Exact Solutions to a Model for Micropolar Fluid Flows with Rayleigh Energy Dissipation

Polynomial exact solutions of the Navier–Stokes equations for describing micropolar incompressible fluid flows with energy dissipation are reported. The transformation of mechanical energy into thermal energy is taken into account. The heat equation for the Rayleigh function contains the sum of the squares of the components of the Cauchy velocity tensor (the main component for the dissipative function). Unidirectional homogeneous and non-homogeneous fluid flows with moment stresses are considered. The solvability of overdetermined systems for studying homogeneous and non-homogeneous shear flows is studied. The paper pays attention to the exact integration of equations for three-dimensional flows. The construction of classes of exact solutions is carried out first using the Lin–Sidorov–Aristov solution family. In other words, the velocity field depends linearly on part of the coordinates. The coefficients of the linear forms of the velocity field depend on the third coordinate and time. The pressure field and the temperature field are quadratic forms with similar functional arbitrariness. In addition, exact solutions for the velocity field with a nonlinear dependence on part of the coordinates are considered.