Drag force acting on a bubble in a cloud of compressible spherical bubbles at large Reynolds numbers

We have derived the expressions for the viscous forces acting on a bubble in a cloud of bubbles by using the approach by Levich. To obtain the dissipation function, an approximate expression for the velocity potential calculated previously by the authors up to order β3 has been used. Here β=b¯/d is a small dimensionless parameter, b¯ is the mean bubble radius and d is the mean distance between bubbles.