Asymptotic analysis of a micropolar fluid flow in a thin domain with a free and rough boundary
The lubrication problem is considered as micropolar fluid flow in a two-dimensional domain with free boundary of small thinness and roughness, described by a PDEs system with respect to the velocity field of the fluid \(u^{\varepsilon}=(u_{1}^{\varepsilon},u_{2}^{\varepsilon}),\) the pressure \(p^{\varepsilon}\) and the angular velocity \(\omega^{\varepsilon}\) of the microrotations of the particles. It consists of the equilibrium equations for momentum, mass and moment of momentum and is considered in the space-time domain \((0,T)\times\Omega^{\varepsilon}\), \(\Omega^{\varepsilon}=\{z=(z_{1},z_{2})\in{\mathbb{R}^{2}}\), \(0<z_{1}<L\), \(0<z_{2}<\varepsilon h^{\varepsilon}(z_{1})\}\), \(h^{\varepsilon}(z_{1})=h(z_{1}\), \(\frac{z_{1}}{\varepsilon})\). Here the form of \(\Omega^{\varepsilon}\) is originated from the important fields of lubrication theory and applications in the mechanical and electromechanical industries.NEWLINENEWLINEIn the relevant functional spaces, the existence and uniqueness of a weak solution to the PDEs system is proved. Then a priori estimates for the velocity, microrotation and pressure independently on \(\varepsilon\) are established. The limit problem when \(\varepsilon\) tends to zero is studied.
scientific article; zbMATH DE number 6070885