Unsteady flow of Bingham fluid in a two dimensional elastic domain (Q6548300)

Let \(\Omega _{0}\), \(\Omega _{1}\subset \mathbb{R}^{2}\) be bounded domains with smooth boundary \(\Gamma _{0}=\partial \Omega _{0}\) and \(\Gamma _{1}=\partial \Omega _{1}\) and \(T\) a finite and positive real number. The boundary \( \Gamma _{1}\) of \(\Omega _{1}\) is decomposed in two closed, non-empty and disjoint parts \(\Gamma _{11}\cup \Gamma _{12}\). \(\Omega _{0}\) represents an elastic domain occupied by a rigid, viscoplastic and incompressible Bingham fluid, and \(\Omega _{1}\) is the thickness of the elastic domain. \N\NThe problem under consideration is: Find the velocity field \(u=(u_{1},u_{2}):\Omega _{0}\times (0,T)\rightarrow \mathbb{R}^{2}\), the stress field \(\sigma _{0}=(\sigma _{0_{i,j}})_{1\leq i,j\leq 2}:\Omega _{0}\times (0,T)\rightarrow \mathbb{S}^{2}\) and \(w=(w_{1},w_{2}):\Omega _{1}\times (0,T)\rightarrow \mathbb{R}^{2}\), such that: \(\frac{Du}{Dt}=(\frac{\partial u }{\partial t}+u\nabla u)=\operatorname{Div}(\sigma _{0})+f_{0}\), \(\operatorname{div}u=0\), in \(\Omega _{0}\times (0,T)\), \(\frac{\partial ^{2}w}{\partial t^{2}}=\operatorname{div}\sigma _{1}+f_{1}\), in \(\Omega _{1}\times (0,T)\), \(\widetilde{\sigma }_{0}=2\mu \varepsilon (u)+g\frac{\varepsilon (u)}{\left\vert \varepsilon (u)\right\vert }\), if \(\left\vert \varepsilon (u)\right\vert \neq 0\), \( \left\vert \widetilde{\sigma }_{0}\right\vert \leq g\), if \(\left\vert \varepsilon (u)\right\vert =0\), in \(\Omega _{0}\times (0,T)\), \(\sigma _{1_{i,j}}=\sum_{k,l=1}^{2}A_{i,j,k,l}\varepsilon _{k,l}\), in \(\Omega _{1}\times (0,T)\), where \(A\) is the fourth-order Cauchy tensor and \( \varepsilon _{k,l}\) is the strain tensor in \(\Omega _{1}\). \N\NThe boundary and transmission conditions: \(u=\frac{\partial w}{\partial t}=w^{\prime }\) on \( \Gamma _{0}\times (0,T)\), \(\sigma _{1}-\sigma _{0}=\frac{1}{2} (\sum_{i=1}^{2}u_{i}\cos(\eta _{i}))\times u_{i}\) on \(\Gamma _{0}\times (0,T) \), \(\sigma _{1}\eta =g\) on \(\Gamma _{11}\times (0,T)\), \(w=0\) on \(\Gamma _{12}\times (0,T)\), are imposed, together with the initial conditions: \( u(0)=u_{0}\) in \(\Omega _{0}\times (0,T)\), \(w(0)=w_{0}\), \(w^{\prime }(0)=w_{1}\) in \(\Omega _{1}\times (0,T)\). Here \(\eta \) is the outward normal to \(\Gamma _{0}\) oriented toward the exterior of \(\Omega _{0}\). The authors prove that the above problem is equivalent to find \((u,\Phi )\in L^{2}(0,T,V_{1})\cap L^{\infty }(0,T,L^{2}(\Omega _{0})^{2})\times L^{\infty }(0,T;L^{2}(\Omega _{1})^{2})\) solution to some variational inequality, where \(\Phi =\frac{\partial w}{\partial t}\) and \(V_{1}=\{v:v\in (H^{1}(\Omega _{0}))^{2}\), \(\operatorname{div}v=0\) in \(\Omega _{0}\}\). \N\NThe main result proves that the variational inequality has at least one solution \((u,\Phi )\) for any finite \(T\), such that \ \(u\in L^{2}(0,T,V_{1})\cap L^{\infty }(0,T,L^{2}(\Omega _{0})^{2})\), \(u^{\prime }\in L^{2}(0,T,V_{1}^{\prime })\), \(\Phi \in L^{2}(0,T;V_{2})\cap L^{\infty }(0,T;L^{2}(\Omega _{1})^{2})\), and \(\Phi ^{\prime }\in L^{2}(0,T,V_{2}^{\prime })\), where \(V_{2}=\{v:v\in (H^{1}(\Omega _{1}))^{2}\), \(v=0\) on \(\Gamma _{12}\}\). For the proof, the authors use the Faedo-Galerkin method, they also prove uniform estimates on the approximate solution which allow passing to the limit.