Melting and heat absorption effects in boundary layer stagnation … · Melting and heat absorption...

Ain Shams Engineering Journal (2016) xxx, xxx–xxx

Ain Shams University

Ain Shams Engineering Journal

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ENGINEERING PHYSICS AND MATHEMATICS

Melting and heat absorption effects in boundary

layer stagnation-point flow towards a stretching

sheet in a micropolar fluid

* Corresponding author.

E-mail address: [email protected] (K. Singh).

Peer review under responsibility of Ain Shams University.

Production and hosting by Elsevier

http://dx.doi.org/10.1016/j.asej.2016.04.0172090-4479 � 2016 Faculty of Engineering, Ain Shams University. Production and hosting by Elsevier B.V.This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Please cite this article in press as: Singh K, Kumar M, Melting and heat absorption effects in boundary layer stagnation-point flow towards a stretching smicropolar fluid, Ain Shams Eng J (2016), http://dx.doi.org/10.1016/j.asej.2016.04.017

Khilap Singh *, Manoj Kumar

Department of Mathematics, Statistics and Computer Science, G.B. Pant University of Agriculture and Technology,Pantnagar, Uttarakhand, India

Received 17 October 2015; revised 29 March 2016; accepted 12 April 2016

KEYWORDS

Boundary layer;

Heat absorption;

Melting heat transfer;

Micropolar fluid;

Stagnation-point;

Stretching sheet

Abstract The present investigation deals with the study of fluid flow and heat transfer character-

istics occurring during the melting process due to a stretching surface in micropolar fluid with heat

absorption. The governing equations representing fluid flow have been transformed into nonlinear

ordinary differential equations using similarity transformation. The equations thus obtained have

been solved numerically using Runge–Kutta–Fehlberg method with shooting technique. The effects

of the melting parameter, micropolar parameter heat absorption parameter and stretching param-

eter on the fluid flow and heat transfer characteristics have been tabulated, illustrated graphically

and discussed in detail. Results show that heat transfer rate decreases with melting parameter

and heat absorption parameter significantly at the fluid-solid interface.� 2016 Faculty of Engineering, Ain Shams University. Production and hosting by Elsevier B.V. This is an

open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

The micropolar fluids are those which contain micro-constituents that can undergo rotation, the presence of which

can affect the hydrodynamics of the flow. It has many practicalapplications, for example, analyzing the behavior of exoticlubricants, colloidal suspensions, solidification of liquid crys-

tals, extrusion of polymer fluids, cooling of metallic plate in

a bath, animal blood, body fluids and so more. The flow andheat transfer characteristic induced by a stretching surface ina non-Newtonian fluid is of great important because it occurs

in many manufacturing processes such as in the polymerindustry. Eringen [1–3] has introduced the theory of micropo-lar fluids that is capable to describe those fluids by taking into

account the effect arising from local structure and micromo-tions of the fluid elements. Das [4] studied the effect of partialslip on steady boundary layer stagnation point flow of an elec-trically conducting micropolar fluid impinging normally

through a shrinking sheet in the presence of a uniform trans-verse magnetic field. The unsteady MHD boundary layer flowof a micropolar fluid near the stagnation point of a two-

dimensional plane surface through a porous medium has beenstudied by Nadeem et al. [5]. Ishak et al. [6] investigated heattransfer over a stretching surface with variable heat flux in

heet in a

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Nomenclature

a, c constant (m�1)

cp specific heat at constant pressure (J kg�1 K�1)cs heat capacity of the solid surfaceCf skin friction coefficientf dimensionless stream function

g dimensionless angular velocityj micro inertia density (m2)K micropolar or material parameter

k thermal conductivity of the fluid (W m�1 K�1)l reference lengthM melting parameter

N component of micro-rotation (rad s�1)n constantNux Nusselt numberPr Prandtl number

Q0 coefficient of heat absorptionqw surface heat fluxRex Reynolds number

e stretching parameterT temperature (K)Ts temperature of the solid medium

u, v velocities in x; y-directions, (m s�1)

x, y axial and perpendicular co-ordinates (m)

Greek symbols

w stream functiona thermal diffusivityc spin-gradient viscosity (N s)

l dynamic viscosity (Pa s)g non-dimensional distancej vortex viscosity

t kinematic viscosity (m2 s�1)q fluid density (kg m�3)sw surface heat fluxh non-dimensional temperature

k latent heat of the fluid

Subscripts1 free stream condition

m condition at the melting surface

Superscript0 derivative with respect to g

2 K. Singh, M. Kumar

micropolar fluid. Wang [7] investigated the shrinking flowwhere velocity of boundary layer moves toward a fixed point

and they found an exact solution of Navier-Stokes equations.A good list of references for micropolar fluids is available inŁukaszewicz [8]. Tien and Yen [9] investigated the effect of

melting on forced convection heat transfer between a meltingbody and surrounding fluid. Epstein and Cho [10] analyzedthe melting heat transfer of the steady laminar flows over a flat

plate. The steady laminar boundary layer flow and heat trans-fer from a warm, laminar liquid flow to a melting surface mov-ing parallel to a constant free stream have been studied byIshak et al. [11]. Rosali et al. [12] studied micropolar fluid flow

towards a permeable stretching/shrinking sheet in a porousmedium numerically. Yacob et al. [13] investigated a modelto study the heat transfer characteristics occurring during the

melting process due to a stretching/shrinking sheet and theyhave studied the effects of the material parameter, meltingparameter and the stretching/shrinking parameter on the

velocity, temperature, skin friction coefficient and the localNusselt number. Cheng and Lin [14] analyzed the meltingeffect on transient mixed convective heat transfer from a verti-cal plate in a liquid saturated porous medium. Das [15] ana-

lyzed the MHD flow and heat transfer from a warm,electrically conducting fluid to melting surface moving parallelto a constant free stream in the presence of thermal radiation

and found that fluid temperature and the thermal boundarylayer thickness decrease for increasing thermal radiation, melt-ing parameter and magnetic field whereas reverse effect occurs

for moving parameter. The effects of thermal radiation usingthe nonlinear Rosseland approximation are investigated byCortell [16]. Mahmoud and Waheed [17] presented the effect

of slip velocity on the flow and heat transfer for an electricallyconducting micropolar fluid through a permeable stretchingsurface with variable heat flux in the presence of heat genera-

Please cite this article in press as: Singh K, Kumar M, Melting and heat absorptionmicropolar fluid, Ain Shams Eng J (2016), http://dx.doi.org/10.1016/j.asej.2016.04.0

tion (absorption) and a transverse magnetic field. They foundthat local Nusselt number decreased as the heat generation

parameter is increased with an increase in the absolute valueof the heat absorption parameter. Bataller [18] proposed theeffects of viscous dissipation, work due to deformation, inter-

nal heat generation (absorption) and thermal radiation. It wasshown internal heat generation/absorption enhances or dampsthe heat transformation. Ravikumar et al. [19] studied

unsteady, two-dimensional, laminar, boundary-layer flow ofa viscous, incompressible, electrically conducting andheat-absorbing, Rivlin–Ericksen flow fluid along a semi-infinite vertical permeable moving plate in the presence of a

uniform transverse magnetic field and thermal buoyancyeffect. They observed that heat absorption coefficient increaseresults a decrease in velocity and temperature. Khan and

Makinde [20] studied the bioconvection flow of MHD nano-fluid over a convectively heat stretching sheet in the presenceof gyrotactic microorganisms. Khan et al. [21] observed heat

and mass transfer in the third-grade nanofluids flow over aconvectively-heated stretching permeable surface. The dualsolution of stagnation-point flow of a magnetohydrodynamicPrandtl fluid through a shrinking sheet was made by Akbar

et al. [22]. Akbar et al. [23] analyzed numerical solution ofEyring Powell fluid flow towards a stretching sheet in the pres-ence of magnetic field. Ahmed and Mohamed [24] studied

steady, laminar, hydro-magnetic, simultaneous heat and masstransfer by laminar flow of a Newtonian, viscous, electricallyconducting and heat generating/absorbing fluid over a contin-

uously stretching surface in the presence of the combined effectof Hall currents and mass diffusion of chemical species withfirst and higher order reactions. They found that for heat

source, the velocity and temperature increase while the concen-tration decreases, however, the opposite behavior is obtainedfor heat sink.

effects in boundary layer stagnation-point flow towards a stretching sheet in a17


Melting and heat absorption effects in boundary layer stagnation-point flow 3

In the present work, we consider the boundary layerstagnation-point flow and heat transfer in the presence of melt-ing process and heat absorption in a micropolar fluid towards

a stretching sheet. We reduced the non-linear partial differen-tial equations to a system of ordinary differential equations, byintroducing the similarity transformations. The equations thus

obtained have been solved numerically using Runge–Kutta–Fehlberg method with shooting technique. The effectsof the melting parameter, micropolar parameter heat absorp-

tion parameter and stretching parameter on the fluidflow and heat transfer characteristics have been tabulated,illustrated graphically and discussed in detail.

2. Mathematical formulation

The graphical model of the problem has been given along with

flow configuration and coordinate system (see Fig. 1). The sys-tem deals with two dimensional stagnation point steady flow ofmicropolar fluid towards a stretching surface with heat absorp-tion. The velocity of the external flow is ueðxÞ ¼ ax and the

velocity of the stretching surface is uwðxÞ ¼ cx, where a andc are positive constants, and x is the coordinate measuredalong the surface. It is also assumed that the temperature of

the melting surface and free stream condition is Tm and T1,where T1 > Tm. In addition, the temperature of the solid med-ium far from the interface is constant and is denoted by Ts

where Ts < Tm. The viscous dissipation has assumed to be neg-ligible. Under these assumptions, the governing equations rep-resenting flow are as follows:

@u

@xþ @v

@y¼ 0 ð1Þ

u@u

@xþ v

@u

@y¼ ue

@ue@x

þ lþ jq

� �@2u

@y2þ jq@N

@yð2Þ

u@N

@xþ v

@N

@y¼ c

qj@2N

@y2� jqj

2Nþ @u

@y

� �ð3Þ

u@T

@xþ v

@T

@y¼ a

@2T

@y2� Q0

qcpðT� TmÞ ð4Þ

with the following boundary conditions:

u ¼ uwðxÞ ¼ cx; N ¼ �n @u@y; T ¼ Tm; at y ¼ 0

u ¼ ueðxÞ ! ax; N ! 0; T ! T1; as y ! 1

Figure 1 Flow configuration and coordinate system.


and

k@T

@y

� �y¼0

¼ q½kþ csðTm � TsÞ�vðx; 0Þ ð5Þ

Here u and v are the velocity components along the x and yaxis respectively. Further, l is dynamic viscosity, j is vortexviscosity, r is electrical conductivity of the fluid, q is fluid

density, T is fluid temperature, j is micro inertia density, N ismicrorotation, c is spin gradient viscosity, a is thermal diffusiv-ity, Q0 is coefficient of heat absorption, k is the thermal con-

ductivity of fluid, k is the latent heat of the fluid, cp is

specific heat at constant pressure and cs is the heat capacity

of the solid surface. We note that n is a constant such that0 6 n 6 1. The case when n ¼ 0, is called strong concentrationwhich indicates no microrotation near the wall. In case

n ¼ 1=2 it indicates the vanishing of anti-symmetric part ofthe stress tensor and denotes weak concentration and the casen ¼ 1 is used for the modeling of turbulent boundary layer

flows by Yacob et al. [13]. c ¼ lþ j2

� �l ¼ lð1þ K=2Þl, where

K ¼ j=l is the micropolar or material parameter and l ¼ t=aas reference length. The total spin N reduces to the angularvelocity.

The continuity Eq. (1) is satisfied by introducing the streamfunction w such that

u ¼ @w@y

; v ¼ � @w@x

: ð6Þ

Eqs. (2), (3) and (4) can be transformed into a set of

non-linear ordinary differential equations by using thefollowing similarity transformations:

w ¼ ðatÞ1=2xfðgÞ; N ¼ xaa

t

� �12

gðgÞ;

hðgÞ ¼ T� Tm

T1 � Tm

; g ¼ a

t

� �1=2

y ð7Þ

The transformed ordinary differential equations are asfollows:

ð1þ KÞf 000 þ ff 00 þ 1� f 02 þ Kg0 ¼ 0 ð8Þ

ð1þ K=2Þg00 þ fg0 � f 0g� Kð2gþ f 00Þ ¼ 0 ð9Þ

h00 þ Prðfh0 �HhÞ ¼ 0 ð10ÞThe boundary conditions (5) become

f 0ð0Þ ¼ e; gð0Þ ¼ �nf 00ð0Þ; Prfð0Þ þMh0ð0Þ ¼ 0; hð0Þ ¼ 0;

f 0ð1Þ ! 1; gð1Þ ! 0; hð1Þ ! 1

ð11Þwhere primes denote differentiation with respect to g andPr = t/a is Prandtl number. e ¼ c=a is the stretching (e > 0)

parameter, M is the dimensionless melting parameter and His heat absorption parameter which are defined as

M ¼ cpðT1 � TmÞkþ csðTm � TsÞ ; H ¼ Q0

aqcpð12Þ

The physical parameters of interest are the skin frictioncoefficient Cf and the local Nusselt number Nux which are

defined as

Cf ¼ swqu2e

; Nux ¼ xqwkðT1 � TmÞ ð13Þ



Table 1 Comparison of values of f 00ð0Þ and �h0ð0Þ for various values of M, K, e when H= 0.

H e M K Ref. [11] Ref. [7] Ref. [13] Present results

f 00ð0Þ f 00ð0Þ f 00ð0Þ �h0ð0Þ f 00ð0Þ �h0ð0Þ0 0 0 1.2326 1.232588 1.232588 �0.570465 1.232586 �0.570465

1 1.006404 �0.544535 1.006405 �0.544535

1 0 1.037003 �0.361961 1.037000 �0.361962

0 1 0.879324 �0.347892 0.879324 �0.347892

0.5 0 0 0.71330 0.713295 �0.692064 0.713294 �0.692065

1 0.582403 �0.680176 0.582403 �0.680175

1 0 0.599090 �0.438971 0.599090 �0.438971

1 0.506342 �0.432443 0.506341 �0.432443

Table 2 Values of f 00ð0Þ and �h0ð0Þ for various values ofM, K

and H when e = 0.5.

e M K H f 00ð0Þ �h0ð0Þ0.5 0 1 0.1 0.582377 �0.556085

0.5 1 1 0.1 0.516407 �0.372190

0.5 2 1 0.1 0.482978 �0.288090

0.5 1 0 0.1 0.613757 �0.378680

0.5 1 2 0.1 0.516411 �0.367390

0.5 1 1 0 0.506334 �0.432411

0.5 1 1 0.2 0.525239 �0.320230

Figure 2 Skin friction coefficient f 00ð0Þ with e for several valuesof M when K = 1 and H = 0.3.

Figure 3 Heat transfer coefficient �h0ð0Þ with e for several valuesof M when K= 1 and H= 0.3.

Figure 4 Skin friction coefficient f 00ð0Þ with e for several valuesof K when M= 1 and H= 0.3.


where sw and qw are the surface shear stress and the surfaceheat flux, which are given by

sw ¼ lþ jð Þ @u

@y

� �� y¼0

; qw ¼ �k@T

@y

� �y¼0

ð14Þ

Hence using (6), we get

Re1=2x Cf ¼ ½1þ ð1� nÞK�f 00ð0Þ; Re1=2x Nux ¼ �h0ð0Þ ð15Þwhere Rex ¼ ueðxÞx=t is the local Reynolds number.

Please cite this article in press as: Singh K, Kumar M, Melting and heat absorption effects in boundary layer stagnation-point flow towards a stretching sheet in amicropolar fluid, Ain Shams Eng J (2016), http://dx.doi.org/10.1016/j.asej.2016.04.017



3. Method of solution

To solve transformed Eqs. (8)–(10) with boundary conditions(11) as an initial value problem, the initial guess values of

f 00(0), g0(0) and h0(0) are chosen and Runge–Kutta–Fehlbergmethod is applied to obtain a solution. Then the calculatedvalues of f 0(g), g(g) and h(g) at g = g1 (g1 is the sufficient

large value of g) are compared with the given boundary condi-tions f 0(g1) = 1, g(g1) = 0 and h(g1)= 1. The missing val-

ues of f 00ð0Þ, g0(0) and h0ð0Þ, for some values of the

micropolar parameter K, melting parameter M, heat absorp-tion parameter H and the stretching parameter e are adjustedby shooting method, while the Prandtl number Pr is fixed to

unity and we take n = 0.5 (weak concentration). We ran thecomputer program written in MATLAB for different valuesof step size Dg and found that there is a negligible, change in

Figure 5 Heat transfer coefficient �h0ð0Þ with e for several valuesof M when K = 1and H= 0.3.

Figure 6 Skin friction coefficient f 00ð0Þ with e for several valuesof H when M= 1 and K = 1.


the velocity, temperature, skin friction coefficient and localNusselt number for value of Dg> 0.001. Hence in the presentpaper the authors have set step-size Dg = 0.001.

4. Results and discussion

In order to validate the numerical results obtained, we com-

pare our results with that reported by Wang [7], Ishak et al.[11], and Yacob et al. [13] as shown in Table 1, and they arefound to be in a good agreement.

It is seen from Table 2, Figs. 2 and 4 that both the maltingand material parameters show similar effect on the skin fric-tion coefficient, i.e. increasing K and M is to decrease the val-

ues of f 00(0), in absolute sense. From Fig. 6, it is clear that anincrease in heat absorption parameter leads to increase in the

value of f 00ð0Þ in absolute sense. The values of f 00ð0Þ are posi-

Figure 7 Heat transfer coefficient �h0ð0Þ with e for several valuesof H when M= 1 and K = 1.

Figure 8 Velocity profiles f 0ðgÞ for several values of M when

K = 1, H = 0.1 and e = 0.5.




tive when e < 1, and become negative when e > 1. Physically,

positive value of f 00ð0Þ means the fluid exerts a drag force on

the solid surface and negative value means the solid surfaceexerts a drag force on the fluid. The zero skin friction whene= 1, since for this case the stretching velocity is equal to

the free stream velocity.It is observed from Table 2 and Figs. 3 and 7 that the heat

transfer rate at the fluid-solid interface �h0ð0Þ decreases as themelting and heat absorption parameters increase. From Fig. 5,it is seen that for increasing value of K the heat transfer rate at

the fluid-solid interface �h0ð0Þ decreases for e< 1, but

increases for e > 1. The negative values of �h0ð0Þ presentedin Figs. 3, 5 and 7 show that the heat is transferred from the

warm fluid to cool solid surface.Figs. 8 and 9 depict effects of melting parameters M on the

velocity and temperature profiles respectively. It is clear from

Figure 9 Temperature profile hðgÞ for several values of M when

K= 1, H= 0.1 and e = 0.5.

Figure 10 Velocity profiles f 0ðgÞ for several values of K when

M = 1, H= 0.1 and e = 0.5.


Figs. 8 and 9 that velocity and temperature of the fluiddecrease with increasing value of melting parameter M.

Figs. 10 and 11 show the effects of micropolar or

material parameter K on velocity and temperaturerespectively. It is seen from Fig. 10 that velocity decreases asK increases, but temperature of the fluid decreases slightly as

K increases.Figs. 12 and 13 represent the effects of heat absorption

parameter H on velocity and temperature profile respectively.

It is evident from Fig. 12 that velocity increases slowly withincreasing value of H. From Fig. 13 it is depicted that temper-ature of the fluid decreases withH. Finally, from all Figs. 8–13,it can be easily seen that the far field boundary conditions (10)

are satisfied asymptotically and it signifies the correctness ofthe numerical scheme used.

Figure 11 Temperature profile hðgÞ for several values of K when

M= 1, H = 0.1 and e= 0.5.

Figure 12 Velocity profiles f 0ðgÞ for several values of H when

M= 1, K = 1 and e = 0.5.



Figure 13 Temperature profile hðgÞ for several values of H when

M= 1, K = 1 and e = 0.5.


5. Conclusion

We have studied the effects of the melting parameter, materialparameter, heat absorption parameter and stretching parame-ter on skin friction coefficient, local Nusselt number (which

represents the heat transfer rate at the surface) for the steadylaminar boundary layer stagnation point flow and heat trans-fer from a warm micropolar fluid to a melting solid surface of

the same material. Using the similarity variables, the governingnon-linear partial differential equations are transformed into asystem of coupled non-linear ordinary differential equations

and the solved numerically by using Runge–Kutta–Fehlbergmethod with shooting technique. Both the material and themelting parameter decrease the friction and the heat transferrate at the fluid-solid interface. The heat absorption parameter

increases the skin friction coefficient, but decreases the heattransfer rate at the fluid-solid interface.

Acknowledgements

Research was supported by UGC (Grant No. F. 17-97/2008

(SA-I)) Government of India under the scheme of UGC-Junior research fellowship in science for Joint CSIR-UGCNET (JRF) test qualified students, and gratefully acknowl-

edged by first author. The authors wish to express their verysincere thanks to referees for their valuable comments and sug-gestions for the improvement of the manuscript.

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http://refhub.elsevier.com/S2090-4479(16)30056-9/h0005









































































Khilap Singh is a research scholar in Depart-

ment of Mathematics, Statistics and Com-

puter Science in G.B. Pant University of

Agriculture and Technology, Pantnagar,

Uttarakhand, India. He obtained his M.Sc.

(Mathematics) degree from H.N.B. Garhwal

University, Uttarakhand, India in 2010. His

research interest is Computational Micropolar

Fluid Dynamics. He published about 8 papers

in National and International Journals. He

attended several conferences/Seminars.


Manoj Kumar received his Ph.D. degree from

the University of Roorkee, India, and is cur-

rently, Professor of Mathematics in the

Department of Mathematics, Statistics and

Computer Science at G.B. Pant University of

Agriculture and Technology, Pantnagar,

India. He has supervised many students at

Ph.D. and M.S. degree levels. His current

research interest mainly include: nonlinear

science, industrial engineering and computa-

tional fluid dynamics.



Melting and heat absorption effects in boundary layer stagnation … · Melting and heat absorption...

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