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On the determination of point-forces on a Stokes system. (English) Zbl 1075.76020
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 allowable functional spaces in \(\Omega\), the authors prove the possibility of reducing the reciprocity gap functional to a point-force reciprocity gap function \({\mathcal G}_b (y)=\int_{\partial\Omega}[{\mathbf t_\ast\cdot U}(x-y)\cdot{\mathbf b}-{\mathbf v_\ast\cdot T}({\mathbf U}(x-y){\mathbf\cdot b},{\mathbf P}(x-y){\mathbf\cdot b}){\mathbf\cdot n}]\,ds\) taking into account the equality \(\sum_{k=1}^N{\mathbf a}_k{\mathbf\cdot U}(y-s_k){\mathbf\cdot b}=-{\mathcal G}_b (y)\), \(\forall y\in {\mathbb R}^n\backslash\bar{\Omega}\), where the pair \(({\mathbf U}(x-y){\mathbf\cdot b},{\mathbf P}(x-y){\mathbf\cdot b})\) is the flow due to a stoklets located at \(y\), b is a freely chosen but fixed strength, \(({\mathbf U},{\mathbf P})\) is the known fundamental singular solution of Stokes equations corresponding to concentrated forces along the coordinate axes, and \({\mathbf a}_k,s_k\) are the strength and location of \(N\) point forces to be recovered as best fit parameters in the course of a discrete nonlinear least-squares minimization procedure applied to \(\sum_{y\in Y_L}| {\mathcal G}_b(y)-\sum_{k=1}^M{\mathbf c}_k{\mathbf\cdot U}(y-z_k){\mathbf\cdot b}| ^2\). Here \(Y_L\) is a discrete set of \(L\gg M\geq N\) randomly chosen points in the exterior of \(\Omega\). For regularizing of noise effects on the boundary data, the authors recommend the use of the vectorial reciprocity gap function which components are obtained for \({\mathbf b}\) equal to the respective unit vectors. Also, for a single point force an exact analytical recovering procedure is given.
Three numerical examples for the two-dimensional case illustrate the suppression of noise effects, the effect of the size of \(({\mathbf v}_R,p_R)\) relative to \(({\mathbf v}_s,p_s)\) and the influence of variation of \(M\) on the accuracy of the recovering results. As the paper is not quite self-contained and is written in a rather terse style, working out the examples will help much to understanding the text. There are some innocuous typographical errors and several sign mistakes in the paper, which the potential reader will easily detect.
Relative little elaboration is devoted to the possible application of the results of this investigation. In the introduction the authors only mention that the point forces may stand for the presence of small particles in the fluid, each of which can be assumed no larger than a single mass point exerting a force on the fluid. The inverse problem calculation should then give the number and the location, or possibly predict other characteristics of the particles. If so, several questions arose to this reviewer as for the adequacy and meaning of some assumptions and statements made in the paper.

MSC:
76D07 Stokes and related (Oseen, etc.) flows
35Q30 Navier-Stokes equations
35R30 Inverse problems for PDEs
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