Free convection boundary layer flow of a micropolar fluid past slender cones

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FREE CONVECTION BOUNDARY LAYER FLOWOF A MICROPOLAR FLUID PAST SLENDER CONES H.S. Takhar Simon Engineering Laboratories, University of Manchester, Manchester, England M139PL. Rama Subba Reddy Gorla, Professor and Chairman Department of Mechanical Engineering, Cleveland State University Cleveland, Ohio 44115 William R. Schoren, Research Assistant Department of Mechanical Engineering, Cleveland State University, Cleveland, Ohio 44115 (Received 16 April 1987; accepted for print 5 April 1988)

Abstract Using the theory of micropolar fluids developed by Eringen, the transverse curvature effects on axisymmetric free convection boundary layer flow of a miropolar fluid past slender vertical cones are investigated. The case of constant surface heat flux is considered in this paper. Using perturbation techniques, the governing equations for momentum, angular momentum and energy have been solved numerically. Graphical representations for the velocity, angular velocity and thermal functions are presented for various physical and fluid property parameters. Introduction Free convection flow is caused by buoyancy forces which arise from density differences in a fluid. These density differences are a consequence of temperature gradients within the fluid. Free convection flow is a significant factor in several practical applications which include, for example, chemical processes and related equipment and the cooling of rotating machinery and electronic components. Eringen [ i ] has developed the theory of micropolar fluids and has extended this theory to include thermomicropolar fluids [2]. A comprehensive review of research involving micropolar fluid mechanics was provided by Ariman, et. al. [3,4]. The laminar free convection boundary layer flow of a thermomicropolar fluid past a non-isothermal vertical flat plate was 167

168

H.S.

TAKHAR,

R.S.R.

GORLA

a n d W.R.

SCHOREN

i n v e s t i g a t e d by Jena and Mathur [ 5 ] . They derived a s i m i l a r i t y s o l u t i o n by p r e s c r i b i n g the plate temperature as a l i n e a r function of the streamwise coordinate. The present work has been undertaken in order to study the influence of transvere curvature on laminar free convective boundary layer flow of a thermomicropolar f l u i d past a slender v e r t i c a l cone. The case of a constant surface heat f l u x from the cone is i n v e s t i g a t e d in t h i s paper.

G0verningE_q_u_ationA In the analysis that f o l l o w s ,

three general assumptions are made: ( I )

temperature of the cone everywhere exceeds that of the ambient f l u i d . The cone angle '@' has a constant p o s i t i v e value less than 45 ° . fluid

(3)

The

(2) All

properties are constant except those causing the buoyancy e f f e c t . Although the reverse case for the f i r s t

mathematically

identical,

assumption can be shown to be

the consequence of the f i r s t

free convection flow is in the upward d i r e c t i o n (x = O) w i l l

be pointed downward.

(Figure I ) .

assumptions allows the use of the f o l l o w i n g equations for an incompressible,

assumption is that the

and the vertex of the cone The second and t h i r d

partial

differential

axisymmetric, steady, laminar free

convection boundary layer flow of a micropolar f l u i d : Mass : ~_ (ru) + ~__ ( r v ) = 0 Bx ~)r

(I)

Momentum : uDU + D x va~U =~r

E v + Kp_--I --Ir T r

--rDU] + < IN + ~T~] + GB(T-T )cos$ _ ~i

~

(2) Angular Momentum: u DN + v aN = ~ a-xDr pj

{L ~_[--rDN~ - N } _ ~ _ {Du + 2N} r BrL_ a r ] ~ PJ ~-r

(3)

Energy: uaT + vaT = ~11 a l--raT-] ax D--ff r a--r-I_ ~rl J

The use of these governing equations l i m i t s the a p p l i c a t i o n of t h i s analysis to slender cones because these equations do not account for the

(4)

M I C R O P O L A R B O U N D A R Y L A Y E R PAST C O N E S

buoyancy e f f e c t s in the ' r '

or radial d i r e c t i o n .

169

The boundary conditions for

the v e l o c i t y f i e l d are: At r = xtan¢:

u = v = O,

L~r_1

2 As r ÷ -:

(5)

N : _1 BF-I A-l

u + O, N +0

The boundary conditions for the temperature f i e l d are: At r = x t a n ¢ : As r + -:

(6)

T = Tw = T® + H 3(m h ( x )

T + T~

Analysis In solving the continuity equation or the equation of the conservation of mass, a stream function ¢ ( x , r ) is usually defined such t h a t : 1~,i, u

:

= ~

r @r

;

v

:

(7) r @x

In order to f a c i l i t a t e the solution of the other governing equations, the following coordinate transformation are defined: : 2 ~ x Ctan¢

_[m+3l -4--

; n : r2 - x2 tan2¢ {x 2 tan2¢

where 1/4 : x ;

L ¢(xr)

C : [-GSHL3c°s¢l-~ L_ u2 ]

: vLC t a n ¢ ~

m+7 4 f({n)

[m+l] N = 2v2C2 P ~ T g ( { n ) r{[ h(x)

: e({,n)

: T-T=

HR'"

170

H.S.

= ~ ;

TAKHAR,

X =

Substituting

~ uj

R.S.R.

;

GORLA

and

W.R.

SCHOREN

B = x2tan2¢ J

the expressions

(8)

in (7) and (8) i n t o

equations

(2)-(4)

yields

the f o l l o w i n g :

+ e = ~F-m+3~

I-4 I_

4~(I+~n)2 g,,

._

-

2

_

~f-" a~f- f

I_

a__~f'_-}

(9)

]

A2B(I+~n)2 ~2 f,, _ 2B(l+~n)~2g

-

.~

~n + (3m + 5-_ f ' g + ( l + ~ n ) ( m + 7 ) f g '

(I0)

-

_ --2-

~__--oaf- f' ~_0_ (ii)

A prime denotes d i f f e r e n t i a t i o n

with

equations.

The boundary c o n d i t i o n s

coordinates

are:

f(~,O)

= f'(~,O)

= O;

g(~,O)

= -~f"(~,O); 2

f'(&,-)

r e s p e c t to

'n'

f o r the v e l o c i t y

o n l y in the above field

in the t r a n s f o r m e d

= 0

(12)

The boundary c o n d i t i o n s

g(~,-)

= 0

f o r the t e m p e r a t u r e

in the t r a n s f o r m e d

coordinates

are: 0(~,0)

The f o l l o w i n g

= h(x)

= 1 + ~ anon; n=O

@(~,~) = 0

power s e r i e s expansions are assumed f o r the v e l o c i t y ,

(13)

angular

MICROPOLAR

BOUNDARY

LAYER PAST CONES

v e l o c i t y and thermal functions in order to solve the p a r t i a l equations (9),

171

differential

(I0) and ( I I ) : : f0(n) + ~ f l ( n )

+ ~2f2(n) + . . . .

(14)

g(C,n) : go(n) + { g l ( n )

+ {2g2(n) + . . . .

(15)

e(C,n) : e0(n ) + Cel(n) + C2B2(n) + . . . .

(16)

f(~,n)

Substituting (14), (15) and (16) into equations (9), (10) and ( I i ) and equating equal powers of { to zero yields a set of ordinary differential equations governing the momentum, angular momentum and energy fields.

These

details are not shown in the interest of conserving space. Although various wall temperatures are allowed by equation (13), the case of constant wall heat f l u x w i l l local

be considered here.

Using Fourier's Law, the

heat t r a n s f e r rate from the cone surface i s :

qw

: -kl -G~H5 cos¢_ I / 4 ~ [5m-l] T(@e) L

(17)

" :o

In order for the heat f l u x from the cone surface 'qw' to be maintained constant, the following conditions must hold t r u e : m:0.2 e'(0) : 0 for n > 0 n Therefore :

1/4

L T]

qw : -kF-GBH5 COS¢--I

06(0)

(18)

Since 'qw' is constant, an expression for 'H' can be found from equation (18) and s u b s t i t u t e d i n t o the expression for e(~,n) from equation (8).

Rearranging

t h i s expression and noting that T = Tw at the wall y i e l d s an equation for the wall temperature: T : T = T +~- v2qw~ w l k4GBcoso

-B~ (

Z en(O){ n n=0

(19)

172

H.S.

TAKHAR,

R.S.R.

GORLA

and W.R.

SCHOREN

Applying the conditions for constant surface flux yields the following equations: 0.4 [ ( i + _A) f" Cf x = 2C3 -#

NUx

=

(20)

-HCx %(O) (Tw_T)

Nux : -[h(x)] Grx-~

-5/4

(21)

O' 0(0)

Numerical solutions to equations governing f j ( n ) ,

(22)

gj(n) and ej(n) have been

found using A, ~, B and Pr as parameters. A step size for n of 0.001 was used along with double precision arithmetic for all computations. Shooting techniques were used to determine unknown values for the v e l o c i t y , velocity,

angular

and thermal functions at the cone surface.

From Figure 2, i t can be observed that the second term of the power series expansion for the velocity function, namely f ' has negative values. We note that the f i r s t term f '

represents the velocity function when

transverse curvature effects are ignored.

This implies that the velocity of

the convective f l u i d decreases as the transverse curvature increases.

This

conclusion is in concurrence with the previous statement since smaller temperature gradients give rise to smaller f l u i d velocities.

In free

convection flow, the driving force for fluid motion is density differences due to fluid temperature gradients.

Therefore the f l u i d velocity will decrease as

the temperature gradient in the f l u i d decreases with increasing transverse curvature.

Figures 3 and 4 represent numerical solutions for the angular

velocity and thermal functions respectively.

Concluding Remarks In this paper, the theory of thermomicropolar fluids formulated by Eringen has been used to derive a set of boundary layer equations for axisymmetric free convective flow past slender vertical cones. The case of constant surface heat flux has been considered.

Perturbation techniques have

MICROPOLAR

BOUNDARY

LAYER

PAST C O N E S

173

been used to generate numerical solutions for the governing equations for momentum, angular momentum, and energy. Graphical representations for the velocity, angular velocity, and thermal functions are presented for various dimensionless material properties of the micropolar fluid. surface friction factor and Nusselt number are given.

Equations for the

The numerical results

obtained have clearly demonstrated that the surface temperature of the cone increases less rapidly in the streamwise direction with increasing transverse curvature.

I t also has been shown that the fluid velocity decreases with

increasing transverse curvature. Nomenclature a

coefficient for the surface temperature function

B

dimensionless physical parameter

C

dimensionless constant

Cfx

local friction coefficient

f(¢,n)

-

dimensionless velocity function

g(~,n)

-

dimensionless microrotation function

G

acceleration due to gravity

Grx

local Grashof number

h(x)

surface temperature function

H

temperature constant

J

micro-inertia per unit mass

k

thermal conductivity

K

microrotation viscosity parameter

L

reference length

m

exponent for the surface temperature

N

angular velocity

Nu

Nusselt number

Pr

Prandtl number

q

heat flux

r

radial coordinate

Re

Reynolds number

T

temperature

U

velocity in the x-direction velocity in the y-direction

V

174

H.S.

x

TAKHAR,

R.S.R.

GORLA

and W.R.

-

d i s t a n c e along the cone surface

x

-

dimensionless v a r i a b l e

y

-

c o o r d i n a t e in the r a d i a l

~,n

-

dimensionless coordinates

~,~

-

dimensionless f l u i d

-

coefficient

-

dynamic v i s c o s i t y

-

kinematic v i s c o s i t y

p

-

d e n s i t y of f l u i d

¥

-

material

-

thermal d i f f u s i v i t y

O(C,n)

-

dimensionless temperature f u n c t i o n

-

conditions

-

ambient c o n d i t i o n s

SCHOREN

in the x - d i r e c t i o n direction

properties

of thermal expansion

p r o p e r t y of the f l u i d

Subscripts w

at the cone surface

References [1]

A.C. Eringen, "Theory of M i c r o p o l a r F l u i d s " ,

Journal of Mathematics and

Mechanics, 16, pp. 1-18. (1966). [2]

A.C. Eringen,

"Theory of Thermomicrofluids",

Analysis and A p p l i c a t i o n s , [3]

T. Ariman, M.A. Turk and N.D. S y l v e s t e r , -

A Review", I n t e r n a t i o n a l

pp. 905-930. [4]

38, pp. 480-496.

"Microcontinuum F l u i d Mechanics

(1973).

F l u i d Mechanics", I n t e r n a t i o n a l

[5]

(1972).

Journal of Engineering Science, 11,

T. Ariman, M.A. Turk and N.D. S y l v e s t e r ,

pp. 273-293.

Journal of Mathematical

"Applications

of Microcontinuum

Journal of Engineering Science, 12,

(1974).

S.Ko Jena and M.N. Mathur, " S i m i l a r i t y

Solutions f o r Laminar Free

Convective Flow of a Thermomicropolar F l u i d Past a Non-lsothermal Vertical

Flat P l a t e " ,

International

19, pp. 1431-1439. (1981).

Journal of Engineering Science,

MICROPOLAR

BOUNDARY

LAYER

PAST C O N E S

175

G

v~<~

Figure I .

I

r

Geometrical Configuration for Flow Past Cone

0,8 Pr=lO.O0 =20.00 0.6

A =10.00

B =0.10 m =0.20

0.4

0.2

0 2

3

4

5

B

-0.2

-0.4

-0,6

-0.8

-1.0

-1.2

Figure 2.

Distribution for the Velocity Functions

176

H.S.

TAKHAR,

R.S.R.

GORLA

and W.R.

SCHOREN

0~ Pr=lO.O0 ~20~00

06

I/

A=10.00 El=0.10

%

oI

+ ~4-

-0.4 ~ /

~

-0.6 ~

- -~----.__j_~. --+====~.~>_-~

gl XlO' g~

-0.8 I

10 } i

-12 t Figure

3.

Distribution

for the Angular Velocity

\ 0.4

Functions

Pr=lO.O0 =20.00 A=10.00 m=o,2o

0+2o

-0.2 -0.4

-0.6 -0.8

-1.2

F i g u r e 4.

Distribution

f o r t h e Thermal

Functions