Multiscale modelling couples patches of two-layer thin fluid flow

The multiscale gap-tooth scheme uses a given microscale simulator of complicated physical processes to enable macroscale simulations by computing only only small sparse patches. This article develops the gap-tooth scheme to the case of nonlinear microscale simulations of thin fluid flow. The microscale simulator is derived by artificially assuming the fluid film flow having two artificial layers but no distinguishing physical feature. Centre manifold theory assures that there exists a slow manifold in the two-layer fluid film flow. Eigenvalue analysis confirms the stability of the microscale simulator. This article uses the gap-tooth scheme to simulate the two-layer fluid film flow. Coupling conditions are developed by approximating the values at the edges of patches by neighbouring macroscale values. Numerical eigenvalue analysis suggests that the gap-tooth scheme with the developed two-layer microscale simulator empowers feasible computation of large scale simulations of fluid film flows. We also implement numerical simulations of the fluid film flow by the gap-tooth scheme. Comparison between a gap-tooth simulation and a microscale simulation over the whole domain demonstrates that the gap-tooth scheme feasibly computes fluid film flow dynamics with computational savings.