Applied Mathematics and Computation 217 (2010) 685–688

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate/amc

Computational treatment of free convection effect on flow ofelastico-viscous fluid past an accelerated plate with constant heat flux

J. Singh a, S.K. Gupta a,*, Srinivasan Chandrasekaran b

a Hydraulics & Water Resources Engineering, Department of Civil Engineering, Institute of Technology, BHU, Varanasi 221005, UP, Indiab Department of Ocean Engineering, Indian Institute of Technology Madras, Chennai, India

a r t i c l e i n f o a b s t r a c t

Keywords:Free convectionVisco-elastic fluidLaplace transform

0096-3003/$ - see front matter � 2010 Elsevier Incdoi:10.1016/j.amc.2010.06.005

* Corresponding author.E-mail address: sunkmg@gmail.com (S.K. Gupta)

Effect of free convection on the visco-elastic fluid (Walter – B’ type) flow past an infinitevertical plate accelerating in its own plane with constant heat flux is examined analytically.It is found that for given values of Grashof number, Prandtl number and Newtonian param-eter; flow velocity at any point increases with the increase in time and non-Newtonianparameter, however, it decreases with both, the heating and cooling of the plate.

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1. Introduction

Several industrial applications involve the flow of non-Newtonian fluids, and thus the flow behaviour of such fluids finds agreat relevance. Molten metals, plastic, pulps, emulsions, slurries and raw materials in fluid state are some examples to men-tion. Non-Newtonian flow also finds practical applications in bio-engineering, wherein blood circulation in human/animal ar-tery is explained by an appropriate visco-elastic fluid model of small elasticity. The study of a visco-elastic pulsatile flow helpsin understanding the mechanism of dialysis of blood through an artificial kidney. The constitutive equations of certain class ofnon-Newtonian fluids with short memories have been proposed by Walters [1] and Beard and Walters [2] for elastic-viscousfluid, referred to as Walters Liquid B’. The flow of viscous incompressible fluid past an impulsively started infinite horizontalplate in its own plane was first studied by Stokes [3]. Raptis et al. [4], using these equation, studied the influence of free con-vection and mass transfer on flow through a porous medium. Raptis and Predikis [5] studied the influence of free convectionand mass transfer on oscillatory flow through a porous medium. The flow of visco-elastic incompressible and electrically con-ducting fluid past an infinite plate in presence of a transverse magnetic field, when the plate executes simple harmonic motionparallel to itself has been discussed by Sheriet and Ezzat [6]. The effects of suction, free oscillations and free convection cur-rents on flow have been studied by Soundalgekar and Patil [7]. Singh [8] studied the mass transfer effect on the flow past anaccelerated vertical plate with constant heat flux. Singh and Singh [9] studied the transient hydromagnetic free convectionflow past an impulsively started vertical plate. Singh [10] studied the flow of elasto-viscous fluid past an accelerated porousplate. Recently, researchers [11–15] studied the several problems related to the flow of Walter’s Liquid B’.

The present study investigates the effect of free convection on the flow of Walters Liquid B’ past an accelerating (in itsown plane) infinite vertical plate subjected to the constant heat flux.

2. Problem formulation and solution

We consider unsteady free convection flow of an incompressible elastic-viscous fluid past an infinite vertical plate. The x0-axis is taken along the plate in the upward direction and y0-axis is taken normal to it. Initially the plate is at rest but at time

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686 J. Singh et al. / Applied Mathematics and Computation 217 (2010) 685–688

t0 = 0 the plate is accelerated with a velocity u0 = U t0 in its own plane and heat is also supplied to the plate at the constantrate. The equations which govern the free convection flow of an elastic-viscous fluid under the usual Boussinesq’s approx-imations in dimensionless form are given below:

@u@t¼ @

2u@y2 þ Grh�

@3u@y2@t

ð1Þ

@h@t¼ 1

Pr

@2h@y2 ð2Þ

where, h is the non-dimensional temperature; Pr, is Prandtl number; Gr is the Grashof number.The initial and boundary conditions for velocity field and temperature field are

t 6 0; u ¼ 0; h ¼ 0 for all y

t > 0;u ¼ t; h

y ¼ �1 at y ¼ 0

u ¼ 0; h ¼ 0 as y!1

(ð3Þ

The non-dimensional equations introduced in the above equations are defined as,

y ¼ y0Um2

� �1=3

; t ¼ t0Um

� �1=3

; u ¼ u0

ðmUÞ1=3

h ¼ kU1=3ðT 0 � T 01Þqm2=3

2=3

; Gr ¼m2=3qgb

kU4=3

Pr ¼lCp

k; K ¼ K 0

hUm2

� �2=3

ð4Þ

where, q is the constant heat flux per unit area, m is kinematic viscosity, K is elastic parameter, u is flow velocity. The solutionof Eq. (2) under the boundary condition (3) has been obtained by Soundalgekar and Patil [7] and Georgantopoulos et al. [16].We will now solve Eq. (1) under the boundary condition (3). Since Eq. (1) is third order differential equation when K – 0; andfor K = 0, it reduces to an equation governing the Newtonian fluid flow. Mathematically, we need three boundary conditionsto solve the third order differential equation for a unique solution. But there are only two boundary conditions. Therefore, toovercome this difficulty, following Beard and Walters [2], u may be taken as:

u ¼ u0 þ Ku1 ð5Þ

Substituting Eq. (5) in Eq. (1) and equating the coefficients of different powers of K and neglecting the powers of K2, we get

@u0

@t¼ @

2u0

@y2 þ Grh ð6Þ

@u1

@t¼ @

2u1

@y2 �@3u0

@y2@tð7Þ

The corresponding boundary conditions are:

t 6 0; u0 ¼ 0; u1 ¼ 0 for all y

t > 0;u0 ¼ t; u1 ¼ 0 at y ¼ 0u0 ¼ 0; u1 ¼ 0 as y !1

�ð8Þ

Applying the Laplace transform technique, the solution of (6) and (7) under the boundary conditions (8) are:

u0 ¼ tð1þ 2g2ÞerfcðgÞ � 2gtffiffiffipp expð�g2Þ

þ Grð4tÞ3=2

ðPr�1ÞffiffiffiffiPr

p 1þg2

6ffiffiffipp expð�g2Þ � g

12 ð3þ 2g2ÞerfcðgÞn oh

� 1þg2Pr6ffiffiffipp expð�g2PrÞ � g

12 ð3þ 2g2PrÞffiffiffiffiffiPrp

erfc gffiffiffiffiffiPrp� �n oi ð9Þ

u1 ¼2Gr

ffiffiffiffiffiPrtp

ðPr�1Þ21ffiffiffipp expð�g2Þ � g erfcðgÞn oh

� 1ffiffiffipp expð�g2PrÞ � g

ffiffiffiffiffiPrp

erfc gffiffiffiffiffiPrp� �n oi

� gffiffiffipp expð�g2Þ � Grg

ffiffitp

ðPr�1ÞffiffiffiffiPr

p erfcðgÞ

ð10Þ

Fig. 1. Velocity profiles for uniformly accelerated plate.

Fig. 2. Velocity profiles for uniformly accelerated plate under cooling and heating conditions.

J. Singh et al. / Applied Mathematics and Computation 217 (2010) 685–688 687

Now using Eq. (5), we get

u ¼ tð1þ 2g2ÞerfcðgÞ � 2gtffiffiffiffipp expð�g2Þ

þ Grð4tÞ3=2

ðPr � 1ÞffiffiffiffiffiPrp 1þ g2

6ffiffiffiffipp expð�g2Þ

� ���

g12ð3þ 2g2ÞerfcðgÞþ

� 1þ g2Pr

6ffiffiffiffipp expð�g2PrÞ

�� g

12ð3þ 2g2PrÞ

ffiffiffiffiffiPr

perfc g

ffiffiffiffiffiPr

p ��

þ K2Gr

ffiffiffiffiffiffiffiPrtp

ðPr � 1Þ21ffiffiffiffipp expð�g2Þ � g erfcðgÞ� �"

� 1ffiffiffiffipp expð�g2PrÞ � g

ffiffiffiffiffiPr

perfc g

ffiffiffiffiffiPr

p �� �

� gffiffiffiffipp expð�g2Þ � Grg

ffiffitp

ðPr � 1ÞffiffiffiffiffiPrp erfcðgÞ

�ð11Þ

where, g ¼ y2

ffiffitp

.

Fig. 3. Velocity profiles for uniformly accelerated plate under heating condition at different Prandtl numbers.

688 J. Singh et al. / Applied Mathematics and Computation 217 (2010) 685–688

3. Results and discussion

For the purpose of discussing the results the numerical calculations are carried out for different values of t, Gr, Pr and K,respectively. Fig. 1 shows that for fixed values of t, Gr, Pr, the velocity at any point decreases with the increase of non-New-tonian parameter K. It is also clear from Fig. 1 that for fixed values of Gr, Pr and K, flow velocity increases with the increase intime parameter t. Fig. 2 reveals that flow velocity decreases with the increase in non-Newtonian parameter K for both heat-ing and cooling of the plate (i.e. Gr < 0 or Gr > 0). It is also clear from Fig. 2 that, in case of relatively greater heating of theplate, velocity decreases; whereas it increases with the greater cooling of the plate. Fig. 3 shows that for fixed values of Gr, tand K, the velocity at any point increases with the increase in Pr.

4. Conclusions

(1) The velocity of visco-elastic fluid (Walter’s fluid B’) is less in comparison to that of Newtonian fluid.(2) The flow velocity increases with the increase in time.(3) The flow velocity increases with the increase in Prandtl number.

References

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