Using the theory of micropolar fluids due to Eringen, asymptotic boundary layer solutions are presented to study the combined convection from a vertical semi-infinite plate to a micropolar fluid. Consideration is given to the region close to the leading edge as well as the region far away from the leading edge. Numerical results are obtained for the velocity, angular velocity and temperature distribution. The missing wall values of the velocity, angular velocity and thermal functions are tabulated. Micropolar fluids display drag reduction and reduced surface heat transfer rate when compared to Newtonian fluids

Gorla, Rama Subba Reddy

Lin, Paul P.

Yang, An-Jen J.

English

EngagedScholarship @ Cleveland State University

Cleveland State University
EngagedScholarship@CSU
Mechanical Engineering Faculty Publications Mechanical Engineering Department
1990
Asymptotic Boundary-Layer Solutions for Mixed
Convection from a Vertical Surface in a Micropolar
Fluid
Rama Subba Reddy Gorla
Cleveland State University, r.gorla@csuohio.edu
Paul P. Lin
Cleveland State University, p.lin@csuohio.edu
An-Jen J. Yang
Cleveland State University
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publication. A definitive version was subsequently published in International Journal of Engineering
Science, 28, 6, (01-01-1990); 10.1016/0020-7225(90)90054-M
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Original Citation
Gorla, R., Lin, P., , & An-Jen, Y. (1990). Asymptotic boundary layer solutions for mixed convection from a vertical surface in a
micropolar fluid. International Journal of Engineering Science, 28(6), 525-533. doi:10.1016/0020-7225(90)90054-M
ASYMPTOTIC BOUNDARY LAYER SOLUTIONS FOR  
MIXED CONVECTION FROM A VERTICAL SURFACE IN  
A MICROPOLAR FLUID  
RAMA SUBBA REDDY GORLA, PAUL P. LIN and AN-JEN YANG 
Department of Mechanical Engineering, Cleveland State University, Cleveland, OH 44115, U.S.A. 
Abstrad-Using the theory of micropolar fluids due to Eringen, asymptotic boundary layer solutions 
are presented to study the combined convection from a vertical semi-infinite plate to a micropolar 
fluid. Consideration is given to the region close to the leading edge as well as the region far away from 
the leading edge. Numerical results are obtained for the velocity, angular velocity and temperature 
distribution. The missing wall values of the velocity, angular velocity and thermal functions are 
tabulated. Micropolar fluids display drag reduction and reduced surface heat transfer rate when 
compared to Newtonian fluids. 
INTRODUCTION 
Mixed convection from a semi-infinite vertical surface in Newtonian fluids has been analyzed by 
many investigators [1-3]. The two important circumstances that arise are due to the buoyancy 
forces aiding or opposing the forced flow. In the case of buoyancy-opposed regime, an adverse 
pressure gradient is caused by the buoyancy force. As a result, the external flow separates and 
reverses direction at some location downstream. Since similarity solutions are not possible for 
this and for many mixed convection flows, perturbation and finite difference methods are often 
used. 
Although the previously mentioned investigations have been carried out for the Newtonian 
fluids, there exist relatively fewer studies concerning non-Newtonian fluids with microstructures 
such as polymeric additives, colloidal suspensions, animal blood, liquid crystals, fluid with 
additives etc. Eringen [4] has developed the theory of micropolar fluids which show 
microrotatation effects as well as micro-inertia. The theory of thermomicropolar fluids was 
developed by Eringen [5] by extending the theory of micropolar fluids. Gorla [6] studied the 
thermal boundary layer of a micropolar fluid at a stagnation point using the theory of 
micropolar fluids due to Eringen. The forced convective heat transfer characteristics of the 
micropolar fluid flow over a flat plate have been investigated by Gorla [7]. Jena and Mathur [8] 
obtained a similarity solution for the laminar free convective flow of a thermomicropolar fluid 
along a vertical flat plate. 
In this paper, we have considered the mixed convection in the micropolar boundary layer 
flow on a vertical flat plate maintained at a uniform temperature Tw. A uniform laminar, 
incompressible, steady, micropolar fluid stream at velocity U"" and temperature T", flows 
parallel to the surface. A Blausius flow dominates near the leading edge where the buoyancy 
effects appear only as a perturbation of the imposed flow U"". Far downstream of the leading 
edge, for aiding effects, the flow is dominated by the buoyancy forces and the forced flow effect 
appears as a perturbation. Asymptotic boundary layer solutions are derived for the two 
regions, near as well as far away from the leading edge. Numerical results are presented for a 
range of values of the dimensionless material properties and Prandtl number of the fluid. 
ANALYSIS 
The governing equations for a steady, laminar, incompressible, micropolar fluid over a 
semi-infinite vertical flat plate with variable micro-inertia may be written within boundary layer 
approximation as: 
au av 
-+-=0 (1)ax ay 
525 
2 
au au) a u aN p (u- + v-;- = (tt + K)-2 + K-+ pgf3(t - Too) (2)ax oy ay ay 
2 
pj(u aN + v aN) = _ K(2N + au) + y a N (3)
ax ay ay ay2 
2
aT a~ a TpCp [u-+v- =Kr-2 (4)
ax ay ay 
The boundary condition for the velocity field may be written as 
y = 0: u = v = 0, N = 0 
(5)
y_oo: u- U"" 
For the temperature field, we have 
y =0: T= Tw 
(6) 
Proceeding with the analysis, we define a stream function 1jJ such that 
a1jJ 
u=­
ay
and 
a1jJ 
v= --
ax 
Near the leading edge solution 
In this region, forced flow dominates since the buoyancy force has acted over a small vertical 
distance. We now define the following dimensionless variables: 
Uoo )112
T}= - y( 2vx 
f: =gf3(Tw- Too)x = Grx 
'=' U~ Re; 
1jJ = (2VUooX)1I2f(g, T}) 
U )112
N = ( 2v: .U,,· g(g, T}) 
(7) 
8= T-T.., 
Tw - Too 
After substituting expressions in equation (7) into equations (2)-(4), we have: 
(1 + il.)!", + ff" + il.g' ± 2g8 = 2;[f' af - f" af] (8) 
ag ag 
AGg" - 2A(2g +f") + G(f'g + fg') = 2g;[g' af - f' ag ] (9)a; a; 
8" + f8' =2;[f' a8 _ 8' af] (10)
Pr a; a; 
In the above equations, a prime denotes differentiation with respect to TJ only. The transformed 
boundary conditions are given by: 
f(;, 0) = f'(;, 0) = 0, f'(;, 00) = 1 
g(;, 0) = g(;, 00) = 0 (11) 
8(;,0) =1, 8(;,00) =0 
In order to solve the governing conservation equations, we assume that 
f(~, 1]) = /0(1]) + ~ ..ft(1]) + ~2. 12(1]) + .. . 
g(~, 1]) = go(1/) + ~. g\(11) + ~2. g2(1]) + .. . (12) 
0(;,1]) = 00(11) +;·0\(1]) +;2. O2(1]) + .. . 
When the expressions in equation (12) are substituted into equations (8), (9) and (10) and 
terms involving equal powers of ~ and equated to zero, one obtains a set of ordinary 
differential equations governing the momentum, angular momentum and energy fields. These 
details are not shown here in order to conserve space. Equations governing /;( '/]), g;( '/]) and 
O;('/]) have been solved on an IBM 370 computer using the fourth order Runge-Kutta method 
of numerical integration procedure. The double precision arithmetic was used in all the 
computations. A step size of ~'/] = 0.001 was selected. The missing wall values 17(0), g;(O) and 
8:(0) were determined by shooting techniques. 
The shear stress at the surface of the plate is given by: 
11 )112
iw = flUc>O ( ~,; . (1 + 1\).1"(;,0) (13) 
The friction coefficient is given by 
Cf = (P~~/2) = (1 +~) . Vi, 1"(;,0) . Re;1I2 (14) 
where Rex = (pUoox / fl). The local heat flux may be written by Fourier's law as 
aT (Pl1 )112
qw(X) = - Kf • ay = - Kf • (Tw - Too) 2'; . 8'(~, 0) (15) 
The local heat transfer coefficient is given by 
(16)  
The local Nusselt number can be written as 
(17)  
Figures 1-3 display the numerical results obtained for the distribution of functions 
representing velocity, angular velocity and temperature within the boundary layer. The values 
1.0 r--------=------------, 
A=O.l 
G =1 
.\ =0.5 
Pr= 10 
4 6 8 
'1 
Fig. 1. Distribution of velocity functions (near the leading edge region). 
0.2r---------------------------------------~ 
0.1 
o 
-0.1 
11 = 0.1 
G =1 ,,= 0.5 
Pr = 10 
-O.2L-__~~__~__~L____L____~__~L____L__~ 
o 2 4 6 8 
'1 
Fig. 2. Distribution of angular velocity functions (near the leading edge region). 
0.15 r-------------------------. 
11 = 0.1 
G = 10.10 ,,= 0.5 
Pr =10 
o 	 2 4 6 8 
'1 
Fig. 3. Distribution of temperature functions (near the leading edge region). 
Table 1. Values of I~(O). r;(O). 1:;(0) and 1'3(0) (near the leading edge solution) 
!!. 
0.00 
0.00 
0.00 
0.00 
0.10 
0.10 
0.10 
0.10 
0.15 
0.15 
0.15 
0.15 
G 
1. 
2. 
1. 
2. 
1. 
2. 
1. 
2. 
1. 
2. 
1. 
2. 
}. 
0.5 
0.5 
5.0 
5.0 
0.5 
0.5 
5.0 
5.0 
0.5 
0.5 
5.0 
5.0 
1:;(0) 
0.469250 
0.469530 
0.469540 
0.469540 
0.438820 
0.439990 
0.440000 
0.440000 
0.425000 
0.426650 
0.426088 
0.427028 
nO) 
0.974180 
0.975056 
0.975046 
0.975046 
0.908811 
0.898568 
0.907281 
0.909303 
0.879736 
0.859106 
0.877778 
0.880046 
[2(0) 1;(0) 
-0.130830 0.112333 
-0.140198 0.122020 
-0.140951 0.119485 
-0.154389 0.133200 
-0.124322 0.110093 
-0.184024 0.176984 
-0.124868 0.103236 
-0.138013 0.116560 
-0.108626 0.089230 
-0.100808 0.079257 
-0.118327 0.096894 
-0.130874 0.109387 
Table 2. Values of g7.(O). g~(O). g;(O) and g;(O) (near the leading edge solution) 
a G A. g~(O) g;(O) g,(O) g;(O) 
0.00 I. 0.5 -0.482950 -0.403250 0.00 0.00 
0.00 2. 0.5 -0.347348 -0.603250 0.00 0.00 
0.00 I. 5.0 -0.453295 -0.397105 0.102971 -0.090995 
0.00 2. 5.0 -0.307613 -0.207842 0.066252 -0.060152 
0.10 1. 0.5 -0.463894 -0.389483 0.00 0.00 
0.10 2. 0.5 -0.383425 -0.384325 0.00 0.00 
0.10 I. 5.0 -0.435376 -0.379945 0.09346 -0.080537 
0.10 2. 5.0 -0.295924 -0.200578 0.061013 -0.054186 
0.15 I. 0.5 -0.454758 -0.382245 0.093466 -0.080537 
0.15 2. 0.5 -0.329922 -0.382245 0.093466 -0.080537 
0.15 I. 5.0 -0.426355 -0.372491 0.089584 -0.076455 
0.15 2. 5.0 -0.290637 -0.197197 0.058640 -0.051518 
Table 3. Values of O~(O). 0;(0). 0;(0) and 0;(0) (near the leading edge solution) 
a G A. O~(O) 0;(0) 0;(0) 0;(0) 
0.00 I. 0.5 -1.024740 -0.070756 0.005053 -0.055079 
0.00 2. 0.5 -1.024365 -0.070894 0.054542 -0.055247 
0.00 I. 5.0 -1.024375 -0.063622 0.047328 -0.048119 
0.00 2. 5.0 -1.024375 -0.063057 0.044775 -0.045969 
0.10 I. 0.5 -1.007875 -0.069413 0.053271 -0.053723 
0.10 2. 0.5 -1.007675 -0.128509 0.084433 -0.093211 
0.10 1. 5.0 -1.008250 -0.061911 0.044327 -0.043702 
0.10 2. 5.0 -1.006875 -0.061425 0.042253 -0.042257 
0.15 I. 0.5 -1.000000 -0.068714 0.050676 -0.048300 
0.15 2. 0.5 -0.999688 -0.066956 0.046181 -0.041736 
0.15 I. 5.0 -1.000125 -0.061065 0.043063 -0.041952 
0.15 2. 5.0 -0.998844 -0.060634 0.041077 -0.040564 
of [~(O), nCO), [2(0) and 13(0) which are proportional to the friction factor have been tabulated 
in Table 1. Table 2 displays values of g;(O) for i = 0, 1, 2 and 3 which should be useful in the 
calculation of wall couple stress. The values of eh(O), ei(O), e~(O) and e~(O) which are 
proportional to the Nusselt number have been tabulated in Table 3. As /::.. increases, it may be 
observed that both the friction factor and the Nusselt number decrease. The results clearly 
demonstrated that the micropolar fluids display drag reduction as well as heat transfer rate 
reduction, when compared to Newtonian fluids (/::.. = 0). As the buoyancy parameter ; 
increases, the heat transfer rate is augmented. 
Far [rom the leading edge solution 
Far downstream from the leading edge, buoyancy forces dominate the flow. For this case, we 
define 
Gr)m _1J1 =4v 4 x [Po(t]) + ;-1I2~(t]) + ;-1 log ; Fz(t]) + ;-lFz(t]) + ...J(  
N =4U~. ;[So(t]) + ;-1I2S1(t]) + ;-1 log ; Sz(t]) + ;-lSzCt]) + ...J  v 
e=~ -=:. ~ = [Ho(t]) + ;- lI2H1(t]) + ;-1 log ~H2(t]) + ;-lH2(t]) + ...J 
_ x 3gP(Tw - Too)Grx - 2 
V 
Re = Uoox 
x (18)
V 
The overbars indicate variables which are associated with the above transformation are to be 
applied for downstream where the buoyancy effects are dominant. 
After substituting expressions in (18) into equations (2)-(4) we obtain the following ordinary 
differential equations: 
(1 + ~)F'~ - 2(F(J)2 + 3F()F~ + ~S:J + Ho =0 (19) 
(1 + ~)F"i + 3FoF'{ - 2FbF; + F;F3 + ~S; + HI = 0 (20) 
(1 + ~)F'~ + 3FoF~ - F3F2 + ~S; + H2 = 0 (21) 
(1 + ~)F~ + 3FoF~ - F'OF2 + FIF'; + ~S~ + H2 = 0 (22) 
AS'O - (~Al + 4F;)So - ~A2F'; + 3FoSb = 0 (23) 
AS'; - (~AI + 2F~)Sl - 4F;S(1 - ~A2F'; + 3F()S; + F;S(l = 0 (24) 
AS~ - ~A1S2 - 4F;So ~A2F~ + 3F(lS~ F2 S() = 0 (25) 
AS~ - ~AlS2 - 4F~So ~A2F~ + 3F(}S~ - F2 S(1 = 0 (26) 
m; + 3 Pr FoH() = 0 (27) 
H'; + Pr(2F:R, + 3FoH1 + F;H(I) =0 (28) 
H~ + Pr(3Fofi~ F2H() + 4Fbfi2) =0 (29) 
H~ + Pr(F1H; + 2F;Hl FiH;) + 3F1H~ + 4F(R2):::::: 0 (30) 
The appropriate boundary conditions are given by 
F()(O) = F;(O) =Fi(O) = Fi(O) =0 
Fb(O) :::::: Fi(O) :::::: F2(0) = F~(O) 0 
So(O) :::::: SI(O) SzeO) :::::: S2(0) =0 
Ho(O) =1, H1(O) =H2(0) = H2(O) = 0  
Fb(oo) =0, F;(oo) = 1/2, F;(oo) = F;(oo) = 0  
So(oo) = S1(00) S2(00) = Sioo) 0  
Ho(oo) = H 1(00):::::: H2(00) = Hz(oo) = 0 (31) 
The local shear stress and heat transfer at the surface are obtained as: 
411(1 + ~)v (Gr )3/4
i",:::::: I""' x 4 x [F3(0) + ~-1I2F';(0) + ~-Ilog ~ F~(O) + ~-IF2(0) + .. ·1 (32)2 
(33) 
Figures 4-6 show the results for the velocity, angular velocity and temperature functions for 
this region. Tables 4-6 display the missing wall functions for the velocity, angular velocity and 
temperature functions. These quantities are useful in the evaluation of wall shear stress, wall 
couple stress and the surface heat transfer rate. The results indicate that the friction factor and 
heat transfer rate decrease as ~ increases. Micropolar fluids display reduction in drag as well as 
heat transfer rate when compared to Newtonian fluids. As the buoyancy parameter increases, 
heat transfer rate gets augmented. 
In order to provide a matching between the two asymptotic solutions, one valid for the 
leading edge proximity and the other valid for far away distances from the leading edge region, 
we have proposed a graphical method. Figure 7 displays the friction factor vs ~ in which the 
two asymptotic solutions from equations (14) and (32) are plotted. Results for the intermediate 
values of ~ are obtained by the proposed matching of the two solutions by a smooth solid 
curve. Figure 8 shows the results for the Nusselt number versus ;. Solutions for small ; and 
large; are obtained from equations (17) and (33), respectively. The solid curve denotes the 
matching between these two solutions. 
0.6~------------------------------------------, 
0.4 
0.8 1.2 
tJ. = 0.1 
" = 0.5 
Pr = 10 
1.6 2.0 
.,  
Fig. 4. Distribution of velocity functions (far away from the leading edge). 
0.2r---------------------------------------, 
5, 
.0.2 
11.=0.5 
Pr = 10 
'O.4~--~--~--~----L---J---~--~--~~--~~ 
o 0.4 0.8 1.2 1.6 2.0 
.,  
Fig. 5. Distribution of angular velocity functions (far away from the leading edge). 
1.0r-----------------------, 
'O.2~_~_~_~_ _L_ _L_ _J_~L__~_~~ 
o 0.4 0.8 1.2 1.6 2.0 
.,  
0.6 
0.2 
tJ. = 0.1 
A = 0.5 
Pr = 10 
Fig. 6. Distribution of temperature functions (far away from the leading edge). 
Table 4. Values of F~(O), F';(O), F;(O) and "';(0) (far away from the leading edge solution) 
/';. ). F~(O) F';(O) F;(O) "';(0) 
0.00 0.5 0.4151000 0.2460000 0.2467601 -4.219E - 04 
0.00 5.0 0.4150000 0.2460005 0.2467601 -4.219E- 04 
0.10 0.5 0.3784970 0.6840744 0.4407601 -3.619E - 04 
0.10 5.0 0.3784970 0.6840745 0.4406601 -3.619E - 04 
0.15 0.5 0.3745670 0.2510004 0.2086601 -3.869E-04 
0.15 5.0 0.3745670 0.2514004 0.2086601 -3.869E-04 
Table 5. Values of S~(O), S;(O), S;(O) and 5;(0) (far away from the leading edge solution) 
/';. ). S~(O) S;(O) S~(O) 5;(0) 
0.00 0.5 -4.154E-10 -3.377E-06 -3.424E - 06 -2.243E-08 
0.00 5.0 -4.154E - 10 -3.377E-06 -3.424E - 06 -2.243E-OS 
0.10 0.5 -1.250E- 03 -S.15SE-03 -2.262E-03 2.500E - 06 
0.10 5.0 -1.245E - 04 -S.551E - 04 -2.235E - 04 2.877E - 04 
0.15 0.5 -1.935E - 04 -S.OOSE-03 -2.406E-03 4.165E - 06 
0.15 5.0 -1.877E -04 -8.719E-04 -2.375E-04 4.325E- 07 
Table 6. Values of H~(O), H;(O), H;(O) and H;(O) (far away from the leading edge solution) 
/';. ). H~(O) H;(O) H;(O) H;(O) 
0.00 0.5 -1.1466373 -7.917E - 02 0.6894727 -1.352E-03 
0.00 5.0 -1.1466373 -7.917E - 02 0.6894727 -1.352E - 03 
0.10 0.5 -1.1046667 1.073500 2.1962214 -1.352E - 03 
0.10 5.0 -1.1046667 1.0735000 2.4191546 -1.352E - 03 
0.15 0.5 -1.1112000 -4.7ooooE - 02 0.2086601 -1.352E - 03 
0.15 5.0 -1.1112000 -4.7ooooE - 02 0.6553018 -1.352E-03 
12 
10 
W 
? B 
<1 
+ 
~ 
<II 6-...... 
1 ~ 
1<1) 
~ 
- 4~ U 
2 
0 
Pr = 1 
- - - - small ~approximation 
-'- large~approximation 
-- proposed curve joining 
I 
the asymptotes \ 
\1 
I  
I  
\ \  
I I  
\ I  
\ \  
\ \  
\ \ /:/
\ \ '/
\ \ 
\ \ 
ij' 
\ \ 
\ ,, , , , 
" " " ',' ~=O II' 
7  
' ...."' .... , (/ , , ,', ~=5  
10- 3 10- 2 10-1 10 10 2 10 3 10' 
Fig. 7. Friction factor vs g. 
CONCLUDING REMARKS 
The theory of micropolar fluids due to Eringen is used to formulate a set of boundary layer 
equations for the mixed convective flow of an incompressible micropolar fluid along a vertical 
plate. Asymptotic solutions are presented for two regions, namely, the region close to the 
leading edge and the region far away from the leading edge. Numerical results are presented to 
illustrate the flow and heat transfer characteristics and their dependence on the material 
properties. The missing values of the velocity, angular velocity and thermal functions are 
tabulated for a range of values of the dimensionless material properties and Prandtl number of 
the fluid. 
10 , ,
'- ,, , Pr = 1 
" , ~ , - - - - small ~approximation 
.... ::: ::: " _. - large ~approximation 
, """ ~ -- proposed curve joining 
,  the asymptotes 
"-~ /d=O 
..  ~=5/~:::::::::_ 
z" ..........:~.  .",
0.1 ": ' , 	 , 
'-
0.01 L...-____'--____'--____'--____-'--____.L: 
10- 3 10- 2 10- 1 	 10 10 2 
Fig. 8. Heat transfer rate vs ;. 
Acknowledgements-The authors are grateful to the reviewers for their constructive comments which have helped in 
the improvement of the quality of the paper. Thanks are due to Professor A. C. Eringen for his helpful advice. 
REFERENCES 
[IJ E. M. SPARROW and J. L. GREG, J. Appl. Mech. 81, 133 (1959).  
[2J A. A. SZEWCZYK, J. Heat Transfer 86, 501 (1964).  
[3J J. H. MERKIN, J. Fluid Mech. 35,439 (1969).  
[4] A. C. ERINGEN, J. Math. Mech. 16, 1 (1966). 
[5J A. c. ERINGEN, J. Math. Anal. Appl. 38,480 (1972). 
[6] R. S. R. GORLA, Int. J. Engng Sci. 18, 611 (1980).  
[7J R. S. GORLA, Int. J. Engng Sci. 21, 791 (1983).  
[8J S. K. JENA and M. M. MATHUR, Int. J. Engng Sci. 19,1431 (1981).  
(Received 30 October 1989) 
NOMENCLATURE 
{,F dimensionless velocity u~ velocity at the boundary layer edge 
g,S dimensionless microrotation x distance along the surface 
Gr Grashoff number y distance normal to the surface 
h 
j 
heat transfer coefficient 
microinertia per unit mass 
1/J 
Il 
stream function 
viscosity coefficient 
L 
N 
characteristic length 
angular velocity 
P 
; 
density of the fluid 
buoyancy parameter 
Nu Nusselt number (J, H dimensionless temperature 
p pressure L\,,1., G dimensionless material properties 
Pr Prandtl number 
q surface heat flux 
Re 
T 
u 
Reynolds number 
temperature 
velocity in x-direction 
Subscripts 
w surface conditions 
v velocity in y-direction 00 conditions far away from the surface 


Asymptotic Boundary-Layer Solutions for Mixed Convection from a Vertical Surface in a Micropolar Fluid

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Asymptotic Boundary-Layer Solutions for Mixed Convection from a Vertical Surface in a Micropolar Fluid

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