Numerical solution of axisymmetric, unsteady free-boundary problems at finite Reynolds number. I. Finite-difference scheme and its application to the deformation of a bubble in a uniaxial straining flow

A brief description of a numerical technique suitable for solving axisymmetric, unsteady free-boundary problems in fluid mechanics is presented. The technique is based on a finite-difference solution of the equations of motion on a moving orthogonal curvilinear coordinate system, which is constructed numerically and adjusted to fit the boundary shape at any time. The initial value problem is solved using a fully implicit first-order backward time differencing scheme in order to insure numerical stability. As an example of application, the unsteady deformation of a bubble in a uniaxial extensional flow for Reynolds numbers is considered in the range of 0.1≤R≤100. The computation shows that the bubble extends indefinitely if the Weber number is larger than a critical value (W>W_c). Furthermore, it is shown that a bubble may not achieve a stable steady state even at subcritical values of Weber number if the initial shape is sufficiently different from the steady shape. Finally, potential-flow solutions as an approximation for R→∞ have also been obtained. These show that an initially deformed bubble undergoes oscillatory motion if W<W_c, with a frequency of oscillation (based on the surface tension time scale) that decreases as Weber number increases and equals zero at the critical Weber number beyond which steady solutions could not be obtained in the earlier numerical work of Ryskin and Leal [J. Fluid Mech. 148, 37 (1984)]. This clearly indicates that the point of nonconvergence of the steady-state numerical problem is, in fact, a limit point in the branch of steady-state solutions.