Miller Mendoza Jimenez
ETH Zürich, Suisse
Many systems in Nature present either spatial curvature (e.g. curved space due to presence of stars or flow on soap films) or geometric confinement constraining the degrees of freedom of particles moving on such media, e.g. solar photosphere, flow between two rotating cylinders and spheres, and hemodynamics through deformable vessels, to name but a few. However, up to now, the study of such systems has been limited to relatively simple geometries, because once the curved space becomes more complicated, the use of one common underlying coordinate system (e.g. Cartesian, spherical, or cylindrical) leads to an expensive use of computational resources and a poor approximation of the spatial curvature and the boundary conditions.
Here, we present a new lattice kinetic scheme that can handle flows in virtually arbitrary complex manifolds in a very natural and elegant way, by resorting to a covariant formulation of the lattice Boltzmann (LB) kinetic equation in general coordinates, in flat and curved spaces. This idea can also be applied to model wave propagation phenomena in general geometries.
As an additional feature, complex boundary conditions related with a specific geometry, e.g. surface of a sphere, or more sophisticated ones, like Moebius bands and the Klein bottle, can be treated exactly by cubic cells in the contravariant coordinate frame, thereby avoiding the staircase approximations which would result from the use of cubic cells associated with Euclidean geometry.