Stokes flow

Abstract: The forces on a small rigid sphere in a nonuniform flow are considered from first prinicples in order to resolve the errors in Tchen’s equation and the subsequent modified versions that have since appeared. Forces from the undisturbed flow and the disturbance flow created by the presence of the sphere are treated separately. Proper account is taken of the effect of spatial variations of the undisturbed flow on both forces. In particular the appropriate Faxen correction for unsteady Stokes flow is derived and included as part of the consistent approximation for the equation of motion.

Abstract: Steady-State Solutions of the Navier-Stokes Equations: Statement of the Problem and Open Questions.- Basic Function Spaces and Related Inequalities.- The Function Spaces of Hydrodynamics.- Steady Stokes Flow in Bounded Domains.- Steady Stokes Flow in Exterior Domains.- Steady Stokes Flow in Domains with Unbounded Boundaries.- Steady Oseen Flow in Exterior Domains.- Steady Generalized Oseen Flow in Exterior Domains.- Steady Navier-Stokes Flow in Bounded Domains.- Steady Navier-Stokes Flow in Three-Dimensional Exterior Domains. Irrotational Case.- Steady Navier-Stokes Flow in Three-Dimensional Exterior Domains. Rotational Case.- Steady Navier-Stokes Flow in Two-Dimensional Exterior Domains.- Steady Navier-Stokes Flow in Domains with Unbounded Boundaries.- Bibliography.- Index.

Abstract: Stokes flow through a rigid porous medium is analyzed in terms of the method of volume averaging. The traditional averaging procedure leads to an equation of motion and a continuity equation expressed in terms of the volume-averaged pressure and velocity. The equation of motion contains integrals involving spatial deviations of the pressure and velocity, the Brinkman correction, and other lower-order terms. The analysis clearly indicates why the Brinkman correction should not be used to accommodate ano slip condition at an interface between a porous medium and a bounding solid surface.