Image systems for regularised Stokeslets at walls and free surfaces

Flow around, and hydrodynamic forces on, moving objects can be determined for low Reynolds numbers by superposing elemental Stokeslets to enforce velocity boundary conditions. The method of regularised Stokeslets provides a computationally-practical approach, replacing a sum over singular point elements by one over finite distributed sources. Flow disturbances extend a considerable distance in low-Re flow (typically falling off in magnitude as the reciprocal of distance) and the presence and nature of a nearby boundary cause a profound change to forces on moving objects. Representing boundaries by additional Stokeslets requires significantly more computational resources, so it is common to represent single planar boundaries by the method of images. Here we formally state the image systems for regularised Stokeslets in the case of plane no-slip walls and free surfaces in a form that is independent of orientation and in canonical six-term forms (for velocity) and three-term forms (for pressure) that are convenient for computation. We validate our implementation using exact solutions (sphere moving in unbounded flow and parallel or perpendicular to slip or no-slip boundaries) and previous simulations (translating and rotating helical filament). We then present new comparisons with experimental data for velocity profiles and forces on objects translating (towed cube in a confined tank) and rotating (helical micro-robot). Our simulations illustrate competing physical effects near a boundary. For an object moving, or propelling fluid, toward or away from a boundary, whether slip or no-slip, an increase in thrust is required to redirect fluid through 90°; viscous drag on a no-slip boundary increases the resistance coefficient further. For an object moving parallel to a boundary, however, opposing effects with different range come into play: the absence of fluid above the boundary reduces the required overall thrust on the fluid over a wide range of gap sizes, but enhanced viscous drag near a no-slip wall exceeds this at small gap sizes.