1128 Technical Notes

due to the combined buoyancy of heat and mass diffusion in a thermally stratified medium, J. Heat Tran,~/br II I, 657 663 (1989).

7. K. lnouye and A. Tate, Finite-difference version ofquasi- linearization applied to boundary layer equations. AIAA J. 12, 558 560(1974).

8. R. E Bellman and R. E, Kalaba, Quasilmearization and

Nonlinear Boumhtrv I"ahw Prohh,ms. American Elsevier, New York (1965).

9. R. S. Varga. Matrix lteratire Anall'sL~', p. 194. Prentice- Hall, Englewood Cliffs. New Jersey 11962).

10. S. W. Churchill and H. S. S. Chu, Correlating equations for laminar and turbulent free convection from a vertical plate, hit. J. Heat Mass Tram/or 18, 1323 1329 (1975).

Int. J. lit,lit :lla,~t l'r, msh'r. Vol. 3(~. N o 4, pp. 1128 1131. 1993 Printed in Great Britain

0(117 93 I(} 93 $6.00 + 0.00 ( 1993 Pergamon Press Lid

Heat transfer from a stretching surface

N o o a AFZAL

Department of Mechanical Engineering, Aligarh Muslim University. Aligarh 202002. India

(Received 4 April 1991 and in final Jorm I [ May 1992)

1. I N T R O D U C T I O N

THE HEAT transfer from a stretching surface is of interest in polymer extrusion processes where the object, after passing through a die, enters the fluid for cooling below a certain temperature. The rate at which such objects are cooled has an important bearing on the properties of the final product. In the cooling fluids the momentum boundary layer for linear stretching of sheet U c z x was first studied by Crane [1], whereas power law stretching U z x" was initially described by Afzal and Varshney [2].

Heat transfer from a linearly stretched surface U :c x based on the above work [1] has attracted the attention of several workers. The case of constant wall temperature has already been the subject of study [1.3]. Similarly, for a non-uniform wall temperature closed form solution in terms of special functions has also been reported [5]. The case of uniform sheet velocity (zero stretching) is also well documented [7, 8].

The present work deals with heat transfer from an arbi- trarily stretching surface U w~ x m for investigating the effects of non-uniform surface temperature. Several closed form solutions for specific values of m including their numerical solutions are presented in this technical note.

2. E Q U A T I O N S

Let a polymer sheet emerging out of a slit at origin (.v = 0) be moving with non-uniform velocity U(x) in an ambient

fluid at rest. The coordinate systems shown in Fig. 1, where coordinate x is the direction of motion of the sheet and I' is the coordinate normal to it. The u and c are velocity components in the x and y directions, respectively. Further, v is the molecular kinematic viscosity and a the Prandtl number of the fluid. The boundary layer equations of mass, momentum and energy for two-dimensional constant pres- sure flow in usual notations are as follows :

u~+r , = 0 (I)

uu, +cu,. = vu,, (2)

u T , + v T , = a 'vT,,.. (3)

The boundary conditions for the flow induced by stretching sheet (issuing from the slit x = 0) moving with non-uniform surface speed U(x) in quiescent environment are :

y = 0 . u = U(x), c = O , T = T, (x) (4)

.t'/6---, :~, u ~ O , T-- , T . . (5)

Introducing the similarity variables

V. l ' U x

= ,%/(2,~)/(q), '1 = ,%/(2c.')" ¢ = ,,(m~ IV

T = T, + ( T , , - T . ) O ( q )

2m U = L:,,x", T~ = T~ +Cx" , [ } - (6)

1 + m

T==

- U(x) _- Uox m

Tw(x) = T=o+ Cx n

FIG. 1. Coordinate system for the flow induced by a polymer sheet moving with non-uniform surface speed in an ambient fluid at rest.

Technical Notes 1129

the m o m e n t u m boundary layer equat ions reduce to

f " + f f " - f l f " = 0

.I(0) = o, f ' ( 0 ) = l, f ' ( ,zo) = 0 (7)

and thermal boundary layer equat ions to

a - I O " + f O ' - n ( 2 - f l ) . f ' O = 0

0(0) = 1, 0(oo) = 0. (8)

Equat ions (7) describing the flow on a stretching sheet were first proposed by Afzal and Varshney [2] and the solutions for - 2 < fl < 2 were also reported. The later investigations [10-13] connected with these equat ions (7) have not cited the original paper [2], while the bulk of earlier results [2] have been entirely reproduced/adopted in these papers. On the other hand Aziz and Na [14] in their book referred to equat ions (7) with reference to the original work [2].

3. M O M E N T U M B O U N D A R Y L A Y E R

When [1 is large the solution can be obtained by the method of matched asymptotic expansions [9]. In the outer layer (r/ fixed fl --* ~,~) equat ions (7) show that to order I/fl the func- tion f i s constant , which fails to satisfy the no slip condition. In the inner variables [9]

F ( Z ) = J'(q)lr, Z = ,11~, ~: = (6//1) ~ • (9)

the boundary layer equat ions (7) become

F ' " - 6 F ~= = -e ' -FF"

F(0) = 0 , F'(O) = I.

Using the inner expansion

F = Fo+e,~-Fi + . . . (10)

the solution to equat ions for F,, and F~ which matches with the outer solution is

Z F o -

I + Z

5 I

The velocity gradient at the wall is given by

f " ( O ) = - 1 + 5/7 + ~ + - . - . (12)

4. T H E R M A L B O U N D A R Y L A Y E R

For fl = 1 closed fo rm so]ut ion o f the thermal bounda ry layer equat ions (8) has been reported [5]. The solutions for other specific values of fl are described here.

(a) fl = - 1 : based on the solution of m o m e n t u m bound- ary layer equat ions (7)

.f(q) = (2X) 1/2, X = tanh=(q/`]2) (13)

the thermal boundary layer equat ions (8) after some manipu- lation can be expressed as

X(I - X ) h ~ x + [ c - ( a + b + l )X]h , . -abh = 0 (14)

h(O) = i, h(1) = 0 (15)

where h(X) = 0(q) and constants a, b and c are given by

a + b = ( I - 2 t r ) / 2 , ab = 3ha~2, c = 1/2. (16)

Equat ion 114) is the well-known hypergeometric equation [15] and its solution satisfying boundary condit ions (15) is

h = F ( a , b , ~ , X ) + D ( 2 X ) ~ I ' - F ( ½ + a , ~ + b , ~ , X ) (17)

D = - ` ] 2 F ( I - - a ) F ( I - b ) / [ F ( ~ - a ) F ( ~ - b ) ] (18)

where F(a, b, c, X) is the hypergeometric function and F(-) the complete gamma function [15]. Fur ther constants a and h from (16) are given by

(a,b) = [ l - 2 t rT [ ( I -2 t y )2 -2 4 n t y ]~" - ] /4 . (19)

The temperature gradient on the sheet is

0'(0) = D. (20)

Solution (17) for specific values o f n can be expressed into a simple function. I f n = - 1 it becomes

0 = (1 - X ) ° = sech 2° 0//`]2) (21)

and if n = 0 then

0 = l -B.~(~ ,a) /Bl (~ ,a)

= _ ( - ) I t , - ) V(~) - -

w h e r e B,(n, m) is the incomplete beta funct ion defined by j-, B,(m,n) = - " - I ( l - - - ) " Id- . (23)

0

If cr = 1 solution (22) is further reduced to

0 = 1 - ` ] X , 0'(0) = - 1/.]2. (24)

(b) For e = I, n = [ t / (2 -B) the closed form solution of energy equation (8) is

0' = f (25)

which shows that classical Reynolds analogy between m o m e n t u m and heat transfer is also valid for flow on stretch- ing sheet having non-uni form surface temperatures.

(c) I fn (2- - f l ) = - - I the closed form solution o f (8 ) is

( ;o) 0(q) = exp - a . f d t 1 , 0'(0) = 0. (26)

(d) For general values of fl and n solutions of energy equation (8), for limiting Prandtl numbers have been obtained by the method of matched asymptotic expansions. The algebra is complicated and for brevity, only final results are given here. Fo r a ---, ~ thermal boundary layer thickness is of the order of a ~ 2 a n d h e a t transfer rate is given by

0'(0) = - \/(2a):~ +f"(O)[(2K+ 1 )~x-" - 2Kzq/3 + " "

where

(")/('+1 a = F ~ - + l F T " K = n ( 2 - f l ) . (27)

For a --, 0, thermal boundary layer thickness is of the order o f o and the heat transfer rate is

0'(0) = - ( l + K ) . f ( m ) o - + ( l +K)~-[KJ,_--f(m)J.]a~-+ . . .

J,,, = f * [ f ( q ) - .11~.)]" dq. 128)

(e) Fo r fl -+ oe the analysis by matched asymptotic expan- sions shows that the solution exists only for n < 0 when the heat transfer rate is given by

o ' ( o ) = i ~ 5 ~ ` ] ( 6 ~ ) , 7 - + o ( [ J - :) , ~ < o

2a = ( 1 - 24ha)ii_, _ 1. (29)

4. R E S U L T S A N D D I S C U S S I O N

The asymptot ic solution (12) of m o m e n t u m boundary layer for fl --+ cc is displayed in Fig. 2. The predictions are very good even for fl = 1 where it predicts - 1.079 against exact value o f - 1. The series (12) can further be improved through conventional Euler t ransformat ion by recasting it

II30 Technical Notes

('(0) 0

-2

-2

. . . . . . . Merkin

Exact Solution

Present Work : Improved Series 130)

112)

S ~ S

I I I I I I

-I 0 1 2 3 4

FiG. 2. A compar ison of velocity gradient f " (0 ) at the stretching sheet f rom asymptot ic solution at large // with available solut ions: - - , improved asymptotic series (30); , Merkin 's [12] series solut ion;

- - - - - , exact results; , three term solution (12) for/3 ---, o':J.

in terms of variable E = I / ( f l+3 ) and noting the fact that as fl ---, - 2 , (2+f l )s '4 /"(0) approaches a constant [I I] to get

r, ,i ] (2+#)~'"f"(°) = - \ ~ / L ~ - 4 0 + 4ooo E + " (30)

As E --* 1 (~ ---* - 2), the right-hand side of (30) predicts 0.694 whereas the exact result [I 1] is 0.743. Relation (30) f o r / / = I.

-I

0

n : 0

-1

0'(o) -2

FIG. 3. Heat transfer rate on a non-isothermal stretching sheet for ~ = 0.72.

0 and - 1 predicts - 1.0134, - 0 . 6 4 9 6 and - 0 . 0 4 9 whereas correspondingly exact results are represented by - 1 . 0 , -0 .6275 and 0.0. The series solution of Merkin [12] as dis- played in Fig. 3 holds good for - 1.5 < /3 < 2 and fails when // becomes larger than 2 or approaches - 2 . In conclusior,. relation (30) predicts f ' ( 0 ) in the entire domain o f / / r a n g i n g from infinity to - 2 .

Numerical solutions to thermal boundary layer equat ions (8) have been obtained for various values o f / / and n per- taining to two values of Prandtl number a = 0.72 (air) and 7 (water). The wall temperature gradient 0'(0) is displayed in Fig. 1 for o -= 0.72. For n > 0, or more precisely

-I 0

n:O -2

if(o) ! -4 -

-6

-8

0 1 2

FIG. 4. Heat transfer rate on a non-isotherma] stretching sheet for a = 7_0.

Technical Notes 1131

n(2- /3 ) > 0 the temperature gradient 0'(0) is negative when the heat flows from a surface to the ambient fluid. For a fixed value of n the magnitude of heat transfer rate decreases as/3 increases. For a fixed ~ the magnitude of heat transfer increases as n increases. For o" = 7 (water), the numerical solutions are displayed in Fig. 4. For n = - I the heat trans- fer rate changes sign through an infinite discontinuity, for a = 7 it occurs around p = 0. This behaviour is also shown by closed form solutions, but such a behaviour is physically unrealistic and corresponds to breakdown of boundary layer theory. The closed form solutions presented in Section 3 agree with the derived numerical solutions.

REFERENCES

I. L. J. Crane, Flow past a stretching plate, Z A M P 21, 645-647 (1970).

2. N. Afzal and I. S. Varshney, The cooling of a low heat resistance stretching sheet moving through a fluid, Wiirme- und Stoffiibertra 9. 14, 289-293 (1980). See also rejoinder, Wiirme- und Stoffiibertrag. 17, 217-219 (1983).

3. J. Vleggaar, Laminar boundary layer behaviour on con- t inuous accelerated surface, Chem. Enyn 9 Sct. 32, 1517- 1525 (1977).

4. P. S. Gupta and A. S. Gupta, Heat and mass transfer on stretching sheet with suction and blowing, Can. J. Chem. Engng 55, 744-746 (1977).

5. L.J . Grubka and K. M. Bobba, Heat transfer character- istics of a cont inuous stretching surface with variable temperature, J. Heat Transfer 107, 248-250 (1985).

6. F. K. Tsou, E. M. Sparrow and R. J. Goldstein, Flow and heat transfer in boundary layer on a continuous moving surface, Int. J. Heat Mass Transfer 10, 219 (1967).

7. L. J. Crane, Heat transfer on continuous solid surface, In 9. Arch. 43, 5203 5214 (1974).

8. V. M. Soundalgekar and T. V. R. Murthy, Heat transfer in flow past a continuous moving plate with variable temperature, Wiirme- und Stoff~ihertraf¢. 14, 9I 93 (1980).

9. N. Afzal and V. K. Luthra, Highly accelerated boundary layer at moderately large Reynolds number, AIAA J. 12, 529 532 (1974).

10. H. K. Kuikin, On boundary layer in fluid mechanics that decay algebraically along stretches of wall that are not vanishingly small, IMA J. Appl. Math. 27, 387~405 (1981).

I 1. W. H. H. Banks. Similarity solutions of the boundary layer equations for stretching wall, J. Mec. Theo. Appl. 2, 375-392 (1983).

12. J. H. Merkin, A note on the solution of differential equation arising in boundary layer theory, J. Enyn 9 Math. 18, 31-36 (1984).

13. W. H. H. Banks and M. B. Zaturska, On eigenvalue problem for boundary layer on a stretching sheet. IMA J. AppL Math. 36, 263-273 (1986).

14. A. Aziz and T. Y. Na. Perturbation Methods in Heat Trans/~,r. p. 182. Hemisphere, New York (1983).

15. M. Abramowitz and 1. A. Stegun, Handbook c~/ Math- ematical Functions. Dover, New York (1970).