Acta Mechanica 101, 161-174 (1993)

ACTA MECHANICA 9 Springer-Verlag 1993

Free convection in the boundary layer flow of a micropolar fluid along a vertical wavy surface C.-P. Chiu a n d H . - M . Chou, Taiwan, R e p u b l i c of C h i n a (Received October 7, 1991; revised July 17, 1992)

Summary.A cubic sptine collocation numerical method and a simple transposition theorem have been used to study the free convection in the flow of a micropolar fluid along irregular vertical surfaces. A sinusoidal surface is used to elucidate the amplitude wavelength ratio effects on the free convection in a micropolar boundary layer. The effects of micropolar parameter R and geometries on the velocity and temperature fields have been graphically studied. The skin friction stress on the wall has also been studied and discussed. It is observed that the frequency of the local heat transfer rate is twice that of the wavy surface, irrespective of whether the fluid is a Newtonian fluid or micropolar fluid; the same result is also obtained for the skin friction on the wall.

List of symbols d B Cp

Cf g Gr h j Ks L N Nu Nu p Pr R s T u, v x, y

amplitude dimensionless material parameter, Eq. (8) specific heat of the fluid at constant pressure skin-friction coefficient gravitational acceleration Grashof number heat transfer coefficient micro-inertia density thermal conductivity wave length microrotation local Nussett number defined in Eq. (20) total Nusselt number defined in Eq. (21) pressure Prandtl number micropolar parameter, z/# distance measured along the surface from the leading edge temperature velocity components coordinates

Greek ~ymbols

fi 7 G

amplitude-wavelength ratio (d/L) thermal expansion coefficient spin-gradient viscosity dimensionless parameter surface geometry function

162 0 X

z V3

C.-R Chiu and H.-M. Chou dimensionless temperature, (T-- T~)/(Tw -- T~) vortex viscosity dimensionless material parameter absolute viscosity microrotation component dimensionless parameter density of fluid

Superscripts

dimensional quantity nondimensional quantity derivative with respect to x Subscripts

w oo

surface conditions conditions far away from the surface

1 Introduction The concept of micropolar fluids introduced by Eringen [1] deals with a class of fluids which exhibit certain microscopic effects arising from the local structure and micromotions of the fluid elements. These fluid contain dilute suspensions of rigid macromolecules with individual motions which support stress and body moments and are influenced by spin-inertia. The theory of micropolar fluid and its extension to thermomicropolar fluids [2] may form suitable non-Newtonian fluid models which can be used to analyze the behavior of exotic lubricants [3], [4], colloidal suspensions or polymeric fluids [5], liquid crystals [6], [7] and animal blood [8]. Some theoretical studies [6]-[8] have been compared and favorably agree with experimental measurement. Furthermore, Kolpashchikov et al. [9] have devised a way to measure micropolar parameters experimentally. However, more experimental and theoretical work is still required in this area. A thorough review of the subject and application of micropolar fluid mechanics was provided by Ariman et al. [10], [11]. Studies of the flows of heat convection in micropolar fluids have focused mainly on flat plates [12]-[18] or regular surfaces [19]-[23]. Few studies have considered the effects of complex geometries on natural convection in micropolar fluids: these have been restricted to flow along constant curvature surfaces [24] and convex surfaces [25], and the case of forced convection was considered. Prandtl's transposition theorem [26] was used to transform a complex geometry into a simple shape in this investigation. The gist of the theorem is that the flow is displaced by the amount of the vertical displacement of an irregular solid surface, and the vertical component of the velocity is adjusted according to the slope of the surface. The form of the boundary layer equations is invariant under the transformation, and the surface conditions can therefore be applied on a transformed flat surface. Though the transformation itself is quite simple, it can handle very complex geometries. The effect required to solve the transformed equations numerically is about the same as that for the original forms, although the equations in transformed space are more complicated than their original forms. In the present paper, the free convection in the boundary layer flow of a micropolar fluid along a vertical wavy surface is investigated. Of interest are the effects of the micropolar

Free convection in the boundary layer flow

163

parameter R and the amplitude wavelength ratio a on velocity, Nusselt number, and temperature fields. These results are compared with the corresponding flow problem for a Newtonian fluid and the skin friction on the wall is also studied.

2 Mathematical

formulation

The governing equations for a steady, laminar, incompressible, micropolar fluid flow along a semi-infinite vertical surface of arbitrary shape with variable micro-inertia may be written within Boussinesq approximation as

0~

0~

--

+

o

~~ + ~~

c,x

Uy

= 0

(1)

0 t ~ + ~

= - ~ + (# + ~) \yd~ + o~U + og~(:r- r~o) + ~ \ 02 /

(2)

=-0~

02J

(3)

835 2v3 +7\022 + 022]

(4)

oJ~u~-+~@/ ocp ~ U~ + ~ ~

+ (# + z ) \ 8 2 2 + 822] + ~ \

\dX

= u.s \022 + o2~ j ,

where fi and ~ are the components of velocity along the 2, 2 directions; T, p, and g are the temperature, pressure, and gravitational constant; 0,/~, and fi are the density, viscosity, and the thermal expansion coefficient of the fluid; j, ~, and 7 are the micro-inertia density, vortex viscosity, and spin-gradient viscosity; K s and Cp are the thermal conductivity and specific heat of the fluid at constant pressure; v3 is the component ofmicrorotation whose direction of rotation is in the (x - y) plane. The appropriate boundary conditions are: (a) On the wavy surface (2 = 0): T = T~

(6.1)

= ~5= 0

(6.2) (6.3)

(Antisymmetric part of stress vanishes at the wall) (b) Matching with the quiescent free stream (y --, oo): (6.4) 123 - - + 0 ,

and

T - , T~.

The boundary condition (6.3) means the microrotation equals one half of the fluid vorticity at the boundary (or angular velocity). This type of boundary condition has been used and explained

164

C.-R Chiu and H.-M. Chou

by Ahmadi [12] and Ramachandran et al. [24]. In addition, it should be noticed that ~ is not normal to the wavy surface, but is normal to the s axis. To consider a semi-infinite vertical wavy plate embedded in a micropolar fluid, the surfaces of the plate can be described by y = #(2).

(7)

The temperature of the wavy surface is held at T~, which is warmer than the ambient temperature Too. The above governing equations (1)-(5) may be nondimensionalized by using the following nondimensionalizations: 2 = L'

r/-

p -- # L Gr*/4'

11 Gr ~/4

11 G r 1/2'

0-

pL20 /) - 112 Gr

T - Too , T~,- Too

d# d2

17'-

z

,

d17 , dx

-

11 G r 3/4 o-"-

de dx

(8)

L 2

R=--, 11

B-

J Gr 1/2'

P r - 11Cp and

2=-11j

Gr = gfl(T~- Too)~2L3

Kf '

I12

The nondimensionalized equations, after ignoring terms of small orders in Gr, are: aft' & 0~ + N = 0

f a ~ + fiar/

(9)

a~ +

~Or*/4 +0+(I+R)

f217" _[_ 0"'0 = 17' ap - (1 + r

a_N

0N

(1+17'2)~j

+R

~

~a/3 Or,/~

[ - ( 1 + 17'2)~&i- 2N] + 2(1 + #2) (ae2v) k ar/2)

f~+O~=RB

QO 00 1 a20 f ~ + t3 a~ = Pr (1 + 17,2)- - . ar/2

(10.1)

(10.2)

(11)

(12)

Equation (10.2) indicates that ~ is O(Gr-1/4). This implies that the lowest-order ~ can be determined from the inviscid flow solution. For the current problem this pressure gradient is zero. Elimination of ~ between Eqs. (10.1) and (10.2) results in the following equation: h ~ + ,3

Or/

-

1 -}-

17 ,2

( - 0 + fi2a'17") + (1 + R) (1 + 17,2) c~2f ar/~ + R ~0~-) '

(13)

Free convection in the boundary layer flow

165

Equations (9) and (11)-(13) in the parabolic coordinates (x, y) are:

Ou Ou 2u+4x~-yT-+7uy Ox

Ov oy

=0

(14)

4XU ~x + (v - uy) - + 2 + u203' k 1 -}- 0"'2/

-1 @ 0-,2 0 + ( I + R ) ( I + a

'2) 05,2 + R

(15) 4xu g~ + (v -- uy) ~

+ uN = RB(4x) 1/2 - ( 1 + a '2) gxx -- 2N

(~0

00

1

+ ,~(1 + a '2) \ (~y2 ]

020

4xu - - + (v - uy) - - = - - (1 + ~r'2) - ,yx 03' Pr @2,

(16)

(17)

where 2 x = ff = ~ ,

17 _ l ~ -- 8 Grl/4 Y - (4x) t/4 (4x) 1/4 L

/i u - (4xfl/2

1 /iQL (4x)1/2/~ Grl/2

(18) v = (4x) 1/4 ~ = (4x) 1/4 (/~ - cr'fi) 0L /l Gr I,'4 N-

1 v30L 2 (4x) T M # Or 3!4

(4x) 1;4

The corresponding b o u n d a r y conditions are: (a') y = 0 :

0 = 1, N =

(b') y ~ oo:

u=v=0,

(19.1)

1 (1 + a '2) 5u

0 --~ 0,

u -~ 0,

and

(19.2) N --, 0.

(19.3)

Condition (19.2) is the dimensionless form of Eq. (6.3), after ignoring terms of small orders in Gr (as shown in the Appendix). The local Nusselt n u m b e r is defined as

OT ~n L T,~-T~-

hL_ NU=-Kr

(Gr)l/4 a,2)~/2(~0~ , \4x] (I+ \oy/y=o

(20)

8 where ~ represents differentiation along t h e normal to the surface. The total Nusselt n u m b e r m a y be obtained by integration of the above equation. It is

Or ';*

s

L-G--~

~

=o

dx

(21)

0

and s = i (1 + or'z) 1/2 dx is the distance along the surface, measured from the leading edge. 0

C.-R Chiu and H.-M. Chou

166 The shearing stress on the surface is

%=

#

+

+~

+v3

The skin-friction coefficient

C:

(22)

y=o"

is defined by

2"Cw

(23)

eU,2,

C:-

# Gr 1/2 where U, - - is a characteristic velocity. ~oL In terms of the non-dimensional quantities, we have

c:

=

(2 + R) (1 -

(24)

Uyy =o'

3 Numerical method

The coupled nonlinear partial differential equations (14)-(17), combined with the appropriate boundary conditions (19), have been solved by using the cubic spline collocation method and finite difference approximation. The main advantages of using a cubic spline collocation are that: (i) The governing matrix system is always tridiagonal, so, well developed and highly efficient inversion algorithms, such as the Thomas algorithm, are applicable. (ii) The matrix system obtained may be reduced to a scalar set of equations that either contains values of the function itself, its first derivative or its second derivative at the node points while maintaining a tridiagonal formulation. (iii) The spatial accuracy of the spline approximation has shown that the first derivative has fourth-order accuracy for a uniform mesh and third-order accuracy for a nonuniform mesh. The second derivative has second-order accuracy for a uniform as well as a nonuniform grid. (iv) Since the values of first or second derivatives may be evaluated directly, boundary conditions containing derivatives may be directly incorporated into the solution procedure thus avoiding the difficulty that exists with conventional finite difference schemes. The cubic spline integration technique has been applied to solve problems in fluid flow and heat transfer by Rubin and Graves [27], Rubin and Khosla [28], Wang and Kahawita [29], [30], and Char and Chert [31]. Equations (14)-(17) using false transient technique in discretized form are U.n.+l

2u..+l ,,s

un~-I z,j

,,j

+ 4xi

__ un+l

Ax

_ _ U .n . l,j

At

/).n+l

i-i.s

- - Yil""+l +

,,s

__ /)n+l

Ay

i-i,s _ 0

(25)

U n . __ U n t,j i - 1,j

-

- 4xiur4

Ax

(vi"4 -- ylui".S) l." + 1

Or:+1 t,J

--

2+

l+a,2J

~ ,.a: + 1- +- a '2 + (1 + R) (1 + #2) L . + I + R l u . + l

(26)

167

Free convection in the b o u n d a r y layer flow

N ~-+~ ~,d - N~z,j At

N ~~,J, - N"i -

- 4xlu~d

l,j

Ax

(vi"j -- ylu~"4)

IN" +

- u~"4N~4 + R B ( 4 x i ) ~/2 [ - ( 1 + a 'a) 1," - 2N~o] + 2(1 + o-'2) L N n + l

0 .~.+1

"

"J At- Old = - 4 x i u p , j

0~.

1

"

(1)inj -- yiuin'j) 10"+1 -~- -~ (1 -b if,i) Lo.+l,

''J -3xOi-l'J

(27) (28)

where A x = xi - x i - 1 ,

A y = y~ - yf_~

~lA

~21d

~N Oy '

~2N @2

O0

020

lo = -@- ,

Lo -

@2,

and A t = t "+~ - t ~ represents the false time step. After some rearrangement, Eqs. (25)-(28) may be expressed in the following form:

•bi,j"+t

:

Fi,j @

G i,j 1~i,j "+~

-}-

S i,j L "+1 ~i,d '

(29)

where ~b represents the functions u, N, and 0. This equation represents the relation between the function and its first two derivatives. The quantities F~d, G~o, and S~ d are known coefficients evaluated at previous time steps (Table 1). It should be noted that Eq. (29) is of a general nature and does not depend on the choice of method used for the spatial integration. Equation (29) combined with cubic spline relations described in [29] may be written in the following tridiagonal form: .rfl~+1 a i , j q,,qn . i , j+1 _ 1 -}- b .~,j.g~,j

(30)

~ n + l 1 = di,j, Jr- Ci,jq)i,j+

Table 1. Coefficients of Eqs. (29) 4

~:

u

,.~ + A t ue.

E

-

N

n

N "~4 . q- A t

I -4x~ur, j

s

--dt(v'~j, -- Yi u"i,j)"

At(1 + R) (1 + G'2)

n

-4xlu74 ui,~ --A Xu i - ~ o

2 + 1--~,~/(u,,j)

G

+ 1 + G'~ + R1N'<

N ~ ' - - NT-I,j t,r -ax

ui"iNinj

u"

- - A t ( v ~ j -- Yi i,j)

At2(1 + a '2)

+ R B ( 4 x : ) :/2 [--(1 + a '2) t,," - 2 N ~ i ] ]

0

~,j + 0~.

At

[

- 4x~u,."s

1

,,s --~xxi t,j"

~ - - A t ( v ~ j _ Yi u id)

At t'r

168

C.-R Chiu and H.-M. Chou

g I

Tw

T~ I

I = ~ sin (21T'~/L)

Fig. 1. Physical model and coordinates where r represents the function (u, N, 0) or its first two derivatives. Equation (30) can be easily solved by use of the Thomas algorithm (Anderson et al. [32]). The singularity at x = 0 has been removed by the scaling. The computation, therefore, can be started at x = 0, and then marched downstream. At every x-station, the computations are iterated until the solution ceases to change significantly, i.e.

i,j - •

-I

(31)

where superscript " denotes time step and r refers to u, N, and 0. The grid dependence of the solution has been tested. In this analysis, the y-grid size was fixed at 0.01, and the x-grid size at 0.025. In this article, numerical results are presented for ~r = c~sin (2~x) to elucidate the geometric effects on the natural convection flow of a micropolar fluid. By simply inputting the appropriate function for a, a similar computation can also be carried out for other shapes of the surface.

4 Numerical results and discussion In order to verify the accuracy of the computer program used in this study, the results obtained for Newtonian fluid (i.e. R = 2 = B = 0) are compared with those computed by Yao [33], [34] first. For the case of the flat plate (i.e. c~ = 0), the value of local heat transfer rate is the same at every x-station and equal to 0.5708 which is in good agreement with 0.5671 obtained by Yao. The behaviours of u, v and 0 for c~ = 0.1, furthermore, are also in favorable agreement with the results yielded by Yao.

169

Free convection in the boundary layer flow 0.30 t

o.2 !

a =O.I a =0.2

A

* Yao

[33]

0.20 sNewtonian, 0.15

x=1.75

R=I, x=t.75 /

0.10

jR=5,

", /

x=l.75 ~R=5.

x=I.50

0.05

0.00

5.00

10.00

15.00

20.00

Y Fig. 2. Axial velocity profiles for various values of the micropolar parameter R and amplitude-wavelength ratio a with 2 = 5 and Pr = I N u m e r i c a l results have been o b t a i n e d for the surface described by cr = ~ sin (2nx) for a m p l i t u d e - w a v e l e n g t h ratios of 0.1 a n d 0.2 (i.e. c~ = 0.1 a n d 0.2). The velocity a n d t e m p e r a t u r e d i s t r i b u t i o n for various values of R, x, a n d c~are calculated t a k i n g the m a t e r i a l p a r a m e t e r s B = 1 a n d P r a n d t l n u m b e r P r = 1 t h r o u g h o u t the study. 1.00 a =0.1 a =0.2 *

Yao

[33]

0.75

x=1.75 x=1.75 0.50

0.25

x=1.75

~ '

~

\x x

0.00 0.00

2.00

4.00

6.00

8.00

-r'r-~, 10.00

Y Fig. 3. Temperature distribution for various values of the micropolar parameter R and amplitudewavelength ratio ~ with 2 = 5 and Pr = 1

170

C.-R Chiu and H.-M. Chou 3.00

x=l.5 a =0.i a =0.2 * Yao [33]

2.00

I

/,~/R=5

J..00

R=I

o

i.o fluid >

0.00

t

~,\~//\//

\\U, ,r

-1.00

-2.00

-3.00

10.00

0.00

30.00

20.00

Y 1.00 x=1.75 a =0.1 a =0.2 Yao

[33]

0.00 ,/~Newtonian

fluid

R=I

,oo \ ~, ~ - ~ -2.00

l\ /

~/,?-

-

=

=

=

-

.....

/ /

1

/

/ \

/

-3.00 ......... O .00

-~-, . . . . . . . . i0.00

~ ......... 20.00

~

b

30.00

Y Fig. 4. Normal velocity profiles for various values of the micropolar parameter R and amplitude-wavelength ratio c~with 2 = 5 and Pr = 1; a node b trough It is worthwhile to point out first that the behavior between the profiles at x = 1.5 (node) and x = 2.0 (node) is similar in this study, and so is between the profiles at x -- 1.75 (trough) and x = 2.25 (crest). F o r convenience, therefore, the cases x = 1.5 and x = 1.75 are only described in this paper. The variations in the velocity c o m p o n e n t u, and temperature profiles with the m i c r o p o l a r p a r a m e t e r R and amplitude wavelength ratio c~ are presented in Figs. 2 and 3, respectively. It is shown that both the h y d r o d y n a m i c and thermal b o u n d a r y layer are thicker near the nodes than near the trough and increase on increasing c~.It should be noticed that at the

Free convection in the boundary layer flow

171

trough station, the geometry effects on the fluid are not very significant. It is also observed that the micropolar fluid has a thicker hydrodynamic and thermal b o u n d a r y layer. The reason for this behavior is that owing to the influence of vortex viscosity, the velocity of micropolar fluid is small, which gives rise to the less heat transfer rate, thus increasing the hydrodynamic and thermal b o u n d a r y layer thicknesses. The effects of micropolar parameter and amplitude wavelength ratio e on the velocity component, v, are described in Figs. 4 a and 4 b. It is clear from Figs. 4 a and 4 b that the velocity component v near the surface is numerically larger at the trough than at the node, especially for a high micropolar parameter R. This is the main reason why the local heat transfer rate near the

Table 2. Local heat transfer rate x

Nu/[Gr\l/4 /~x-x) for 2 = 5 and Pr = 1

Newtonian fluid

1.5000 1.6250 1.7500 1.8750 2.0000 2A250 2.2500 2.375 0 2.5000 2.6250 2.7500 2.8750 3.0000

(node) (trough) (node) (crest) (node) (trough) (node)

R= 1

R= 5

= 0.1

c~= 0.2

c~= 0.1

c~= 0.2

c~= 0.1

c~= 0.2

.535 919 .555911 .562036 .546929 .534797 .557163 .562414 .544 776 .533628 .558064 .562696 .542824 .532453

.464 354 .500265 .523690 .472925 .460098 .499669 .521798 .466142 .455783 .498469 .519818 .459938 .451472

.495136 .515946 .527941 .510805 .494146 .516771 .527990 .508 842 .493093 .517330 .527994 .507056 .492018

.417 692 .452275 .488187 .433311 .413416 .450488 .485029 .426 700 .409076 .448312 .481945 .420620 .404735

.416 704 .437508 .457126 .438241 .415780 .437601 .456665 .436 628 .414871 .437619 .456260 .435163 .413973

.343 070 .374879 .423994 .365157 .338868 .371814 .419431 .359 201 .334739 .368697 .415157 .353661 .330674

a =O.i a =0.2 !.50 - N e w t o n i a n fluid

"7

6

/-a=5

i.O0

Z

H 0.00

~ .........

0.00

j .........

i.O0

i .........

2.00 X

b,,h

3.00

Fig. 5. Averaged heat transfer rate for 2 = 5 and Pr = 1

......

l

4.00

172

C.-R Chiu and H.-M. Chou 2.50

J 4 4

R = 1.0 ............... R = 5.0 . . . . . . . New4co~ia~ fluid

-i~

X-5

X=I

2.00 !

X=10

,

A

1.50 1

1.00

i

0.50

i

0.00

1.00

2.00

~

i

3.00

i

E

i

i

~

h

,

,

i-'l

4.00

Y Fig. 6. Skin friction coefficient for c~= 0.1 and Pr = 1 trough is larger than near the node and the difference is more pronounced for micropolar fluids (as shown in Table 2). It should be noticed again that v is not normal to the wavy surface, but is normal to the x-axis. The total Nusselt number expressed in Eq. (21) is shown in Fig. 5. It is observed that as the micropolar parameter R and amplitude wavelength ratio c~increase, the total Nusselt number decreases constantly. The wavy variation of the total Nusselt number can be observed only near the leading edge, and disappears downstream gradually for c~ = 0.2. The local heat transfer rate defined in Eq. (20) is shown in Table 2 for various values of R and ~. It is clear from Table 2 that the wavelength of local heat transfer rate is only half of that of the wavy surface and the amplitude of local heat transfer rate increases with an increase in ~ or R and an increase in R or e yields a decrease in the local heat transfer rate. In addition, the Newtonian fluid is found to have higher local and averaged heat transfer rates than a micropolar fluid. The skin friction coefficient defined in Eq. (24) is plotted in Fig. 6. As pointed out earlier, under the influence of vortex viscosity, the skin friction increases uniformly with an increase in R, as compared to the Newtonian fluid, in spite of ~. It is found that an increase in )o leads to a decrease in the skin friction, but the sensitivity of 2 to the skin friction coefficient is lessened at nodes and heightened at crest or trough. Further, the effects of R are much more significant than those of 2 to the skin friction. It is clear from Fig. 6 that the wavelength of the skin friction is also only half of that of the wavy surface.

5 Conclusions In this study, the free convection in micropolar boundary layer flow along a vertical wavy surface has been solved by a simple transposition theorem and cubic spline collocation numerical method. It has been found that the effect of increasing amplitude wavelength ratio e results in increasing the hydrodynamic and thermal boundary layer thicknesses and decreasing the local

Free convection in the boundary layer flow

173

heat transfer rate. In addition, the effects of the amplitude wavelength ratio are not very prominent near the trough, irrespective of the values of R and ~. As the micropolar parameter R increases, the heat transfer rate decreases, while the hydrodynamic and thermal boundary layer thicknesses and the skin friction coefficient increase. It is observed that the frequency of the local heat transfer rate is twice that of the wavy surface, irrespective of whether the fluid is a Newtonian fluid or a micropolar fluid. Furthermore, the wavelength of the skin friction is only half of that of the wavy surface.

Appendix F r o m the nondimensionalized Eq. (8), the velocity gradients in Eq. (6.3) can be written as 62 c~ - eL/ 121 G r l / 4 ~ ~ + r ' Grl/2 (0~

~Y~fft-I~oL2 ( -

(3~) - (cr')2 Gr3/4 ~~3~ + ~a,, Grl/2 1

Gr3/r 0 ~ ) .

(A1)

(12)

Introducing Eqs. (A 1 ) - ( 1 2) and Eq. (8) into Eq. (6.3), and ignoring terms of small orders in Gr, gives 1 0fi ,<; = - - ~ (i + a 'z) ~ .

(A3)

Substituting Eq. (18) into Eq. (A 3), we have N = - F _1

(1 + a '2) 0u

References [1] Eringen, A. C.: Theory of micropolar fluids. J. Math. Mech. 16, 1 - 1 8 (1966). [2] Eringen, A. C.: Theory of thermomicrofluids. J. Math. Anal. Appl. 38, 480-496 (1972). [3] Khonsari, M. M.: On the self-excited whirl orbits of a journal in a sleeve bearing lubricated with micropolar fluids. Acta Mech. 81, 235-244 (1990). [4] Khonsari, M. M., Brewe, D.: On the performance of finite journal bearings lubricated with micropolar fluids. STLE Tribology Trans. 32, 155-160 (1989). [5] Hudimoto, B., Tokuoka, T.: Two-dimensional shear flows of linear micropolar fluids. Int. J. Eng. Sci. 7, 515--522 (1969). [6] Lockwood, F., Benchaita, M., Friberg, S.: Study of tyotropic liquid crystals in viscometric flow and elastohydrodynamic contact. ASLE Tribology Trans. 30, 539-548 (1987). [7] Lee, J. D., Eringen, A. C.: Boundary effects of orientation of nematic liquid crystals. J. Chem. Phys. 55, 4509-4512 (1971). [8] Ariman, T., Turk, M. A., Sylvester, N. D.: On steady and pulsatile flow of blood. J. Appl. Mech. 41, l - 7 (1974). [9] Kolpashchikov, V., Migun, N. R, Prokhorenko, R R: Experimental determinations of material micropolar coefficients. Int. J. Eng. Sci. 21, 405-411 (1983). [10] Ariman, T., Turk, M. A., Sylvester, N. D.: Microcontinuum fluid mechanics - a review. Int. J. Eng. Sci. 11, 905-930 (1973). [11] Ariman, T., Turk, M. A., Sylw ster, N. D.: Applications ofmicrocontinuum fluids mechanics. Int. J. Eng. Sci. 12, 273-293 (1974).

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C.-P Chiu and H.-M. Chou: Free convection in the boundary layer flow

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Authors' address: Prof. C.-R Chiu and graduate student H.-M. Chou, Department of Mechanical Engineering, National Cheng Kung University, Tainan, Taiwan, Republic of China