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Nonlinear aspects of the theory of lubrication. (English) Zbl 0565.76064

The author considers the problem of hydrodynamical lubrication, where the unknown pressure P satisfies the equation: \(div(\rho H^ 3\mu^{- 1}\text{grad} P)=6U(\rho H)_ x\) in the domain: \(D=\{(x,y,z)\in {\mathbb{R}}^ 3\), \(X=(x,y)\in \Omega\), \(\theta\leq z\leq H(X)\}\), where \(\Omega\) is an open, bounded subset of \({\mathbb{R}}^ 2\) with regular boundary \(\partial \Omega\), and H(X) is a given function of class \(C^ 1({\bar \Omega})\) such that \(H_ 1\geq H(X)\geq H_ 0>0\). The plane containing \(\Omega\) is moving with constant velocity U in the x- direction; \(\rho\) is the density, and \(\mu\) the viscosity. The boundary condition is \(P=G\) on \(\partial \Omega\), with G a given function. Two cases are studied in the paper: in the first case a perfect gas with constant temperature and viscosity is considered, thus a linear relation between pressure and density \(\rho =\alpha P\) holds where \(\alpha\) is a given positive constant. In the second case an incompressible fluid is assumed, whose viscosity depends on the temperature in a given way, a consequence of the energy equation. Some results on the existence and uniqueness of the solution for the above non-linear elliptic boundary value problems are demonstrated.
Reviewer: S.Nocilla

MSC:

76Nxx Compressible fluids and gas dynamics
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