The Plane Parallel Flow of a Density-Stratified Fluid With and Without a Suspension of Particles

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Abstract
The response of a viscous incompressible nondiffusive fluid containing a dilute suspension of small spherical particles and occupying a semi-infinite region bounded by an infinite plate, when the latter is slowly moved parallel to itself in a prescribed manner normal to the direction of stratification, is studied. In particular, when the system density decays exponentially with distance from the plate, the problem is equivalent to the axial flow of the same system in a long tube. When the particle density vanishes, the problem reduces to the unsteady diffusion of vorticity in a long cylinder, with the “diffusivity” a function of the stratification length, whose solution is well known. A degeneracy occurs in the solution as the density approaches uniformity and this is examined using asymptotic expansions and some exact and asymptotic results for an impulsively moved plate are given. It is found that, for a given stratification, the ratio of the wall stress with particles to that without, increases first and then, after the non-equilibrium overshoot, it continues to increase with time; this is in contrast to the case of uniform density, when the ratio goes asymptotically to a constant value. When the particle density vanishes and the kinematic viscosity of the fluid goes to infinity at very large distances from the plate, the fluid there acquires, in a finite time, a finite velocity that strongly depends on the stratification length.

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