The finite points model of the~Stokes--Leibenson problem for the Hele-Shaw flow

O. A. Vasilieva ; A. S. Demidov

Fundamentalʹnaâ i prikladnaâ matematika, Tome 5 (1999) no. 1, pp. 67-84.

Résumé

A model destined for investigation of the causes of some peculiarities of the classical Stokes–Leibenson problem, in particular, the requirement of analyticity of the initial contour for the solvability of the problem (for the case of a sink as well as for a source) is described. The essence of the model is the following. The movement of the contour is imitated by the movement of a finite number of points that belong to some quasicontour. Its movement inherits the law of the movement of the contour in the classical sense. The existence of convex quasicontours and appropriate position of the source-sink is proved, for which the problem is unsolvable in the classical sense. An obstacle for the existence of the classical solution is the presence of points of the quasicontour where the tangent velocity assumes the values $\pm\infty$, oscillating infinitely rapidly in the case of the source and conserving the sign in the case of a sink. In the case of a source this determines a physically justified movement even of a “irregular” initial contour, and in the case of a sink this clarifies the necessity of high smoothness of the initial curve.