Modeling and analysis of megneto-Carreau fluid with radiative heat flux: Dual solutions about critical point

Abstract

In this article, we aim to analyze the dual solutions for the flow of non-Newtonian material (Carreau fluid) over a radially shrinking surface. Magnetohydrodynamics fluid is considered. Concept of Stefan Boltzmann constant and mean absorption coefficient is used in the mathematical modeling of energy expression. Mass transfer is discussed. The upper and lower branch solutions for the Sherwood number, skin friction coefficient, and Nusselt number are calculated for different pertinent flow variables. Appropriate transformation variables are employed for reduction of partial differential equations system into ordinary differential equations. Dual solutions are obtained for the non-dimensional concentration, temperature, velocity, gradient of concentration, gradient of temperature, and gradient of velocity. The critical values for each upper and lower solutions are obtained for the case of gradient of velocity, gradient of temperature, and gradient of concentration. It is formed that concentration and temperature fields display same impact regarding both upper and lower branch solutions for velocity ratio and temperature ratio parameters.