Viscous heating in nonisothermal die flows of viscoplastic fluids with wall slip

Analytical solutions are developed for the viscous beating in slit and cylindrical tube die flows of viscoplastic fluids subject to slip at the wall. Closed-form expressions for the velocity and temperature distributions, as well as the bulk temperature, valid from the entrance to the exit of the die are presented. The problem of solving the equation of conservation of energy is transformed to that of solving an eigenvalue problem using the finite integral transform technique; as many eigenvalues and eigenfunctions as needed are determined using the sign count method. The accuracy of the solutions is established and the effects of slip parameter, apparent yield stress, and Griffith number on viscous heating are examined. The solutions can be used for estimating pressure drop, and temperature rise in die flows of viscoplastic fluids with or without wall slip occurring at the walls.