How the Reynolds equation is related to the Stokes equations. (English) Zbl 0538.76038

Let the Stokes equation being obeyed in a two-dimensional layer bounded by a plane \(y=0 (\Gamma_ 1)\), \(y=\epsilon h(x) (\Gamma_ 3)\) and straight lines \(\Gamma_{2,4}\), \(x=\pm L\). As usually in the boundary layer problem we demand \(\Phi_ y=const\). on \(\Gamma_ 1\), \(\partial \Phi /\partial u=0\) on \(\Gamma_ 3\) where \(\Phi\) is the stream function which automatically obeys equation of continuity. Changing the variables one obtains for the stream function the equation \(\epsilon^ 4\Phi_{xxxx}+2\epsilon^ 2\Phi_{xxyy}+\Phi_{yyyy}=0\). The degenerate problem, \(\epsilon =0\), can be solved explicitely provided the boundary conditions on \(\Gamma_ 2\) and \(\Gamma_ 4\) are chosen in an appropriate form. The main point of the paper is the theorem which gives estimates on w, w being the difference between the full and degenerate solution. The proof uses estimations in the Sobolev space.
In spite of heavy machinery the paper is written clearly and one can object only the lack of physical interpretations of main results.
Reviewer: R.Herczynski


76D10 Boundary-layer theory, separation and reattachment, higher-order effects
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