Plane-parallel flows of an incompressible fluid in a bounded domain with minimum mean square vorticity are considered. The flow function is biharmonic function. Such flows include, for example, the stationary solution of 2D Stocks problem with a potential righthand side. If the velocity on the boundary is specified, then definition of the flow is reduced to the solution of the boundary value problem of the biharmonic equation. The projection algorithm for solving boundary value problems for the biharmonic equation in complicated domains is presented.