The effect of hydrodynamic coupling on the axisymmetric drainage of thin films

The effect of hydrodynamic coupling of adjacent phases on the axisymmetric drainage of thin films is examined using a prototype model of coalescence. For long times, pressure forces in the film dominate flow in all three regions, and finally all move effectively as one, whereas for short times, profiles are sharp and initial flow differences in the three regions can dominate pressure effects. For intermediate times, temporal evolution of velocity profiles depends in a complicated way on the kinematic viscosity ratio and the parameter $\text{R = (?}{}_{\mathrm{A}}\text{μ}{}_{\mathrm{A}}\text{/?}{}_{\mathrm{B}}\text{/μ}{}_{\mathrm{B}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$, as well as on initial conditions and pressure gradient. Generally speaking, the initial flows have less of an effect on overall drainage time than the presence of induced circulation in adjacent phases. Analytical solutions are plotted for a range of systems and representative initial conditions and pressure gradients. In a subsequent article, film-thinning equations are solved using this information.