Analytical approximations for the orientation distribution of small dipolar particles in steady shear flows

Analytical approximations are obtained to solutions of the steady Fokker-Planck equation describing the probability density function for the orientation of dipolar particles in a steady, low-Reynolds-number shear flow and a uniform external field. Exact computer algebra is used to solve the equation in terms of a truncated spherical harmonic expansion. It is demonstrated that very low orders of approximation are required for spheres but that spheriods introduce resolution problems in certain flow regimes. Moments of orientation probability density function are derived and applications to swimming cells in bioconvection are discussed. A separate symptotic expansion is performed for the case in which spherical particles are in a flow with high vorticy, and the results are compared with the truncated spherical harmonic expansion. Agreement between the two methods is excellent.