Unsteady non-Newtonian fluid flows with boundary conditions of friction type: the case of shear thinning fluids (Q6564649)

The present paper deals with the incompressible shear-thinning fluids flow governed by the \(p\)-Laplacian (\(p < 2\)) non-stationary Stokes system \(\partial_t \mathbf{u} -\nabla \cdot\left(\mu(\theta, \mathbf{u}, |D\mathbf{u}|) |D\mathbf{u}|^{p-2}D\mathbf{u} \right) +\nabla \pi = \mathbf{f} \) and \( \nabla\cdot \mathbf{u} = 0\) in \(]0,T[ \times \Omega\). \NHere \(\Omega = \{(x',x_3)\in \omega\times \mathbb{R}: \ 0<x_3 <h(x')\}\), where \(\omega\) is a bounded Lipschitz domain of \(\mathbb{R}^2\) and \(h\in C^{0,1}(\bar\omega)\), a given function \(\theta\in L^{\tilde p}(0,T; L^{\tilde q}(\Omega))\), with \(\tilde p,\tilde q \geq 1,\) a given function \( \mathbf{f}\in L^{p'}(0,T; \mathbf{L}^2(\Omega))\), with \(p' =p/(p-1) \) being the conjugate exponent to \(p>1\), and the viscosity \(\mu\) is a bounded continuous function, monotone in the last argument. Moreover, the Tresca friction law is considered on \(]0,T[\times\Gamma_0\) where \(\Gamma_0= \partial\Omega\cap \{x_3=0\}\), for some positive friction threshold \(k\in L^{p'}(0,T;L^{p'}(\Gamma_0))\). \N\NDefining \(V^p_{0,\operatorname{div}} = \{\mathbf{v} \in \mathbf{W}^{1,p}(\Omega): \nabla\cdot \mathbf{v} = 0 \mbox{ in }\Omega, v_3=0\mbox{ on } \Gamma_0, \ \mathbf{v}=0 \mbox{ on }\partial\Omega \setminus\Gamma_0 \}\) as the test function space, the authors reformulate the problem as a variational inequality under homogeneous initial condition. \NThe authors prove the existence of a weak solution \(\mathbf{u}\in C([0,T]; \mathbf{L}^2(\Omega))\cap L^p(0,T;V^p_{0,\operatorname{div}})\), such that \(\partial_t\mathbf{u}\in L^{p'}(0,T;(V^p_{0,\operatorname{div}})')\) and \(\pi \in H^{-1}(0,T; L^{p'}_0(\Omega))\), of the variational inequality if some extra assumptions on the initial and boundary conditions are provided. \N\NThe proof relies on the Schauder's fixed point argument and a vanishing viscosity technique by considering a sequence of approximate problems where the term \(2\epsilon |D\mathbf{u}|^{p'-2}D\mathbf{u}\) is added to the original stress tensor, with \(0<\epsilon<1\). For any \(\epsilon >0\), the already known existence of weak solution to the incompressible shear-thickening fluid flow \((p'>2)\) is applied. Thus, by establishing qualitative a priori estimates, with universal constant \(C>0\) and by using monotonicity properties and compactness argument, the authors firstly pass to the limit as \(\epsilon\) tends to zero. Then, the existence of a fluid velocity vector \(\mathbf{u}\) results from the fixed point technique while the pressure \(\pi\) results from de Rham's theorem.