Acta Physica Academiae Scientiarum Hungaricae, Tomus 45 (1), pp. 81--86 (1978)
APPLICATION OF G.P.D.P. TO T H E R M A L B O U N D A R Y LAYER By P. SINGH*
and D. K.
BHATTACHARYA
DEPARTMENT OF MATHEMATICS, INDIAN INSTITUTE OF TECHNOLOGY, KHAKAGPUR, INDIA
(Received 4. VII. 1978)
The governing principle of dissipative processes is applied to study the steady state thermal boundary layer along a semi-infinite flat plate when plate temperature differs from that of free stream. The thermal boundary--layer thickness is obtained using a third degree profile. It is found that the tate of heat transfer from the plate to the fluid computed for various Prandtl numbers using present variational method is quite close to the already known exact results. The results of the present investigation are fairly better than those of well-known Karman-Pohlhausen solution.
Introduction I n the p r e s e n t i n v e s t i g a t i o n our m a i n airo is to s t u d y t h e a p p l i c a b i l i t y of t h e g o v e r n i n g principle of dissipative processes to t h e r m a l b o u n d a r y l a y e r along a semi-infinite f l a t p l a t e w h e n ah incompressible viscous fluid flows over it. T h e t e m p e r a t u r e of the p l a t e Tw is u n i f o r m a n d it differs f r o m free s t r e a m t e m p e r a t u r e T w (T~ > Tw). I t is well k n o w n t h a t in the f o r m u l a t i o n of GYARMATI'S v a r i a t i o n a l principle the various b a l a n c e equations p l a y a basic role [1, 2]. I n the p r e s e n t case the e n e r g y b a l a n c e w i t h c o n v e c t i v e t e r m is r e q u i r e d to s t u d y the t e m p e r a t u r e d i s t ¡ inside t h e b o u n d a r y l a y e r if the viscous b o u n d a r y l a y e r is a l r e a d y known. I f T denotes the t e m p e r a t u r e the e n e r g y b a l a n c e e q u a t i o n w i t h o u t viscous dissipation is
0%
~T
A- 0% " ( v T ) -4- VIq = 0,
(1)
Ot
where v denotes t h e v e l o c i t y r e c t o r a n d [q is the h e a t c u r r e n t density. ~ and ct ate d e n s i t y a n d the specific h e a t of the fluid, r e s p e c t i v e l y . W e shall f o r m u l a t e GYARMATI'S v a r i a t i o n a l principle in t h e e n e r g y picture. T h e energy dissipation for t h e s y s t e m in the e n e r g y p i e t u r e is [2, 3] Tg :
--Iq 9 vlnT.
(2)
* Present address: Department of Mathematics, Indian Institute of Technology, Kanpur, India. 6
Acta Physica Academiae Scientiarum ttungaricae 45, 1978
82
P. SINGH and D. K. BHATTACHARYA
Here the state variable In T is used for T. The linear constitutive relation in energy pieture is given b y v(ln T ) , (3)
Iq = --L~
where" L~ is the phenomenologieal coefficient and represents the conductivity in the linear Onsager theory. In the present formulation L ~ is given b y
Lxx= KT,
(4)
where K is the coefficient of thermal conductivity of the fluid. The dissipation potentials can now be defined in the energy picture as ~* = 1L~~(V In T) ~ , 2
O* ---- ~ I~. 2L~
(5)
Finally, the CrYARMATI'S prineiple in energT picture 6 fv (Ta -- ~* -- O*)dV = 0
(6)
takes the following form for the system under consideratiou
(~fv[_iq.vlnT_ L~~, (vlnT) ~ 2
l~I~]dV=O,
2 Lxx
(7)
where d V denotes the element of the volume V of the system.
Solution o f t w o dimensional thermal boundary layer Consider the two dimensional steady flow of an incompressible fluid along a flat plate. The essence of boundary layer theory is to presume that irreversible processes of heat and momentum transfer ate confined in very thin layer adjacent to the plate. If x measures the distance along the plate from x ~ 0 to x ~ c~ and y normal to it and u and v are velocity components along x and y directions respectively, the EcI. (1) reduces to
( 0r ~v0Tl This equation describes the steady state temperature distribution inside the boundary layer region. The constitutive equation in this particular case has the forro 0 lnT Iql : --Lx~ ~ (9)
0y
.r
Physir Academiae Scientiarum Hungarir
45, 1978
APPLICATION OF G.P.D.P. TO THERMAL BOUNDARY LAYER
in the energy picture. The p¡
83
(7) takes now the simple form
Oy
2 t~)
2L~ ~i. dxdy=
O,
(I0)
d T denotes the thermal b o u n d a r y layer thickness. The prineiple (10) eontains two unknown Iql, and T which are eonnected b y the exaet constitutive relation (9). In the dual field m e t h o d we assume the t h e r m o d y n a m i e eurrent Iql in t e r m of ah approximate eonstitutive relation where
[4, 5, 61 0 In T*
Iql = - - ~ - - , oy
(II)
where T* is ah approximate t e m p e r a t u r e field and satisfies the same conditions as T. In the exact theory T = T* and the Lagrangian density is zero. Introducing (11) in (8) and (10), we get oT u--+
ox
fo'f£
[ oT* oT Oy
L~YY
v
oT
oy
o~T * = ~--,
of
1t~~/~
1/~~*/~l~~~z=0,
~tWyj Y ~ oyjj
(12)
(13)
where ~ denotes the thermal diffusivity of fluid. Using similarity transformations r/= y
u=
, u~f(~),
~x 1/-
1 ]/ vu| v=-2 V x
(~f'--f)
(la)
the equation (12) and principle (13) yields
d2T_____~_ * Pr
dT
= 0,
(15)
d~ 2 -4- -~" f d~
~o
[dT dT" J0
1 (dTle
1 (dT']~]
2 [-~-~ ) -- -2- [-~--~ ) J d~ ---= 0,
(16)
Pr is P r a n d t l number. To determine T* we need the veloeity profile inside the b o u n d a r y layer over a flat plate. We have already obtained a variational solution of viseous
where
6*
Acta Physica Academiae Scientiarum Hungaricae 45, 1978
84
P. SINGH and D. K. BHATTACHARYA
boundary layer with the help of GYARMATI'S principle (see SI~GH and BHATTACHARYA [7]). The third degree polynomial for veloeity profile is u =f'--
37
1 Ta
u=
2d
2 da
(17)
where d denotes the viseous boundary layer thiekness, h s value is d = 4 9 696.
(18)
We shall use the expression (17) in the ealeulation of thermal boundary layer thiekness from the present variational formulation. To determine d T the thermal boundary layer thiekness we assume the following third degree polynomial for T--T= _ 1 T w - - T~
3 7 q_ 1 7 3 2 dT 2 d~r
(19)
whieh satisfies the boundary conditions
at
~---- 0,
d2T
T = Tw,
--0~
d7 ~
7=dT,
dT
T = T=,
(20)
--0.
d7 In (19) d r is the variational parameter whieh is to be determined from the prineiple (16). From Eq. (15) with (17) and (19) we get dT* 3 PI1 77 d-----~-- 16- d T l.14 d3d~r
375 5dd~.
1 ?~5 9~3 1 d~ 10 d 3 + d -1- 35 d 3
2 d~] -5- --d-j ' (21)
whieh satisfies the condition dT*/drI = 0 at the edge of thermal boundary layer. Substitution of (21) and (19) in prineiple (16) and partial integration w.r. to 7 gives
~o~[/ " dx Pr
1 1 7 - 1 8 d~ _ _ 9 . 3 7
d
{ dT" - - P r2 0" 018 d 8 - - 0 . 4 5 7
dT, d4 9 1
Acta Physica Academiae Scientiarura ttungaricae 45, 1978
.~) 120o d3 dT dT')]
84 d 2
=0.
(22)
APPLICATION
OF G.P.D.P. TO THERMAL
BOUNDARY
LAYER
85
The E u l e r - - L a g r a n g e ' s e q u a t i o n of (22) is A1 ~
s§
6-{-
11 9 475
65 9 487
/15
-
•r
-
A3
Pr
15 9 179
P~r
0,
(23)
where
A - - dT d E q u a t i o n (23) is solved using the N e w t o n - - R a p h s o n m e t h o d to get its roots. T h e v a l u e of t h e r m a l b o u n d a r y layer thickuess depends on P r a n d t l n u m b e r Pr" W e can now calculate the h e a t t r a n s f e r at the plate which is ah i m p o r t a n t physical characteristic of the problem. The nondimensional heat t r a n s f e r o b t a i n e d with the help of the f o r m u l a
q=
KfOTI t--~-Y] y = 0
is
q* = - - K I
vx
~
~7= 0.
This a p p r o x i m a t e value of q* is quite close to e x a c t values [8] as can be seeu f r o m T a b l e I. T h e heat t r a u s f e r o b t a i n e d with the help of K a r m a n - P o h l h a u s e u t e c h n i q u e is also c o m p a r e d with the present values and it is fouud t h a t the preseut m e t h o d which is based on souud p h y s i c a l r e a l i t y can be used as a p p r o x i m a t e variational t e c h n i q u e for this t y p e of problems. Table I
Heat transfer at the plate Pr
0.6 0.8 1.0 1.1 7.0 10.0 15.0
Approximate value from G. P. D. P.
0.266 0.295 0.319 0.330 0.623 0.705 0.806
Exact value
0.276 0.307 0.332 0.344 0.645 0.730 0.835
! Karman-P omh leht ha uo sde n i'
0.266 0.295 0.319 0.330 0.621 0.701 0.803
The difference between the a p p r o x i m a t e and the e x a c t values of heat transfer at the plate is less t h a n 4 % . The results can be i m p r o v e d b y taking more v a r i a t i o n a l p a r a m e t e r s in the t e m p e r a t u r e profile and performing first variaActa Physica Academiae Scientiarum Hungaricae 45, 1978
8~
P. SINGH and D. K. BHATTACHARYA
tions of each of t h e p a r a m e t e r s i n d e p e n d e n t l y . F o r c i n g all these v a r i a t i o n s to v a n i s h simultaneously provides us a set of equations 0F
=0,
(i=l,
2,...n),
Oci
where c i are n v a r i a t i o n a l p a r a m e t e r s , These equations can be solved to o b t a i n t e m p e r a t u r e profile. I n more complicated flow field such analysis m a y be v e r y helpful in a p p r o a c h i n g the e x a c t result w h i c h is n o t k n o w n a priori.
Acknowledgement One of the authors (P.S.) is thankful to Prof. I. GYAn~L~TI,Hungarian Academy of Sciences, Budapest, for bis valuable suggestions in the Ÿ of this paper. D. K. BHATTACHARYAis thankful to C.S.I.R., New Delhi, for financial assistance.
REFERENCES 1. I. GYAllMATI,Ann. Phys., 23, 353, 1969. 2. I. GYAI~MATI,Non Equilibrium Thermodynamics Field Theory and Variational Principles, Springer, Berlin, 1970. 3. H. FARKAS,Z. Phys. Chem., 239, 124, 1968. 4. A. STARK, Ann. Phys., 27, 53, 1974. 5. P. SIrr Int. J. tteat and Mass Transfer, 19, 571, 1976. 6. P. SINCH, J. Non-Equilib. Thermodynamics, 1, 105, 1976. 7. P. SINCHand D. K. BHATTACHARYA,Application of Gyarmati Principle to Boundary Layer Flow, Acta Mech., 1977 (in press). 8. H. SCHLICHTI~G,Boundary Layer Theory, Mc-Graw Hill Co., ~qew York, 1968.
.Acta Phydea Academiae Seientiarum Hun8aricae 45, 1978
