on the relative velocity between the moving fluid and the moving sheet to certain values. Energy conversion for conjugate convection, conduction, and radiation analysis have been performed for free convective heat transfer with radiation by Hsiao [9]. e results show that the free convection effect will produce a larger heat transfer effect better than the forced convection. Raguraman et al. [10] investigated the effects of some important design parameters for coal-water slurry Hindawi Publishing Corporation Journal of Engineering Volume 2014, Article ID 137426, 6 pages http://dx.doi.org/10.1155/2014/137426

Transcript of Research Article Free Convection in Heat Transfer Flow...

Page 1: Research Article Free Convection in Heat Transfer Flow ...downloads.hindawi.com/journals/je/2014/137426.pdf · V =0 + V = 1 nf + nf 2 1/22 + nf nf 2 ( ) nf + V = nf 2 2, where and

Research ArticleFree Convection in Heat Transfer Flow over a Moving Sheet inAlumina Water Nanofluid

Padam Singh and Manoj Kumar

Department of Mathematics Statistics and Computer Science GB Pant University of Agriculture and TechnologyPantnagar Uttarakhand 263145 India

Correspondence should be addressed to Padam Singh tomarpadamgmailcom

Received 24 May 2014 Accepted 4 October 2014 Published 2 November 2014

Academic Editor Oronzio Manca

Copyright copy 2014 P Singh and M Kumar This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The present paper deals with study of free convection in two-dimensional magnetohydrodynamic (MHD) boundary layer flowof an incompressible viscous electrically conducting and steady nanofluid The governing equations representing fluid flow aretransformed into a set of simultaneous ordinary differential equations by using appropriate similarity transformationThe equationsthus obtained have been solved numerically using adaptive Runge-Kutta method with shooting technique The effects of physicalparameters like magnetic parameter temperature buoyancy parameter on relative velocity and temperature distribution profileshear stress profile and temperature gradient profile were depicted graphically and analyzed Significant changes were observeddue to these parameters in velocity and temperature profiles

1 Introduction

Applications of heat transfer through moving material in amoving fluid medium have wide range of applications in realworld Various researchers studied the concepts of movingsheet in a flowing fluid medium Numerical and analyticalsolution for momentum and energy in laminar boundarylayer flow over continuously moving sheet was discussedby Zheng and Zhang [1] The authors found that the effectof velocity ratio parameter and other parameters on heattransfer were significant Al-Sanea [2] studied the flow andthermal characteristic of a moving vertical sheet of extrudedmaterial close to and far downstream from the extrusionslot Regimes of forced mixed and natural convection havebeen delineated in buoyancy flow as a function of Reynoldsand thermal Grashof numbers for various values of Prandtlnumber and buoyancy parameter Seddeek [3] analyzed theeffect of magnetic field on the flow of micropolar fluid pasta continuously moving plate Patel et al [4] suggested anew model for thermal conductivity of nanofluids which isfound to agree excellently with a wide range of experimentalpublished data The momentum and heat transfer flow ofan incompressible laminar fluid past a moving sheet based

on composite reference velocity was studied by Cortell [5]The author found that the direction of the wall shear stresschanges in such an interval and an increase of the relativevelocity parameter yields an increase in temperature Ishaket al [6] studied the magnetohydrodynamic boundary layerflow due to a moving extensible surface It was found thatthe dual solutions exist for the flow near 119910-axis wherethe velocity profiles show a reversed flow Bachok et al[7] investigated steady laminar boundary layer flow of ananofluid past a moving semi infinite flat plate The resultsindicate that dual solutions exist when the plate and thefree stream move in the opposite directions Habib andEl-Zahar [8] proposed a model for determining the heattransfer between a moving sheet and flowing fluid Theauthors found that the heat transfer depends on the relativevelocity between the moving fluid and the moving sheet tocertain values Energy conversion for conjugate convectionconduction and radiation analysis have been performedfor free convective heat transfer with radiation by Hsiao[9] The results show that the free convection effect willproduce a larger heat transfer effect better than the forcedconvection Raguraman et al [10] investigated the effectsof some important design parameters for coal-water slurry

Hindawi Publishing CorporationJournal of EngineeringVolume 2014 Article ID 137426 6 pageshttpdxdoiorg1011552014137426

2 Journal of Engineering

in agitated vessel used in coal gasification such as stirrerspeed location of stirrer 119863119889 ratio and coal-water ratioConfirmation test verified that the Taguchi method achievedoptimization of heat transfer coefficient in agitated vesselThe objective of the present study is to analyze the effect ofmagnetic field and temperature buoyancy effect on the heattransfer flow between a moving sheet and moving aluminawater nanofluid environment Hsiao [11 12] studied two-dimensional conjugate heat transfer with Ohmic dissipationmixed convection of an incompressible Maxwell fluid ona stagnation point and also investigated energy conversionproblems of conjugate conduction convection and radiationheat andmass transfer with viscous dissipation andmagneticeffects The results have shown that it should produce greaterheat transfer effect with larger values of Prandtl number freeconvection parameter and conduction-convection numberOn the other hand the Eckert number and magnetic param-eter will reduce the heat transfer effect For heat convec-tion energy conversion aspect some importance parametersapplied to the system such as buoyancy parameters radiativeenergy parameter boundary proportional parameter andPrandtl number which can produce positive effects for largervalues of those parameters Idusuyi et al [13] formulated acomputational model for the heat generation and dissipationin a disk brake during braking and the following releaseperiod The results show similarity in thermal behaviourat the contact surface for the asbestos and aramid brakepad materials with a temperature difference of 18 K after 10seconds

The objective of the present study is to analyze the effectofmagnetic field and temperature buoyancy effect on the heattransfer flow between a moving sheet and moving aluminawater nanofluid environment

2 Mathematical Description

The graphical model of the problem considered has beengiven to depict the flow configuration and coordinate systemThe system deals with two-dimensional MHD boundarylayer incompressible viscous electrically conducting andsteady fluid flow (Figure 1) The governing equations repre-senting flow are the following

120597119906

120597119909+120597V120597119910

= 0 (1)

119906120597119906

120597119909+ V

120597119906

120597119910= minus

1

120588nf

120597119901

120597119909+ 120592nf

1205972119906

1205971199102

+ 119892 (120573119879)nf (119879 minus 119879infin) minus

120590nf1198612(119909)

120588nf119906

(2)

119906120597119879

120597119909+ V

120597119879

120597119910= 120572nf

1205972119879

1205971199102 (3)

where 119906 and V are the velocity components in 119909 and 119910 direc-tion respectively 120592nf (120573119879)nf 120588nf 120590nf and 120572nf are the kine-matic viscosity thermal expansion coefficient density elec-trical conductivity and thermal diffusivity of the nanofluid

x

Tinfin

Uinfin

Us

L

B

Figure 1 Flow configuration and coordinate system

119892 is the gravitational acceleration and 119879 119879119908 and 119879

infinare the

temperature of the nanofluid inside the thermal boundarylayer the plate temperature and the nanofluid temperatureat free stream respectively 119861 (119909) = (1radic119909) 119861

0denotes the

magnetic field imposed in the transverse direction to thefluid

The appropriate boundary conditions for the abovemen-tioned problem are as follows (Table 1)

119906 = 119880119904 V = 0 119879 = 119879

119904 at 119910 = 0

119906 997888rarr 119880infin 119879 997888rarr 119879

infin as 119910 997888rarr infin

(4)

3 Method for Solution

To solve the governing equations (1) (2) and (3) with theboundary conditions (4) the following similarity transforma-tion has been introduced

120595 = [120592nf (119880infin minus 119880119904) 119909]12

119891 (120578)

120579 =119879 minus 119879infin

119879119904minus 119879infin

120578 = [119880infinminus 119880119904

120592nf119909]

12

119910

(5)

Here 119880infin

and 119880119904are free stream velocity and moving sheet

velocity respectively 120595 is the stream function which satisfies(1) with 119906 = 120597120595120597119910 and V = minus120597120595120597119909 Thus (5) gives

119906 = (119880infinminus 119880119904) 1198911015840(120578)

V = minus1

2(120592nf (119880infin minus 119880119904)

119909)

12

(119891 (120578) minus 1205781198911015840(120578))

(6)

Now using (5) and (6) in (2) and (3) we get the followingnonlinear differential equations

119891101584010158401015840(120578) +

1

2119891 (120578) 119891

10158401015840(120578) minus119872119891

1015840(120578) + 119892

119905120579 (120578) = 0

12057910158401015840+1

2Prf (120578) 1205791015840 (120578) = 0

(7)

Journal of Engineering 3

Table 1 Nanofluid properties

Dynamicviscosity [14] 120583nf = 120583119891(1 minus 0)

25

Density [15] 120588nf = (1 minus 0) 120588119891 + 0120588119904Specific heatcapacity [16] (119862

119901)nf= (1 minus 0) (119862

119901)119891+ 0(119862

119901)119904

Thermalconductivity [4]

(119896119904119896119891) + (119899 minus 1) minus 0(1 minus (119896

119904119896119891))119896nf

= 119896119891(119896119904119896119891) + (119899 minus 1) minus (119899 minus 1) 0(1 minus (119896

119904119896119891))

Kinematicviscosity [17] 120592nf = 120583nf120588nf

Thermaldiffusivity 120572nf = 119896nf120588nf(119862119901)nf

Empirical shapefactor 119899 = 3120595 120595 = 1 for spherical particles

the boundary conditions (4) will become

119891 (0) = 0 1198911015840(0) =

1

|1 minus 120572|

11989110158401015840(0) = unknown 120579 (0) = 1

1205791015840(0) = unknown

1198911015840(120578) 997888rarr

120572

|1 minus 120572| 120579 (120578) 997888rarr 0 as 120578 997888rarr infin

(8)

where 120572 = 0 represent the state of moving surface instationary fluid (119880

infin= 0) and 0 lt 120572 lt infin is the condition

of moving surface in moving fluid with 119880infin

gt 119880119904 The

dimensionless parameters introduced in (7) are defined asfollows

119872 = 120590nf1198612

0120588nf(119880infin minus119880119904) is the magnetic parameter 119866

119903=

119892 (120573119879)nf (119879119904 minus 119879infin) 119909

31205922

nf is the local temperature Grashofnumber Re = 119880

infin119909120592nf is the local Reynolds number 119892

119905=

119866119903Re2 is the temperature buoyancy parameter Pr = 120592nf120572nf

is the Prandtl number and 120572 = 119880infin119880119904is the relative velocity

parameterTo solve the set of nonlinear differential equations (7)

with subject to the boundary conditions (8) adaptive Runge-Kuttamethodwith shooting technique has been appliedThismethod is based on the discretization of the problem domainand the calculation of unknown boundary conditions Thedomain of the problem is discretized and the boundaryconditions for 120578 = infin are replaced by 1198911015840 (120578max) = 120572abs(1 minus120572) and 120579 (120578max) = 0 where 120578max is sufficiently large value of120578 comparing to step size at which the boundary conditions(8) for 1198911015840(120578) and 120579(120578) are satisfied On account of theconsistency and to fulfill stability criteria 120578max = 10 and stepsize Δ120578 = 001 have been taken To solve the problem thenonlinear equations (7) are first converted into first orderordinary linear differential equations as follows

1198911015840(120578) = 119910 (2) 119891

10158401015840(120578) = 119910 (3)

119891101584010158401015840(120578) = [minus

1

2119910 (1) 119910 (3) + 119872119910 (2) minus 119892

119905119910 (4)]

1205791015840= 119910 (5) 120579

10158401015840(120578) = [minus

1

2Pr119910 (1) 119910 (5)]

(9)

Table 2 Values of 11989110158401015840(0) and 1205791015840(0) for Pr = 66556406 and 0 = 005obtained by shooting method

Magneticparameter(119872)

Temperaturebuoyancy parameter

(119892119905)

Wall shearstresses11989110158401015840(0)

Skin frictioncoefficient1205791015840(0)

00 02 minus0664300 minus197611

02 00 minus1297800 minus190991

02 05 minus1157890 minus191991

02 10 minus1016890 minus192921

02 30 minus0479190 minus196421

05 02 minus1768590 minus185411

10 02 minus2269990 minus179411

30 02 minus3605449 minus165431

0 2 4 6 8 10

00

05

10

15

20

gt = 02

Pr = 66556406 at 120601 = 05

M = 05

M = 0

M = 1

M = 3

f998400 (120578)

120578

Figure 2 Effect of magnetic parameter on velocity

There are three conditions on the boundary 120578 = 0 and twoconditions at 120578 = infin as given in (8) Shooting techniquehas been used to find required missing initial conditionsThe value of unknowns 11989110158401015840(0) and 1205791015840 (0) has been tabulatedin Table 2 Adaptive Runge-Kutta method estimates thetruncation error at each integration step and automaticallyadjusts the step size to keep the error within prescribed limitsThe formula used to measure the actual conservative erroris defined by 119890 (ℎ) = ((15)sum

5

119894=11198642

119894(ℎ))12

here 119864119894(ℎ) is the

measure of error in the dependent variable 119910119894 Per-step error

control is achieved by adjusting the increment ℎ by ℎ119894+1

=

09ℎ119894(120576119890(ℎ

119894))15 119894 = 1 2 3 4 5 so that it is approximately

equal to the prescribed tolerance 120576 = 10minus6 The assumed

initial step size is ℎ = 001

4 Result and DiscussionThe computations have been made for velocity shear stressconcentration and concentration gradient profiles corre-sponding to fixed values of magnetic parameter (119872) 120572 andtemperature buoyancy parameter (119892

119905) The value of Pr =

66556406 is taken corresponding to 005 volume fraction ofthe alumina water nanofluid at 20∘C The parameter 119892

119905has

been introduced to represent the temperature buoyancy effecton flow field Figures 2ndash6 exhibit the velocity shear stress

4 Journal of Engineering

0 2 4 6 8 10

0

1gt = 02

(120578 = 293290777 f998400(120578) = minus00311476673)

M = 0 02 05 1 3

120578

f998400998400(120578) minus1

minus2

minus3

minus4

Pr = 66556406 at 120601 = 05

Figure 3 Effect of magnetic parameter on shear stress

00 05 10 15 2000

02

04

06

08

10

gt = 02

Pr = 66556406 at 120601 = 05

M = 0 02 05 1 3

120578

120579(120578)

Figure 4 Effect of magnetic parameter on temperature

concentration and concentration gradient profiles respec-tively for various values of magnetic parameter (ie119872 = 002 05 1 3) It is noticed that in Figures 2 and 3 that the veloc-ity decreases and the shear stress decreases up to (293290777minus00311476673) and then increases with an increase in themagnetic parameter The temperature increases with anincrease in the magnetic parameter temperature gradi-ent increases up to (0524497748 minus0835081441) and thendecreases with an increment in the values of magneticparameter as shown in Figures 4 and 5 Figures 6 and 7show the effect of convection parameter on velocity shearstress concentration and concentration gradient profilesrespectively for 119892

119905= 0 02 05 1 It has been observed that

the velocity increases with an increase in the convectionparameter as shown in Figure 6 and shear stress increases upto (0433856945 minus103343467) with an increase in the con-vection parameter and then decreases as shown in Figure 7The effect of convection parameter on temperature gradientand temperature also has been studied through Figures 8 and9 It is noticed that the temperature and temperature gradient

00 05 10 15 20

00

M = 0 02 05 1 3

gt = 02

Pr = 66556406 at 120601 = 05

(120578 = 0524497748 120579998400(120578) = minus0835081441)

minus05

minus10

minus15

minus20

120579998400 (120578)

120578

Figure 5 Effect of magnetic parameter on temperature gradient

0 2 4 6 8 10

06

08

10

12

14

16

18

20

M = 02

gt = 0 02 05 1

Pr = 66556406 at 120601 = 05

Zoom

f998400 (120578)

120578

Figure 6 Effect of temperature buoyancy parameter on velocity

0 2 4

00

02

Pr = 66556406 at 120601 = 05

M = 02

gt = 0 02 05 1

(120578 = 0433856945 f998400(120578) = minus103343467)

120578

minus02

minus04

minus06

minus08

minus10

minus12

minus14

f998400998400(120578)

Figure 7 Effect of temperature buoyancy parameter on shear stress

Journal of Engineering 5

00 02 04 06 08 10 12 14

00

Pr = 66556406 at 120601 = 05

M = 02

gt = 0 02 05 1

minus05

minus10

minus15

minus20

Zoom

120579998400 (120578)

120578

Figure 8 Effect of temperature buoyancy parameter on temperaturegradient

00 02 04 06 08 10 12 14

00

02

04

06

08

10

Pr = 66556406 at 120601 = 05

M = 02

gt = 0 02 05 1

Zoom

120579(120578)

120578

Figure 9 Effect of temperature buoyancy parameter on tempera-ture

show significant changes with an increase in the temperaturebuoyancy parameter

5 Conclusion

Free convective heat transfer in an electronically conduct-ing nanofluid over a moving sheet with magnetic field isconsidered The nonlinear partial differential equations aretransformed into a system of nonlinear ordinary differentialequations by using similarity transformation and then solvednumerically using the Runge-Kutta fifth order method along

with the shooting technique From the numerical calculationsof the skin friction coefficient and wall shear stresses thefollowing is concluded

(i) An increase in magnetic parameter decreases thevelocity and wall shear stress but enhances the tem-perature skin friction coefficient The shear stressdecrease up to a fixed value of magnetic parameterand after that goes to increases For the magneticparameter temperature gradient shows the oppositebehaviour to the shear stress

(ii) As the temperature buoyancy parameter increases itincreases the velocity wall shear stress and decreasesskin friction coefficient and temperature gradient

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] L C Zheng and X X Zhang ldquoSkin friction and heat transferin power-law fluid laminar boundary layer along a movingsurfacerdquo International Journal of Heat and Mass Transfer vol45 no 13 pp 2667ndash2672 2002

[2] S A Al-Sanea ldquoConvection regimes and heat transfer char-acteristics along a continuously moving heated vertical platerdquoInternational Journal of Heat and Fluid Flow vol 24 no 6 pp888ndash901 2003

[3] M A Seddeek ldquoFlow of a magneto-micropolar fluid past acontinuously moving platerdquo Physics Letters Section A GeneralAtomic and Solid State Physics vol 306 no 4 pp 255ndash257 2003

[4] H E Patel T Sundararajan T Pradeep A Dasgupta N Das-gupta and S K Das ldquoA micro-convection model for thermalconductivity of nanofluidsrdquo PramanamdashJournal of Physics vol65 no 5 pp 863ndash869 2005

[5] R Cortell ldquoFlow and heat transfer in a moving fluid overa moving flat surfacerdquo Theoretical and Computational FluidDynamics vol 21 no 6 pp 435ndash446 2007

[6] A Ishak R Nazar and I Pop ldquoMHD boundary-layer flowdue to a moving extensible surfacerdquo Journal of EngineeringMathematics vol 62 no 1 pp 23ndash33 2008

[7] N Bachok A Ishak and I Pop ldquoBoundary-layer flow ofnanofluids over a moving surface in a flowing fluidrdquo Interna-tional Journal of Thermal Sciences vol 49 no 9 pp 1663ndash16682010

[8] H M Habib and E R El-Zahar ldquoMathematical modeling ofheat-transfer for a moving sheet in a moving fluidrdquo Journal ofApplied Fluid Mechanics vol 6 no 3 pp 369ndash373 2013

[9] K-L Hsiao ldquoEnergy conversion conjugate conduction-convection and radiation over non-linearly extrusion stretchingsheet with physical multimedia effectsrdquo Energy vol 59 pp494ndash502 2013

[10] C M Raguraman A Ragupathy and L Sivakumar ldquoExper-imental determination of heat transfer coefficient in stirredvessel for coal-water slurry based on the taguchi methodrdquoJournal of Engineering vol 2013 Article ID 719296 8 pages2013

6 Journal of Engineering

[11] K-L Hsiao ldquoConjugate heat transfer for mixed convection andMaxwell fluid on a stagnation pointrdquoArabian Journal for Scienceand Engineering vol 39 no 6 pp 4325ndash4332 2014

[12] K-L Hsiao ldquoNanofluid flow with multimedia physical featuresfor conjugate mixed convection and radiationrdquo Computers ampFluids vol 104 pp 1ndash8 2014

[13] N Idusuyi I Babajide O K Ajayi and T T Olugasa ldquoAcomputational study on the use of an aluminium metal matrixcomposite and aramid as alternative brake disc and brake padmaterialrdquo Journal of Engineering vol 2014 Article ID 494697 6pages 2014

[14] A Einstein Investigation on the Theory of the Brownian Move-ment Dover New York NY USA 1956

[15] D A Drew and S L PassmanTheory ofMulticomponent FluidsSpringer New York NY USA 1999

[16] Y Xuan and W Roetzel ldquoConceptions for heat transfer corre-lation of nanofluidsrdquo International Journal of Heat and MassTransfer vol 43 no 19 pp 3701ndash3707 2000

[17] E Mamut ldquoCharacterization of heat and mass transfer proper-ties of nanofluidsrdquoRomanian Journal of Physics vol 51 pp 5ndash122006

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Page 2: Research Article Free Convection in Heat Transfer Flow ...downloads.hindawi.com/journals/je/2014/137426.pdf · V =0 + V = 1 nf + nf 2 1/22 + nf nf 2 ( ) nf + V = nf 2 2, where and

2 Journal of Engineering

in agitated vessel used in coal gasification such as stirrerspeed location of stirrer 119863119889 ratio and coal-water ratioConfirmation test verified that the Taguchi method achievedoptimization of heat transfer coefficient in agitated vesselThe objective of the present study is to analyze the effect ofmagnetic field and temperature buoyancy effect on the heattransfer flow between a moving sheet and moving aluminawater nanofluid environment Hsiao [11 12] studied two-dimensional conjugate heat transfer with Ohmic dissipationmixed convection of an incompressible Maxwell fluid ona stagnation point and also investigated energy conversionproblems of conjugate conduction convection and radiationheat andmass transfer with viscous dissipation andmagneticeffects The results have shown that it should produce greaterheat transfer effect with larger values of Prandtl number freeconvection parameter and conduction-convection numberOn the other hand the Eckert number and magnetic param-eter will reduce the heat transfer effect For heat convec-tion energy conversion aspect some importance parametersapplied to the system such as buoyancy parameters radiativeenergy parameter boundary proportional parameter andPrandtl number which can produce positive effects for largervalues of those parameters Idusuyi et al [13] formulated acomputational model for the heat generation and dissipationin a disk brake during braking and the following releaseperiod The results show similarity in thermal behaviourat the contact surface for the asbestos and aramid brakepad materials with a temperature difference of 18 K after 10seconds

The objective of the present study is to analyze the effectofmagnetic field and temperature buoyancy effect on the heattransfer flow between a moving sheet and moving aluminawater nanofluid environment

2 Mathematical Description

The graphical model of the problem considered has beengiven to depict the flow configuration and coordinate systemThe system deals with two-dimensional MHD boundarylayer incompressible viscous electrically conducting andsteady fluid flow (Figure 1) The governing equations repre-senting flow are the following

120597119906

120597119909+120597V120597119910

= 0 (1)

119906120597119906

120597119909+ V

120597119906

120597119910= minus

1

120588nf

120597119901

120597119909+ 120592nf

1205972119906

1205971199102

+ 119892 (120573119879)nf (119879 minus 119879infin) minus

120590nf1198612(119909)

120588nf119906

(2)

119906120597119879

120597119909+ V

120597119879

120597119910= 120572nf

1205972119879

1205971199102 (3)

where 119906 and V are the velocity components in 119909 and 119910 direc-tion respectively 120592nf (120573119879)nf 120588nf 120590nf and 120572nf are the kine-matic viscosity thermal expansion coefficient density elec-trical conductivity and thermal diffusivity of the nanofluid

x

Tinfin

Uinfin

Us

L

B

Figure 1 Flow configuration and coordinate system

119892 is the gravitational acceleration and 119879 119879119908 and 119879

infinare the

temperature of the nanofluid inside the thermal boundarylayer the plate temperature and the nanofluid temperatureat free stream respectively 119861 (119909) = (1radic119909) 119861

0denotes the

magnetic field imposed in the transverse direction to thefluid

The appropriate boundary conditions for the abovemen-tioned problem are as follows (Table 1)

119906 = 119880119904 V = 0 119879 = 119879

119904 at 119910 = 0

119906 997888rarr 119880infin 119879 997888rarr 119879

infin as 119910 997888rarr infin

(4)

3 Method for Solution

To solve the governing equations (1) (2) and (3) with theboundary conditions (4) the following similarity transforma-tion has been introduced

120595 = [120592nf (119880infin minus 119880119904) 119909]12

119891 (120578)

120579 =119879 minus 119879infin

119879119904minus 119879infin

120578 = [119880infinminus 119880119904

120592nf119909]

12

119910

(5)

Here 119880infin

and 119880119904are free stream velocity and moving sheet

velocity respectively 120595 is the stream function which satisfies(1) with 119906 = 120597120595120597119910 and V = minus120597120595120597119909 Thus (5) gives

119906 = (119880infinminus 119880119904) 1198911015840(120578)

V = minus1

2(120592nf (119880infin minus 119880119904)

119909)

12

(119891 (120578) minus 1205781198911015840(120578))

(6)

Now using (5) and (6) in (2) and (3) we get the followingnonlinear differential equations

119891101584010158401015840(120578) +

1

2119891 (120578) 119891

10158401015840(120578) minus119872119891

1015840(120578) + 119892

119905120579 (120578) = 0

12057910158401015840+1

2Prf (120578) 1205791015840 (120578) = 0

(7)

Journal of Engineering 3

Table 1 Nanofluid properties

Dynamicviscosity [14] 120583nf = 120583119891(1 minus 0)

25

Density [15] 120588nf = (1 minus 0) 120588119891 + 0120588119904Specific heatcapacity [16] (119862

119901)nf= (1 minus 0) (119862

119901)119891+ 0(119862

119901)119904

Thermalconductivity [4]

(119896119904119896119891) + (119899 minus 1) minus 0(1 minus (119896

119904119896119891))119896nf

= 119896119891(119896119904119896119891) + (119899 minus 1) minus (119899 minus 1) 0(1 minus (119896

119904119896119891))

Kinematicviscosity [17] 120592nf = 120583nf120588nf

Thermaldiffusivity 120572nf = 119896nf120588nf(119862119901)nf

Empirical shapefactor 119899 = 3120595 120595 = 1 for spherical particles

the boundary conditions (4) will become

119891 (0) = 0 1198911015840(0) =

1

|1 minus 120572|

11989110158401015840(0) = unknown 120579 (0) = 1

1205791015840(0) = unknown

1198911015840(120578) 997888rarr

120572

|1 minus 120572| 120579 (120578) 997888rarr 0 as 120578 997888rarr infin

(8)

where 120572 = 0 represent the state of moving surface instationary fluid (119880

infin= 0) and 0 lt 120572 lt infin is the condition

of moving surface in moving fluid with 119880infin

gt 119880119904 The

dimensionless parameters introduced in (7) are defined asfollows

119872 = 120590nf1198612

0120588nf(119880infin minus119880119904) is the magnetic parameter 119866

119903=

119892 (120573119879)nf (119879119904 minus 119879infin) 119909

31205922

nf is the local temperature Grashofnumber Re = 119880

infin119909120592nf is the local Reynolds number 119892

119905=

119866119903Re2 is the temperature buoyancy parameter Pr = 120592nf120572nf

is the Prandtl number and 120572 = 119880infin119880119904is the relative velocity

parameterTo solve the set of nonlinear differential equations (7)

with subject to the boundary conditions (8) adaptive Runge-Kuttamethodwith shooting technique has been appliedThismethod is based on the discretization of the problem domainand the calculation of unknown boundary conditions Thedomain of the problem is discretized and the boundaryconditions for 120578 = infin are replaced by 1198911015840 (120578max) = 120572abs(1 minus120572) and 120579 (120578max) = 0 where 120578max is sufficiently large value of120578 comparing to step size at which the boundary conditions(8) for 1198911015840(120578) and 120579(120578) are satisfied On account of theconsistency and to fulfill stability criteria 120578max = 10 and stepsize Δ120578 = 001 have been taken To solve the problem thenonlinear equations (7) are first converted into first orderordinary linear differential equations as follows

1198911015840(120578) = 119910 (2) 119891

10158401015840(120578) = 119910 (3)

119891101584010158401015840(120578) = [minus

1

2119910 (1) 119910 (3) + 119872119910 (2) minus 119892

119905119910 (4)]

1205791015840= 119910 (5) 120579

10158401015840(120578) = [minus

1

2Pr119910 (1) 119910 (5)]

(9)

Table 2 Values of 11989110158401015840(0) and 1205791015840(0) for Pr = 66556406 and 0 = 005obtained by shooting method

Magneticparameter(119872)

Temperaturebuoyancy parameter

(119892119905)

Wall shearstresses11989110158401015840(0)

Skin frictioncoefficient1205791015840(0)

00 02 minus0664300 minus197611

02 00 minus1297800 minus190991

02 05 minus1157890 minus191991

02 10 minus1016890 minus192921

02 30 minus0479190 minus196421

05 02 minus1768590 minus185411

10 02 minus2269990 minus179411

30 02 minus3605449 minus165431

0 2 4 6 8 10

00

05

10

15

20

gt = 02

Pr = 66556406 at 120601 = 05

M = 05

M = 0

M = 1

M = 3

f998400 (120578)

120578

Figure 2 Effect of magnetic parameter on velocity

There are three conditions on the boundary 120578 = 0 and twoconditions at 120578 = infin as given in (8) Shooting techniquehas been used to find required missing initial conditionsThe value of unknowns 11989110158401015840(0) and 1205791015840 (0) has been tabulatedin Table 2 Adaptive Runge-Kutta method estimates thetruncation error at each integration step and automaticallyadjusts the step size to keep the error within prescribed limitsThe formula used to measure the actual conservative erroris defined by 119890 (ℎ) = ((15)sum

5

119894=11198642

119894(ℎ))12

here 119864119894(ℎ) is the

measure of error in the dependent variable 119910119894 Per-step error

control is achieved by adjusting the increment ℎ by ℎ119894+1

=

09ℎ119894(120576119890(ℎ

119894))15 119894 = 1 2 3 4 5 so that it is approximately

equal to the prescribed tolerance 120576 = 10minus6 The assumed

initial step size is ℎ = 001

4 Result and DiscussionThe computations have been made for velocity shear stressconcentration and concentration gradient profiles corre-sponding to fixed values of magnetic parameter (119872) 120572 andtemperature buoyancy parameter (119892

119905) The value of Pr =

66556406 is taken corresponding to 005 volume fraction ofthe alumina water nanofluid at 20∘C The parameter 119892

119905has

been introduced to represent the temperature buoyancy effecton flow field Figures 2ndash6 exhibit the velocity shear stress

4 Journal of Engineering

0 2 4 6 8 10

0

1gt = 02

(120578 = 293290777 f998400(120578) = minus00311476673)

M = 0 02 05 1 3

120578

f998400998400(120578) minus1

minus2

minus3

minus4

Pr = 66556406 at 120601 = 05

Figure 3 Effect of magnetic parameter on shear stress

00 05 10 15 2000

02

04

06

08

10

gt = 02

Pr = 66556406 at 120601 = 05

M = 0 02 05 1 3

120578

120579(120578)

Figure 4 Effect of magnetic parameter on temperature

concentration and concentration gradient profiles respec-tively for various values of magnetic parameter (ie119872 = 002 05 1 3) It is noticed that in Figures 2 and 3 that the veloc-ity decreases and the shear stress decreases up to (293290777minus00311476673) and then increases with an increase in themagnetic parameter The temperature increases with anincrease in the magnetic parameter temperature gradi-ent increases up to (0524497748 minus0835081441) and thendecreases with an increment in the values of magneticparameter as shown in Figures 4 and 5 Figures 6 and 7show the effect of convection parameter on velocity shearstress concentration and concentration gradient profilesrespectively for 119892

119905= 0 02 05 1 It has been observed that

the velocity increases with an increase in the convectionparameter as shown in Figure 6 and shear stress increases upto (0433856945 minus103343467) with an increase in the con-vection parameter and then decreases as shown in Figure 7The effect of convection parameter on temperature gradientand temperature also has been studied through Figures 8 and9 It is noticed that the temperature and temperature gradient

00 05 10 15 20

00

M = 0 02 05 1 3

gt = 02

Pr = 66556406 at 120601 = 05

(120578 = 0524497748 120579998400(120578) = minus0835081441)

minus05

minus10

minus15

minus20

120579998400 (120578)

120578

Figure 5 Effect of magnetic parameter on temperature gradient

0 2 4 6 8 10

06

08

10

12

14

16

18

20

M = 02

gt = 0 02 05 1

Pr = 66556406 at 120601 = 05

Zoom

f998400 (120578)

120578

Figure 6 Effect of temperature buoyancy parameter on velocity

0 2 4

00

02

Pr = 66556406 at 120601 = 05

M = 02

gt = 0 02 05 1

(120578 = 0433856945 f998400(120578) = minus103343467)

120578

minus02

minus04

minus06

minus08

minus10

minus12

minus14

f998400998400(120578)

Figure 7 Effect of temperature buoyancy parameter on shear stress

Journal of Engineering 5

00 02 04 06 08 10 12 14

00

Pr = 66556406 at 120601 = 05

M = 02

gt = 0 02 05 1

minus05

minus10

minus15

minus20

Zoom

120579998400 (120578)

120578

Figure 8 Effect of temperature buoyancy parameter on temperaturegradient

00 02 04 06 08 10 12 14

00

02

04

06

08

10

Pr = 66556406 at 120601 = 05

M = 02

gt = 0 02 05 1

Zoom

120579(120578)

120578

Figure 9 Effect of temperature buoyancy parameter on tempera-ture

show significant changes with an increase in the temperaturebuoyancy parameter

5 Conclusion

Free convective heat transfer in an electronically conduct-ing nanofluid over a moving sheet with magnetic field isconsidered The nonlinear partial differential equations aretransformed into a system of nonlinear ordinary differentialequations by using similarity transformation and then solvednumerically using the Runge-Kutta fifth order method along

with the shooting technique From the numerical calculationsof the skin friction coefficient and wall shear stresses thefollowing is concluded

(i) An increase in magnetic parameter decreases thevelocity and wall shear stress but enhances the tem-perature skin friction coefficient The shear stressdecrease up to a fixed value of magnetic parameterand after that goes to increases For the magneticparameter temperature gradient shows the oppositebehaviour to the shear stress

(ii) As the temperature buoyancy parameter increases itincreases the velocity wall shear stress and decreasesskin friction coefficient and temperature gradient

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] L C Zheng and X X Zhang ldquoSkin friction and heat transferin power-law fluid laminar boundary layer along a movingsurfacerdquo International Journal of Heat and Mass Transfer vol45 no 13 pp 2667ndash2672 2002

[2] S A Al-Sanea ldquoConvection regimes and heat transfer char-acteristics along a continuously moving heated vertical platerdquoInternational Journal of Heat and Fluid Flow vol 24 no 6 pp888ndash901 2003

[3] M A Seddeek ldquoFlow of a magneto-micropolar fluid past acontinuously moving platerdquo Physics Letters Section A GeneralAtomic and Solid State Physics vol 306 no 4 pp 255ndash257 2003

[4] H E Patel T Sundararajan T Pradeep A Dasgupta N Das-gupta and S K Das ldquoA micro-convection model for thermalconductivity of nanofluidsrdquo PramanamdashJournal of Physics vol65 no 5 pp 863ndash869 2005

[5] R Cortell ldquoFlow and heat transfer in a moving fluid overa moving flat surfacerdquo Theoretical and Computational FluidDynamics vol 21 no 6 pp 435ndash446 2007

[6] A Ishak R Nazar and I Pop ldquoMHD boundary-layer flowdue to a moving extensible surfacerdquo Journal of EngineeringMathematics vol 62 no 1 pp 23ndash33 2008

[7] N Bachok A Ishak and I Pop ldquoBoundary-layer flow ofnanofluids over a moving surface in a flowing fluidrdquo Interna-tional Journal of Thermal Sciences vol 49 no 9 pp 1663ndash16682010

[8] H M Habib and E R El-Zahar ldquoMathematical modeling ofheat-transfer for a moving sheet in a moving fluidrdquo Journal ofApplied Fluid Mechanics vol 6 no 3 pp 369ndash373 2013

[9] K-L Hsiao ldquoEnergy conversion conjugate conduction-convection and radiation over non-linearly extrusion stretchingsheet with physical multimedia effectsrdquo Energy vol 59 pp494ndash502 2013

[10] C M Raguraman A Ragupathy and L Sivakumar ldquoExper-imental determination of heat transfer coefficient in stirredvessel for coal-water slurry based on the taguchi methodrdquoJournal of Engineering vol 2013 Article ID 719296 8 pages2013

6 Journal of Engineering

[11] K-L Hsiao ldquoConjugate heat transfer for mixed convection andMaxwell fluid on a stagnation pointrdquoArabian Journal for Scienceand Engineering vol 39 no 6 pp 4325ndash4332 2014

[12] K-L Hsiao ldquoNanofluid flow with multimedia physical featuresfor conjugate mixed convection and radiationrdquo Computers ampFluids vol 104 pp 1ndash8 2014

[13] N Idusuyi I Babajide O K Ajayi and T T Olugasa ldquoAcomputational study on the use of an aluminium metal matrixcomposite and aramid as alternative brake disc and brake padmaterialrdquo Journal of Engineering vol 2014 Article ID 494697 6pages 2014

[14] A Einstein Investigation on the Theory of the Brownian Move-ment Dover New York NY USA 1956

[15] D A Drew and S L PassmanTheory ofMulticomponent FluidsSpringer New York NY USA 1999

[16] Y Xuan and W Roetzel ldquoConceptions for heat transfer corre-lation of nanofluidsrdquo International Journal of Heat and MassTransfer vol 43 no 19 pp 3701ndash3707 2000

[17] E Mamut ldquoCharacterization of heat and mass transfer proper-ties of nanofluidsrdquoRomanian Journal of Physics vol 51 pp 5ndash122006

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International Journal of

Page 3: Research Article Free Convection in Heat Transfer Flow ...downloads.hindawi.com/journals/je/2014/137426.pdf · V =0 + V = 1 nf + nf 2 1/22 + nf nf 2 ( ) nf + V = nf 2 2, where and

Journal of Engineering 3

Table 1 Nanofluid properties

Dynamicviscosity [14] 120583nf = 120583119891(1 minus 0)

25

Density [15] 120588nf = (1 minus 0) 120588119891 + 0120588119904Specific heatcapacity [16] (119862

119901)nf= (1 minus 0) (119862

119901)119891+ 0(119862

119901)119904

Thermalconductivity [4]

(119896119904119896119891) + (119899 minus 1) minus 0(1 minus (119896

119904119896119891))119896nf

= 119896119891(119896119904119896119891) + (119899 minus 1) minus (119899 minus 1) 0(1 minus (119896

119904119896119891))

Kinematicviscosity [17] 120592nf = 120583nf120588nf

Thermaldiffusivity 120572nf = 119896nf120588nf(119862119901)nf

Empirical shapefactor 119899 = 3120595 120595 = 1 for spherical particles

the boundary conditions (4) will become

119891 (0) = 0 1198911015840(0) =

1

|1 minus 120572|

11989110158401015840(0) = unknown 120579 (0) = 1

1205791015840(0) = unknown

1198911015840(120578) 997888rarr

120572

|1 minus 120572| 120579 (120578) 997888rarr 0 as 120578 997888rarr infin

(8)

where 120572 = 0 represent the state of moving surface instationary fluid (119880

infin= 0) and 0 lt 120572 lt infin is the condition

of moving surface in moving fluid with 119880infin

gt 119880119904 The

dimensionless parameters introduced in (7) are defined asfollows

119872 = 120590nf1198612

0120588nf(119880infin minus119880119904) is the magnetic parameter 119866

119903=

119892 (120573119879)nf (119879119904 minus 119879infin) 119909

31205922

nf is the local temperature Grashofnumber Re = 119880

infin119909120592nf is the local Reynolds number 119892

119905=

119866119903Re2 is the temperature buoyancy parameter Pr = 120592nf120572nf

is the Prandtl number and 120572 = 119880infin119880119904is the relative velocity

parameterTo solve the set of nonlinear differential equations (7)

with subject to the boundary conditions (8) adaptive Runge-Kuttamethodwith shooting technique has been appliedThismethod is based on the discretization of the problem domainand the calculation of unknown boundary conditions Thedomain of the problem is discretized and the boundaryconditions for 120578 = infin are replaced by 1198911015840 (120578max) = 120572abs(1 minus120572) and 120579 (120578max) = 0 where 120578max is sufficiently large value of120578 comparing to step size at which the boundary conditions(8) for 1198911015840(120578) and 120579(120578) are satisfied On account of theconsistency and to fulfill stability criteria 120578max = 10 and stepsize Δ120578 = 001 have been taken To solve the problem thenonlinear equations (7) are first converted into first orderordinary linear differential equations as follows

1198911015840(120578) = 119910 (2) 119891

10158401015840(120578) = 119910 (3)

119891101584010158401015840(120578) = [minus

1

2119910 (1) 119910 (3) + 119872119910 (2) minus 119892

119905119910 (4)]

1205791015840= 119910 (5) 120579

10158401015840(120578) = [minus

1

2Pr119910 (1) 119910 (5)]

(9)

Table 2 Values of 11989110158401015840(0) and 1205791015840(0) for Pr = 66556406 and 0 = 005obtained by shooting method

Magneticparameter(119872)

Temperaturebuoyancy parameter

(119892119905)

Wall shearstresses11989110158401015840(0)

Skin frictioncoefficient1205791015840(0)

00 02 minus0664300 minus197611

02 00 minus1297800 minus190991

02 05 minus1157890 minus191991

02 10 minus1016890 minus192921

02 30 minus0479190 minus196421

05 02 minus1768590 minus185411

10 02 minus2269990 minus179411

30 02 minus3605449 minus165431

0 2 4 6 8 10

00

05

10

15

20

gt = 02

Pr = 66556406 at 120601 = 05

M = 05

M = 0

M = 1

M = 3

f998400 (120578)

120578

Figure 2 Effect of magnetic parameter on velocity

There are three conditions on the boundary 120578 = 0 and twoconditions at 120578 = infin as given in (8) Shooting techniquehas been used to find required missing initial conditionsThe value of unknowns 11989110158401015840(0) and 1205791015840 (0) has been tabulatedin Table 2 Adaptive Runge-Kutta method estimates thetruncation error at each integration step and automaticallyadjusts the step size to keep the error within prescribed limitsThe formula used to measure the actual conservative erroris defined by 119890 (ℎ) = ((15)sum

5

119894=11198642

119894(ℎ))12

here 119864119894(ℎ) is the

measure of error in the dependent variable 119910119894 Per-step error

control is achieved by adjusting the increment ℎ by ℎ119894+1

=

09ℎ119894(120576119890(ℎ

119894))15 119894 = 1 2 3 4 5 so that it is approximately

equal to the prescribed tolerance 120576 = 10minus6 The assumed

initial step size is ℎ = 001

4 Result and DiscussionThe computations have been made for velocity shear stressconcentration and concentration gradient profiles corre-sponding to fixed values of magnetic parameter (119872) 120572 andtemperature buoyancy parameter (119892

119905) The value of Pr =

66556406 is taken corresponding to 005 volume fraction ofthe alumina water nanofluid at 20∘C The parameter 119892

119905has

been introduced to represent the temperature buoyancy effecton flow field Figures 2ndash6 exhibit the velocity shear stress

4 Journal of Engineering

0 2 4 6 8 10

0

1gt = 02

(120578 = 293290777 f998400(120578) = minus00311476673)

M = 0 02 05 1 3

120578

f998400998400(120578) minus1

minus2

minus3

minus4

Pr = 66556406 at 120601 = 05

Figure 3 Effect of magnetic parameter on shear stress

00 05 10 15 2000

02

04

06

08

10

gt = 02

Pr = 66556406 at 120601 = 05

M = 0 02 05 1 3

120578

120579(120578)

Figure 4 Effect of magnetic parameter on temperature

concentration and concentration gradient profiles respec-tively for various values of magnetic parameter (ie119872 = 002 05 1 3) It is noticed that in Figures 2 and 3 that the veloc-ity decreases and the shear stress decreases up to (293290777minus00311476673) and then increases with an increase in themagnetic parameter The temperature increases with anincrease in the magnetic parameter temperature gradi-ent increases up to (0524497748 minus0835081441) and thendecreases with an increment in the values of magneticparameter as shown in Figures 4 and 5 Figures 6 and 7show the effect of convection parameter on velocity shearstress concentration and concentration gradient profilesrespectively for 119892

119905= 0 02 05 1 It has been observed that

the velocity increases with an increase in the convectionparameter as shown in Figure 6 and shear stress increases upto (0433856945 minus103343467) with an increase in the con-vection parameter and then decreases as shown in Figure 7The effect of convection parameter on temperature gradientand temperature also has been studied through Figures 8 and9 It is noticed that the temperature and temperature gradient

00 05 10 15 20

00

M = 0 02 05 1 3

gt = 02

Pr = 66556406 at 120601 = 05

(120578 = 0524497748 120579998400(120578) = minus0835081441)

minus05

minus10

minus15

minus20

120579998400 (120578)

120578

Figure 5 Effect of magnetic parameter on temperature gradient

0 2 4 6 8 10

06

08

10

12

14

16

18

20

M = 02

gt = 0 02 05 1

Pr = 66556406 at 120601 = 05

Zoom

f998400 (120578)

120578

Figure 6 Effect of temperature buoyancy parameter on velocity

0 2 4

00

02

Pr = 66556406 at 120601 = 05

M = 02

gt = 0 02 05 1

(120578 = 0433856945 f998400(120578) = minus103343467)

120578

minus02

minus04

minus06

minus08

minus10

minus12

minus14

f998400998400(120578)

Figure 7 Effect of temperature buoyancy parameter on shear stress

Journal of Engineering 5

00 02 04 06 08 10 12 14

00

Pr = 66556406 at 120601 = 05

M = 02

gt = 0 02 05 1

minus05

minus10

minus15

minus20

Zoom

120579998400 (120578)

120578

Figure 8 Effect of temperature buoyancy parameter on temperaturegradient

00 02 04 06 08 10 12 14

00

02

04

06

08

10

Pr = 66556406 at 120601 = 05

M = 02

gt = 0 02 05 1

Zoom

120579(120578)

120578

Figure 9 Effect of temperature buoyancy parameter on tempera-ture

show significant changes with an increase in the temperaturebuoyancy parameter

5 Conclusion

Free convective heat transfer in an electronically conduct-ing nanofluid over a moving sheet with magnetic field isconsidered The nonlinear partial differential equations aretransformed into a system of nonlinear ordinary differentialequations by using similarity transformation and then solvednumerically using the Runge-Kutta fifth order method along

with the shooting technique From the numerical calculationsof the skin friction coefficient and wall shear stresses thefollowing is concluded

(i) An increase in magnetic parameter decreases thevelocity and wall shear stress but enhances the tem-perature skin friction coefficient The shear stressdecrease up to a fixed value of magnetic parameterand after that goes to increases For the magneticparameter temperature gradient shows the oppositebehaviour to the shear stress

(ii) As the temperature buoyancy parameter increases itincreases the velocity wall shear stress and decreasesskin friction coefficient and temperature gradient

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] L C Zheng and X X Zhang ldquoSkin friction and heat transferin power-law fluid laminar boundary layer along a movingsurfacerdquo International Journal of Heat and Mass Transfer vol45 no 13 pp 2667ndash2672 2002

[2] S A Al-Sanea ldquoConvection regimes and heat transfer char-acteristics along a continuously moving heated vertical platerdquoInternational Journal of Heat and Fluid Flow vol 24 no 6 pp888ndash901 2003

[3] M A Seddeek ldquoFlow of a magneto-micropolar fluid past acontinuously moving platerdquo Physics Letters Section A GeneralAtomic and Solid State Physics vol 306 no 4 pp 255ndash257 2003

[4] H E Patel T Sundararajan T Pradeep A Dasgupta N Das-gupta and S K Das ldquoA micro-convection model for thermalconductivity of nanofluidsrdquo PramanamdashJournal of Physics vol65 no 5 pp 863ndash869 2005

[5] R Cortell ldquoFlow and heat transfer in a moving fluid overa moving flat surfacerdquo Theoretical and Computational FluidDynamics vol 21 no 6 pp 435ndash446 2007

[6] A Ishak R Nazar and I Pop ldquoMHD boundary-layer flowdue to a moving extensible surfacerdquo Journal of EngineeringMathematics vol 62 no 1 pp 23ndash33 2008

[7] N Bachok A Ishak and I Pop ldquoBoundary-layer flow ofnanofluids over a moving surface in a flowing fluidrdquo Interna-tional Journal of Thermal Sciences vol 49 no 9 pp 1663ndash16682010

[8] H M Habib and E R El-Zahar ldquoMathematical modeling ofheat-transfer for a moving sheet in a moving fluidrdquo Journal ofApplied Fluid Mechanics vol 6 no 3 pp 369ndash373 2013

[9] K-L Hsiao ldquoEnergy conversion conjugate conduction-convection and radiation over non-linearly extrusion stretchingsheet with physical multimedia effectsrdquo Energy vol 59 pp494ndash502 2013

[10] C M Raguraman A Ragupathy and L Sivakumar ldquoExper-imental determination of heat transfer coefficient in stirredvessel for coal-water slurry based on the taguchi methodrdquoJournal of Engineering vol 2013 Article ID 719296 8 pages2013

6 Journal of Engineering

[11] K-L Hsiao ldquoConjugate heat transfer for mixed convection andMaxwell fluid on a stagnation pointrdquoArabian Journal for Scienceand Engineering vol 39 no 6 pp 4325ndash4332 2014

[12] K-L Hsiao ldquoNanofluid flow with multimedia physical featuresfor conjugate mixed convection and radiationrdquo Computers ampFluids vol 104 pp 1ndash8 2014

[13] N Idusuyi I Babajide O K Ajayi and T T Olugasa ldquoAcomputational study on the use of an aluminium metal matrixcomposite and aramid as alternative brake disc and brake padmaterialrdquo Journal of Engineering vol 2014 Article ID 494697 6pages 2014

[14] A Einstein Investigation on the Theory of the Brownian Move-ment Dover New York NY USA 1956

[15] D A Drew and S L PassmanTheory ofMulticomponent FluidsSpringer New York NY USA 1999

[16] Y Xuan and W Roetzel ldquoConceptions for heat transfer corre-lation of nanofluidsrdquo International Journal of Heat and MassTransfer vol 43 no 19 pp 3701ndash3707 2000

[17] E Mamut ldquoCharacterization of heat and mass transfer proper-ties of nanofluidsrdquoRomanian Journal of Physics vol 51 pp 5ndash122006

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Page 4: Research Article Free Convection in Heat Transfer Flow ...downloads.hindawi.com/journals/je/2014/137426.pdf · V =0 + V = 1 nf + nf 2 1/22 + nf nf 2 ( ) nf + V = nf 2 2, where and

4 Journal of Engineering

0 2 4 6 8 10

0

1gt = 02

(120578 = 293290777 f998400(120578) = minus00311476673)

M = 0 02 05 1 3

120578

f998400998400(120578) minus1

minus2

minus3

minus4

Pr = 66556406 at 120601 = 05

Figure 3 Effect of magnetic parameter on shear stress

00 05 10 15 2000

02

04

06

08

10

gt = 02

Pr = 66556406 at 120601 = 05

M = 0 02 05 1 3

120578

120579(120578)

Figure 4 Effect of magnetic parameter on temperature

concentration and concentration gradient profiles respec-tively for various values of magnetic parameter (ie119872 = 002 05 1 3) It is noticed that in Figures 2 and 3 that the veloc-ity decreases and the shear stress decreases up to (293290777minus00311476673) and then increases with an increase in themagnetic parameter The temperature increases with anincrease in the magnetic parameter temperature gradi-ent increases up to (0524497748 minus0835081441) and thendecreases with an increment in the values of magneticparameter as shown in Figures 4 and 5 Figures 6 and 7show the effect of convection parameter on velocity shearstress concentration and concentration gradient profilesrespectively for 119892

119905= 0 02 05 1 It has been observed that

the velocity increases with an increase in the convectionparameter as shown in Figure 6 and shear stress increases upto (0433856945 minus103343467) with an increase in the con-vection parameter and then decreases as shown in Figure 7The effect of convection parameter on temperature gradientand temperature also has been studied through Figures 8 and9 It is noticed that the temperature and temperature gradient

00 05 10 15 20

00

M = 0 02 05 1 3

gt = 02

Pr = 66556406 at 120601 = 05

(120578 = 0524497748 120579998400(120578) = minus0835081441)

minus05

minus10

minus15

minus20

120579998400 (120578)

120578

Figure 5 Effect of magnetic parameter on temperature gradient

0 2 4 6 8 10

06

08

10

12

14

16

18

20

M = 02

gt = 0 02 05 1

Pr = 66556406 at 120601 = 05

Zoom

f998400 (120578)

120578

Figure 6 Effect of temperature buoyancy parameter on velocity

0 2 4

00

02

Pr = 66556406 at 120601 = 05

M = 02

gt = 0 02 05 1

(120578 = 0433856945 f998400(120578) = minus103343467)

120578

minus02

minus04

minus06

minus08

minus10

minus12

minus14

f998400998400(120578)

Figure 7 Effect of temperature buoyancy parameter on shear stress

Journal of Engineering 5

00 02 04 06 08 10 12 14

00

Pr = 66556406 at 120601 = 05

M = 02

gt = 0 02 05 1

minus05

minus10

minus15

minus20

Zoom

120579998400 (120578)

120578

Figure 8 Effect of temperature buoyancy parameter on temperaturegradient

00 02 04 06 08 10 12 14

00

02

04

06

08

10

Pr = 66556406 at 120601 = 05

M = 02

gt = 0 02 05 1

Zoom

120579(120578)

120578

Figure 9 Effect of temperature buoyancy parameter on tempera-ture

show significant changes with an increase in the temperaturebuoyancy parameter

5 Conclusion

Free convective heat transfer in an electronically conduct-ing nanofluid over a moving sheet with magnetic field isconsidered The nonlinear partial differential equations aretransformed into a system of nonlinear ordinary differentialequations by using similarity transformation and then solvednumerically using the Runge-Kutta fifth order method along

with the shooting technique From the numerical calculationsof the skin friction coefficient and wall shear stresses thefollowing is concluded

(i) An increase in magnetic parameter decreases thevelocity and wall shear stress but enhances the tem-perature skin friction coefficient The shear stressdecrease up to a fixed value of magnetic parameterand after that goes to increases For the magneticparameter temperature gradient shows the oppositebehaviour to the shear stress

(ii) As the temperature buoyancy parameter increases itincreases the velocity wall shear stress and decreasesskin friction coefficient and temperature gradient

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] L C Zheng and X X Zhang ldquoSkin friction and heat transferin power-law fluid laminar boundary layer along a movingsurfacerdquo International Journal of Heat and Mass Transfer vol45 no 13 pp 2667ndash2672 2002

[2] S A Al-Sanea ldquoConvection regimes and heat transfer char-acteristics along a continuously moving heated vertical platerdquoInternational Journal of Heat and Fluid Flow vol 24 no 6 pp888ndash901 2003

[3] M A Seddeek ldquoFlow of a magneto-micropolar fluid past acontinuously moving platerdquo Physics Letters Section A GeneralAtomic and Solid State Physics vol 306 no 4 pp 255ndash257 2003

[4] H E Patel T Sundararajan T Pradeep A Dasgupta N Das-gupta and S K Das ldquoA micro-convection model for thermalconductivity of nanofluidsrdquo PramanamdashJournal of Physics vol65 no 5 pp 863ndash869 2005

[5] R Cortell ldquoFlow and heat transfer in a moving fluid overa moving flat surfacerdquo Theoretical and Computational FluidDynamics vol 21 no 6 pp 435ndash446 2007

[6] A Ishak R Nazar and I Pop ldquoMHD boundary-layer flowdue to a moving extensible surfacerdquo Journal of EngineeringMathematics vol 62 no 1 pp 23ndash33 2008

[7] N Bachok A Ishak and I Pop ldquoBoundary-layer flow ofnanofluids over a moving surface in a flowing fluidrdquo Interna-tional Journal of Thermal Sciences vol 49 no 9 pp 1663ndash16682010

[8] H M Habib and E R El-Zahar ldquoMathematical modeling ofheat-transfer for a moving sheet in a moving fluidrdquo Journal ofApplied Fluid Mechanics vol 6 no 3 pp 369ndash373 2013

[9] K-L Hsiao ldquoEnergy conversion conjugate conduction-convection and radiation over non-linearly extrusion stretchingsheet with physical multimedia effectsrdquo Energy vol 59 pp494ndash502 2013

[10] C M Raguraman A Ragupathy and L Sivakumar ldquoExper-imental determination of heat transfer coefficient in stirredvessel for coal-water slurry based on the taguchi methodrdquoJournal of Engineering vol 2013 Article ID 719296 8 pages2013

6 Journal of Engineering

[11] K-L Hsiao ldquoConjugate heat transfer for mixed convection andMaxwell fluid on a stagnation pointrdquoArabian Journal for Scienceand Engineering vol 39 no 6 pp 4325ndash4332 2014

[12] K-L Hsiao ldquoNanofluid flow with multimedia physical featuresfor conjugate mixed convection and radiationrdquo Computers ampFluids vol 104 pp 1ndash8 2014

[13] N Idusuyi I Babajide O K Ajayi and T T Olugasa ldquoAcomputational study on the use of an aluminium metal matrixcomposite and aramid as alternative brake disc and brake padmaterialrdquo Journal of Engineering vol 2014 Article ID 494697 6pages 2014

[14] A Einstein Investigation on the Theory of the Brownian Move-ment Dover New York NY USA 1956

[15] D A Drew and S L PassmanTheory ofMulticomponent FluidsSpringer New York NY USA 1999

[16] Y Xuan and W Roetzel ldquoConceptions for heat transfer corre-lation of nanofluidsrdquo International Journal of Heat and MassTransfer vol 43 no 19 pp 3701ndash3707 2000

[17] E Mamut ldquoCharacterization of heat and mass transfer proper-ties of nanofluidsrdquoRomanian Journal of Physics vol 51 pp 5ndash122006

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Page 5: Research Article Free Convection in Heat Transfer Flow ...downloads.hindawi.com/journals/je/2014/137426.pdf · V =0 + V = 1 nf + nf 2 1/22 + nf nf 2 ( ) nf + V = nf 2 2, where and

Journal of Engineering 5

00 02 04 06 08 10 12 14

00

Pr = 66556406 at 120601 = 05

M = 02

gt = 0 02 05 1

minus05

minus10

minus15

minus20

Zoom

120579998400 (120578)

120578

Figure 8 Effect of temperature buoyancy parameter on temperaturegradient

00 02 04 06 08 10 12 14

00

02

04

06

08

10

Pr = 66556406 at 120601 = 05

M = 02

gt = 0 02 05 1

Zoom

120579(120578)

120578

Figure 9 Effect of temperature buoyancy parameter on tempera-ture

show significant changes with an increase in the temperaturebuoyancy parameter

5 Conclusion

Free convective heat transfer in an electronically conduct-ing nanofluid over a moving sheet with magnetic field isconsidered The nonlinear partial differential equations aretransformed into a system of nonlinear ordinary differentialequations by using similarity transformation and then solvednumerically using the Runge-Kutta fifth order method along

with the shooting technique From the numerical calculationsof the skin friction coefficient and wall shear stresses thefollowing is concluded

(i) An increase in magnetic parameter decreases thevelocity and wall shear stress but enhances the tem-perature skin friction coefficient The shear stressdecrease up to a fixed value of magnetic parameterand after that goes to increases For the magneticparameter temperature gradient shows the oppositebehaviour to the shear stress

(ii) As the temperature buoyancy parameter increases itincreases the velocity wall shear stress and decreasesskin friction coefficient and temperature gradient

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] L C Zheng and X X Zhang ldquoSkin friction and heat transferin power-law fluid laminar boundary layer along a movingsurfacerdquo International Journal of Heat and Mass Transfer vol45 no 13 pp 2667ndash2672 2002

[2] S A Al-Sanea ldquoConvection regimes and heat transfer char-acteristics along a continuously moving heated vertical platerdquoInternational Journal of Heat and Fluid Flow vol 24 no 6 pp888ndash901 2003

[3] M A Seddeek ldquoFlow of a magneto-micropolar fluid past acontinuously moving platerdquo Physics Letters Section A GeneralAtomic and Solid State Physics vol 306 no 4 pp 255ndash257 2003

[4] H E Patel T Sundararajan T Pradeep A Dasgupta N Das-gupta and S K Das ldquoA micro-convection model for thermalconductivity of nanofluidsrdquo PramanamdashJournal of Physics vol65 no 5 pp 863ndash869 2005

[5] R Cortell ldquoFlow and heat transfer in a moving fluid overa moving flat surfacerdquo Theoretical and Computational FluidDynamics vol 21 no 6 pp 435ndash446 2007

[6] A Ishak R Nazar and I Pop ldquoMHD boundary-layer flowdue to a moving extensible surfacerdquo Journal of EngineeringMathematics vol 62 no 1 pp 23ndash33 2008

[7] N Bachok A Ishak and I Pop ldquoBoundary-layer flow ofnanofluids over a moving surface in a flowing fluidrdquo Interna-tional Journal of Thermal Sciences vol 49 no 9 pp 1663ndash16682010

[8] H M Habib and E R El-Zahar ldquoMathematical modeling ofheat-transfer for a moving sheet in a moving fluidrdquo Journal ofApplied Fluid Mechanics vol 6 no 3 pp 369ndash373 2013

[9] K-L Hsiao ldquoEnergy conversion conjugate conduction-convection and radiation over non-linearly extrusion stretchingsheet with physical multimedia effectsrdquo Energy vol 59 pp494ndash502 2013

[10] C M Raguraman A Ragupathy and L Sivakumar ldquoExper-imental determination of heat transfer coefficient in stirredvessel for coal-water slurry based on the taguchi methodrdquoJournal of Engineering vol 2013 Article ID 719296 8 pages2013

6 Journal of Engineering

[11] K-L Hsiao ldquoConjugate heat transfer for mixed convection andMaxwell fluid on a stagnation pointrdquoArabian Journal for Scienceand Engineering vol 39 no 6 pp 4325ndash4332 2014

[12] K-L Hsiao ldquoNanofluid flow with multimedia physical featuresfor conjugate mixed convection and radiationrdquo Computers ampFluids vol 104 pp 1ndash8 2014

[13] N Idusuyi I Babajide O K Ajayi and T T Olugasa ldquoAcomputational study on the use of an aluminium metal matrixcomposite and aramid as alternative brake disc and brake padmaterialrdquo Journal of Engineering vol 2014 Article ID 494697 6pages 2014

[14] A Einstein Investigation on the Theory of the Brownian Move-ment Dover New York NY USA 1956

[15] D A Drew and S L PassmanTheory ofMulticomponent FluidsSpringer New York NY USA 1999

[16] Y Xuan and W Roetzel ldquoConceptions for heat transfer corre-lation of nanofluidsrdquo International Journal of Heat and MassTransfer vol 43 no 19 pp 3701ndash3707 2000

[17] E Mamut ldquoCharacterization of heat and mass transfer proper-ties of nanofluidsrdquoRomanian Journal of Physics vol 51 pp 5ndash122006

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Page 6: Research Article Free Convection in Heat Transfer Flow ...downloads.hindawi.com/journals/je/2014/137426.pdf · V =0 + V = 1 nf + nf 2 1/22 + nf nf 2 ( ) nf + V = nf 2 2, where and

6 Journal of Engineering

[11] K-L Hsiao ldquoConjugate heat transfer for mixed convection andMaxwell fluid on a stagnation pointrdquoArabian Journal for Scienceand Engineering vol 39 no 6 pp 4325ndash4332 2014

[12] K-L Hsiao ldquoNanofluid flow with multimedia physical featuresfor conjugate mixed convection and radiationrdquo Computers ampFluids vol 104 pp 1ndash8 2014

[13] N Idusuyi I Babajide O K Ajayi and T T Olugasa ldquoAcomputational study on the use of an aluminium metal matrixcomposite and aramid as alternative brake disc and brake padmaterialrdquo Journal of Engineering vol 2014 Article ID 494697 6pages 2014

[14] A Einstein Investigation on the Theory of the Brownian Move-ment Dover New York NY USA 1956

[15] D A Drew and S L PassmanTheory ofMulticomponent FluidsSpringer New York NY USA 1999

[16] Y Xuan and W Roetzel ldquoConceptions for heat transfer corre-lation of nanofluidsrdquo International Journal of Heat and MassTransfer vol 43 no 19 pp 3701ndash3707 2000

[17] E Mamut ldquoCharacterization of heat and mass transfer proper-ties of nanofluidsrdquoRomanian Journal of Physics vol 51 pp 5ndash122006

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Page 7: Research Article Free Convection in Heat Transfer Flow ...downloads.hindawi.com/journals/je/2014/137426.pdf · V =0 + V = 1 nf + nf 2 1/22 + nf nf 2 ( ) nf + V = nf 2 2, where and

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of


Mercury Chamber Considerations V. Graves IDS-NF Target Studies July 2011.

Mercury Chamber Considerations V. Graves IDS-NF Target Studies July 2011.

SCOPE OF ACCREDITATION TO ISO/IEC 17025:2005 ... - … · scope of accreditation to iso/iec 17025:2005 ... 0.6 % + 0.01 nf 0.6 % + 0.01 nf 0.3 % + 0.1 nf ... 0.96 % + 50 v

SCOPE OF ACCREDITATION TO ISO/IEC 17025:2005 ... - … · scope of accreditation to iso/iec 17025:2005 ... 0.6 % + 0.01 nf 0.6 % + 0.01 nf 0.3 % + 0.1 nf ... 0.96 % + 50 v

High Voltage PolyTerm Ceramic Capacitors · 2021. 1. 26. · 1.0 nF 1.2 nF 1.5 nF 1.8 nF 2.2 nF 2.7 nF 3.3 nF 4.7 nF 5.6 nF 6.8 nF 8.2 nF 10 nF 12 nF 15 nF 18 nF 22 nF 27 nF 33 nF

High Voltage PolyTerm Ceramic Capacitors · 2021. 1. 26. · 1.0 nF 1.2 nF 1.5 nF 1.8 nF 2.2 nF 2.7 nF 3.3 nF 4.7 nF 5.6 nF 6.8 nF 8.2 nF 10 nF 12 nF 15 nF 18 nF 22 nF 27 nF 33 nF

Bu ild ab eP rc s nF h v - buzzardsbay.org

Bu ild ab eP rc s nF h v - buzzardsbay.org

IQ - GC America€¦ · GC Initial™ IQ Lustre Pastes NF V-Shades GC Initial IQ Lustre Pastes NF V-Shades EN Intended Use and Shelf Life DE Bestimmungsgemäßer Gebrauch und Haltbarkeit

IQ - GC America€¦ · GC Initial™ IQ Lustre Pastes NF V-Shades GC Initial IQ Lustre Pastes NF V-Shades EN Intended Use and Shelf Life DE Bestimmungsgemäßer Gebrauch und Haltbarkeit

2.sit i nf. bb2

2.sit i nf. bb2

Your excellent helper in cable test! INSTRUCTION MANUALkonturm.ru/catalogy/paspeng/nf-308.pdfMain tester(NF-308) Receiver(NF-308-S) Remote identifier NF-308 1.wiremap 2.pair&length

Your excellent helper in cable test! INSTRUCTION MANUALkonturm.ru/catalogy/paspeng/nf-308.pdfMain tester(NF-308) Receiver(NF-308-S) Remote identifier NF-308 1.wiremap 2.pair&length

NF 533 Reporting NF 533 Reporting NF 533 Reporting NF 533 Reporting NF 533 Reporting

NF 533 Reporting NF 533 Reporting NF 533 Reporting NF 533 Reporting NF 533 Reporting

UM10466 UBA2015P reference design 230 V (AC) · 2.2 nF 250 V F1 FUSE HOLDER 20 × 5 mm F3 fuse holder cover, 20 × 5 mm F2 T1A V1 710 V C2 220 nF C21 47 nF C15 100 nF C18 220 nF C20

UM10466 UBA2015P reference design 230 V (AC) · 2.2 nF 250 V F1 FUSE HOLDER 20 × 5 mm F3 fuse holder cover, 20 × 5 mm F2 T1A V1 710 V C2 220 nF C21 47 nF C15 100 nF C18 220 nF C20

Certification Rules of the Mark - LCIE · 2015. 8. 26. · NF EN 50086-1 by NF EN 61386-1 and NF EN 50086-2-4 by NF EN 61386-24 : conduit systems for cable management . NF EN 50146

Certification Rules of the Mark - LCIE · 2015. 8. 26. · NF EN 50086-1 by NF EN 61386-1 and NF EN 50086-2-4 by NF EN 61386-24 : conduit systems for cable management . NF EN 50146

D Rm R A HBW Sn MPa 5 m.K W/ NEW · ue12p ue9p uz39 uz40 uz23 uz19 ua9 ua10n upb10 nf en1982 nf en1982 nf en1982 nf en198 nf en1982 nf en1982 nf en1982 nf en1982 nf en1982 nf l14-705

D Rm R A HBW Sn MPa 5 m.K W/ NEW · ue12p ue9p uz39 uz40 uz23 uz19 ua9 ua10n upb10 nf en1982 nf en1982 nf en1982 nf en198 nf en1982 nf en1982 nf en1982 nf en1982 nf en1982 nf l14-705

ppication ote - LYTSwitch-4 Family - Power Integrations · ppication ote - LYTSwitch-4 Family December 2014 ... C15 100 nF 50 V C14 10 nF 50 V C9 56 µF 50 V LYTSwitch-4 U1 LYT4317E

ppication ote - LYTSwitch-4 Family - Power Integrations · ppication ote - LYTSwitch-4 Family December 2014 ... C15 100 nF 50 V C14 10 nF 50 V C9 56 µF 50 V LYTSwitch-4 U1 LYT4317E

NF GUIDE...Enumeraion of presumpive Bacillus spp. NF EN 15784 V 18-237 page 106 Enumeraion of Enterococcus (E. faecium) spp. NF EN 15788 V 18-232 page 107 Enumeraion of Lactobacillus

NF GUIDE...Enumeraion of presumpive Bacillus spp. NF EN 15784 V 18-237 page 106 Enumeraion of Enterococcus (E. faecium) spp. NF EN 15788 V 18-232 page 107 Enumeraion of Lactobacillus

Mercury Chamber Update V. Graves NF-IDS Meeting October 4, 2011.

Mercury Chamber Update V. Graves NF-IDS Meeting October 4, 2011.

Commentary USP–NF 2021 Issue 2 - USP–NF | USP-NF

Commentary USP–NF 2021 Issue 2 - USP–NF | USP-NF

Fisheries Case (UK v. Norway) NF

Fisheries Case (UK v. Norway) NF

Lab 10: Oscillators - Physics & Astronomy+ 10 nF 3.3 k 1 k 10 nF 1 k 10 nF 1 k R1 V IN V OUT C1 R2 R3 C2 C3 Analysis of feedback circuit is difficult because of loading RC filters

Lab 10: Oscillators - Physics & Astronomy+ 10 nF 3.3 k 1 k 10 nF 1 k 10 nF 1 k R1 V IN V OUT C1 R2 R3 C2 C3 Analysis of feedback circuit is difficult because of loading RC filters

RECOMMENDED BOOK LIST€¦ · Fossil Fish Found Alive: Discovering the Coelacanth, Sally M. Walker F S F F S NF NF F S F S NF NF NF F NF NF F F S NF NF. Created Date: 6/1/2016 11:55:48

RECOMMENDED BOOK LIST€¦ · Fossil Fish Found Alive: Discovering the Coelacanth, Sally M. Walker F S F F S NF NF F S F S NF NF NF F NF NF F F S NF NF. Created Date: 6/1/2016 11:55:48

Isolating converter. - GHM GROUP · Ci 1 nF 1 nF Li 240 nH 240 nH IIB / IIIC C0 667 nF 769 nF L0 200 mH 8 mH IIC C0 85 nF 99 nF L0 100 mH 2 mH ... EN 60079-11 300 V AC/DC according

Isolating converter. - GHM GROUP · Ci 1 nF 1 nF Li 240 nH 240 nH IIB / IIIC C0 667 nF 769 nF L0 200 mH 8 mH IIC C0 85 nF 99 nF L0 100 mH 2 mH ... EN 60079-11 300 V AC/DC according

Asian VNL: NTC: V NF

Asian VNL: NTC: V NF