In this paper, we consider the nonstationary shear flow of a compressible, viscous, and heat-conducting micropolar fluid. The mathematical model is set up in Lagrangian description in the form of initial-boundary problem with inhomogeneous boundary conditions for velocity and standard homogeneous boundary conditions for microrotation and heat flux. Under the assumptions that this problem has a generalized solution and that the initial mass density, temperature, the velocity, and microrotation vectors are smooth enough functions, we prove that this solution is unique. KEYWORDS micropolar fluid, shear flow, uniqueness of the solution MSC CLASSIFICATION 35D99; 35G61; 35Q35; 76N99
INTRODUCTIONThe theory of micropolar fluids was first introduced by Ahmed Cemal Eringen in the 1960s. 1 Besides classical hydrodinamical variable, such as mass density and velocity field, to be able to describe microphenoma, he introduced the new vector filed called microrotation velocity. In other words, he gave a generalization of classical Navier-Stokes model, which allowed the mathematical analysis of physical phenomena at the micro level. Today, many potential applications of this model can be found: micropolar fluids are used in the modeling of liquid crystals with rigid molecules, magnetic fluids, clouds with dust, muddy fluids, some biological fluids, etc. 2 For some specific applications, we refer to Dražić. 3 Assumptions for the model in this paper are that the flow is isotropic, viscous, and heat-conducting, as well as that the fluid is in the thermodinamical sense perfect and polytropic. Such flow was first considered by Mujaković. 4 For recent progress in the mathematical analysis of this problem in one-dimensional case, we refer to Dražić and Simčić, 5 in spherically symmetric case to Dražić, 6 and for the cylindrically symmetric case to Huang and Dražić. 7 A kind of flow that is analyzed in this paper is the flow between two parallel thermo-insulated horizontal plates, with the upper one moving irrotationally. This kind of flow is called a shear flow and it has a great potential for applications, especially in lubrication theory, for example, in lubrication of magnetic disks. The corresponding initial-boundary problem was derived in Dražić et al,8,and in 9 it was proved that the problem has a generalized solution locally in time. Let us note that this is a problem with nonhomogeneous boundary conditions for velocity vector, which demands a different approach in some stages of its analysis, compared with analysis of a model with homogeneous boundary conditions.Here, we will prove the uniqueness of a generalized solution for a shear flow problem. We use the method where we form an auxiliary system for the difference of two solutions and prove that this system has a unique zero solution. This method was applied, for example, in Mujaković and Dražić 10 for the problem with spherical symmetry.