The shape oscillations of a two-dimensional drop including viscous effects

Abstract

The implementation of a boundary integral method for potential flow is presented for the case of a two-dimensional drop freely oscillating in a vacuum. Calculations using a standard boundary integral formulation and a double-layer potential boundary integral formulation are compared for the case of an inviscid drop with a clean interface. Additional comparisons are made using a least-squares spectral transform method for interpolating and differentiating versus more common methods using cubic splines and central differences. The boundary integral method for potential (i.e. inviscid) flow is extended in two viscous examples to approximate (i) the weak viscous effects in the bulk fluid far from the clean interface, or (ii) the surface viscous effects in an inviscid drop arising from an interface that is highly contaminated with an insoluble surfactant. The addition of an incompressibility constraint, implemented in a least-squares sense, to the standard boundary integral formulation is shown to significantly improve its ability to preserve the conserved quantities of volume and total energy. Nevertheless, the double-layer potential boundary integral formulation, despite its more complicated form, is found to be computationally more efficient than the standard formulation. The use of the least-squares spectral transform method is shown to be more accurate, and in certain conditions more efficient, than using cubic splines and sixth-order central differences for time-evolution of this system. Simulations approximating the damping effects of clean viscous drop are found to be consistent with small deformation theory while the calculations incorporating the damping effects of surface dilatational viscosity are shown to dissipate the energy of the oscillations at a rate that is neither exponential nor algebraic.