Identification of obstacles immersed in a stationary Oseen fluid via boundary measurements

Abstract

This is a repository copy of Identification of obstacles immersed in a stationary Oseen fluid via boundary measurements.

The method of fundamental solutions for Brinkman flows. Part II. Interior domains

“…As far as the physical problem is concerned, we model a scenario where an unknown defect, flaw or fault (modelled as a rigid inclusion D) contained in the porous medium Ω is to be detected from Cauchy data fluid (velocity, traction) measurements (2.3) and (2.4) at the boundary of the porous medium container. Similar formulations have previously been considered in the context of stationary Stokes flow of slow incompressible fluids, [2,3,8], the Oseen flow, [16], or the full Navier-Stokes equations, [4,9].…”

“…where µ 1 and µ 2 are positive regularization parameters, which can be prescribed either by trial and error or by using some criterion such as the L-surface method, see [5,12], or the L-curve method [11,6] if we take µ 1 = µ 2 = µ, or µ 1 = 0 and vary µ 2 , or µ 2 = 0 and vary µ 1 , see [16].…”

“…In addition, partial boundary data is also considered. A similar method has recently been developed by the authors for the identification of a rigid obstacle immersed in a stationary Oseen fluid flow from boundary measurements, [16].…”

The identification of obstacles immersed in a steady incompressible viscous fluid

“…Alternatively, one could measure the fluid velocity u out at the outlet boundary Γ out , [6]. The measurement (11) was also considered in [2] for the simpler linear case of slow viscous Stokes fluid flow. The uniqueness of solution (u, p, D) of the inverse problem ( 1)-( 5) and ( 11) under the assumptions that Ω 0 is connected, u in ̸ ≡ 0 and ν is sufficiently large could follow along the lines of the proof of Theorem 1.2 of [7] for the Dirichlet problem (in which the Neumann boundary condition ( 5) is replaced by the homogeneous Dirichlet boundary condition u = 0 on Γ out ).…”

“…In such a situation, inverse modeling offers a valuable approach to overcome these challenges by utilizing observed effects or indirect measurements to infer the properties, shapes, sizes and locations of submerged obstacles. There are several papers about the identification of obstacles immersed in some different types of fluids such as, potential [5], Stokes [18,3,2], Oseen [15,11], Brinkman [12,13] or Navier-Stokes [1,7]. The studies in [1,7] were mainly theoretical and they dealt with the simpler Dirichlet boundary conditions for which the existence and uniqueness of solution theory is available [23].…”