Heat transfer from a stretching surface

1128

Technical Notes

Nonlinear Boumhtrv I"ahw Prohh,ms. American Elsevier, New York (1965). 9. R. S. Varga. Matrix lteratire Anall'sL~', p. 194. PrenticeHall, 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).

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 ofquasilinearization applied to boundary layer equations. AIAA J. 12, 558 560(1974). 8. R. E Bellman and R. E, Kalaba, Quasilmearization and

Int. J. lit,lit :lla,~t l'r, msh'r. Printed in Great Britain

Vol. 3(~. N o 4, pp. 1128 1131. 1993

0(117 93 I(} 93 $6.00 + 0.00

1993 Pergamon Press Lid

(

Heat transfer from a stretching surface Nooa

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 arbitrarily 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.

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 pressure flow in usual notations are as follows : u~+r, = 0

(I)

uu, +cu,. = vu,,

(2)

uT,+vT, =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 : u = U(x),

y=0.

.t'/6---, :~,

c=O,

u~O,

T= T,(x)

T--, T . .

(4) (5)

Introducing the similarity variables V.l'

= ,%/(2,~)/(q),

Ux

'1 = ,%/(2c.')" ¢ = ,,(m~ IV

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

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

U = L:,,x",

-

T~ = T~ + C x " ,

[}-

2m 1+ m

U(x) _- Uox m Tw(x) = T=o+ Cx n

T==

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

(6)

Technical Notes the m o m e n t u m b o u n d a r y layer equations reduce to f" + ff"-

.I(0) = o,

flf"

where F(a, b, c, X) is the hypergeometric function and F(-) the complete g a m m a function [15]. F u r t h e r constants a and h from (16) are given by

= 0

f ' ( 0 ) = l,

f'(,zo) = 0

(7)

and thermal b o u n d a r y layer equations to a-IO"+fO'-n(2-fl).f'O 0(0) = 1,

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

0'(0) = D. (8)

Equations (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 equations (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 equations (7) with reference to the original work [2].

0 = (1 - X ) ° = sech 2° 0//`]2)

Z = ,11~,

~: = (6//1) ~ •

(9)

the b o u n d a r y layer equations (7) become

(21)

and if n = 0 then 0 = l-B.~(~,a)/Bl(~,a) =

(-)

_

It,-)

V(~)

--

B,(n, m) is the incomplete beta function defined by

LAYER

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

(20)

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

where

BOUNDARY

F ( Z ) = J'(q)lr,

(19)

The temperature gradient on the sheet is

= 0

0(oo) = 0.

3. M O M E N T U M

1129

B,(m,n) =

j-,

-"-I(l---)"

Id-.

(23)

0

If cr = 1 solution (22) is further reduced to 0 = 1-`]X,

0'(0) = - 1/.]2.

(24)

(b) F o r e = I, n = [ t / ( 2 - B ) the closed form solution of energy equation (8) is 0' = f

F ' " - 6 F ~= = - e ' - F F " F(0) = 0 ,

F'(O) = I.

Using the inner expansion F = Fo+e,~-Fi + . . .

(10) 0(q) = exp

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

5

I

The velocity gradient at the wall is given by -

1 + 5/7 + ~

+-.-

.

(12)

BOUNDARY

.fdt 1 ,

0'(0) = 0.

(26)

(")/('+1

a= F ~-+l 4. T H E R M A L

-a

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

f"(O)=

(;o)

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

Z I+Z

Fo-

(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 stretching sheet having n o n - u n i f o r m surface temperatures. (c) I f n ( 2 - - f l ) = - - I the closed form solution o f ( 8 ) is

LAYER

F o r fl = 1 closed f o r m so]ution o f the thermal b o u n d a r y

F T

" K=n(2-fl).

(27)

F o r a --, 0, thermal b o u n d a r y layer thickness is of the order o f o and the heat transfer rate is

layer equations (8) has been reported [5]. The solutions for other specific values of fl are described here.

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

(a) fl = - 1 : based on the solution of m o m e n t u m boundary layer equations (7)

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

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

(13)

the thermal b o u n d a r y layer equations (8) after some m a n i p u lation can be expressed as X(I - X ) h ~ x + [ c - ( a + b + h(O) = i,

l)X]h,.-abh = 0 h(1) = 0

(14) (15)

128)

(e) F o r fl -+ oe the analysis by matched asymptotic expansions 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 - :) ,

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

~ < o

(29)

where h ( X ) = 0(q) and constants a, b and c are given by a+b = (I-2tr)/2,

ab = 3ha~2,

c = 1/2.

(16)

Equation 114) is the well-known hypergeometric equation [15] and its solution satisfying b o u n d a r y conditions (15) is h = F(a,b,~,X)+D(2X)~I'-F(½+a,~+b,~,X)

(17)

D = -`]2F(I--a)F(I

(18)

-b)/[F(~-a)F(~-b)]

4. R E S U L T S

AND

DISCUSSION

The asymptotic solution (12) of m o m e n t u m b o u n d a r y 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 transformation by recasting it

II30

Technical Notes

.......

Merkin Exact Solution

('(0) 0 Present

Work : Improved Series 130)

S~ S

112)

-2 -2

I

I

I

I

I

I

-I

0

1

2

3

4

FiG. 2. A c o m p a r i s o n of velocity gradient f " ( 0 ) at the stretching sheet from asymptotic solution at large // with available solutions: - - , improved asymptotic series (30); , Merkin's [12] series solution; - - - - - , 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+fl)s'4/"(0) approaches a constant [I I] to get

(2+#)~'"f"(°) = - \ ~ /

r,L~ - 4 0,i + 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.

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 . 6 2 7 5 and 0.0. The series solution of Merkin [12] as displayed 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 d o m a i n o f / / r a n g i n g from infinity to - 2 . Numerical solutions to thermal b o u n d a r y layer equations (8) have been obtained for various values o f / / and n pertaining to two values of Prandtl n u m b e r a = 0.72 (air) and 7 (water). The wall temperature gradient 0'(0) is displayed in Fig. 1 for o - = 0.72. F o r n > 0, or more precisely

-I

0 n:0

-I

0

n:O

-2

-1

0'(o)

if(o)! -4-

-2 -6

-8 0

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

1

2

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

Technical Notes 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 transfer 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 continuous accelerated surface, Chem. Enyn 9 Sct. 32, 15171525 (1977). 4. P. S. G u p t a 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. G r u b k a and K. M. Bobba, Heat transfer characteristics of a continuous stretching surface with variable temperature, J. Heat Transfer 107, 248-250 (1985).

1131

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, In9. 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, I M A 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~/ Mathematical Functions. Dover, New York (1970).