An Inverse Problem for Slow Viscous Incompressible Flows

This paper considers an inverse boundary value problem associated to the Stokes equations which govern the motion of slow viscous incompressible ows of uids. The determination of the under-speci ed boundary values of the normal uid velocity is made possible by utilising within the analysis additional pressure measurements which are available from elsewhere on the boundary. The inverse boundary value Stokes problem has been numerically discretised using a boundary element method (BEM). Then the resulting ill-conditioned system of linear algebraic equations solved in a circular domain using the Tikhonov regularization method with the choice of the regularization parameter based on the L-curve criterion. In addition, an investigation into the stability of the numerical solution has been made by adding a random small perturbation to the input data. NOMENCLATURE A;B;C;D;E; F;G;H In uence matrices I Identity matrix Kkl; Lk Kernel functions M Number of boundary elements R Radius of the circle f1; f2; f1; f2 Stress force components n Outward normal p; p Pressure (r; ) Polar coordinates ulk; q k Fundamental solutions v Fluid velocity vector v1; v2; v1; v2; vr; v Components of the uid velocity w Fluid vorticity (x1; x2); (x1; x2) Cartesian coordinates 0; ? Underand over-speci ed portions of the boundary, respectively Solution domain Percentage of noise Kronecker delta symbol Gaussian random variables Coe cient function ; ; Given functions Regularization parameter Streamfunction Standard deviation INTRODUCTION The basic equations governing the incompressible creeping ow are the Stokes equations, namely, r2v = rp or vi;jj = p;i in r v = 0 or vi;i = 0 in (1) where the quantities v and p denote the dimensionless uid velocity and the associated dimensionless pressure, respectively. In two-dimensions, the introduction of a streamfunction reduces the Stokes eqns (1) to the biharmonic equation