Viscous drop in compressional Stokes flow

The dynamics of the deformation of a drop in axisymmetric compressional viscous flow is addressed through analytical and numerical analyses for a variety of capillary numbers, Ca, and viscosity ratios, λ . For low Ca, the drop is approximated by an oblate spheroid, and an analytical solution is obtained in terms of spheroidal harmonics; whereas, for the case of equal viscosities ( λ = 1), the velocity field within and outside a drop of a given shape admits an integral representation, and steady shapes are found in the form of Chebyshev series. For arbitrary Ca and λ , exact steady shapes are evaluated numerically via an integral equation. The critical Ca, below which a steady drop shape exists, is established for various λ . Remarkably, in contrast to the extensional flow case, critical steady shapes, being flat discs with rounded rims, have similar degrees of deformation ( D∼ 0. 75) for all λ studied. It is also shown that for almost the entire range of Ca and λ , the steady shapes have accurate two-parameter approximations. The validity and implications of spheroidal and two-parameter shape approximations are examined in comparison to the exact steady shapes.