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On boundary conditions for velocity potentials in confined flows: Application to Couette flow. (English) Zbl 0703.76028
Summary: The representation of solenoidal fields by means of two scalar potentials can be a very useful method for a wide range of problems, in particular for the incompressible Navier-Stokes equations, though in finite containers boundary conditions may not be easily handled. The differential equations for the potentials are of an order higher than the original Navier-Stokes ones. As a consequence additional boundary conditions are needed to solve them. These differential equations and the corresponding boundary conditions for any geometry have been derived and the equivalence with the original problem has been proved. Special emphasis has been laid on domains with nontrivial geometry in which integral boundary conditions appear. As an example, the results have been applied to the periodic Couette flow. In this case the integral boundary conditions can be avoided by an appropriate change of variables, hence reducing the order of the equations obtained.

MSC:
76D10 Boundary-layer theory, separation and reattachment, higher-order effects
76D05 Navier-Stokes equations for incompressible viscous fluids
35Q30 Navier-Stokes equations
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