engleski

Mathematical model for compressible viscous micropolar fluid flow

With the classical Navier-Stokes equations physical phenomena of fluid flow were only considered at the macro level. However, the rapid development of science, which among other areas, is focused on nanotechnology requires an analysis of phenomena at the micro level as well. The first considerations that go in the direction of studying micro phenomena appear in the works of brothers Cosserat created at the beginning of the last century. But because of its complexity this theory remained neglected for many years until in the 1960s A. C. Eringen introduced the concept of the micropolar fluid. Eringen's theory assumes that each particle of fluid has a finite size and contains a microstructure. The general Eringen's approach introduces the microdeformation tensor, which he calls director. In this way, we obtain the model which is too complicated for practical use and Eringen proposed a simpler model where microdeformations are not allowed. Therefore, the behavior of the fluid at the microlevel is described by using one new vector field - microrotation velocity. Here we analyze the compressible flow of an isotropic, viscous and heat conducting micropolar fluid, which is in the thermodynamical sense perfect and polytropic. The boundaries of the spatial domain present solid thermally insulated walls, initial density as well as initial temperature are strictly positive, and corresponding problem has homogeneous boundary conditions. We present the models with different dimensions of spatial domain whereby we analyze the existence and uniqueness of the solution, as well as the asymptotic behavior of the solution.

compressible micropolar fluid, generalized solution, homogeneous boundary conditions

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