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Dynamic theory of suspensions with Brownian effects. (English) Zbl 0543.76133
This paper considers a suspension of particles in a fluid settling under the influence of gravity and dispersing by Brownian motion. A mathematical description is provided by the Stokes equation and a Fokker-Planck equation for the one-particle phase space (position and velocity) density. This nonlinear system depends on a number of empirically determined functions of the spatial concentration of the particles. The stability and asymptotic behavior of the system is analyzed under various scalings. In particular under special assumptions about the particle diffusion coefficient, a Cahn-Hilliard type theory is derived [J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system III, J. Chem. Phys. 31, 688-701 (1959)]. The results are applied to explanation of an observed instability in suspensions.

MSC:
76T99 Multiphase and multicomponent flows
76E30 Nonlinear effects in hydrodynamic stability
76M99 Basic methods in fluid mechanics
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