Three-dimensional laminar free convection on a circular cylinder

J. Electroanal. Chem., 100 (1979) 673--686

673

Elsevier Sequoia S.A., Lausanne -- Printed in The Netherlands

THREE.DIMENSIONAL LAMINAR FREE CONVECTION ON A CIRCULAR CYLINDER

FRAN~OISE GENNEVIEVE, ARYADI SUWONO and MICHEL DAGUENET Laboratoire de Thermodynamique et Energdtique, Centre Universitaire, Avenue de Villeneuve, 65025 Perpignan Cedex (France)

(Received 27th November 1978) ABSTRACT The three-dimensional laminar free convection problem on an inclined circular cylinder is studied. The boundary-layer equations are written with new variables as a system of three, coupled, partial differential equations. These equations are represented by series where the inclination angle of the cylinder appears. The series are numerically solved and the velocity and temperature profiles are given for the zero-order terms.

(I) INTRODUCTION O n l y a f e w a u t h o r s have s t u d i e d t h r e e - d i m e n s i o n a l free c o n v e c t i o n . S t e w a r t [1] gives results w i t h an integral m e t h o d f o r large P r a n d t l or S c h m i d t n u m b e r s ; P e u b e a n d Blay [2] using t h e h y p o t h e s i s o f a f l o w prevailing a c c o r d i n g t o t h e s t e e p e s t a s c e n t lines c o m e t o a t w o - d i m e n s i o n a l p r o b l e m , We s h o w here, f r o m t h e circular c y l i n d e r , a m e t h o d d e r i v e d f r o m t h e t w o d i m e n s i o n a l o n e o f Saville a n d Churchill [3] a n d e x t e n d e d t o t h e t h r e e - d i m e n sional case. This m e t h o d p e r m i t s t h e r e s o l u t i o n o f t h e b o u n d a r y - l a y e r e q u a t i o n s w i t h i n t h e t h r e e - d i m e n s i o n a l l a m i n a r f r e e c o n v e c t i o n as a f u n c t i o n o f t h e angle o f i n c l i n a t i o n o f t h e c y l i n d e r w i t h r e s p e c t t o t h e vertical d i r e c t i o n . NOMENCLATURE C, F g gx, gz G

h e a t c a p a c i t y at c o n s t a n t pressure d i m e n s i o n l e s s s t r e a m f u n c t i o n d e f i n e d in eqns. (14) a c c e l e r a t i o n o f gravity gravitational a c c e l e r a t i o n c o m p o n e n t s in x a n d z d i r e c t i o n s d i m e n s i o n l e s s s t r e a m f u n c t i o n d e f i n e d in eqns. (14) Gr = g[JL3(Tp - - T~)/p2; G r a s h o f n u m b e r f o r h e a t or mass t r a n s f e r k t h e r m a l c o n d u c t i v i t y or mass d i f f u s i v i t y k g e o m e t r i c a l c o n f i g u r a t i o n f a c t o r d e f i n e d in eqns. (11) L c h a r a c t e r i s t i c length (length o f t h e c y l i n d e r ) y=o

(Tp - - T~) - local Nusselt (or S h e r w o o d ) n u m b e r o n a c y l i n d e r

of length x

674

P

X,y,Z

geometrical configuration factor defined in eqns. (18) Prandtl n u m b e r (pCp/k) or Schmidt n u m b e r (v/k) geometrical configuration factor defined in eqns. (18) radius of the cylinder geometrical configuration factor defined in eqns. (18) geometrical configuration factor defined in eqns. (11) temperature or concentration dimensionless temperature or concentration defined in eqns. (6) velocity components in x, y and z directions dimensionless velocity components in x, y and z directions defined in eqns. (6) curvilinear coordinates chosen for the cylinder (see Fig. 1) dimensionless curvilinear coordinates defined in eqns. (6)

0t

thermal diffusivity ( ~ ) o r

P~ Q ro R

T Vx ~ Uy ~ Uz Vx~ Vy, Uz x, y, z

mass diffusivity

thermal expansion coefficient supposed constant

~

p ®

angle of inclination of cylinder with respect to vertical direction integration variable defined in eqns. (14) dimensionless temperature or concentration defined in eqns. (14) rational n u m b e r dynamical viscosity kinematic viscosity density stream function defined in eqns. (13) stream function defined in eqns. (13) integration variable defined in eqns. (14) integration variable defined in eqns. (14)

7 0 X P /J

P

¢

Su bscrip ts p 0, 0 0, n j, 0 j, n

on the surface at infinity zero, zero-order terms zero, n-order terms j, zero-order terms j, n-order terms

Superscripts -

refers to adimensionalisation

(II) G O V E R N I N G E Q U A T I O N S

Consider an inclined cylinder (Fig. 1) of length L and radius r0 in an infinite newtonian fluid at rest. We study the transfer of heat or the transfer of mass from the cylinder surface

675

Fig. 1. Coordinate system chosen for the inclined circular cylinder.

of uniform, constant temperature or concentration Tp to the fluid of undisturbed temperature or concentration T~, by steady laminar free convection. All fluid properties are taken as constants except for the density charges in the b u o y a n c y term. When the b o u n d a r y layer approximation is assumed to be possible, the equations of continuity, motion and energy are respectively: aVx + avy + av~ = 0 ax ay az ~Vx

Vx ~

aVz

v x -~

(1)

aVx

av x

~Vz

~Vz

+ vy -~y + vz - ~

+ vy -~y + Vz ~

aT

aT

a2Vx

= g x ~ ( T - - Too) + v -ay - 2

~2v z

= g z f J ( T - - Too) + ~, -ay - 2

aT

a2T

(2)

(3)

(4)

ay: The boundary conditions appropriate to the problem are: y=0,

vx = v y = v z = 0 ,

T=Tp

y -~ o%

v., Vz -* 0,

T-* T~

(5)

It is convenient to write the equations in a dimensionless form using the following ratios -2 = X .

L ,

~-x-

-- vlr/4y__ .

-Y

12x

T - - Too Tp -- Too

L

;

,

-

-

_

vy

z

-5= ~

Vy

,

,

~z-

Vz

(6)

676 Application of these dimensionless ratios to eqns. (1)--(5) yields

a~+ a~+a~= a~ ay a~ 0

(7)

~ + _vy a-~-y+ T _vz a~~ =-~[~]T + aaY~ vx a-~-

(8)

a~ _ a~ a~ a ~ - + vy -~- + ~ ~-= = ~[~]T + oy o~

(9)

a~ a T + _ a T _ 1 a~T v--~~-+~-~y a~ vz a~ P, a~2

(lO)

with

~[~1 = gz; g

s-[~] = gx g

(11)

and =0,

Vx =vy = v z = 0 ,

T=I

-~ ~,

vx, Vz -~ 0,

T -, 0

(12)

(III) ANALYTICAL RESOLUTION The stream functions which satisfy identically the continuity eqn. (7) are written in terms of velocity:

V x - aa~~'.

~. v-;--ay,

~=-

( aa~ ~+ ~¢)

(13)

By introducing new variables like those used by Saville and Churchill [3] in the two-dimensional case, that is

---- : [S-(X)]1/3d~; ~'=Z; 0

[~-(z)]'3y

~(~, y, ~) F(}, 7, ~) = 4

3,4 4f)314

_ -~ q~tx, ~, ~) G(~, r/, ~') - [~( ~-)],/3(4 ~.)3/4

(14)

0(}, ~, D = T(z, y , ~ ) The set of eqns. (8)--(10) transforms to the form

F'"+ GF"---~V'f'+ ( ~ ) { F F " - - 4 P ( } ) F ' 2 ) ta(F', G) a(f',F)t = ( ~ ) ~ a-~,~ +(~o a~:,~ )

+0

(15)

677

:G'" + GG" --2-a'2 + (-~)(FG" 3~ = (_~)

_

a(G', G)

+

---~4 Q(~)F , V , } + R(~)k(~)O

(_~) a ( V , , F ) l

la(O,G)

_

p~1 0" + {V + (~')F}0' = (4~.) 3 ( ~

(16)

a(0,F)}

+ (~}) a(}, 7?)

(17)

where following notations are introduced 1 1 P(}) = 2 + 3

}

a [Y(Z)]

[Y(~)]

at

Q(~) = 2[P(~) - ~]

(18)

4~

R(}) =

3[y(~)]+,3

Primes indicate derivatives with respect to the variable ~?, and a ( , ) / 3 ( , ) denotes the Jacobian. The boundary conditions are = 0, ~7- ~ ,

F = F' = G = G' = 0,

0=1

F',G'-~0,

0-~0

(19)

It should be noted that one of the principal functions, P(~), is identical to the one that appears in the transformed equations for two-dimensional planar flows given by Saville and Churchill [3]. Accordingly, we expand P(~) in series in ~ as p(~) = ~ pj~Xj

(20)

i=0

In order to use the Goertler-type series we expand R(~) and Q(~) in the same manner R(~) = ~ Rj~XJ;

Q(~) = ~ Qi~Xj

j=O

j=o

(21)

According to the expansion of the principal functions we assume for solutions of (15)--(17) convergent series of the form F(~, 7, ~) = ~ Fj0?, ~-)~xj j=0

G(~, 77, ~) = ~

j=0

V+07, ~)~x+

c~

0(~, rh f) = ~ Oi(~, ~-)~,i i=O

(22)

Substituting eqns. (20), (21) and (22) in eqns. (15), (16) and (17) and comparing

678 the powers of ~ we obtain a sequence of sets of equations. For j -- 0 Fo'" + GoFo"

~_' ~-,' _4 ,, ~ . ~ o -~,'2~ o J + Oo = (~') a(F~,Go) _~,.,o:.o+ (3~-){FoFo __4D a (i", n)

__

2

,, _ 4 ,~, ~o Go,,, + GoGo--:G'o2+(4-~){FoG'~

o,-,o~+Rok(~')Oo

F,~,~

=

+

4 . (~.)a(Go, Go) ,7)

1 ,, 4 , ~ )3(0o, Go) ~-~0o + {Go + (-~)Fo)Oo = ( - ~ a(~,~?)

(23) (24)

(25)

and forj = 1, 2, 3 Pt ,

+ GoF~' -- 2 ~t,~o-'j , ~ , , , , + FoGj) + Fo. Gj + Oj

+ (~ ~){FoFi" + (1+ +~kj')Fo"Fj -- ~(2Po + + k '"~F' Jo~j ) _ (G,o_~+GjaF~ , aF~a~

--Fo

'+°+-~-

F~

aGo

(26)

=Lj-I

. . GoG i --.~ GoGj + Rok(~)Oj G i. . . + ' , + GoG~ ,, /

4

{

"

4 ~ "x ~,"v,,~

__4

+ (~ ~*) FoGi + (1 + ~ J Jt-o~i __ 4 ~ ~ , ~,,

G'o

- ' j Pr Oj" + GoO j' + OoG

+

--sXjFoOj-

Gi

+ (~')

(G'o

•

-- G;

FoOi'

+Gj a0o -~--

'

t

(Qo + XJ)FoGi

0o+

-- G i

= Mi_l

(27)

+ ( 1 + -~;~)O'oFj 4 • ,

- - Oo -aGj ~ - - - - Oj '

(28)

=Vj-1

with j--1 k=l

j--1

G~Fi,_..k _ ~

GkF~,__k + (43 ~) [~ {PJF~ 2 + F~) ~

k=l

j--1 k=l

j--1

Pj_kF{

k=l i--1

(Po + kk)F~Fj'---k + ~

k=l

k

Pj--k ~

i--1

(1 + ~ kk)FkFi"---k

F.~F~_~} -- ~

i=l

k=l

j--1 k=l

Gk aFi'--k a~" j--1

Mi-1

=2 ~ k=l

Fi,, k ~__k)]

(29)

j--1

G'kG;--k

-- ~ G k V j -" -k k=l

+ (~')

OuO I-~ L'c~j ~'~ F,~_,

+

679 j-1 +F;

5-I

~_J Qj_kG~

k=l

j--1

(Qo +kk)GkFs'-k

+~ k=l

k

5--1

+ ~J Qs--k ~ k=l

i=l

F'~G'k-i} -- ~ (1 + ~ k k ) F k G j ' k k=l

(

j--I

k=l

j--k-~-]_j

a~ j--I

k=l

j--1 k=l

~ Rk0j__k ) k=l

(30)

j--I

N5-1 =--~-J GkO;--k + (I~') [ ~ k ~ --~

--k(f)(Rs0o-I-

k=l

j--1

( l + ~ k, k ) F k 0 j _, k + ~

k=l

k0kF;_ k

( G~ a05-~

, k aGk~l 05_

"~-/..J

~"

(31)

with boundary conditions:

for j = 0 ~? = O,

Fo = F o' --G o = G o -' 0-,

~? -~ oo,

Fo, G~-~ 0,

0o = 1

t

0o~0

(32)

for j = 1, 2, 3 ?? =

t

0,

t

F 5 = Fj = G 5 = G 5 =

-, oo,

0,

Fj, G;-~ 0,

05 =

0

0j -* 0

(33)

Further to transform the above partial differential equations into ordinary differential equations let us suppose that the functions in ~', ~(~') can be represented in terms of convergent power series. oo

nffi0

Accordingly with this expansion we assume thesolutions of the form oo

oo

Gj = ~

Fj = n~__oFj,n~'n;

n=0

=

oo

Gj,n~

Oj = ~_J 05,n~n n=0

(35)

Substituting eqn. (35) into the system (23)--(28) and comparing powers of ~, each set of equations will give a sequence of sets of ordinary differential equations: f o r j = 0, n = 0 /'2_

b- ' l '

__

Fo:o + uo.o,.o,o

2

t

r

~ Vo.oFo, o + 0o,0 = 0

Go:o + Go.oG;,o -- ~ G'o2o + RokoOo,o = 0 tt

t

Pr 0o, o + Go, oOo,o

0

(36) (37)

(38)

680

f o r j = 0 and n = 1, 2, 3 n r

tr

t

~

2 ]~t

f~_t

Fo,n + Go, oFo, n -- -~(1 + 2n)Go,oF~,. -- ~., o,o.~o,. 4

It

(39)

+ (1 + ~n)Fo, oGo,. + 0o,. = L o , . - 1 rl I

[2_

fJ"

GO,n + "~0,0"~0,n

__ 4

I

l

4

tt

i (1 + n)Go, oGo,. + (1 + ~n)Go,oGo,.

+ RokoOo,n = M o . . - 1 1

.

t

(40)

4

t

4

t

Prr 0o,n + Go, o0o,n --~nGo,o0o,n + (1 + ~n)0o,oGo,n =No,n-1

(41)

f o r j = 1, 2, 3 a n d n = 0 fF

t

t

t

t

tl

F;:o + Go.oFi.o --]-(Go.oFi.o + Fo.oGj.o) + Fo.oGi.o + 0j.o = L i - i , o

(42)

G;'o + G o . o G ; ' o - -~Go.oGLo 4 , . + Go.oGi.o , + RokoOi.o = Mj-l.o

(43)

I

tf

t

F

(44)

Pr Oi.o + Go.oOj.o + Oo.oGj.o = Nj-I.O f o r j = 1, 2, 3 and n = 1, 2, 3 ' ' _ _ 2~--0.0~j.n ~ , ~, F~:" + Go.oF~: n - - ~ ( 1 + 2n)Go.oFj.n tt + (1 + ~4 n)Gi.nFo.o + 0j.n = ri--l.n J'~j,n--1 m'

"

4

t

(45)

t

4

"

Gj.n + Go.oGj., - - ~ ( 1 + n)Go.oGj.n + (1 + ~n)Go.oGj., + Roko0j, 1

n -- M ~j:..n- 1- -, 1n

.

t

(46)

4

Pr 0i'n 4- Go.o0j.n

r

w

"--1 ,n

~ nGo,o0i, n + (1 + ~n)Gi,n0o. o = N],n_ 1

(47)

with Lo,n_l

= ~

t t ( i4 PoFo.oFo.n-1 -- Fo.oF~.n-1

n--1 4

+iPo ~

lffil

n--1 '

'

Fo,lFo,n_l_l

__~

n--1

+ ~ I=1

(3

+

1=1

,,

l)Go,lFo,n-1

n--1

(_I 2 + I)F~,IG~,n_I__ ~

Fo,IF~,n_I_I }

(48)

1=1

--

4

4

P

t

3

--

.

l~]0.n--1 - ~ {~ QoFo.oGo.n-1 -- zRoknOo.o -- Fo.oGo.n-1 n--1 4

+ "~Qo ~

lffil

n--1 t

~,

F~.lGo.n-l-1

__

n--1

+ ~ l=l

-~ 3 Ro ~

k'110o n--]

1=1 n--1

(12 + 1)G~,lG~,n_l__

l=l

(~ + 1)Go.IG~.n_ 1

n--1

-- ~

l=l

Fo.IG~.n__I__I}

(49)

681

n--1 N O n - - I = __4§ F o , o 0 o' , n - 1

--

•

n--1

(1 + 4 ' g l)Go.10O.n-1

~ I=1

n--1

(50)

+4{1~1= IO°'IG~)'n-1-- 1=1 ~ F°'lO°'n-l-1} j--1 Li_I, 0

j--1

2

=

,

k=l

j--1

i--1 __ ~ , G k , 0 G ; - - k , 0 k=l

0 = ~2 k~=l G k' , 0 G j -'- k , 0

Mi-1 •

(51)

D

al,oF;-k,ok=l

=

i--1

(52)

-- ko(RiOo,o + ~ RkOj--k,O) k=l

(52')

j--1

N~-l,o = --Z; Gk,00;--k,0 k=l

n--1

LJ-l,n i,n-1 = ] ~

1=0

n

, , (I + 21)Fj,lGo,n-1 + 2"~ ~

(1 + 21)F;.,Gi.n-,

1=1

n--I

n

4

-- ~

/~',.

(1 + ~l)Go,l--j,n-1

1=1

n--I

__

~ 1=0

" (1 + 4~I)Gj,IFo,n-I

n--I

---411~ ° F 0,1 F"j , n - - l - - 1 3 =

+ ~

( 1 + ~4X j ) •F j , I F 0 , n "- - I - 1

l=0

n--I

n--I

4

r

J'

I

i

--~(2Po + Xj) ~ Fo.,Fi.n-,-l] + ~Pj ~ Fo,IFo.n-I-1 1=0

1=0

n

j--1

n

j--1

4 _" + -~l)Gk.lF~_k.n_l + ~2 ~

~

(21 + 1)F~.,Gi_k.n-,

l=0 k=l

1=0 k = l

n--1 j--1

n--1 j--1 4

-g

"

(1 + ~Xk)Fk:Fj_k..--,--1 + ~ ~ ~

1=0 k = l

1=0 k = l

r

t

(Po + ?~k)Fk,,Fi--k,---1--1

n - - 1 j--1 1=0 k = l n--1

D ~ k ]~r "l' 0,1 F F] - - k , n - - l - - 1

j--1 j--i

1=0 i = l

~

k=l

n--1 ],n--1 M•-l,n

-1=1

(53)

PiF~.lF;-k-i,n-,-1 n

(1 + 21)Gi,IG;,n-, + ] ~ (1 + 21)G~).,G~..-,

---- 2 l=0

l=l

" 1 -- ~ (1 + 4gl)Go,lGj,n-

1=0

(1.1.45I)Gj,IGo," n-I

682 TABLE 1 Initial values F"(0), G'(0), ~'(0) for the (0, 0) order 9'

Pr

15 30 45 60 75

=

0,72

Pr

Pr = 10

= 1

" Fo,o(0)

" Go,o(0)

0~,o(0)

F "o,o~'0 J~

" Go,o(0)

0~,o(0)

" F~,o(0)

" Go,o(0)

1.7488 1.4434 1.2582 1.0968 0.9053

0.1171 0.2083 0.3145 0.4749 0.8446

--0.1951 --0.2363 --0.2711 --0.3110 --0.3768

1.6613 1.3712 1.1953 1.0419 0.8599

0.1113 0.1979 0.2988 0.4511 0.8023

--0.2192 --0.2656 --0.3047 --0.3496 --0.4235

1.0844 0.8951 0.7802 0.6801 0.5613

0.0726 0.1292 0.1951 0.2945 0.5237

t

is ,,, \

,

.

'

'\\,

\

~

'

\\\

0.5

',,

,,, ,\ \

,,

I

I

I

I

~

I

1

3

5

7

9

11

~ t

r

13

15

v

1~

Fig. 2. Dimensionless tangential stream function for Pr = 0.72 for different values of the angle of inclination (o).

683

Pr

=

100

Pr

=

1000

0~,o(0)

F~r o(0)

G~,o(0)

0~,0(0)

F~t o(0)

G~.o(0)

0~,o(0)

--0.4520 --0.5476 -0.6283 --0.7207 --0.8732

0.6511 0.5374 0.4685 0.4083 0.3370

0.0436 0.0776 0.1171 0.1768 0.3144

--0.8471 --1.0263 --1.1774 --1.3507 --1.6365

0.3749 0.3095 0.2698 0.2351 0.1941

0.0251 0.0447 0.0674 0.1018 0.1811

--1.5329 --1.8572 --2.1035 --2.4442 --2.9612

J

i

\

\ \ \ \

~

\

\ ~

~

~

1()

Fig. 3, Dimensionless stream function parallel to the axis of the cylinder for Pr = 0.72 and different values of the angle of inclination (o).

684 n--1 __

n--1

4

Ip

~[~

4

Fo,]Gi,n_I_I + ~

1=0

II

"

(1 + ~Xj)Fj,IG0,n_I_ 1

1=0

n--I

n--I

_ 4~ ~ QoG,o,IF;,n_I_I __ 4_~~ (Qo + Xj)GI,IF;,n_I_I] I=0

J=O

n--1

+ ~-Qj ~ F~,IG'o,n-]-I --Ro ~ 1=0

1=1

n

n

--Rj ~

~

+ ~2

~

(1 +~l)Gk,lG~k,n_ 1

l=0 k=l

n--I j--I

j--I

~

(1 + 21)G~,lGi_k,n_ l --4~ ~

1=0 k = l n

j--1

~ 0 o , n - I -- ~

1=0

~lOi,n--1

j--1

-- ~

~

l=0 k = l

Rkkl---6j--k,n--I+ ~ ~

~

(Qo + Xk)Gi,lF[--k,a-l-1

1=0 k = l

1=0 k = l

n--1 j--1 j--1 Q k~-J0,1 /Y F 'j - - k , n - - l - - 1 + ~

n--I " - - l . n -- 4 I=1

(54)

n

n--I 4

I

(1 + ~l)Go,10j,n_]

-- ~ 1=1

-- ~ 1=0

n--1

4

r

(1 + ~l)Gij0o,n_ 1

n--1

'~.

I

~=o Fo:Oj,.-1-1--4 ~ =

~

4

•

(1 + $ ~,j)Fj ,100,n_l_l

l=0

n--1 + ~

~ ~ ~ V"°gi ' k.1 f ' j - - k - - i , n - - 1 - - 1 1=0 i=1 k = l

1Oj,lGo,.-i ' , + ~4 ~ = 10o,lGj,.-1

n

---3

(1 + ~4 kk)Fk:F~_k,n--]_~ ,,

n - - I j--1

n--1 j--1

4

~

l=0 k = l

~

j--I

~ ~ ,•j 0 j , l F o' , n - 1 - - 1 + 4 ~ ~ 10k,lVj_k,n_ 1 l=0 1=o k = l n

j--1

n--1 j--1 4

--~ ~ (1 + 3 1 ) G k , l O j _ k , n _ l I=0 k = l

__4 ~

~

1=0 k = l

( l + 4 ~"~ . k i F ,, _- ~ , 1

0i - - k , n - - l - - 1

n--1 j--1

+~ ~

~ Xk0k,IF;--k,n--1--1

1=0 k=l

(55)

(IV) N U M E R I C A L R E S O L U T I O N

Numerical solutions of differential equations (36)--(55) with boundary conditions (32)--(33) can be obtained with Runge-Kutta's integration method. We give in Table 1 the initial values F~,0(0), G~,o(0), 0~,o(0) for the (0, 0) order system for several values of Prandtl (or Schmidt) number. As an example, velocity and temperature (or concentration) profiles are repre. sented for Pr = 0.72 in Figs. 2--4. (For a circular cylinder ko = sin 7 and R0 = ro/L cos 7.)

685

~JL eoo(~ )

\ \ \ \ \

~\\', 15

03.

,

\,",\ ',, \

',r.n\

\

\

Fig. 4. Dimensionless temperature or concentration for Pr = 0.72 and for different values of the angle of inclination.

(V) VELOCITIES AND NUSSELT (OR SHERWOOD) NUMBER T h e v e l o c i t y c o m p o n e n t s are

v~= vy =

(~4 ~)112(4 ~)1/2Ft

G~ [s(x)~ _

v 1/4 [~(~)] ,~ G~ - ("- W'" (-() 1'3¢4~)3'4IF+ (-~ ~) -aF~ 3

vz =

Z-) ]''~ (~" t ' ) " G' P Gz1,2 [~ (~ )(4_ ~)ln 3

F o r t h e l o c a l N u s s e l t (or S h e r w o o d ) n u m b e r w e write: y--o

Tp - - T ~

(56)

686

or with dimensionless coordinates

NUx

= _~1/4

-r

~

ov,a~=o

Nux = - - G I / 4 x ~ ' ( 0 )

.

4

[ ~ ( ~ ) ] 1/3

1/4 4 - - 1/4

(57)

(VI) DISCUSSION

We have presented here an original and very general m e t h o d for three-dimensional free convection from a classical particular case: the inclined circular cylinder one. We shall show later the relative importance of the higher order terms to the zero-order terms calculated here and we shall check experimentally our calculus in the mass convection case with the electrochemical m e t h o d developed in our laboratory [4--9]. Finally we shall show h o w from this particular case of the inclined circular cylinder we can generalize this theory to other surfaces. REFERENCES 1 2 3 4 5 6 7 8 9

W.E. Stewart, Int. J. Heat Mass Transfer, 14 (1971) 1013. J.L. Peube and D. Blay, Int. J. Heat Mass Transfer, 21 (1978) 1125. D.A. SaviUe and S.W. Churchill, J. Fluid Mech., 29 (2) (1967) 391. A. Suwono, Th~se, Centre U n i v e r s i t a i ~ de Perpignan, 1977. A. Suwono, M. Daguenet and D. Bodiot, Int. J. Heat Mass Transfer, 19 (1976) 239. A. Suwono and M. Daguenet, J. Chim. Phys., 73 (9--10) (1976) 887. L. Monet, P. Dumargue0 M. Daguenet and D. Bodiot, Electrochim. Acta, 19 (1974) 841, F. Aimeur, M. Daguenet, F. Ken~iche and M. Meklati, Electrochhn. Acta, 18 (1973) 87. M. Daguenet, Int. J. Heat Mass Transfer, 11 (1968) 1581.