Numerical studies of cusp formation at fluid interfaces in Stokes flow

Abstract

Numerical studies are performed addressing the development of regions of high curvature and the spontaneous occurrence of cusped interfacial shapes in two-dimensional and axisymmetric Stokes flow. In the numerical simulations, the velocity field is computed using a boundary-integral method, and the evolution of the concentration of an insoluble surfactant over an evolving interface is computed using an implicit finite-volume method. Three configurations are considered in detail, and the results are used to elucidate three different aspects of cusp formation. In the first series, the deformation of a two-dimensional bubble immersed in a family of straining flows devised by Antanovskii, and of an axisymmetric bubble immersed in an analogous family of flows devised by Sherwood, are examined. The numerical results indicate that highly elongated and cusped two-dimensional shapes, and pointed or cusped axisymmetric shapes, are unstable and should not be expected to occur in practice. In the second series of studies, the role of an insoluble surfactant on the transient deformation of bubbles subject to the Antanovskii or Sherwood flow is investigated. Under certain conditions, the reduced surface tension at the tips raises the local curvature to high values and causes the ejection of a sheet or column of gas by means of tip streaming. In the third series of studies, the coalescence of a polygonal formation of five viscous columns of a fluid placed in an arrangement that differs only slightly from one proposed recently by Richardson is examined. The numerical results confirm Richardson's predictions that transient cusps may occur at a finite time in the presence of surface tension. The underlying physical mechanism is discussed on the basis of reversibility of surface-driven Stokes flow and with reference to the regularity of the motion driven by negative surface tension. Replacing the inviscid ambient gas with a slightly viscous fluid whose viscosity is as low as one hundredth the viscosity of the cylinders suppresses the cusp formation.