Modelling of non-isothermal flow abnormally viscous fluid in the channels with various geometry of boundaries

In this paper, we analyzed the flat non-isothermal stationary flow of abnormally viscous fluid in the channels with asymmetric boundary conditions and an unknown output boundary. The geometry of the channels in which the problem is considered, is such regions, that at the transition to bipolar a system of coordinates map into rectangles. This greatly simplifies the boundary conditions, since it is possible to use an orthogonal grid and boundary conditions are given in its nodes. Fields of this type are often found in applications.
The boundary conditions are set as follows: the liquid sticks to the boundaries of the channels, which rotate at different speeds and have different radius and temperature; moreover, temperature at the entrance to deformation is known, while on the boundary with the surface the material has the surface temperature; the pressure on the enter and exit of the region becomes zero. The rheological model only takes into account the anomaly of viscosity. The material is not compressible.
This process can be described by a system consisting of continuity equations, the equations of conservation of momentum and an energy equation: $\nabla_{i} v^{i} = 0, \quad \rho v^{i} \nabla_{i} v^{i} = -g^{ij}\nabla_{i} P + \nabla_{i} \tau^{ij}$,$\lambda\nabla^i\nabla_i T - \rho c_v v^i \nabla_i T + \tau^{ij} e_{ij} = 0$, rheological properties of the liquid are described by the equation: $ P^{ij} = -g^{ij}P + \tau^{ij}, \quad \tau^{ij} = \mu' e^{ij}$ where $ u,v $ — coordinates of environmental speeds, $ P $ — the hydrostatic pressure, $ T $ — temperature, $ c_v $ — specific heat, $ \rho $ — density, $ \lambda $ — thermal conductivity, $ \tau^{ij} $ — a viscous stress tensor, $ P^{ij} $ — a stress tensor, $ e^{ij} $ — a rate of the deformation tensor, $ g^{ij} $ — the metric tensor.
In this paper, we propose an algorithm for calculating a non-isothermal flow for an arbitrary continuous function that describes the flow curve.