Asymptotic Analysis of the Linearized Navier–stokes Equation on an Exterior Circular Domain: Explicit Solution and the Zero Viscosity Limit

Abstract

In this paper we study and derive explicit formulas for the linearized Navier-Stokes equations on an exterior circular domain in space dimension two. Through an explicit construction, the solution is decomposed into an inviscid solution, a boundary layer solution and a corrector. Bounds on these solutions are given, in the appropriate Sobolev spaces, in terms of the norms of the initial and boundary data. The correction term is shown to be of the same order of magnitude as the square root of the viscosity.

Well-Posedness of the Boundary Layer Equations

“…In the linear case it has been possible to prove the convergence of the linearized Navier-Stokes equations to the corresponding inviscid equations for Sobolev-type initial data. The asymptotic analysis has been successfully performed for the Stokes equations on the half space (Sammartino [16]) and on the exterior of a disk (Lombardo, Caflisch, and Sammartino [11]). Similar results were achieved for the Oseen equations, i.e., the Navier-Stokes equations linearized around a nonzero flow, on a strip (see Lombardo and Sammartino [12] and Temam and Wang [23]).…”

Viscous-Inviscid Interactions in a Boundary-Layer Flow Induced by a Vortex Array

“…Due to the no-slip boundary condition, the convergence of Navier-Stokes (NS) solution to Euler solution, as Re → ∞, fails, and Prandtl boundary layer equations must be used to resolve the flow close to the boundary. We mention the papers [36,37,3,24] where, for analytic initial data, the authors prove the convergence of NS solutions to Euler and Prandtl. The zero viscosity limit was also considered in [18,40,20], where the authors introduce criteria based on a priori estimates of energy dissipation in a viscous sub layer, and in [29] where the author employs the assumption that, initially, the vorticity close to the boundary is zero.…”

On the vanishing viscosity limit in a disk

“…Also, without assuming that the initial vorticity is radially symmetric, the argument in the proof of Theorem 6.1 can be applied to solutions to the Stokes problem (the linearized Navier-Stokes equations) to show that they converge in the vanishing viscosity limit to a solution to the linearized Euler equations (which is just the steady state solution u = u 0 ). This would be more interesting, though, if time-varying Dirichlet boundary conditions, for instance, could be incorporated, as in [13].…”

“…In [13], the authors consider the Stokes problem (linearized Navier-Stokes equations) external to a disk with time-varying Dirichlet boundary conditions, showing that the vanishing viscosity limit holds. In fact, they do much more than this, giving an explicit construction of the solution to the Stokes problem and showing that it can be decomposed into the sum of the solution to the linearized Euler equations, the solution to the associated Prantdl equations, and a small correction term.…”