Generalized Analytic Functions in an Extensional Stokes Flow with a Deformable Drop

An approach of generalized analytic functions of a complex variable to the axially symmetric problem of a deformable drop in an extensional viscous incompressible flow under the zero Reynolds number assumption is presented. The velocity field and pressure in and out of the drop are represented in terms of four generalized analytic functions. When the drop and ambient flow have equal viscosities, integral representations for the velocity, vorticity, and pressure balancing the velocity and stress boundary conditions are obtained based on the generalized Cauchy integral formula. In this case, steady shapes of the drop for different capillary numbers are found by minimizing the norm of the normal velocity over the interface and are shown to have rounded endpoints. The obtained steady shapes satisfy the kinematic condition with high accuracy and are very close to spheroidal shapes. The second part of this work constructs an analytical solution to the problem with the deformable spheroidal drop that balances the velocity and stress boundary conditions for arbitrary viscosity ratio. In this case, the four generalized analytic functions are represented by infinite series in the spheroidal coordinates, and series coefficients are determined analytically from a system of first-order difference equations. Spheroidal shapes near steady state are found for various capillary numbers and viscosity ratios by minimizing the norm of the normal velocity over the interface. Also, critical values for the capillary number are estimated based on the series-form solution for the spheroidal drop. They are much closer to the values obtained by the boundary integral equation approach than to the values predicted by the second-order perturbation theory.