A family of solutions of the plane flow problem (Q1823092)

In connection with an analysis of the familiar ``Stokes paradox'' the problem of plane flow passing an object has been discussed by various authors. This paradox consists in the fact that a solution of the homogeneous two-dimensional Stokes system which is equal to zero on the surface of the object and equal to a given constant different from zero at infinity had not been found. In fact, if the boundary of the object is smooth then this problem has no solution. It is then natural to ask what will occur if this boundary is not smooth. In this paper, using the Fourier transform we find out a family of solution for a very particular case of the above mentioned problem, assuming that the object is an interval which is parallel to the speed of flow at infinity. Let (1) \(\Gamma =\{(x,y)\in R^ 2\); \(-l<x<l\), \(y=0\}\) and \(G=R^ 2\setminus \Gamma\). In the region G we consider the following Stokes's system of viscous incompressible flow: \[ (2)\quad \Delta u+\partial p/\partial x=0,\quad (3)\quad \Delta v+\partial p/\partial y=0,\quad (4)\quad \partial u/\partial x+\partial v/\partial y=0, \] with the following boundary conditions: (5) \(\lim_{| x| +| y| \to \infty}u(x,y)=a,\) \(\lim_{| x| +| y| \to \infty}v(x,y)=0,\) (6) \(u(x,y)|_{\Gamma}=0,\quad v(x,y)|_{\Gamma}=0,\) where \(u=u(x,y),\quad v=v(x,y)\) are components of the speed of the flow and \(p=p(x,y)\) is its pressure. Our method for solving the problem (2)-(6) is as follows: First of all we solve the system (2)-(4) in the domain G. For this, we try to find a solution in the whole \(R^ 2\) of the system \[ \delta u+\partial p/\partial x=f'(x)\delta (y),\quad \Delta v+\partial p/\partial y=0,\quad \partial u/\partial x+\partial v/\partial y=0, \] where f(x) is an unknown summable function on \(R^ 1\) with supp \(f\subset [-l,l]\) and \(g(x)\delta(y)\) is a direct product of two distributions \(g(x)\) and \(\delta(y)\), i.e. \(<g(x)\delta(y),\phi(x,y)>= <g(x),\phi(x,0)>\) for \(\phi(x,y)\in S(R^ 2)\). Then we define the function f(x) satisfying the boundary conditions (5), (6). It is very interesting to note that in this case the problem (2)-(6) has more than one solution.