Solvability Analysis Research Articles

The collocation method for mixed boundary value problems on domains with curved polygonal boundaries

We consider an indirect boundary integral equation formulation for the mixed Dirichlet-Neumann boundary value problem for the Laplace equation on a plane domain with a polygonal boundary. The resulting system of integral equations is solved by a collocation method which uses a mesh grading transformation and a cosine approximating space. The mesh grading transformation method yields fast convergence of the collocation solution by smoothing the singularities of the exact solution. A complete stability and solvability analysis of the transformed integral equations is given by use of a Mellin transform technique, in a setting in which each arc of the polygon has associated with it a periodic Sobolev space.

Steady states for viscous fingers with anisotropic surface tension

Pattern selection is considered for the case of viscous fingering in rectangular Hele-Shaw geometry in the presence of anisotropic surface tension, using solvability analysis and a boundary-integral method. We find that anisotropy introduced as a sinusoidal perturbation with a fourfold symmetry is irrelevant for small driving velocities and the usual steady-state finger width in the absence of the anisotropy is obtained. For sufficiently large driving velocities a new steady-state width is selected when the anisotropy makes the local interfacial tension a maximum at the finger tip. This is in agreement with recent experimental observations.

Dynamics of viscous fingers and threshold instability.

We present a detailed weakly nonlinear analysis, along with a solvability analysis, of an instability in viscous fingering which we term the surface-tension-driven instability. This instability occurs when the surface tension is modified in proportion to the local curvature. It is an intrinsically nonlinear instability which always requires a finite-amplitude perturbation to trigger it. It is also distinctively different from those arising via subcritical bifurcation. Numerical simulations reveal that this instability leads to spiky cellular patterns and may suggest a possibility of leading to a finite time singularity.