Hypersingular boundary integrals in cusped two-dimensional free-surface Stokes flow

Abstract

The biharmonic streamfunction is naturally employed in the complex-variable formulation of free-boundary transient problems in two-dimensional Stokes flow. In the event that analytical solutions are not obtainable, biharmonic boundary-integral methods (BBIMs) are frequently used. By using the well-known analytical solution of Hopper (J. Fluid Mech. vol. 213, 1990, pp. 349–375) for the Stokes-flow coalescence of two cylinders, it is demonstrated that the widely used direct BBIM formulation admits hypersingular integrals when solving evolving free-surface problems in the presence of a cusp, irrespective of the degree of piecewise-polynomial shape functions used to represent the curvilinear free surface. It is also shown that the hypersingularity which arises in the dynamic free-surface cusp formation is of the same fundamental form as that arising in both the static-singularity driven-cavity problem and the submergence or withdrawal of a solid plate relative to a free surface (Moffatt, J. Fluid Mech. vol. 18, 1964, pp. 1–18). The hypersingular BBIM integrals do not admit regularization, finite-part integration or Gauss–Chebyshev integration in the normal sense: the natural BBIM is fundamentally ill-posed in the presence of singularities born of boundary motion. In such cases, the Almansi representation should be used in order to guarantee accurate numerical solutions.