Incompressible Flow Around a Small Obstacle and the Vanishing Viscosity Limit

Abstract

Abstract. In this article we consider viscous flow in the exterior of an obstacle satisfying the standard no-slip boundary condition at the surface of the obstacle. We seek conditions under which solutions of the NavierStokes system in the exterior domain converge to solutions of the Euler system in the full space when both viscosity and the size of the obstacle vanish. We prove that this convergence is true assuming two hypothesis:first, that the initial exterior domain velocity converges strongly in L 2 to the full-space initial velocity and second, that the diameter of the obstacle is smaller than a suitable constant times viscosity, or, in other words, that the obstacle is sufficiently small. The convergence holds as long as the solution to the limit problem is known to exist and stays sufficiently smooth. This work complements the study of incompressible flow around small obstacles, which has been carried out in [4,5,6]

The vanishing viscosity limit in the presence of a porous medium

“…The Reynolds number is a dimensionless quantity, which is inversely proportional to the viscosity. When the characteristic local length scale near the boundary becomes small at a faster rate than the viscosity, one can expect that the local Reynolds number stays of order one, preventing the formation of a strong boundary layer, as was observed in [16], where the vanishing viscosity limit was established in the case of one shrinking obstacle. Homogenizing the Navier-Stokes equations in the perforated (periodic) domain with Reynolds number below a critical value yields Darcy's law as in the case of the Stoke's system, while homogenizing the Euler equations gives non-linear filtration laws that depend on the relation between the particle size and the characteristic velocity of the problem.…”

The Inviscid Limit and Boundary Layers for Navier-Stokes Flows

“…Let ǫ be the scale of the obstacle. The vanishing viscosity limit was shown to hold in the exterior of one obstacle diametrically shrinking to a point in [69] by assuming the condition ǫ ≤ C ν for some positive constant C, which depends on the initial data for the Euler equations in R 2 , u 0 , and the shape of the obstacle, and assuming that the initial condition for the Navier-Stokes solution, u ν 0 , extended by zero to the whole plane, converges to u 0 in L 2 (R 2 ), with an optimal rate of convergence of √ ν. (See [81] for the opposite situation of an expanding domain.)…”

“…In the ensuing discussion, the case of unsteady flows in bounded, simply connected domains or a half space will be the main focus. The interesting case of exterior or multiply-connected domains, such as flow outside one or more obstacles, brings in additional difficulties, for example the infinite energy in the vorticity-velocity formulation of the fluid equations in two dimensions (see [69,81,130] and references therein).…”

The Inviscid Limit and Boundary Layers for Navier-Stokes Flows

“…The vanishing viscosity limit was shown to hold in the exterior of one obstacle diametrically shrinking to a point in [69] by assuming the condition ǫ ≤ C ν for some positive constant C, which depends on the initial data for the Euler equations in R 2 , u 0 , and the shape of the obstacle, and assuming that the initial condition for the Navier-Stokes solution, u ν 0 , extended by zero to the whole plane, converges to u 0 in L 2 (R 2 ), with an optimal rate of convergence of √ ν. (See [81] for the opposite situation of an expanding domain.)…”