| Two categories of low Reynolds number interfacial problems are studied: Migration of long gas bubbles due to an imposed axial temperature gradient is discussed for cylinders of both circular and polygonal cross sections. The objective is to determine the steady bubble velocity in terms of the physical properties of the liquid and the shape of the tube. In circular cylinders, the structure of the solution is that of a constant thickness film bounded by constant curvature cap regions, with transition layers in between. The bubble speed has a nonlinear dependence on the modified capillary number that is ultimately connected with the nonlinearity of the fluid dynamics in the transition regions. In polygonal cylinders, the corner regions of the cylinder determine the bubble speed at leading order, and the bubble speed is linear in modified capillary number at leading order. For regular-polygonal tubes with small numbers of sides or low aspect ratio rectangular cross sections, most of the flux passes through the corner regions, while for larger numbers of sides or large aspect ratios, flow in the thin film regions dominates and the results tend toward those for cylindrical tubes.; The gravity driven steady Stokes flow over a topography, and time dependent viscous flow with a contact line are studied using a biharmonic boundary integral method. In the case of the steady flow, the Stokes equations are solved with a pre-assumed free surface profile. The location of the free boundary is then updated by considering the normal stress condition and using an iteration technique. For small capillary numbers, Ca, the free surface develops a ridge before the entrance to a step down and a depression region right before a step up that their characteristics depend on Ca and the step depth. The results agree well with lubrication theory for small Ca.; For time-dependent flows with a moving contact line, the free surface normal stress condition provides another set of equations coupled with boundary integral equations. These are solved for an initial free surface profile to determine the local velocity of free surface nodal points which is then used to track the evolution of the profile in time. The time evolution and fully developed steady solutions of the viscous flow are studied for high Ca, and a wide range of the contact angle. For a range of high contact angles we found steady solutions with an overhang; by increasing the Ca, the overhang grows and rolls down to the plane. We also determine the limiting parameters for which an overhang appears on the profile. Finally the effect of flow parameters on the free surface profile has been examined. |
