Comparison of differential representations for radially symmetricStokes flow

[摘要] Papkovich and Neuber (PN), and Palaniappan, Nigam, Amaranath, andUsha (PNAU) proposed two different representations of thevelocity and the pressure fields in Stokes flow, in terms ofharmonic and biharmonic functions, which form a practical toolfor many important physical applications. One is theparticle-in-cell model for Stokes flow through a swarm ofparticles. Most of the analytical models in this realm considerspherical particles since for many interior and exterior flowproblems involving small particles, spherical geometry provides avery good approximation. In the interest of producingready-to-use basic functions for Stokes flow, we calculate thePNAU and the PN eigensolutions generated by the appropriateeigenfunctions, and the full series expansion is provided. Weobtain connection formulae by which we can transform any solutionof the Stokes system from the PN to the PNAU eigenform. Thisprocedure shows that any PNAU eigenform corresponds to acombination of PN eigenfunctions, a fact that reflects theflexibility of the second representation. Hence, the advantage ofthe PN representation as it compares to the PNAU solution isobvious. An application is included, which solves the problem ofthe flow in a fluid cell filling the space between two concentricspherical surfaces with Kuwabara-type boundary conditions.

[学科分类] 数学（综合）