Non-isothermal steady flow of power-law fluid in a planar/axismetric channel

The results of research on a one-dimensional steady flow of power-law fluid in the planar channel/circular pipe with allowance for viscous dissipation and temperature dependence of consistency factor defined by exponential law are shown. The flow is described by the motion and heat-transfer equations. The constant temperature and no-slip condition are set on the solid walls. The numerical solutions of formulated problems are obtained using the finite-difference method. The effect of medium rheology and dissipative heating on the flow pattern is parametrically investigated. Typical distributions of velocity, temperature, viscosity, and dissipative function in the channel/pipe cross-sections at various governing parameters are obtained. The algorithm defining critical values of parameter in the problem, which separate domains of existence and non-existence of stable stationary solution, is numerically implemented. Exceeding the obtained critical values leads to a hydrodynamic thermal explosion. When the rate of heat generation due to mechanical energy dissipation is higher than that of heat loss through the walls, the unlimited increase in the temperature occurs. The dependencies of parameter on the power-law index defining a stable stationary solution domain are plotted. The calculated results are in a good agreement with analytical solution.