APPLICATION OF THE FICTITIOUS DOMAIN METHOD FOR SOLVING THE NAVIER-STOKES EQUATIONS CONSIDERING THE DOUBLED MEAN CURVATURE

In this paper, we consider an initial-boundary value problem for an unsteady flow of a viscous incompressible fluid in a bounded region, solved using a system of nonlinear Navier-Stokes equations. The equations describe the motion of the fluid considering viscosity, pressure, and mass force, as well as the solenoidality condition of the velocity field. In the general case, finding an analytical solution to the system of equations presents significant difficulties, and it has not yet been proven whether there is always a smooth solution for all possible conditions. In this regard, the fictitious domain method is used to solve the problem, which allows us to reduce the problem by solving a system of differential equations with appropriate boundary conditions. Particular attention is paid to introducing the concept of twice the mean curvature of the surface, which is necessary for the correct application of the fictitious domain method. For this purpose, the article provides a detailed calculation of the mean curvature using surface parameterization and matrices of the first and second forms. Proof of a lemma related to the calculation of twice the mean curvature is also given, which is of great importance for further numerical methods for solving the Navier-Stokes system of equations. The obtained results expand the scope of application of the fictitious domain method in solving hydrodynamic problems, especially in complex geometries, and can be used to develop more efficient numerical algorithms.

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