Coagulation of Bubbles in a Laminar Flow

In a laminar flow, the coagulation constant K is proportional to the capture cross section of bubbles of volume to by a bubble of volume V, while in a turbulent flow, it is proportional to the flux of bubbles of volume w towards the test bubble of volume V. 

Let us restrict ourselves to the case of fast coagulation, assuming that each collision of bubbles results in their coalescence. The study of mutual approach of bubbles in a laminar flow is based on the analysis of trajectories of their relative motion. The equations of non-inertial motion of a bubble of radius a relative to a bubble of radius h in the quasi-stationary approximation are  

Consider the interaction of bubbles with free surface. Since the velocity of a bubble has two components directed along the line of centers and perpendicular to it, then the hydrodynamic force Fj, has also two components. At relatively large distances between bubbles, the force Fj, is not much different from Stokes force -AufiiaU, where U is the velocity of the flow far from the drops. At small distances, the second component still does not significantly deviate from its Stoke-sean counterpart, while the component directed along the line of centers, differs essentially from a Stokesean component. Therefore F/, could be presented as 

Here Fhs is the Stokesean force Fhe is the force of the central approach of bubbles, which differs from the Stokesean force and takes into account the influence 

All the forces and, consequently, the trajectories of bubble motion lie in the plane that passes through the line of centers and is parallel to the flow velocity 

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Note that, as opposed to the coagulation of bubbles in a laminar flow, coagulation in a turbulent flow is impossible in the absence of molecular forces, since Dt- at<5=r a )—>0. Therefore the integral term entering the solution of the diffusion equation has a nonintegrable singularity that can be eliminated only by the introduction of a molecular attraction force growing as at > 0. 

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