Non-isothermal Steady Flow of Non-Newtonian Fluid in an Axisymmetric Channel

Abstract

Non-isothermal steady-state flow of a viscoplastic fluid in an axisymmetric channel is studied with account for viscous dissipation at a specified constant flow rate. The rheology of the medium is described by the Herschel-Bulkley law with a temperature dependence of yield stress and consistency defined by exponential law. On the solid wall, the no-slip boundary condition and the assigned temperature are used. The mathematical statement of the problem includes the dimensionless motion and energy equations and boundary conditions. The problem is solved numerically using a finite-difference approach. The difference equations are solved by sweep method. When applying a shock-capturing method for calculating the flow, the rheological model is regularized in order to eliminate stress singularity in the regions of zero shear rates. The steady-state distributions of velocity, temperature, viscosity, and dissipative function are obtained. A limiting value of pressure drop defining the existence domain of a steady solution is proved to exist. Two problem solutions are obtained at a specified pressure drop, which are referred to as high- and low-temperature flow regimes. As a result of computations, different flow structures with unyielded regions occurring near symmetry line and in the dead zone next to a solid wall are revealed.