Numerical analysis of tips in viscous flow

When a drop (or gas bubble) is placed in a strong viscous flow (e.g., a shear flow), it develops very sharp tips at its ends. Sharp tips are also formed when a viscous fluid is withdrawn from the neighborhood of its interface with the ambient air or with another fluid (selective withdrawal). However, it is observed frequently that there exists a critical flow strength above which the drop transitions toward a jetting state in which a jet comes out from the tip. In this paper, we look numerically for stationary drop shapes, both globally and close to the tip, which we study with very high resolution. To this end we use a boundary integral method to solve the axisymmetric flow equations for arbitrary viscosity ratios in the inertialess (Stokes) limit. Stationary states are solved for using Newton's method. This permits us to find both stable and unstable steady states and to investigate the nature of the jetting transition. The critical parameters for this transition are in reasonable agreement with slender-body theory. Excellent agreement is found with our earlier experiments in the selective withdrawal geometry [S. Courrech du Pont and J. Eggers, Phys. Rev. Lett. 96, 034501 (2006)], for which the viscosity of the phase inside the tip is negligible. We describe a scale invariance of the experimental interface profiles away from the tip. Then we investigate the highly curved tip region not considered previously with comparable precision. We find that the shape near the tip is universal, i.e., independent of the outer flow and of the geometry of the system (drop or selective withdrawal). While the tip curvature becomes extremely large, it always remains finite if surface tension is present.