Qualitative properties and approximation of solutions of Bingham flows: on the stabilization for large time and the geometry of the support
The isothermal and unsteady flow of an incompressible Bingham visco-plastic medium, during the time interval \((0,T)\), is modeled by the following system of equations (clearly of the Navier- Stokes system type): \[ \rho(\partial _t \mathbf{u}+ (\mathbf{u}\cdot \bigtriangledown\nabla)\mathbf{u})= \nabla \cdot \pmb{\sigma}+\mathbf{f}(t,x) \quad\text{in }(0,T)\times \widetilde{\Omega}, \] \[ \nabla \cdot \mathbf{u}=0 \quad\text{in }\quad (0,T)\times \widetilde{\Omega}, \] \[ \pmb{\sigma}=-p\mathbf{I}+\sqrt{2}g\frac{\mathbf{D(u)}}{|\mathbf{D(u)}|}+2\mu\mathbf{D(u)}, \] \[ \mathbf{u}(0)=\mathbf{u}_0 \quad (\text{with }\nabla \cdot\mathbf{u}_0)=0. \] Here \(\mathbf{u}\) and \(p\) represent velocity and pressure, respectively. The positive constants \(\rho, \mu\) and \(g\) represent density, viscosity and plasticity yield of the Bingham medium, respectively. Moreover, \(\mathbf{f}(t,x)\) is a given density of external forces, \(\mathbf{D(v)}=[\nabla \mathbf{v}+(\nabla \mathbf{v})^t]/2(=D_{ij}(\mathbf{v})_{1\leq i,j\leq d}, \forall \mathbf{v} \in (H^1(\widetilde{\Omega}))^d\), and \(|\mathbf{D(v)}|\) is the Frobenius norm of tensor \(\mathbf{D(v)}\), i.e., \[ |\mathbf{D(v)}|=\left( \sum_{1\leq i,j\leq d} |D_{ij}(\mathbf{v})|^2\right)^{1/2}. \] The domain \(\widetilde{\Omega}\) is an open and connected subset of \(\mathbb{R}^d\) (\(d=2\) or \(3\) in most of the applications), \(\widetilde{\Gamma}\) is the boundary of \(\widetilde{\Omega}\) and \(T>0\) is arbitrarily fixed (and possibly \(T=+\infty\)). Only homogeneous Dirichlet boundary conditions are considered. For the two-dimensional flow in a bounded domain, it is shown the extinction in a finite time even under suitable nonzero external forces. Also, the special case of three-dimensional domain given as an infinitely long cylinder of bounded cross section is considered. Sufficient conditions leading to a scalar formulation on the cross section are given. The stabilization of the solutions is proved, when \(t\) goes to infinity, to the solution \(u_{\infty}\) of the associated stationary problem once a suitable convergence on the right hand forcing term is assumed. Some sufficient conditions are given for the extinction in a finite time of solutions of the scalar problem. It is shown that, at least under radially symmetric conditions, when the stationary state is not trivial, \(u_{\infty} \neq 0\), there are cases in which the stabilization to the stationary solution needs an infinite time to take place. The results of numerical experiments are given.
scientific article; zbMATH DE number 5735154
scientific article