Particle reflection from a bubble surface

The determination of the particle velocity after its inelastic collision with a bubble is based on the calculation of the energy losses spent in overcoming the viscous resistance during the approach and thinning of the film formed between the bubble and the particle (cf Section 11.2). This may be estimated by taking the ratio between the particle velocity after collision v and its initial velocity (equal to bubble velocity). 

As in Section 11.1, we consider the reflection of a particle from a flat boundary at normal sedimentation. This assumption can be used for a bubble when we are interested in impacts close to the pole, at ,. Thus, we can simplify the expression for the normal flow of liquid setting cosine to unity and assuming that over whole section of the surface at j the length of recoil is characterized by a constant value. In dimensionless form, the equation for calculating the inertia path change due to the opposite motion of the liquid which has a velocity distribution expressed by a linear relationship (differs from a linear second-order differential equation with constant coefficients only due to a variation of Reynolds number for a retarded particle). It reads 

Taking into account the approximation (11.54), we can solve Eq. (11.55) by assigning a constant coefficient at the second-order derivative (minimum and the maximum values, respectively) and by satisfying the initial conditions 

For a supercritical value of K = 1 /12 the roots of the characteristic equation, corresponding to Eq. (11.55), are imaginary. The solution (11.58) describes a damped oscillation. Once the particle has lost velocity, the opposite motion of liquid brings it back to the surface. Therefore, when the distance of the particle from the surface is maximum at time its velocity approaches zero. Based on (11.58), we obtain 

Substituting this value into Eq. (11.59), gives a very long equation for the length of the inertia path. This equation can be simplified when the path is suffieiently short and Eq/3 can be neglected. 

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DUE TO Joint Action of Particle Reflection from a Bubble Surface and Centrifugal Forces... 

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