Approximating stationary deformation of flat and toroidal drops in compressional viscous flow using generalized Cassini ovals

Viscous drops, subject to a linear flow in an immiscible viscous fluid, deform. When the resulting drop shape is simple the problem can be addressed by an asymptotic approach or by approximating the deformation using known simple shapes. When the resulting deformation is more complex, the problem is usually addressed numerically. In this paper, we address the problem of drops that are deforming in an axisymmetric compressional (bi-axial extensional) flow. Yielded shapes are flat drops, flat drops with dimples and toroidal drops. The latter two are highly unstable. We propose to approximate the solution of this problem, approximating the shapes by using generalized Cassini ovals, defined herein. The analysis reproduced the branches with shapes of stationary stable flat drops and stationary unstable toroidal drops, available from numerical calculation. Furthermore, it predicts the point of loss of stability of the flat drop to exhibit the transition branch that leads into the formation of the toroidal shapes, and shows that this branch shows stationary, yet unstable, flat drops with ever growing dimples up to collapse.