Three-dimensional thin film flow problems solved accurately and efficiently: Rivulet formation, merger and evolution

The evolution of a gravity-driven, thin, continuously-fed liquid film down an inclined planar substrate is modelled using the long-wave approximation with, as observed experimentally, a periodic pattern of finger-like rivulets forming at the unstable advancing front; note that, the film may be perfectly or partially wetting. The required long-time solution, with the merger of neighbouring rivulets a possibility, is obtained using a purpose designed efficient multigrid methodology incorporating complementary error-controlled automatic spatio-temporal adaptivity and mesh devolution strategies. Critical wavelengths are extracted from the various solutions showing the effect of inclination angle. The rivulet pattern formed is explored, revealing very good agreement with existing experimental results; partial wetting is demonstrated to have a strong influence on the evolution of the system.