Steady flows of Jeffrey-Hamel type from the half-plane into an infinite channel. II: Linearization on a symmetric solution.
The steady flow of a Navier-Stokes fluid is analysed in a two-dimensional asymmetric unbounded domain \(\widetilde\Omega\), having two outlets to infinity: a half-plane \(K= \{(x_1, x_2)\in \mathbb{R}^2: x_1>0\}\) and a semi-strip \[ \Pi_-= \{(x_1, x_2)\in \mathbb{R}^2: x_1< 0,\,| x_2|< 1\}. \] Assuming that \(\widetilde\Omega\) can be obtained as a small perturbation of the symmetric domain \(\Omega\) (i.e. \(\widetilde\Omega=k(\Omega)\) and \(k: \mathbb{R}^2\to \mathbb{R}^2\) is a diffeomorphism such that \(k= I\) outside the ball \(B_R\) and \(k- I\) is small in a certain sense), the existence of a unique solution to the Navier-Stokes system in \(\widetilde\Omega\) is shown. The change of variables \(\widetilde\Omega\ni \widetilde x\to x= k^{-1}(\widetilde x)\in \Omega\) turns the Navier-Stokes problem in \(\widetilde\Omega\) into the perturbed Navier-Stokes problem in \(\Omega\), which later is considered as a perturbation of the linear problem. This linearization around the symmetric solution for non-zero flux leads to the absence of compatibility conditions for the convective term, and for negative flux to the domination of nonlinear terms by the linear ones. The paper is completed with examples of non-symmetric domains. The well-posedness and even existence of physically reasonable solutions in a non-symmetric domain with a prescribed positive flux remains still an open question.
scientific article; zbMATH DE number 1975952