heat transfer in boundary layer flow of maxwell fluid … · shrinking sheet with wall mass...

Thermal Energy and Power Engineering TEPE

TEPE Volume 2, Issue 3 Aug. 2013, PP. 72-78 www.vkingpub.com © American V-King Scientific Publishing

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Heat transfer in the boundary layer flow of Maxwell fluid over a permeable shrinking sheet

Krishnendu Bhattacharyya1, Tasawar Hayat2, Rama Subba Reddy Gorla3 1Department of Mathematics, The University of Burdwan, Burdwan-713104, West Bengal, India

2Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan 3Department of Mechanical Engineering, Cleveland State University, Cleveland, OH 44115, USA

[email protected],[email protected]; [email protected]; [email protected]

Abstract – An analysis is carried out to investigate the heat transfer in boundary layer flow of Maxwell fluid over a porous shrinking sheet with wall mass transfer. The governing PDEs are converted into self-similar nonlinear ODEs and then those are solved numerically using shooting method. The study reveals that dual solutions of velocity and temperature fields exist and the steady flow of Maxwell fluid is possible with lesser amount of suction when the Deborah number increases. For both solutions, the viscous boundary layer thickness decreases with Deborah number. Similarly, the thermal boundary layer thickness (in both solutions) decreases with Deborah number and Prandtl number, but thermal boundary layer thickness increases with increasing values of wall mass suction parameter for second solution though it decreases with suction for first solution.

Keywords- Heat transfer; dual solutions; boundary layer flow; Maxwell fluid; permeable shrinking sheet; wall mass transfer

NOMENCLATURE c shrinking constant cp specific heat f non-dimensional stream function f′ dimensionless velocity Pr Prandtl number S wall mass transfer parameter T temperature Tw constant temperature at the sheet T∞ free stream temperature Uw shrinking velocity u velocity component in x direction v velocity component in y direction vw wall mass transfer velocity x distance along the sheet y distance normal to the sheet Greek symbols β Deborah number η similarity variable κ fluid thermal conductivity λ relaxation time µ coefficient of fluid viscosity υ kinematic fluid viscosity θ non-dimensional temperature ρ fluid density ψ stream function Subscripts

w condition at the sheet

I. INTRODUCTION

The heat transfer phenomenon in boundary layer flow over a stretching/shrinking sheet is very important for its day by day increasing industrial applications. The quality of final industrial products strongly depends on the heat transfer characteristics. The flow due to a linearly stretching plate was first studied by Crane [1]. Numerous studies [2-11] have been conducted later to extend the pioneering work of Crane. Opposite to forward stretching, Wang [12] introduced the flow due to linear shrinking of a sheet in it own plane. The flow due to a shrinking sheet is very much unusual. Normally, the steady boundary layer shrinking flow does not occur because the generated vorticity due to shrinking goes outside the boundary layer region. Miklavčič and Wang [13] investigated the viscous flow over a shrinking sheet with suction and found that the steady flow depends on imposed mass suction, i.e., the mass suction delayed the boundary layer separation. This new type of flow due to shrinking sheet is essentially a backward flow as described by Goldstein [14]. In this regards, Hayat et al. [15] gave an analytic solution of magnetohydrodynamic (MHD) flow of a second grade fluid over a shrinking sheet using homotopy analysis method (HAM). Hayat et al. [16] also obtained an analytic HAM solution for MHD rotating flow of a second grade fluid over a shrinking sheet. Muhaimin et al. [17] showed the effects of heat and mass transfer on MHD boundary layer flow over a shrinking sheet subject to suction. A series solution of three-dimensional MHD and rotating flow over a porous shrinking sheet was obtained by Hayat et al. [18] using HAM. Fang and Zhang [19] reported an exact solution of the transformed Navier-Stokes equations for two-dimensional MHD viscous flow over a porous shrinking sheet subjected to wall mass transfer. Fang et al.[20] studied the unsteady viscous flow over a shrinking surface with mass suction. Cortell [21] discussed the MHD viscous flow caused by a shrinking sheet with suction for two-dimensional and axisymmetric cases. Furthermore, some other properties of shrinking sheet flow can be found in the articles [22-30]. On the other hand, Wang [31] studied the stagnation-point flow towards a shrinking sheet for both two-dimensional and axisymmetric cases. Ishak et al. [32] investigated the stagnation-point flow of micropolar fluid



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over a shrinking sheet. Bhattacharyya and Layek [33] explained the effects of suction/blowing on the steady boundary layer stagnation-point flow and heat transfer towards a shrinking sheet in presence of thermal radiation. In additional, some important aspect of stagnation-point flow towards shrinking sheet can also be found in some recently published articles [34-42].

In recent times, due to increasing industrial applications the flows involving non-Newtonian fluids grab significant attention of modern day researchers. Many materials in real field, like, melts, muds, condensed milk, glues, printing ink, emulsions, soaps, sugar solution, paints, shampoos, tomato paste etc. show properties which differs from those of Newtonian fluids. But, the main difficulty is to construct a single constitutive equation which follows all properties of such non-Newtonian fluids. Because of this, various non-Newtonian fluid models [43-49] had been proposed and discussed in the literature. The governing equations of non-Newtonian fluids are highly non-linear and much more complicated than the governing equations of Newtonian fluids. Maxwell fluid model [50,51] is one of non-Newtonian fluid models, where the relaxation phenomena are taken into consideration. This model has applications in viscoelastic problems where the dimensionless relaxation time is small. The linear stability analysis of plane Couette flow of upper convected Maxwell fluid was made by Renardy and Renardy [52]. Olsson and Yström [53] described some important properties of upper convected Maxwell fluid flow. A new exact solution corresponding to the flow of a Maxwell fluid over a suddenly moved flat plate was determined by Fetecau and Fetecau [54]. Hayat et al. [55] reported a series solution for MHD boundary layer flow of an upper convected Maxwell fluid over a porous stretching sheet and they [56] also obtained the series solution of MHD stagnation-point flow of an upper convected Maxwell fluid over a porous stretching surface. Aliakbar et al. [57] discussed the influence of thermal radiation on MHD flow of Maxwellian fluids above stretching sheets. Hayat et al. [58] investigated the mass transfer in the steady two-dimensional MHD boundary layer flow of an upper convected Maxwell fluid past a porous shrinking sheet in the presence of chemical reaction. Motivated by the above studies, in the present paper, the dual solutions of heat transfer in boundary layer flow of Maxwell fluid over a porous shrinking sheet wall mass transfer are obtained. The transformed ODEs are solved by shooting technique using Runge-Kutta method. Then the flow and heat transfer characteristics are analysed.

There are many industrial applications based on non-Newtonian fluids such as material processing, crystal growth, nuclear reactor cooling etc. Among the several models of non-Newtonian fluids we selected the simplest subclass fluids known as Maxwell model.

II. MATHEMATICAL FORMULATION

Consider the steady two-dimensional boundary layer

flow of Maxwell fluid and heat transfer over a permeable shrinking sheet. The governing equations of motion and the energy equation may be written in usual notation as:

0u vx y∂ ∂

+ =∂ ∂

, (1)

2 2 2 22 2

2 2 22u u u u u uu v u v uvx y x yx y y

λ υ ∂ ∂ ∂ ∂ ∂ ∂

+ + + + = ∂ ∂ ∂ ∂∂ ∂ ∂ (2)

and 2

2p

T T Tu vx y c y

κρ

∂ ∂ ∂+ =

∂ ∂ ∂, (3)

where u and v are the velocity components in x- and y-directions respectively, υ(=µ/ρ) is the kinematic fluid viscosity, ρ is the fluid density, µ is the coefficient of fluid viscosity, λ is the relaxation time, T is the temperature, κ is the fluid thermal conductivity and cp is the specific heat.

The boundary conditions are given by ( ) , at 0; 0 asw wu U x cx v v y u y= − = − = = → →∞ (4)

and at 0; aswT T y T T y∞= = → →∞ , (5)

where Uw=cx is the shrinking velocity with c>0 is shrinking constant, Tw is the constant temperature at the sheet and T∞ is the free stream temperature assumed to be constant. Here vw is the wall mass transfer velocity with vw<0 for suction and vw>0 for blowing.

Now, the stream function ψ(x,y) is introduced as:

andu vy xψ ψ∂ ∂

= = −∂ ∂

. (6)

For relations in (6), the equation (1) is identically satisfied and the equation (2) and the equation (3) are reduced to the following forms:

2 22 2 3 3

2 2 3y x y x y xy x y yψ ψ ψ ψ ψ ψ ψ ψλ

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ − + + ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂

3 3

2 32y x x y yψ ψ ψ ψυ

∂ ∂ ∂ ∂− =∂ ∂ ∂ ∂ ∂

(7)

and 2

2p

T T Ty x x y c yψ ψ κ

ρ∂ ∂ ∂ ∂ ∂

− =∂ ∂ ∂ ∂ ∂

. (8)

The boundary conditions in (4) for the velocity components become

, at 0; 0 aswcx v y yy x yψ ψ ψ∂ ∂ ∂

= − = − = → →∞∂ ∂ ∂

. (9)

Next, the dimensionless variables for ψ and T are introduced as:

( ) and ( ) ( )wc xf T T T Tψ υ η θ η∞ ∞= = + − , (10) where η is the similarity variable and is defined as

wUxy υη = .

Using relations in (10) the following nonlinear self-similar ordinary differential equations are obtained:



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2 2(2 ) 0f ff f ff f f fβ′′′ ′′ ′ ′ ′′ ′′′+ − + − = (11) and

0Pr fθ θ′′ ′+ = , (12) where β=λ/c is Deborah number and Pr=µcp/κ is the Prandtl number.

The boundary conditions (9) and (5) reduce to the following forms

( ) , ( ) 1 at 0; ( ) 0 asf S f fη η η η η′ ′= = − = → →∞ (13) and

( ) 1 at 0; ( ) 0 asθ η η θ η η= = → →∞ , (14)

where wS v cυ= − is the wall mass transfer parameter, S>0 (i.e. vw<0) corresponds to wall mass suction and S<0 (i.e. vw>0) corresponds to wall mass blowing.

III. NUMERICAL METHOD FOR SOLUTION

The highly nonlinear coupled self-similar ODEs (11) and (12) along with the boundary conditions (13) and (14) constitute a two point boundary value problem (BVP) and is solved using shooting method, by converting it into an initial value problem (IVP). In this method, it is necessary to choose a suitable finite value of η→∞, say η∞. The following first-order system is set:

2 2

,,

( 2 ) (1 )

f pp qq p fq fpq fβ β

′ =′ = ′ = − − −

(15)

and ,r

r Pr frθ ′ =

′ = − (16)

with the boundary conditions (0) , (0) 1, (0) 1f S p θ= = − = . (17)

To solve (17) and (18) with (19) as an IVP the values for q(0) i.e. f″(0) and r(0) i.e. θ′(0) are must needed but no such values are given. The initial guess values for f″(0) and θ′(0) are chosen and the fourth order Runge-Kutta method is applied to obtain a solution. The most often used method of the Runge-Kutta family is the Fourth-Order one, which extends the idea of the mid-point method, by jumping 1/4th of the way first, then going half-way, then going 3/4th of the way and finally jumping all the way.

The formula for this method looks as follows: 1

2 1

3 2

( , )( 2, 2)( 2, 2)

n n

n n

n n

k f x t tk f x k t t tk f x k t t t

= ∆= + + ∆ ∆= + + ∆ ∆

( )4 3

531 2 41

( , )

( )6 3 3 6

n n

n n

k f x k t t tkk k kx x o t+

= + + ∆ ∆

= + + + + + ∆

Then the calculated values of f′(η) and θ(η) at η∞(=20) is compared with the given boundary conditions f′(η∞)=0 and θ(η∞)=0 and the values of f″(0) and θ′(0) are adjusted using “secant method” to give better approximation for the solution. The step-size is taken as ∆η=0.01. The process is

repeated until we get the results correct up to the desired accuracy of 10−7 level.

IV. RESULTS AND DISCUSSION

The heat transfer characteristics for steady boundary layer flow of Maxwell fluid past a permeable shrinking sheet with suction are obtained numerically by the above method.

Fig. 1 Skin friction coefficient f″(0) vs. S for various values of β.

Fig. 2 Temperature gradient at the sheet −θ′(0) vs. S for various values of

β.

The two-dimensional flow of Newtonian fluid over a porous shrinking sheet is possible only when the wall mass transfer (suction) parameter S satisfies the inequality S≥2 [13]. On the other hand, present investigation shows that for Maxwell fluid the requirement mass suction for steady flow reduces. In fact, with the increase of Deborah number (β) the intrinsic fluidity of a material become lesser i.e., less fluid material and so the vorticity generation due to shrinking for Maxwell fluid is not as much as of Newtonian fluid. The numerical computation explores that for Deborah number, β=0.1 the value of wall mass transfer parameter S needs to satisfy the inequality S≥1.907045 and for β=0.2, S≥1.801921. So, for Maxwell (β>0) fluid the boundary layer separation is delayed and accordingly the similarity solution exist for lesser amount of mass suction through the porous sheet. In this regard, the values of f″(0) and −θ′(0) against S for different values of β are plotted in Fig. 1 and



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Fig. 2, respectively. The dual solutions of temperature distribution in addition to velocity field are obtained. It is clear from the figures that for β=0.1, the similarity solution is dual nature when S≥1.907045 and no similarity solution exists for S<1.907045. Also, for β=0.2, the dual solutions are obtained when S≥1.801921 and boundary layer solution disappears for S<1.801921. Furthermore, from the figure it also observe that the value of −θ′(0) i.e., the rate of heat transfer increases with β for first solution and it decreases initially in second solution but for larger suction it increases.

Fig. 3 Dual velocity profiles f′(η) for various values of β.

Fig. 4 Dimensionless stream function f(η) for various values of β.

Formally, the Deborah number is defined as the ratio of the relaxation time, characterizing the time it takes for a material to adjust to applied stresses or deformations, and the characteristic time scale of an experiment (or a computer simulation) probing the response of the material. It incorporates both the elasticity and viscosity of the material. At lower Deborah numbers, the material behaves in a more fluid like manner, with an associated Newtonian viscous flow. At higher Deborah numbers, the material behavior changes to a non-Newtonian regime, increasingly dominated by elasticity, demonstrating the solid like behavior.

Fig. 5 Dual temperature profiles θ(η) for various values of β.

Fig. 6 Dual temperature gradient profiles θ′(η) for various values of β.

Fig. 7 Dual velocity profiles f′(η) for various values of S.

The dimensionless velocity profiles and dimensionless stream function profiles for different values of Deborah number β are depicted in Fig. 3 and Fig. 4 respectively. The dimensionless dual velocity profile f′(η) shows that due to increase of Deborah number the viscous boundary layer thickness in both solutions reduces and for this reasons crossing over is observed in second solution. Actually, the fluid material in the flow is reduced with the increase in β and for this the boundary layer thickness decreases, which can be confirmed from the profiles f(η) in Fig. 4. The temperature profiles θ(η) and temperature gradient profiles θ′(η) are presented in Fig. 5 and Fig. 6, respectively. The temperature at a point decreases with increase in β except in



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second solution for small η and in both temperature gradient profiles θ′(η) crossing over is found. The thermal boundary layer thickness also reduces with Deborah number for both solutions. This is to be expected because as Deborah number increases, the fluid becomes dominated by elastic properties.

Fig. 8 Dimensionless stream function f(η) for various values of S.

Fig. 9 Dual temperature profiles θ(η) for various values of S.

Fig. 10 Dual temperature gradient profiles θ′(η) for various values of S.

The influences of wall mass suction parameter on the flow and heat transfer are demonstrated in Figs. 7-10. The velocity profiles show opposite effects of mass suction on first and second solutions. Due to increase in suction, the velocity increases for first solution and for second solution

it decreases [Fig. 7]. So, with mass suction the viscous boundary layer thickness become smaller for first solution and larger for second solution, this can also observed in Fig. 8. Similar to the velocity field, converse effects of mass suction are noticed on the first and second solutions of temperature distribution [Fig. 9]. Like, Deborah number effect, the crossing over in temperature gradient profiles is monitored [Fig. 10].

Fig. 11 Dual temperature profiles θ(η) for various values of Pr.

Fig. 12 Dual temperature gradient profiles θ′(η) for various values of Pr.

In Fig. 11 and Fig. 12, the variations in the temperature and temperature gradient profiles are exhibited. For both solutions, the temperature at a point and the thermal boundary layer thickness are found to decrease with increasing Pr. The Prandtl number is inversely proportional to the thermal diffusivity of the fluid and due to this the thermal boundary layer thickness reduces. So, the temperature gradient θ′(η) vanishes quicker for higher values of Pr. Finally, form all the figures it is important to note that here also second solution boundary layer thickness (both viscous and thermal) is larger than that of first solution boundary layer thickness.

V. CONCLUSIONS

The heat transfer in boundary layer flow of Maxwell fluid over a porous shrinking sheet with wall mass transfer is investigated. The converted nonlinear ODEs are solved numerically by shooting technique using 4th order Runge-



77

Kutta method. In the analysis, the dual solutions for velocity and temperature distributions are found. Due to increase of Deborah number, the steady flow of Maxwell fluid is possible with smaller amount of suction. Also, both viscous and thermal boundary layer thicknesses are decreased with Deborah number in both first and second solutions. For increase of mass suction, the temperature at a point decreases for first solution and increases for second solution. At lower Deborah numbers, the material behaves in a more fluid like manner, with an associated Newtonian viscous flow. At higher Deborah numbers, the material behavior changes to a non-Newtonian regime, increasingly dominated by elasticity, demonstrating the solid like behavior.

ACKNOWLEDGMENTS

The authors like to thank the reviewers for their useful comments. One of the authors (K. Bhattacharyya) gratefully acknowledges the financial support of National Board for Higher Mathematics (NBHM), Department of Atomic Energy, Government of India for pursuing this work.

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