Int. Comm. H e a t M a s s Transfer, Vol. 23, No. 2, pp.

23%248, 1996
Copyright © 1996 Elsevier Science Ltd
Pergamon Printed in the USA. All rights reserved
0735-1933/96 $12.00 + .00

PII S0735-1933(96)00009-7

HEAT TRANSFER WITH VARIABLE CONDUCTIVITY IN A

STAGNATION-POINT FLOW TOWARDS A STRETCHING SHEET

T. C. Chiam
Division of Information Engineering
School of Electrical & Electronic Engineering
Nanyang Technological University
Nanyang Avenue, Singapore 2263

(Communicated by J.P. Hartnett and W.J. Minkowycz)

ABSTRACT
This paper considers the heat transfer in a stagnation-point fluid
flow over a flat sheet stretching with a linear velocity. The thermal
conductivity is assumed to vary linearly with temperature. Both
constant and variable wall temperature distributions are studied and
two similarity equations are obtained. The first is then solved by
using the regular perturbation technique and closed form analytical
solutions are obtained up to second order. Both equations are also
solved numerically by using a shooting method.

Introduction

In [1], Arunachalam and Rajappa studied tile steady state laminar thermal
boundary layer in liquid metals with thermal conductivity which was assumed to
vary linearly with temperature. Drawing upon earlier work, they approximated
tile velocity components in tile energy equation by those of the inviscid outer flow
in order to simpli~' the equation. Further, by taking tile potential velocity to be
a power of distance along an isothermal stationary wall, they derived" a non-linear
ordinary differential equation which had also been obtained earlier by Yang [2] while
studying the transient conduction in a semi infinite solid with variable thermal
conductivity. Arunachalam and Rajappa then employed the regular perturbation

239
 240 T.C. Chiam Vol. 23, No. 2

technique and obtained closed form analytical solutions, up to second order, to the
resulting linearized equations. Their work was further extended by G o v i n d a r a j u l u
and T h a n g a r a j [3] who showed that similarity solutions exist for arbitrary outer
flow and non-isothermal walls.
It is the purpose of this work to study a similar problem for a flow impinging
normally on a stretching sheet. This flow is a combination of two well-known exact
solutions of the Navier Stokes equations, i.e. the two-dimensional stagnation point
flow first studied by Hiemenz and the two-dimensional flow caused solely by a
linearly stretching sheet in an otherwise quiescent incompressible fluid (Crane [4]).
The latter flow has a very simple closed fbrm exponential solution. A combination
of these two problems also yields an exact solution of tile Navier Stokes equation
(see Chiam [5]). In fact. the solution turns out to be exactly the same as that of
the inviscid flow.
This simple exact solution will be used in deriving non-linear similarity equa-
tions from the energy equation as was done in [1] and [3]. It should be noted that in
this case we need not approximate the velocity components in the energy equation
as was necessary in [1] and [3]. Closed form p e r t m b a t i o n solutions and mmmrical
solutions will both be presented.

Formulation

Tile physical situation is that of a viscous fluid impinging normally on a flat

sheet which is being stretched so that the speed at any point on the she('t is propor-
tional to its distance from the origin. The Navier-Stokes equations for this steady
two-dimensional flow are
Ou Ov
o~ + ~ : o, (1)

"~ + ~'o~ : " ,,,o,~.~ + ~ / ' (2)

where, u, c are the flow velocities in the .r and g-directions respectively, and u is
the kinematic viscosity.
The appropriate b o u n d a r y conditions are

,,(~:) = c~.. ,, = o ~t :j = o. (3)

 Vol. 23, No. 2 STAGNATION-POINT FLOW TOWARDS A SHEET 241

u-+0, as y --+ oc. (4)

where c is a c o n s t a n t .
T h e s t r e a m f u n c t i o n for the inviscid flow far from the sheet is 'IJ = a x y . Near
the sheet, we a s s u m e t h a t the flow field is given by the s t r e a m f u n c t i o n

~, = ~ x f ( r / ) ,

where 71 = V ~ o / ~ g , a n d a is a c o n s t a n t . T h e velocity c o m p o n e n t s are t h e n

u = ( a / v ) z f ' (rl) v = - ax~f( r1)

S u b s t i t u t i n g into (2) gives

f,,, + f f , , _ ( f , ) 2 + 1 = O. (5)

which is the well-known H i e m e n z ' s e q u a t i o n . As the sheet is a s s u m e d to stretch

with a velocity p r o p o r t i o n a l to the d i s t a n c e from the origin, we let u = ( a / l / ) z .
Hence the b o u n d a r y c o n d i t i o n on f ' at the sheet is i f ( O ) = 1 i n s t e a d of the u s u a l
c o n d i t i o n f ' ( 0 ) - 0 for a s t a t i o n a r y sheet. Tile a p p r o p r i a t e b o u n d a r y c o n d i t i o n s
for (5) are t h u s
f(O) = O, f'(O) = 1, f'(:~c) = 1. (6)

Hence the e q u a t i o n s governing tile s t a g n a t i o n - p o i n t flow towards a s t r e t c h i n g sheet

are (5) a n d (6). T h e a n a l y t i c a l s o l u t i o n is easily verified to be

f(u) = ~. (7)

T h e velocity c o m p o n e n t s b e c o m e

. = (~/~,>, ~, = ,/2/,,,~. (s)

In the absence of viscous d i s s i p a t i o n a n d heat generation, the e n e r g y e q u a t i o n

for the above t w o - d i m e n s i o n a l flow m a y be w r i t t e n

( or o (:cor
pc,, " Z + 5 7 . / = N \ o~/' (9)
242 T.C. Chiam Vol. 23, No. 2

subject to the b o u n d a r y c o n d i t i o n s

T(0) =T~,, T(oc) -: T ~ , (10)

where Tw is the wall t e m p e r a t u r e , T ~ is the ( c o n s t a n t ) t e m p e r a t u r e of the fluid

far from the sheet, Cp is the specific heat capacity a n d A" the t h e r m a l c o n d u c t i v i t y
which is a s s u m e d to be variable here.
For liquid metals, it has b e e n fi)und t h a t the t h e r m a l c o n d u c t i v i t y I f varies
with t e m p e r a t u r e in a n a p p r o x i m a t e l y linear m a n n e r in the r a n g e from 0 ° F to
4 0 0 ° F (See K a y s [6]). As in [1], we a s s u m e t h a t I f takes the form

IC=I(~(1 + cO), (11)

where
T - Too IC,,, I(o~
O -- and e -- (12)

a n d e is a small p a r a m e t e r .
We shall first a s s u m e t h a t the wall t e m p e r a t u r e d i s t r i b u t i o n Tw varies as a
power of the d i s t a n c e z:
T~ = T~ + A z ' , 13)

where r is a c o n s t a n t .
O n u s i n g (8) a n d (13), the energy e q u a t i o n (9) can be w r i t t e n

re- VTrl~r/ : k, pCp] ~ (1 + drlj .

If we set r = 0 so t h a t the wall t e m p e r a t u r e is c o n s t a n t , the above e q u a t i o n simplifies

to
(1 + eS)e" + cr',le' + e(O') 2 -: O, (15)

which is to be solved s u b j e c t to

O(0) : 1, O(~) = 0. (16)

where (7 : pCp/IC~ is the reciprocal of the t h e r m a l diffusivity.

O n the other h a n d , if we take

r =
Vol. 23, No. 2 STAGNATION-POINT FLOW TOWARDS A SHEET 243

equation (14) yields

(1 + e0)0" + err/0' + e(0') 2 - G0 = 0. (17)

It should be noted that the parameter cr can actually be scaled out of (15) and (17)
by introducing the independent variable ~ = v/Tr?. These equations then become,
respectively,
(1 + e0)0" + ~0' + e(0') 2 = 0, (18)

(1 + e0)0" + ~0' + e(0') 2 - 0 = 0, (19)

where O' = dO/d~. Equation (15) is almost identical to that obtained by Arunacha-
lam and Rajappa [1]. The only difference is that the parameter ~ is replaced by
2. However, the physical interpretation of this coefficient in [1] is different from the
case here. On the other hand, (17) or (19) is a special case of the following equation
derived by Govindarajulu and Thangaraj [3]:

(1 + ~H)H" + CI~H' + ~(H') 2 - C2H = O, (20)

where C1 and C2 are related to the external flow velocity and the wall temperature
distribution respectively. In [3], the authors only presented numerical solutions for
the special case C1 = 2 and C2 = 0 for comparison with results in [1]. As can be
seen from (20), it is possible that other physical situations may lead to different
values for ~, i.e., C1. Hence, we shall retain tile parameter ~r in (15) and use it to
derive analytical solutions which might also be useful in other related problems.

Method of Solution

Equation (15) and (17) subject to (16) can be readily integrated by using
a shooting method employing the standard Runge-Kutta algorithms with Newton
iteration to search for the missing slope 0'(0). All computations were done in double
precision on a 486 PC and convergence was readily obtained in all the cases tested.
The results are presented in the next section.
As in [1], we shall also use the regular perturbation method to derive a sequence
of linear differential equations from (15) and obtain closed form analytical solutions
up to second order. "~Velet

0 = 00 + e01 + e202 + ea0a + -.-. (21)

244 T.C. Chiam Vol. 23, No. 2

Substituting (21) into equation (15) and (16), and equating terms with tim same
powers of e, we obtain the following sequence of b o u n d a r y value problems for 0o,
01, 02, and 0a:
0;' + ~ 0 o = 0, (22)

0o(0) = 1, 00(~x;) = 0; (23)

0'~'+ ~,70'~ = -0o0;' -(0;)~, (24)

01(0) : 0, 01(OC) = 0: (25)

O; t Jr- GT]O; = --O00tl t -- 2 0 ; 0 ' 1 -- 00'01, (26)

0~(0) = 0, 0~(~) = o; (27)

03. -L. 0-?703

. . =. __0002 -- 0101/ -- 0 0tt0 2 -- 9At a t -- (0'1) 2,
--~0~2 (2S)

03(0) = 0, 03(oc) = 0. (29)

A r u n a c h a l a m and E/ajappa has o b t a i n e d the closed form analytical solutions of

(22) (27) but with cr = 2. \Ve use the same procedure as in their paper to obtain
the following solutions for 0o, 01, and 02:

0o(,~) = i - e r f ( X / ~ / 2 , 1 ) , (30)

1 02 1,?0o0o- 10o2, (31)

0,~(,/)= - +~+Ts Oo- ~+ °~+2° s~

+ wo4 - 712 Oo 2 4 - 4 + 7?0o - (9r I - arl 3)0

(Y 7TG

4~

where erf and erfc are the error function and the c o m p l i m e n t a r y error function
respectively. Tile corresponding wall t e m p e r a t u r e gradients are

0~(0) = _ 2~, (33)

Vol. 23, No. 2 STAGNATION-POINT FLOW TOWARDS A SHEET 245

0,,0, (11)
4~ + ;~ . (35)

We m a y recover the solutions of A r u n a c h a l a m and R a j a p p a by setting (7 = 2. To get

the corresponding solutions for (18), we simply set cr = 1 and replace 7? by (. The
third and higher order equations were not solved this way because the complexity
of the m a t h e m a t i c a l m a n i p u l a t i o n s increased very r a p i d l y and obtaining 02 already
entailed a long and tedious process.
The p e r t u r b a t i o n b o u n d a r y value problems (22) (29) are all linear and were
also solved numerically by using superposition of solutions. These results, other
t h a n those from (28) and (29), were also checked by numerically evaluating the
analytical solutions (30) -(32). Excellent agreements were obtained.

Results and Discussion

The numerical results are s u m m a r i z e d in Tables 1 and 2, and Figures 1-3.

Table 1 shows the wall t e m p e r a t u r e gradients when r = 0 for a range of values of
the p a r a m e t e r e. Column 2 lists the values of -0'(0) o b t a i n e d by direct numerical
integration of (15) with a = 1. These m a y be c o m p a r e d with corresponding values
in column 3 c o m p u t e d from the p e r t u r b a t i o n equations (22)-(29) and (21). These
values have been checked by using (33)-(35). It can be seen that the p e r t u r b a t i o n
solutions up to third order give very good results especially when e is small in
magnitude. The general t r e n d is that increasing the values of e from - 0 . 5 to 0.5
causes - 0 ' ( 0 ) to decrease steadily.
Plots of the p e r t u r b a t i o n t e m p e r a t u r e profiles Oi, (i = 0, 1,2, 3) are shown in
Figure 1. It is clear that the m a g n i t u d e s of successive profiles decrease fairly quickly
indicating the r a p i d convergence of series (21). Figure 2 gives the t e m p e r a t u r e
profiles calculated from (18) and (16) for e = - 0 . 5 , - 0 . 1 , 0, 0.1 and 0.5. The decrease
in t e m p e r a t u r e with ~ becomes less sharp as e increases. This was already shown
by the t r e n d of wall t e m p e r a t u r e gradient values presented in Table 1.
246 T.C. Chiam Vol. 23, No. 2

TABLE 1
Comparison of the \Vail T e m p e r a t u r e Gradients for a = 0.1, r 0.
o;(o) = -0.7978850, 0~(0) = 0.5439100
o~(o) = -0.5003170, 01(0 ) = 0.4848741

-0.5 1.314964 1.255529

0.4 1.146846 1.126532
0.3 1.024693 1.019178
-0.2 0.9315125 0.9305587
-0.1 0.8578167 0.8577641
0.05 0.8263946 0.8263919
-0.01 0.8033742 0.8033746

0.0 0.7978850 0.7978850

0.01 0.7924955 0.7924955

0.05 0.7718821 0.7718797
0.1 0.7480553 0.7480123
0.2 0.7058743 0.7052367
0.3 0.6696320 0.6666490
0.4 0.6380998 0.6293398
0.5 0.6103712 0.5904000

TABLE 2
\Vail T e m p e r a t u r e Gradients for \:ariable T~.

0'(o) -e'(o)
-0.5 2.057411 0.0 1.253314
-0.4 1.796036 0.01 1.244927
-0.3 1.606109 0.05 1.212845
-0.2 1.461201 0.1 1.175756
-0.1 1.346566 0.2 1.110079
-0.05 1.297678 0.3 1.053628
-0.01 1.261857 0.4 1.004495
0.5 0.961272
Vol. 23, N o . 2 STAGNATION-POINT FLOW TOWARDS A SHEET 247

1.0

O0
0.8
01
02
0.6
03
-~o.4

0.2

0.0 , , ~.l~:~- . . . . . . . --

-0.2
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

FIG. 1

Plots of the Perturbation Temperature Profiles

1.0

~X,\ e = - 0.5
0.8 \t~ e = - O. 1
\~,,~ - ~ = o.o

0.6
'":,,,:,,,\ - e = 0.5
'L,X~
0.4
~,,\
\ "%).
0.2

0.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0

FIG. 2

Temperature P r o f i l e s for c a s e o f r = 0 f o r a r a n g e o f e.
248 T.C. Chiam Vol. 23, No. 2

Table 2 gives the values of -O'(0) o b t a i n e d from a direct numerical solution of

(19) for the same range of values of e as in Table 1. These values are signifieantly
larger t h a n corresponding vahms in Table 1 and follow the same trend of decrease
as e increases. T e m p e r a t u r e profiles for this ease are presented in Fig. 3 for several
values of e. Their behavior is similar to those in Fig. 2.

1.0

i~ e = - 0.5
0.8 \kX e = - 0.1
--'~\"~\ e = 0.0
\>, = o.q
p,,o.6~ ~),,:,\ X e = 0.5

o.4

FIG. 3
T e m p e r a t u r e Profiles for Variable T,~. for a range of e.

References

1. M. A r u n a c h a l a m and N. R. R a j a p p a , Appl. Sci. Res., 34, 179 (1978).

2. K . T . Yang, Trans. ASME. J. Appl. Mech., 2~, 146 (1958).
3. T. G o v i n d a r a j u l u and C. J. T h a n g a r a j , Zeit. angew. Math. Mech., 67, 657
(1987).
4. L . J . Crane, Zeit. angew. Math. Phys.. 21, 645 (1970).
5. T . C . Chiam, J. Phys. Soe. J a p a n . . 63, 2443 (1994).
6. \V. M. Ka3's, Convective Heat arm Mass Transfer, McGraw Hill, p. 362
(1966).
Received July 5, 1995