The effects of inertia on the buoyancy-driven convection flow in settling vessels having inclined walls

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Abstract
The bulk‐averaged equations of motion, which describe the buoyancy driven flow in particle settlers having inclined walls, are solved over a wide range of the relevant parameter space. The solution assumes that the flow is laminar, that the particle Reynolds number is small, that the suspension is monodisperse, and that the settling occurs in a parallel plate vessel whose length to width ratio is not too large. It is shown that, in the asymptotic limit, Λ→∞, where Λ is the ratio of a sedimentation Grashof number to a sedimentation Reynolds number, R, inertial effects in the flow are O(ξ1/6), where ξ is given explicitly in terms of R, Λ, the inclination angle, θ, and the dimensionless distance from the vessel bottom. Using regular and singular perturbation techniques, the asymptotic form of the equations for Λ≫1 are then solved over the entire range of ξ and the solutions are shown to reduce to those given by Acrivos and Herbolzheimer [J. Fluid Mech. 9 2, 435 (1979)] and by Schneider [J. Fluid Mech. 1 2 0, 323 (1982)] in the limits ξ→0 and ξ→∞, respectively. Since typically in practice Λ∼O(106–109), the present solutions give expressions for the velocity profiles and the thickness of the clear‐fluid slit that forms underneath the downward facing vessel wall, which are valid for a wide class of systems of practical interest.

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