Interfacial dynamics in Stokes flow via a three-dimensional fully-implicit interfacial spectral boundary element algorithm

Since the pioneering work of Youngren and Acrivos [G.K. Youngren, A. Acrivos, On the shape of a gas bubble in a viscous extensional flow, J. Fluid Mech. 76 (1976) 433–442] 30 years ago, interfacial dynamics in Stokes flow has been implemented through explicit time integration of boundary integral schemes which require that the time step is sufficiently small to ensure numerical stability. To avoid this difficulty, we have developed an efficient, fully-implicit time-integration algorithm based on a mathematically rigorous combination of implicit formulas with a Jacobian-free three-dimensional Newton method. The resulting algorithm preserves the stability of the employed implicit formula and thus it has strong stability properties, e.g. it is not affected by the Courant condition or by physical stiffness such as that associated with the critical conditions of interfacial deformation. In our work, the numerical solution of our implicit algorithm is achieved through our spectral boundary element method. Our numerical results for free-suspended droplets are in excellent agreement with experimental findings, analytical predictions and earlier computational results at both subcritical and supercritical conditions, and establish the properties of our fully-implicit spectral boundary element algorithm.