The flow fields in and around a droplet moving axially within a tube

Abstract

A solution is presented for the flow field in and around a single spherical droplet or bubble moving axially at an arbitrary radial location, within a long circular tube. In the tube there is viscous fluid flowing with a constant Poiseuillian velocity distribution far from the droplet.The settling velocity of the droplet or bubble is \begin{eqnarray*} U = \frac{2(\rho_i-\rho_e)ga^2}{9\mu_e}\frac{1+\alpha}{\frac{2}{3}+\alpha}\left[1-\frac{2+3\alpha}{3(1+\alpha)}\left(\frac{a}{R_0}\right)f\left(\frac{b}{R_0}\right)\right]+U_0\left[1-\left(\frac{b}{R_0}\right)^2\right.\\ \left. - \frac{2\alpha}{2+3\alpha}\left(\frac{a}{R_0}\right)^2\right] + O\left(\frac{a}{R_0}\right)^3. \end{eqnarray*} This is a general equation and it contains as special cases the familiar solutions of Stokes, Hadamard-Rybczynski, Brenner & Happel, Greenstein & Happel and Haberman & Sayre.The function describing the deviation of the interface from sphericity is solved and an iterative procedure is suggested for obtaining higher order solutions.