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On non-stationary viscous incompressible flow through a cascade of profiles. (English) Zbl 1124.35054
The incompressible Newtonian flows in blade machines (turbines) is studied, related with some recent results of the authors. The flow domain is given by “cells” periodically spaced in one direction. Particular Dirichlet conditions are used at the end of “cells”. A weak formulation is obtained, using Sobolev and Sobolev-Slobodetskii spaces. An existence theorem of approximate solutions is given, using a time semi-discretization. A priori estimations and a limit process are used to obtain the existence of the solution.
However, some used inequalities are not so clear. For example, consider Lemma 1 (page 1914), in the particular case of a constant $$u \in H^1(D)$$, where $$D$$ is a domain in $$\mathbb{R}^N$$. We obtain the following result: there exists a constant $$C$$ such that (length of boundary of $$D)$$ is less than \hbox{$$C\cdot$$(area of $$D)$$.} It is clear that $$C$$ depends on $$D$$. Moreover, for thin domains (with very long boundary and small area), the constant $$C$$ must be very large. Then all subsequent estimates give us a very slow convergence rate.

##### MSC:
 35Q30 Navier-Stokes equations 76D05 Navier-Stokes equations for incompressible viscous fluids
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