Mathematical and numerical analysis of boundary value problems for fluid flow - Seyukova-Rivkind, Ludmila (2002)
The thesis consists of an introduction and four articles. In the first paper, a mathematical model and a numerical method for particulate flows past blunt bodies are presented. The model consists of the inhomogeneous Euler equations coupled with equations governing the transport of solid particles in a continuous gas phase. It is treated by the Godunov method in conjunction with a time-marching algorithm. The obtained simulation results are compared with the experimental data for the drag exerted by the dispersed two-phase flow upon a body present in the flow field. In the second paper, we consider a stationary free boundary problem for the Navier-Stokes equations, governing the effluence of a viscous incompressible liquid out of unbounded nonexpanding at infinity, in general, non-symmetric strip-like domain Ω_ outside which the liquid forms a sector-like jet with free (unknown) boundary and with the limiting opening θ Є (0, π/2). Conditions on the free boundary take account of the capillary forces but external forces are absent. The total flux of the liquid through arbitrary cross-section of Ω is prescribed and assume to be small. Under this condition, we prove the existence of an isolated solution of the problem, which is found in a certain weighted Holder space of functions. In the third paper, we consider a stationary problem for the Navier-Stokes equations in a domain Ω Ϲ R² with a finite number of "outlets" to infinity in the form of infinite sectors. In addition to the standard adherence boundary conditions, we prescribe total fluxes of velocity vector field in each "outlet", subject to the necessary condition that the sum of all fluxes equals zero. Under certain restrictions on the aperture angles of the "outlets", which seems close to being necessary, we prove that, for small fluxes, this problem has a solution which behaves at infinity like the Jeffery-Hamel flow with the same flux, and we prove that this solution is unique in the class of solutions satisfying the energy inequality. We also study the problem with another type of additional condition at infinity, that which involves limiting values of the pressure at infinity in the outlets. Finally, we present a simplified construction of a small Jeffrey-Hamel solution with a given flux based on the contraction mapping principle.