MHD Casson or Carreau fluids flow with microorganisms over a permeable shrinking surface (Journal of Naval Architecture and Marine Engineering, 2023)
The characteristics of dual solutions of the flow of Casson or Carreau fluids mixed with microorganisms over a permeable shrinking surface are investigated considering the effects of magnetic field, heat dissipation, Dufour and thermophoretic diffusivity. By using similarity transformations, the governing equations are converted into a set of ordinary differential equations. These reduced equations are then solved by the well-known shooting technique along with fourth order Runge-Kutta method. A comparison is provided and the present solutions hold a good agreement with available published results. The local skin friction coefficient, local Nusselt number, local Sherwood number, and local density number of the microorganisms are found to increase with the increase of non-Newtonian Casson parameter, Dufour and thermophoretic diffusivity number, however, it decreases with Weissenberg number. The remarkable finding which is not revealed yet is that dual solutions exist in the flow of Casson or Carreau fluids with microorganisms over a permeable shrinking sheet subject to a certain combination of parameters. Moreover, the domain of the occurrence of dual solutions broadens on account of higher values of all physical parameters of the problem except Eckert number.