Secondary flows induced by the rotation of a sphere or coaxial cones in micropolar fluids

The micropolar fluids like Newtonian and Non-Newtonian fluids cannot sustain a simple shearing motion, wherein only one component of velocity is present. They exhibit both primary and secondary motions when the boundaries are subject to slow rotations. The primary motion, as in Non-Newtonian fluids, characterized by the equation due to Rivlin-Ericksen, Oldroyd, Walters etc., resembles that of Newtonian fluid for slow steady rotation. We further notice that the micro-rotation becomes identically equal to the vorticity present in the fluid and the condition b) of "Wall vorticity" can alone be satisfied at the boundaries. As regards, the secondary motion, we notice that it can be determined by the above procedure for a special class of fluids, namely that for which j0(n2-n3)=4 n3/l2. Moreover for this class of fluids, the micro-rotation is identical with the vorticity of the fluid everywhere. Also the stream function for the secondary flow is identical with that for the Newtonian fluid with a suitable definition of the Reynolds number. In contrast with the Non-Newtonian fluids, characterized by the equation due to Rivlin-Ericksen, Oldroyd, Walters etc., this class of micropolar fluids does not show separation. This is in conformity with the statement of Condiff and Dahler (3) that in any steady flow, internal spin matches the vorticity everywhere provided that (i) spin boundary conditions are satisfied, (ii) body torques and non-conservative body forces are absent, and (iii) inertial and spin-inertial terms are either negligible or vanish identically.