On Viscous Fluid Flow in Curvilinear Coordinate Systems

In the present contribution, we discuss the incompressible Navier–Stokes and Poisson equations for a curvilinear coordinate system constructed from the geometry of the physical domain and its boundaries by the use of a diffeomorph conformal transformation. The disadvantage of obtaining larger equations after the coordinate transformation is compensated by the simpler plane parallel boundaries. The sequence of steps to obtain a numerical solution of the aforementioned equations is lined out. Further, two simulations are presented using the dimensionless Navier–Stokes equations in its two-dimensional and three-dimensional form, together with concavities on the top and bottom boundaries. Some results obtained in the simulations are shown, i.e., the speed, the velocity, and the pressure fields are presented in the original Cartesian coordinate system. The quality of the found solutions was evaluated using the residual concept, which allows to conclude that our findings reproduce the fluid flow fairly well. Thus this reasoning of using curvilinear boundaries to define the new coordinate system that is being used to derive the solution is a new aspect for solving more realistic scenarios in fluid flow problems.