Analysis of viscous incompressible flows of micropolar fluidswith thermal convection and mixed boundary conditions

Abstract

AbstractThe Navier‐Stokes equations do not take into account the microstructure of the fluid in the sense that they do not consider the angular momentum of small particles of the fluid due to their rotation. The model of micropolar fluid represents a generalization of the well‐established Navier‐Stokes equations, in such a way that it introduces a new kinematic vector field called microrotation (the angular velocity field of rotation of particles) and adds a new vectorial equation, expressing the conservation of the angular momentum. We will be concerned with the initial boundary value problem for the flow of micropolar heat conducting fluids in a two‐dimensional channel with mixed boundary conditions. The considered boundary conditions are of three types: the Dirichlet boundary conditions on the inflow, the Navier type conditions on solid surfaces and Neumann‐type boundary conditions on the outflow of the channel. The homogeneous Dirichlet boundary conditions on solid surfaces for the microrotation is commonly used in practice. However, imposing such condition is doubtful from the physical point of view. For that reason, more general boundary conditions for the microrotation were proposed throughout the engineering literature to take into account the rotation of the microelements on the solid boundary, linking the velocity and microrotation through the so‐called boundary viscosity. The well‐posedness of problems with different types of boundary conditions for microrotation are completely unexplored. The present contribution is devoted to the analysis of the existence and uniqueness of the solution.