hr. J. Engng Sci. Vol. 22, No. Printed in Great Britain.
I, pp. 47-55,
002&7225/84 S3.00 + .OO 0 1984 Pergamon Press
HEAT TRANSFER CHARACTERISTICS OF A MICROPOLAR BOUNDARY LAYER IN A CROSSFLOW OVER A NON-ISOTHERMAL CIRCULAR CYLINDER RAMA SUBBA REDDY GORLAt Department of Mechanical Engineering, Cleveland State University, Cleveland, OH 44115, U.S.A. Abstract-A boundary layer analysis has been presented for the heat transfer characteristics of a micropolar fluid Bowing past a circular cylindrical surface with non-isothermal boundary conditions. The solution of the energy equation is obtained in the form of a power series of the distance measured from the stagnation point. The results for the Nusselt number have been obtained for several values of the material parameters of the fluid. A discussion has been provided for the effect of non-isothermal wall on the heat transfer rate.
has formulated the theory of micropolar fluids. This theory includes the effects of local rotary inertia and couple stresses and provides a mathematical model for the non-Newtonian behavior observed in polymeric fluids, animal blood, etc. Peddieson and McNitt  applied the micropolar boundary layer theory to the problems of steady stagnation point flow, steady flow over a semi-infinite plate and impulsively started flow past an infinite flat plate. Gorla investigated the steady boundary layer-flow of a micropolar fluid at a two-dimensional stagnation point on a moving wall and demonstrated that the micropolar fluid flow model is capable of predicting results which exhibit turbulent flow characteristics. The theory of thermomicropolar fluids has been developed by Eringen by extending the theory of micropolar fluids[l]. Gorla studied the thermal boundary layer flow of a micropolar fluid at a stagnation point by making use of Eringen’s theory. In the present paper, we have investigated the heat transfer characteristics of a micropolar fluid boundary layer flow past a circular cylindrical surface with non-isothermal wall temperature. The wall temperature is assumed to be a power series of the distance from the stagnation point along the cylindrical surface. The problem corresponding to the isothermal wall has been studied by Mathur et al.. ERINGEN [ I]
The flow development and the coordinate system are shown in Fig. 1. Assuming a steady, incompressible, laminar, micropolar fluid flow with constant properties, the governing conservation laws for a cylinder situated in a uniform crossflow may be written as Mass:
(1+Ky*)2’ax*Z+c?y*2+(1+Ky*)2’~ tProfessor. 47
R. S. R. GORLA
Fig. 1. Flow development and coordinate system.
(2) 1 (,~~~y*~~~~_K~*~+~*~~:-_ .-a + (1 + KY*)~ dx*
(1 + KY*)’
+ ay*2 *
- (I +‘Ky*)’
- (1 + Ky*)“‘dx
azg* ag* .-_
au* .-___ ax*
K~* (1 +Ky*)
1+Re 1 N;
a0 ag* a a0 ag* -.- _.ay* ax* + KY*) [ ax* ay* 1 1 a28 a% -+Pr.Re 1 (1 + Ky*)‘ax*’ + ay*2 +b(i
-(I Here, K = surface curvature.
(1 + K~*)~‘d,i’@
a8+v* (1 -i-Ky*)ax*
--. 1 Re
Angular momentum: 1 *ag* (~+KY*)~ ax*+”
au* K dK dx* + (I+ Ky*)ay*
dK 86 +yKy*)3dx*‘dx’f
Heat transfer characteristics of a micropoiar boundary layer
In the above equations, u * and u * are the velocity components in x * and y * directions, g* the component of microrotation, T the fluid temperature, h the viscosity coefficient, & the vortex viscosity coefficient, yvthe spin gradient viscosity coefficient,j the microinertia density, a* and K, the coefficients of heat conduction, C, the specific heat, p the density, Re the Reynolds number, E the Eckert number, Pr the Prandtl number, the quantities NT, N:, Nf and a the micropolar fluid parameters characterizing vortex viscosity, microinertia, spin gradient viscosity and micropolar heat conduction respectively and To the temperature at the stagnation point. We now define Y =y*,
e = l/,/(Re)
N2 = j/(L’e’),
N, = KIA, a = a * U,I(p,LCp),
Nj = Y”I(AL~~~) E = Um2/C,,(To- T,)
Re = PU,LIA, 8 = (T - Tm)/(To - T,).
Pr = A. CJK,
Substituting expressions in (6) into eqns (l)-(5) and collecting the coefficients of order unity, the governing equations within boundary layer approximation as c-0, may be written as
ae ag _.___._
( ax ay
0 ay a28
a0 ag ay ax (10)
conditions are given by
8 = fgx) g-+0, e+o: AND
The inviscid external velocity U(x) can be expressed as a power series of the coordinate along the cylindrical surface, x in the following way. U(x) = a, +
u3x3+ &X5+ u,x7+ . . .
where a, = 2; a3 = - 2/3!; a5 = 2/5!; a7 = - Z/7!. Proceeding with the analysis, we let
ES Vol. 22. No. I-D
R. S. R. GORLA
8 = CU(zj-
Primes here indicate differentiation with respect to q only. It is assumed that the wall temperature is described by a power series of x in the form
e,(x) = 1 + b2x2 + b4x4 + b,x6 + . . .
For isothermal wall boundary condition, we have b2’= b4 = . . . = 0. We further note that the functionsf,,f,,
g,, g,, t&, 0, and 0, in eqn (8) may be written as
etc. Upon substituting expressions (12)-( 16) into the governing boundary layer eqns (7)-( 10) and equating the coefficients of like powers of x, we obtain a set of ordinary differential equations. These are not reproduced here in the interest of conserving space. The governing boundary conditions for this set of differential equations are given by Velocity field:
h(O)= f;(o) =
f;(a) = 1,
= 0, fW)
g,(a) = 0
= 1, g,(a) = 0
/Is(O) = h;(O) = L,(O) = 0, h;(m) = 1, L,(oo) = 0
K,(O) = K;(O) = M,(O) = 0, K;(a)) = 0, M,(co) = 0 h7(0)
1, &(co) = 0 0,
j,(O) = j;(O) = N,(O) = 0, j;(m) = 0, Temperature
N,(co) = 0.
e,(o) = +, A,(O) = B,(O) = 0, ~~(0)= 1, A,(O) = &(O) = C,(O) = 0, us(O)= 1, u,(O) = 0, e,(a) = A,(W) = ~~(4 = Us = A,(W) = B,(oo) = c,(o0) = tlj(cO)= ~~(00)= 0.
etc. The governing equations for the velocity and temperature functions have been solved on
Heat transfer characteristics of a micropolar boundary layer
F&. 2. Distribution of velocity functions.
the IBM 370 computer using the fourth-order, Runge-Kutta numerical integration procedure in conjunction with shooting techniques. The double prekision arithmetic was used in all the computations. A step size of Aq = 0.001 was selected. We have assumed here that Pr, N,, N2, N3, a and E are prescribable parameters. RESULTS
The numerical results for the distribution of velocity functions, angular velocity functions and temperature functions have been illustrated in Figs. 2-4 respectively with N, = 4.5, N2 = 9 and NS = 13.5 only for the sake of brevity. The local rate of heat transfer from the wall q&t) and the local heat transfer coefficient
Fig. 3. Distribution of microrotation
R. S. R. GORLA
Pr=lO a=0 E=O
Fig. 4. Distribution of thermal functions.
h(x) may be written as qJx*)
= - K;
The local Nusselt number may be written as c.Nu xc
u5x4 A;(O) +-g
+... . +b,.;(0)+-$.6,-4(0) 1
It may be noted in eqn (20) that a, = 2, u3 = - 2/3! and a5 = 2/S!. The values of 0;(O), A;(O), B;(O), u;(O), A;(O), B;(O), C;(O), u;(O) and u;(O) for a wide range of prescribed values of the material parameters and thermal parameters of the fluid have been computed numerically but will not be shown here to conserve space. By setting b2 = b4 = 0 in eqn (20), we obtain the heat transfer rate from the cylinder with isothermal surface. By varying b2 and b4 we will obtain the heat transfer rate under nonisothermal surface boundary conditions. Figures 5-9 illustrate the heat transfer rate variation vs the angle measured from the stagnation point for several combinations of b2and b4. In all these figures, the Prandtl number was set to be equal to 10. It may be seen that by choosing negative values for b,, the heat transfer rate around the cylinder decreases relative to the isothermal wall case (b2 = b4 = 0). Positive values of both b2 and b4 yield heat transfer rates that are higher than the isothermal wall heat transfer. This information is very important in the design of industrial equipment where the augmented heat transfer rates
Heat transfer characteristics of a micropolar boundary layer
Fig. 5. Heat transfer rate around the circular cylinder with non-isothermal
Fig. 6. Heat transfer rate around the circular cylinder with non-~~th~al
.64 . .30 1 0
7. Heat transfer rate around the circular cylinder with non-isothermal
R. S. R. GORLA
N,= 4.5 1.25
Fig. 8. Heat transfer rate around the circular cylinder with non-isothermal
may be desirable. By controlling the surface temperature properly (i.e. by choosing proper values of b2 and b4), it is possible to maintain the desired level of heat transfer rate augmentation. We have set c1= 0.1, E = 0.01 and Pr = 10 in Figs. 5-8. From these figures, we observe that as N, increases the heat transfer rate increases. An increase in the value of N3 tends to reduce the heat transfer rate. Increasing values of N2 tend to reduce the heat transfer rate. For a particular shape of the microelements, the parameters N1, IV2 and N3 only signify the concentration and size of the microelements. In this context, small value of N, means low concentration and large value means high concentration for the given values of N2 and N3. Similarly for a given N,, namely, for a given concentration small values of N2 or N3 are related to small size of the particles and large values to large size of the particles. From the numerical results obtained in this paper, it was noted that, with the increase of N,, namely, with the increase of concentration of particles of a given shape, the skin friction parameter decreases whereas the gradient of microrotation of the surface increases. On the other hand, for a given concentration of the particles that is for a fixed value of N, the increase in particle size which is characterized by the increasing values of N2 or N3 the skin friction parameter increases while the gradient of microrotation on the surface decreases. The results in Fig. 9 correspond to a zero value for the Eckert number. Thus, it is observed that inclusion of the viscous dissipation terms does not significantly alter the heat transfer rate around the cylinder. It has been observed from the results that the effect of the micropolar fluid parameters is to increase the temperature within the boundary layer and to decrease the heat transfer coefficient relative to a Newtonian fluid.
Fig. 9. Heat transfer rate around the circular cylinder with non-isothermal
Heat transfer characteristics of a micropolar boundary layer NUM~NC~AT~R~
ismrt mEnher dimensionless veWty dimensionless microrotation microinertia per unit mass thermal conductivity vczVzx viscosity coefficient materiai parameters Prandtg number heat fmx Reynolds number temperature velocity in x-direction velocity in y-direction distance dong the surface distance ncumal to the surf&e spin gradient viscosity coe&ient [email protected]
) dimensionless coordinate dimensionless temmrature viscosity coefficient density of’the &lid a&e measuzxd fram the stagnation pi&t conditions at the stagnation point surface conditions conditions far away from the surface