On the determination of point-forces on a Stokes system
type: Item
instance of: scholarly article
title: On the determination of point-forces on a Stokes system
zbMATH Open document ID: 1075.76020
author: Carlos J. S. Alves
author: Ana L. Silvestre
published in: Mathematics and Computers in Simulation
publication date: 2004-08-19T00:00:00Z
review text: The authors state and solve an inverse problem to the steady Stokes flow of viscous incompressible fluid in a two- or three-dimensional domain. Inverse problems in fluid dynamics mean finding flow characteristics only on the base of data measured on the boundaries. In the present case the primary (regular) steady Stokes flow in a bounded simply-connected domain \(\Omega\) is perturbed by the action of point forces in \(\Omega\), whose number, locations and intensities are not known and should be calculated only on the base of measured stress \({\mathbf t_\ast}\) and prescribed (known) velocity \({\mathbf v_\ast}\) of the resultant flow on the boundary \(\partial\Omega\). The boundary (not necessarily rigid walls) and the locations of the point forces are assumed to be fixed in the inertial system in which the flow takes place. As is usual in solving inverse problems, the authors first present the direct problem for the resultant Stokes flow \(({\mathbf v},p)\) and its decomposition in \(\Omega\) in a singular part \(({\mathbf v_s},p_s)\) and a regular one \(({\mathbf v_R},p_R), {\mathbf v_R}={\mathbf v_\ast}-{\mathbf v_s}\) on \(\partial \Omega\), and summarize conditions under which these direct Stokes problems are uniquely solvable. For solving the inverse problem, the authors exploit the concept of the so-called reciprocity gap functional \({\mathcal R}({\mathbf u},q)\), which in this case is actually the right-hand side of Green's formula corresponding to Stokes problem, and is valid for any smooth solenoidal vectors \({\mathbf u}\), \({\mathbf v}\) and \(q,p\): \(\int_\Omega[(\nu\Delta{\mathbf v}-\nabla p){\mathbf u}-{\mathbf v}(\nu\Delta{\mathbf u}-\nabla q)=\int_{\partial\Omega}[{\mathbf t_\ast\cdot u}-{\mathbf v_\ast\cdot T}({\mathbf u},q){\mathbf\cdot n}]= {\mathcal R}({\mathbf u},q)\), where \({\mathbf T}({\mathbf u},q)\) is the stress tensor associated with the flow \(({\mathbf u},q)\). Being free in choosing \(({\mathbf u},q)\) to belong to appropriate a…
Mathematics Subject Classification ID: 76D07
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 35R30
zbMATH DE Number: 2092898
zbMATH Keywords: inverse problem
zbMATH Keywords: reciprocity gap functional
zbMATH Keywords: Green's formula
zbMATH Keywords: least-square minimization
reviewed by: Tomislav Zlatanovski
MaRDI profile type: Publication
full work available at URL: j.matcom.2004.02.007
OpenAlex ID: W2091443415
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DOI: 10.1016/J.MATCOM.2004.02.007
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description: scientific article; zbMATH DE number 2092898