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In the present paper, the boundary layer flow of Walters Liquid B Model over a stretching plate has been considered to solve heat flow problem with variable conductivity. First, using similarity transformation, the velocity components of velocity have been obtained. Then, the heat flow problem has been considered in two ways: 1) prescribed surface temperature (PST), and 2) prescribed stretching plate heat flux (PHF) in case of variable conductivity. Due to variable conductivity, temperature profile has its two part- one mean tempera-ture and other temperature profile induced due to variable conductivity. The related results have been dis-cussed with the help of graphs.

Due to number of applications in industrial manufacturing process, the problem of boundary layer flow past a stretching plate has attracted considerable attention of researchers during the past few decades. Examples of such technological process are hot rolling, wire drawing, glassfiber and paper production. In the process of drawing artificial fibers the polymer solution emerges from orifice with a speed which increases from almost zero at the orifice up to a plateau value at which it remains constant. The moving fiber, which is of great technical importance, is governed by the rate at which the fiber is cooled and this, in turn affects the final properties of the yarn. A number of works are presently available that follow the pioneering classical work of Sakiadis [

There are liquid metals whose thermal conductivity varies with temperature in an approximately linear manner in the range from 0˚ F to 400˚ F. In 1996, T. C. Chiam [

In almost all the flow problems over stretching sheet with heat transfer referred so far where closed form solution is obtained, the thermal conductivity of liquid has been taken constant. In this paper, we try to solve the boundary layer flow of Walters Liquid B Model over a stretching plate and heat transfer with variable thermal conductivity in the following two cases:

1) prescribed surface temperature (PST), and 2) prescribed stretching plate heat flux (PHF). The temperature field has been obtained in each case and the effect on it, of Visco-elasticity and Prandtl number has been discussed graphically.

The problem considered here is the steady boundary layer flow due to a moving flat plate in a quiescent Walters Liquid B Model. The flow is two dimensional where x-axis is along the plane of moving plate and y-axis is normal to it, respectively. We assume that the surface is moving continuously with the velocity in the positive x-direction. Under these assumptions, the boundary layer along moving plate is governed by the equations:

where u, the horizontal velocity component; v, the vertical velocity component;, the kinematic viscosity; k_{0}, the coefficient of visco-elasticity The relevant boundary conditions are:

Introducing the dimensionless variables

the Equations (1) and (2) reduce to

where k_{1}=with boundary conditions

where bar has been dropped for convenience.

Setting the similarity solution of the form, we have

Putting u and v in the Equation (2), we have

which is non-linear differential equation of order three.

The boundary conditions (5) reduce to

Boundary conditions suggest that the velocity function may be of the form where r is complex number with positive real part. Thus,

Now, from the Equation (7), we get.

Therefore, the velocity components become as follows:

In absence of viscous dissipation and heat generation, the energy equation for two dimensional heat flow is given by

subject to boundary conditions

where T_{P} is plate temperature, T_{¥} is temperature of surrounding fluid, C_{P} is specific heat at constant pressure and k is thermal conductivity.

Let the surface temperature be of the form

while the temperature out side the dynamic region be. Now, we define the dimensionless temperature by

where.

For liquid metals, it has been found that the thermal conductivity varies with temperature in an approximately linear manner in the range from 0˚ F to 400˚ F. Therefore, we assume k as where.

Now, substituting u and v and changing the independent variable y to, we have

with boundary conditions

From Equation (11), we note that the heat transfer takes place in two parts, that is, one part of heat transfer is due to temperature difference and the other part is due to variable thermal conductivity. We denote the first part by q_{m} and second by q_{v}. Thus, equating the terms independent of e and the terms involving e, we have

and

, as (17)

The solution of the Equation (14) is

Equation (17) is a non-linear differential equation of order two. Let the solution of this equation be of the form

Putting this solution in the Equation (18), we have

The roots of this equation are 0 and 1/2. Therefore,

The general solution (20) of the Equation (16) is real only when. Therefore, the heat transfer due to variable thermal conductively takes place within the dynamic region. Hence, the boundary conditions (17) may be presented as

The solution (20) finally reduces to

The power law heat flux on the surface of stretching plate is considered to be a quadratic power of x in the form

at (23)

, as (24)

where D is a constant, k is the thermal conductivity. Now we define dimensionless temperature by

where

when

and.

Writing the Equation (11) in terms of, we get the following differential equation

together with boundary conditions:

Equating the terms independent of and the terms involving from Equation (26), we get the following two boundary value problems:

, as (28b)

and,

, as (29b)

The solution of the Equation (28a) together with boundary conditions (28b) is given by

The general solution of the Equation (29a) is

where C and D are arbitrary constants to be determined. Here, we observe that the dynamic region for this temperature field is as it was in PST case. Therefore, the boundary conditions may be taken as

Hence finally the solution becomes

A boundary layer flow of visco-elastic fluid (Walters Liquid B Model) over a stretching plate and heat transfer has been analyzed through out this paper. The boundary layer equations of momentum and heat transfer have been solved analytically. Two different analytical expressions have been obtained for dimensionless temperature fields for two general cases of boundary conditions namely 1) PST case, and 2) PHF respectively. The equation governing the induced temperature field due to variable thermal conductivity has been solved with modified boundary conditions in both the cases PST and PHF The related results have been discussed in the subsequent paragraphs in detail.

_{m}(h) with regard to Prandtl number Pr in PST case. We notice that as Prandtl number Pr increases, the mean temperature decreases.

_{v} in PST case. This temperature field is independent of k_{0} and Pr. We see that the induced temperature decreases as we move away the plate. The variation pattern is represented by a parabolic curve having focus at h = 1. q_{v} decreases as we move away the stretching plate and it becomes almost zero at h = 1 which end upper edge of boundary layer thickness.

Equation (32) represents the temperature field induced by variable thermal conductivity in PHF case. The geometry of this equation is parabola having focus at h = 1. This temperature field is independent of k_{1} and Pr. We see that the induced temperature decreases as we move away the plate. It varies from maximum at the plate to zero at the upper end of boundary layer.

Putting k_{0} = 0, we get the results due to Naseem Ahmad, Z. U. Siddiqui and M. K. Mishra [

The boundary layer flow of Walters Liquid B over linear stretching plate has been studied together with heat transfer with variable thermal conductivity. The heat transfer has been studied by considering two cases: PST and PHS. The effect of visco-elasticity k_{0} has been looked upon in the Figures 1 and 2. This paper is an extension of the work due to Naseem Ahmad, Z. U. Siddiqui and M. K. Mishra [