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

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 {eqnarray*} U = 2(ρ_i-ρ_e)ga^2/9μ_e{1+α}{2/3+α}[1-2+3α/3(1+α)(a/R_0)f(b/R_0)]+U_0[1-(b/R_0)^2.\\ . - 2α/2+3α(a/R_0)^2] + O(a/R_0)^3. {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.