Exact and numerical solutions of a fully developed generalized second-grade incompressible fluid with power-law temperature-dependent viscosity

摘要

We compute exact and numerical solutions of a fully developed flow of a generalized second-grade fluid, with power-law temperature-dependent viscosity (μ = θ~(-M) ), down an inclined plane. Analytical solutions are found for the case when M = m + 1, m ≠ 1, m being a constant that models shear thinning (m < 0) or shear thickening (m > 0). The exact solutions are given in terms of Bessel functions. The numerical solutions indicate that both the velocity and the temperature increase with decreasing Froude number and that there is a critical value of Fr below which temperature "overshoots" its free surface value of unity. This phenomena is not reported in the work of Massoudi and Phuoc [Fully developed flow of a modified second grade fluid with temperature dependent viscosity, Acta Mech. 150 (2001) 23-37.] for viscosity that depends exponentially on temperature.