Time-dependent viscous deformation of a drop in a rapidly rotating denser fluid

Abstract

Viscous stretching of a cigar-shaped drop due to the centrifugal pressure field in a surrounding rapidly rotating denser fluid is analysed. Scaling arguments are used to examine the various contributions to the viscous stresses resisting deformation, and a number of asymptotic regimes are identified which are delineated by the relative magnitudes of the aspect ratio, the viscosity ratio and unity. These asymptotic regimes may usefully be described as the bubble, pipe, sliding-rod and toffee-strand limits. Detailed analysis based upon a slenderness assumption combined with an integral representation of Stokes equations is used to derive evolution equations for the shape of the drop as a function of time in the different regimes. In the limit that interfacial-tension effects are negligible, similarity solutions are developed in which the length of the drop is found to increase as t2/5, t1/4, (t ln t)1/4 and t. The analytical results are in good agreement with numerical simulations based upon a boundary-integral solution to the full viscous flow equations.