Thickness of thermal and velocity boundary layers on a mobile surface of a sphere

INT. COMM. HI~kTI~SS T I ~ Vol. 12, pp. 201-208, 1985

0735-1933/85 $3.00 + .00 Ltd. Printed in the United States

@Per@~Press

THICKNESS OF THERMAL AND VELOCITY BOUNDARY LAYERS ON A MOBILE SURFACE OF A SPHEP~

H. Kalman and R. Letan Department of Mechanical Engineering Ben Gurion University of the Negev Beer Sheva, Israel (Commnnicated by J.P. Hartnett and W.J. Minkowycz)

ABSTRACT Thickness of thermal and velocity boundary laye~ was studied on a mobile surface of a hypothetical solid sphere. The surface velocity of the sphere was introduced as a parameter. A relationship of thickneSS ratio - Prandtl number, { = pr-I/5 was obtained for the forward part of the sphere in the range Pr = 2-10.

Introduction The viscous boundary layer over the solid sphere was studied by Tomotika [i].

A momentum integral equation was solved for the two-dimensional case of

a sphere in a uniform stream.

A velocity profile of a quartic form was used.

Lee and Barrow [2] followed Tomotika's procedure in their study of velocity and diffusion boundary layers.

The fourth order polynomials were selected for

both the velocity and concentration profiles.

The conservation integral equa-

tions were solved to yield the boundary layer's thickness on the rigid surface of a solid sphere. Presently, the effect of surface-mobility on the boundary layer thickness is considered on a hypothetical sphere, which has a mobile solid surface.

This

"mathematical sphere" is used to separately analyze parameters, and isolate effects.

Physically, the solid sphere with a mobile surface may be interpreted

as a fluid sphere which mechanically exhibits a solid surface against normal and tangential stresses impressed on it by the flowing fluid.

The surface

movement at a prescribed velocity in the flow direction is assumed to be arbitrarily controlled by forces exerted on the inside of the surface, and independent of the external flow. in

systems of droplets and bubbles

Such questions of surface mobility arise [5]

where

impurities

affect

202

Vol. 12, No. 2

H. Kalman and R. Letan

the magnitude of surface tension forces in a quantitatively unpredicted way [4].

Thickness of the Velocity Boundary Layer The procedure of Tomotika [I] and Lee and Barrow [2] is adopted in the present study of the thermal and velocity boundary layer thickness.

Prescribed

surface velocities ranging from external flow down to an immobile surface are introduced in the solution as a parameter.

A fourth order polynomial is

selected for velocity distribution and Tomotika's boundary conditions are applied at the edge.

However, on the mobile surface of the hypothetical sphere

the velocity is arbitrarily prescribed as a fraction of the external velocity,

_I U

[

where

u y=o = n

1 < n ~ ~

The momentum e q u a t i o n on t h e s u r f a c e [1] t o y i e l d

the velocity

F o l l o w i n g Eq. N

and

=

1--

(i)

provides

an a d d i t i o n a l

boundary condition

profile.

(1) we d e f i n e

parameters

related

to the surface

1

(2)

n

1

M =

(1 = --4- )>, n~

where ~, i s t h e s h a p e f a c t o r

(3)

[5] d e f i n e d

as,

62 dU

=

mobility,

(4)

dx

Further, a dimensionless quantity, Z, is defined, 62 U Z

=

(5)

R

and a r e l a t i o n s h i p

b e t w e e n Z, and t h e shape f a c t o r ,

~, i s o b t a i n e d ,

U ! =

z

--

U.

(6)

Eqs. (5) and (6) are differentiated with respect to O, and together with the velocity profile introduced into the momentum integral equation dZ _ K(~) U + H(~) U" Z2 - K*(~) cos8 U" dO U-~ sin0 U~ where,

to yield, (7)

VOl. 12, NO. 2

K(i)

~

IAYERS ON A SURFACE OF A SPHERE

203

K4(%) K3(X) = KS(X-----T - ~ KS(i-----T

(s)

K2(l)

(9)

H(~) = Kl(1) K*(X) = I - -

(I0)

KS(1)

and,

1 378 KI(~ ) = 2-~- [ - ~ - - N

K2(X) = ~

KS(~,)

:

~

1 2

- 46

1

21

N2

21 + ~M

11

[ i-8M - -~ ( ~ N 378

[ --~--N

-

46

N2

11 (~N

1)

1 M2 - ~-) - T6 ]

(Ii)

M

- __] ~ 21

11

+ -~- M(~-

02) N -

1

~-)

M2

- ~-

]

i (3N - ~2M )

+ i-6

(i3)

i

K4() 0 : 2N + ~ M

(14)

KS(1) : EI Ki(1)

(i5)

- IK2(1)

Eq. iT) i s a g e n e r a l e x p r e s s i o n o f t h e boundary l a y e r t h i c k n e s s o v e r t h e mathematically formulated mobile surface of a s o l i d sphere.

The two l i m i t i n g

cases of the general solution correspond therefore to: The c a s e o f t h e s u r f a c e i n t e r n a l l y Uy: o : U where n : l ,

moved a t t h e v e l o c i t y o f t h e e x t e r n a l f l o w :

(6:0).

The c a s e o f an immobile s u r f a c e o f a s o l i d s p h e r e : Hor t h i s [1].

c a s e Eq. (7) c o n v e r g e s t o t h e p a r t i c u l a r

Thus, f o r n:® Eqs.

Eq. ( 7 ) , f*(l),

K(I), H(t),

(2) - (5) y i e l d N : I , M=k. Then t h e c o e f f i c i e n t s

K*(k), become i d e n t i c a l

( T o m o t i k a ' s Eqs.

Uy: o : 0 where n : ~ .

s o l u t i o n g i v e n by Tomotika

with Tomotika's f(t),

g(1),

of and

(12 a, b, c ) ) r e s p e c t i v e l y .

Solving for a potential

flow f i e l d

o u t s i d e t h e boundary l a y e r Eq. (7)

reforms to : dZ _ 2 K(X) 1 de 3 sine = k

O

5 MiX) s i n e , Z 2 2 1 2 - -~ K*(t) --sine

a t e : 0, and we o b t a i n ,

(16)

204

Vol. 12, No. 2

H. Kalman and R. Letan

K(~ o )

K*(~o)

-

0

=

(17)

The a b o v e e q u a t i o n h a s t o b e s o l v e d f o r s p e c i f i c ~o' i s

further

Zo

n,

(6=0).

15, 21, =.

immobile surface,

point

= Zo ,

~o' Zo' a r e t h e n a p p l i e d

in solution

The n u m e r i c a l

At n = 1 t h e r e

At n = ~, T o m o t i k a ' s

obtain

The s o l u t i o n ,

(18)

as a p a r a m e t e r .

n = 3, 9,

Z(e=0)

o f n.

2 ~ ~o

=

These values with,

used to obtain,

values

solution

is

i s no v e l o c i t y

solution

is obtained

Zo = 5 . 1 4 4 0 and ~o = 4 . 7 1 6 0 1 .

Z = 2.5-5.5

at

of separation

8 = 0.

o f Eq.

(16)

to yield,

illustrated

in Fig.

boundary layer; for the solid

1, f o r

thus,

Z = 0

sphere of

In t h e r a n g e 3 ~ n ~ = we

V a l u e s o f Z = 5-6 a r e r e a c h e d a t

is approached,

Z(O),

the boundary layer

e = 70 ° .

thickness

As t h e

increases

rapidly.

6 Z

n=21

4 2 --

n=3

9

15

_

O0

1

1

1

[ I 20

I

i

I I I 40 e

I

rl=l k/ I I 60

I

I I 80

I

I

I I00 °

FIG. 1 Effect of surface mobility on thickness of velocity boundary layer.

T h i c k n e s s R a t i o o f T h e r m a l and V e l o c i t y Calculation

of the thermal boundary layer

Boundary Layers

thickness

involves

the energy

integral equation, and the temperature profile in the boundary layer. temperature profile expressed in its dimensionless form to the velocity profile.

at y = 6t:

T - To T= _ To

The

is similar

The boundary conditions are:

aT a2T and a t y = 0: T = T=, ~ y = aY2 ,

T = To,

a2T - 0 aY2

(19)

VOl. 12, No. 2

~IAYERSONASURFACEC~ASPHERE

205

The temperature distribution is obtained as, T - T O = 2( ~t ) - 2( ~it 13 + ( ~ ) 4 T - TO

(201

Utilizing it in the energy integral equation leads to, 1

d [rUd HI(%,~) ] 2~ Rd0 = 6-~

(21)

where, g = 6 t / ~ , and, H l ( k , ~ ) , 3 (1 HI(X'E) = TO 1

-

+iTG

N)g +

~

i s d e f i n e d as 1

1M)~2 1 (2N + ~ - 8-4 Mg3

1 140

(3N -

3 M)g4

(N - 1 M)~5

(22)

Differentiation of Eqs. (21) and (22), introduction of the dimensionless variable Z (Eq. 51, and its derivative (dZ/dO), and application of the potential flow field yield, d~ dO

=

i {___LI coso H2(X,~) ~.Pr sin0

HI(~,~).~.

cos0

1

sin0

4

Hl(l,~ )

1 dZ "~'~'d-E-

3 dZ 3 - ~ H 3 ( g ) ~-~cos0 + ~ H 3 ( ~ ) . Z . s i n 0 }

(231

where, H2, and H3, are d e f i n e d as, HZ(X,~) = ~ 3 (I-N)-~ + ~ 1 - --3

70

(N -

1M).~4

T

(2N + ~1 M ) . g 2 - ~-6-M-~ 1 3 +

1

72-

(N -

1 E

M)~S

(24)

1 3 1 4 3 5 i 6 M H3(~) = ( i-~'~ - 168.~ + i-~-~'~ - 2160"~ )-~

(25)

Eq. (231 reduces to, Hl(Xo,~o)-Xo.~ °

=

1 p--~

at e

:

0

(26)

Eq. (231 has to be simultaneously solved with (dZ/d0), Eq. (16), to yield the thickness ratio, $, of the thermal and velocity boundary layers.

Lee and

Barrow [2] obtained the thickness ratio on a rigid sphere using Tomotika's [I] procedure for the velocity boundary layer thickness.

Our solution (Eqs. (16)

and (23)) converges to the particular solution of Lee and Barrow [2] at, n = ~, for the case of ~ > ~t"

206

H. Kalman and R. Letan

Vol. 12, No. 2

The simultaneous solution of Z(O) and ~(0) (Eqs. (16) and (23)) has been numerically conducted for specific values of the parameter, n, and Prandtl numbers.

Fig. 2 plotted with n = 3, 9, 15,

~

at

Pr = 4 illustrates the

moderate slope of ~ vs. 0, from the forward stagnation point down to @ = 70 ° . The slope becomes steeper as the point of separation is approached. tionship is approximately ~Pr I/3 = 0.96 - 1.12.

The rela-

The quantity, ~.Pr 1/3 =i.12,

at, n = ~, coincides with the Lee and Barrow [2] solution for the rigid sphere

1.2=

15

I.I -

~ . pr 1/3

-

n=3

l.O--

i

C 0

t

i

I

i

I

I

I

20

I

I

40

I

I

e

=

I

I

60

I

I

t

I

I

80

I00 °

FIG. 2 Variation of boundary layer thickness ratio with surface mobility.

Lee and B a r r o w ' s [2] s o l u t i o n f o r a r i g i d 3 w i t h P r a n d t l number as p a r a m e t e r .

The r e l a t i o n s h i p

t h e s p h e r e i s a g a i n ~Pr 1/3 = 1.1 - 1.15. 1.2 Rigid

sphere

Pr= 2 ~. Pr

s p h e r e (n==) i s p l o t t e d

I/3

in Fig.

a t t h e forward p a r t o f

j" [2]

4

i..~

IQ I

0

I

J

I

20

J

l

I

I

40

I

I

e

I

J

J

I

60

I

I

80

t

I

I

I

I00 °

FIG. 3 V a r i a t i o n o f boundary l a y e r t h i c k n e s s r a t i o

on a r i g i d

s p h e r e [2].

Fig. 4 illustrates our solution of the thickness ratio, 6, for the mobile

Vo1. 12, NO. 2

surface.

~IAYERSONASUI~ACEOF

Here again ~prl/3

= 0.9

- 1.06,

1.0

A SPHERE

a t n = 3,

207

i n t h e r a n g e Pr = 2 - 10.

4 6

09

i

m m I

0

t

t

t

20

i

I I

40

~ (9

i

I i 60

I

i

[ I 80

I00 °

FIG. 4 yariation

of boundary layer

thickness ratio mobile surface.

with Prandtl

n u m b e r on a

Figs. 2-4 have demonstrated the relationship Eprl/3 = 0.9 - 1.15, in the range of 3 ~ n ~ ~, and 2 ~ Pr ~ I0 over the forward portion of the sphere.

The

practical conclusion is therefore that the thickness ratio of the thermal and velocity boundary layers on a aobile or immobile surface of a solic sphere, and within the investigated range of Prandtl numbers (2-10), may be expressed by the relationship,

~ = Pr -I/3, over the forward part of the sphere.

The

approximation of ±10% is acceptable in view of the uncertainties involved in determining the surface mobility.

The practical applications are to be found

in systems of fluid spheres where impurities control the surface mobility. Conclusions Thickness of the velocity boundary layer, and the thickness ratio of the thermal-velocity boundary layers were obtained as functions of surface mobility and Prandtl number. Prandtl number.

Both decreased with the increased surface velocity and

In the limiting case of an immobile surface the generalized

solution converged to the particular solution of a rigid sphere, as found in the literature.

The thickness ratio over the forward part of the sphere in

the range of Prandtl numbers 2-10, obeyed the relationship of ~ = Pr -I/3. Nomenclature R

radius of the sphere

r

radius of the t r a n s v e r s e

cross-section of the sphere

T

temperature in boundary layer

To=

free stream temperature

To

temperature of sphere surface

208 U

H. Kalrag_n and R. Letan

Vol. 12, No. 2

velocity at edge of boundary layer

U'

= dU/d0

U

velocity far away from the sphere

u

velocity in the boundary layer

x

distance along the sphere from the forward stagnation point

y

distance normal to surface

Pr

Prandtl number (v/a)

a

thermal diffusivity thickness of velocity boundary layer

~t

thickness of thermal boundary layer

O

angle from forward stagnation point kinematic viscosity thickness ratio, (~t/~) References

1.

S. Tomotika, B r i t . Aero. Res. Com. R.M. No. 1678 (1935).

2.

K. Lee and H. Barrow, I n t . J . Heat Mass T r a n s f e r , 11, 1013 (1968).

3.

R. C l i f t , J.R. Grace and M.E. Weber, Bubbles, Drops and P a r t i c l e s , P r e s s , New York (1978).

4.

Y. L e r n e r , H. Kalman and R. L e t a n , 5ymp. Vol. Basic Aspects o f Two Phase Flow and Heat T r a n s f e r , p. 1, 22nd N a t i o n a l Heat T r a n s f e r Conference, Niagara Falls (Aug. 1984).

5.

H. Schlichting, Boundary Layer Theory, p. 239, McGraw Hill, New York (1979)

Academic