Steady-State Motion of Particles and Drops in a Fluid

Steady-State Motion of Particles and Drops in a Fluid 

In chemical technology one often meets the problem of a steady-state motion of a spherical particle, drop, or bubble with velocity U in a stagnant fluid. Since the Stokes equations are linear, the solution of this problem can be obtained from formulas (2.2.12) and (2.2.13) by adding the terms Vr = -U cos6 and V = U[ sin 6, which describe a translational flow with velocity U, in the direction opposite to the incoming flow. Although the dynamic characteristics of flow remain the same, the streamline pattern looks different in the reference frame fixed to the stagnant fluid. In particular, the streamlines inside the sphere are not closed. 

By equating the drag force F of the sphere with the difference wa gAp between the gravity and buoyancy forces, one can estimate the steady-state velocity of relative motion of phases (the velocity at which a spherical drop falls or rises) as 

Relations (2.2.16) and (2.2.17) cover the entire range 0 j3 oo of the phase viscosity ratio. In the limit cases /3 = 0 (a gas bubble in a highly viscous liquid) and j3 - oo (a solid particle in a fluid), these formulas become 

The last expression for Cf is known as the Stokes law for the drag coefficients of solid spherical particles. This law is confirmed by experiments for Re 0.1. The drag law (2.2.18) for spherical bubbles holds only for extremely pure liquids 

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