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Boundary layer flow past a stretching plate with heat transfer and its numerical study.

Introduction

The flow past a stretching plate is of great importance in many industrial applications such as in polymer industry to draw plastic films and artificial fibers. In the process of drawing artificial fibers the polymer solution emerges from an orifice with a speed which increases from almost zero at the orifice up to a plateau value at which it remain constant. The moving fiber, which is of great technical importance, it governs the rate at which the fiber is coded and this, in turn affects the final properties of the yarn. Crane (1970) investigated boundary layer flow past a stretching sheet whose velocity is proportional to the distance from the slit. Carragher P. (1978) reconsidered the problem of Crane (1970) to study heat transfer and calculated Nusselt number for the entire range of Prandtl number Pr. In 1990, N. Ahmad et al. (1990) solved visco-elastic boundary layer flow past stretching plate and heat transfer for Walter's liquid B model. N. Ahmad and A. Mubeen (1995) investigated boundary layer flow and heat transfer for stretching plate with suction considering viscous incompressible fluid.

None of the scholars referred above tried to get numerical solution for the flow past a stretching plate with heat transfer. Hence, we are trying to solve the problem of the boundary layer flow past a stretching plate with heat transfer numerically. The solution has been compared with available analytical solution.

Formulation of the problem

Considering the two dimensional boundary layer flow, where the x-axis is along the stretching plate and y-axis is normal to it. The equations governing the boundary layer flow of viscous incompressible fluid past a stretching plate are

Continuity equation: [partial derivative]u/[partial derivative]x + [partial derivative]v/[partial derivative]y = 0 (1)

Momentum equation: u [partial derivative]u/[partial derivative]x + [partial derivative]u/[partial derivative]y = y [[partial derivative].sup.2]u/[partial derivative] [y.sup.2] (2)

where [gamma] is the kinematic viscosity. The relevant boundary conditions are

y = 0, u = mx, v = 0, m > 0 u = [infinity], u = 0, v = -C (3)

To solve the problem, we define the following dimensionless variables

y' = y/h, u' = uh/[gamma], x' = x/h, v' = vh/[gamma]

Substituting all these dimensionless variable in the equations (1) and (2), we have the following equations in dimensionless form

[partial derivative]u/[partial derivative]x + [partial derivative]v/[partial derivative]y = 0 (4)

u [partial derivative]u/[partial derivative]x + [partial derivative]u/[partial derivative]y = [[partial derivative].sup.2]u/[partial derivative] [y.sup.2] (5)

and boundary conditions are as follows

y = 0, u = mx, v = 0 y [right arrow] [infinity], u = 0, v = -C (6)

where dash has been dropped for our convenience. Setting the similarity solution of the form

U = mxf' (y) (7)

and substituting it into the equations (4) and using boundary condition (6), we have

v = -m{f(y)-f(0)} (8)

using [mu] and v in equation (5), we have

m([f'.sup.2](y)-f(y)[f.sup.w](y)) = [f.sup.m] (9)

with boundary conditions

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

where we take f(0)=0, without any loss of generality.

Our aim is to find out the value of velocity function f(y) at different points [y.sub.[??]], i = 0,1,2, ..., n, by solving non-linear differential equation (9) together with boundary conditions (10). The equation (9) and (10) constitute a non-linear boundary value problem. To get a unique solution of this problem, we have to reduce this problem into initial value problem. Also, a boundary value problem cannot be solved numerically if it is defined in an infinite domain. Therefore, for physical justification, we may redefine our problem as follows

[f.sup.m](y) = m ([f'.sup.2] (y)-f(y)f'(y)) (11)

together with boundary condition

y=0, f=0, f'=1 y=1, f'=0, f(1)=c (12)

Applying shooting method, we reduce it into initial value problem. The guess for [f.sup.m](0) has been given in the following table

Hence f'(0) = -1.23, therefore our boundary value problem is to be solved as the following initial value problem

[f.sup.[??]](y) = [f'.sup.2] (y)-f(y)[f.sup.H](y) (13)

with initial conditions

y=0, f=0, f'=1, [f.sup.w] = -1.23

Heat Transfer for stretching plate

The boundary layer equation governing the flow of heat is

U [partial derivative]T/[partial derivative]x + v [partial derivative]T/[partial derivative]y = [gamma]/Pr [[partial derivative].sup.2]T/Pr[partial derivative][y.sup.2] (14)

where T is temperature, [gamma] is the kinematic viscosity and Pr is Prandtl number. The relevant boundary conditions are:

y=0, T=[T.sub.p] y [right arrow] [infinity], T=[T.sub.S] (15)

where [T.sub.P] and [T.sub.s] are constant temperatures of plate and surrounding. Boundary conditions suggest that the temperature field varies with regard to y only so, [partial derivative]T/[partial derivative]x = 0. Therefore, the equation (14) reduces to

v [partial derivative]T/[partial derivative]y = [gamma]/Pr [[partial derivative].sup.2]T/[partial derivative][y.sup.2] (16)

To transfer the energy equation into an appropriate dimensionless form, we define

[theta](y) = T-[T.sub.5]/[T.sub.P]-[T.sub.S] (17)

so that the equation (16) reduces to

[d.sup.2][theta]/[dy.sup.2] + mPr f(y) d[theta]/dy = 0 (18)

and boundary conditions (15) reduce to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

Equation (18) and boundary conditions (19) constitute a non-linear boundary value problem. The condition y [right arrow] [infinity], [theta] = 0 may be equivalent to y [right arrow] 1, [theta] = 0, outside the boundary layer. So, again reducing this boundary value problem into the following initial value problem:

[d.sup.2][theta]/[dy.sup.2] + m Pr f(y)d[theta]/dy = 0 (20)

[theta](0) = 1, [theta]'(0) = -[[1-m[P.sub.[gamma]].sup.-1] (21)

whose numerical solution has been presented in the following table for m=1 and for Prandtl numbers Pr=0.7,1.0,5.42 and 7.0

Results and discussions

We have solved the problem "Boundary Layer flow past a Stretching Plate and Heat Transfer. Numerically, This problem has also been solved by N. Ahmad et al. (1990) for Walter's liquid B model, analytically. We compare our results with available results of N. Ahmad et al. (1990, reducing to Newtonian fluid by putting [K.sup.[??].sub.0] =0, m=1. We are discussing the results in the following paragraphs.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

* Figure 1 compares two solutions [f.sub.P], [f.sub.N] the present or numerical solution and [f.sub.N], the analytic solution obtained by N. Ahmad et al. [1]. We see that our solution coincides with solution of N. Ahmad in the immediate neighborhood of stretching plate, but our solution deviates from the exact solution obtained by N. Ahmad et al.[1] as we move away from the stretching plate. Thus, our solution is valid within boundary layer.

* Figure 2, shows the comparison between [f'.sup.P] and [f'.sub.N]. In this case, our results are as good as the results of N. Ahmad et al [1]. Hence the method is suitable to solve this type of problems if one takes care of error.

* Figure-3 exhibits the variation of temperature distribution [theta] for different variables numerically. It means that for small Prandtl numbers the flow heat becomes faster, that is Pr acts as one of controllers of heat transfer.

* Temperature gradient [theta]' has been shown in Figure 4. We see that [theta] increases Pr numerically when Pr number increases.

References

Crane, L.J., 1970. "Flow past a stretching plate", ZAMP, 21: 645

Carragher, P., 1978. "Boundary layer flow and heat transfer the stretching plate", Ph.D. Thesis, University of Dublin, Chapter, 4: 41

Ahmad, N. et.al., 1990. "Visco-elastic boundary layer flow past a stretching plate and heat transfer", ZAMP, 41: 294.

Ahmad, N. and A. Mubeen, 1995. "Boundary layer flow and heat transfer for stretching plate with suction", International Communications in Heat and Mass Transfer, 22(6): 895.

Corresponding Author: Naseem Ahmad, Department of Mathematics and Statistics, University of Maiduguri, Maiduguri, Borno State, Nigeria E-mail: <naseem_mt@yahoo.om>

(1) Naseem Ahmad, Z. U. Siddiqui and (2) M.S. Patil

(1) Department of Mathematics and Statistics, University of Maiduguri, Maiduguri, Borno State, Nigeria

(2) Department of Mathematics, PDA Engineering College, Gulbarga, India

Naseem Ahmad, Z.U.Siddiqui and M.S.Patil,: Boundary Layer Flow past a Stretching Plate with Heat Transfer and its Numerical Study, Adv. in Nat. Appl. Sci., 1(1): 15-20, 2007
Iteration Approximate of [f.sup.H] Error

1[m.sub.0] 0.6 1.83
2[m.sub.1] 0.7 1.93
3[m.sub.2] -1.2297 .0003
4[m.sub.4] -1.2308 0.0

Using Runge-Kutta method of order four, we get the results
shown in the following table:

y f [f.sup.t] [f.sup.H]

0 0 1 -1.23
0.1 0.094 0.8818 -1.1359
0.2 0.1766 0.7724 -1.0527
0.3 0.2487 0.6709 -0.9790
0.4 0.3110 0.5763 -0.9136
0.5 0.3642 0.4879 -0.8554
0.6 0.4088 0.4045 -0.8033
0.7 0.44453 0.3265 -0.7566
0.8 0.4742 0.2530 -0.7143
0.9 0.4964 0.1841 -0.6760
1.0 0.5111 0.1183 -0.6406

y Pr=0.7 Pr=1.0

 [theta] [theta]' [theta] [theta]'

0.0 1.000 -1.080 1.0 -1.124
0.2 0.7850 -1.066 0.776 -1.103
0.4 0.5750 -1.03 0.561 -1.049
0.6 0.3890 -0.983 0.359 -0.979
0.8 0.198 -0.924 0.169 -0.900
1.0 0.019 -0.904 0.020 -0.560

y Pr=5.42 Pr=7.0

 [theta] [theta]' [theta] [theta]'

0.0 1.0 -2.47 1.0 -4.3
0.2 0.7760 -2.241 0.521 -3.79
0.4 0.561 -1.714 0.123 -2.722
0.6 0.39 -1.1570 -0.163 -1.642
0.8 0.169 -0.7132 -0.374 -0.88
1.0 0.020 -0.4172 -0.236 -0.44
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Title Annotation:ORIGINAL ARTICLE
Author:Ahmad, Naseem; Siddiqui, Z.U.; Patil, M.S.
Publication:Advances in Natural and Applied Sciences
Date:Sep 1, 2007
Words:1743
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