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A mathematical analysis is performed to study the flow and heat transfer phenomena of Casson based nanofluid with effects of the porosity parameter and viscous dissipation over the exponentially permeable stretching and shrinking surface. The considered nanofluid comprises Casson as a base fluid that contains silver (Ag) and copper (Cu) solid nanoparticles. The system of the nonlinear governing partial differential equations (PDEs) are converted into ordinary differential equations (ODEs) by applying similarity transformation. The obtained ODEs are solved by using shooting technique in Maple software. Numerically obtained results reveal dual solutions for various values of pertinent parameters. Due to occurrence of dual solutions, the stability analysis is done in order to find stable solution. Positive signs of smallest eigenvalues point out that the first solution is stable and second unstable. The variation of the velocity and the temperature profiles with coefficient of the skin friction and the Nusselt number are shown graphically. Both temperature profiles and its boundary layer thicknesses increase as volume fraction of nanoparticles of Ag and Cu are increased in the Casson fluid. Velocity profiles and corresponding boundary layer thicknesses decrease by suspension of nanoparticles of silver and copper, whereas the silver Ag nanoparticles show the greater rate of heat transfer enhancement as compared to copper Cu nanoparticles when suspended in Casson fluid.

A numerical investigation is made to determine the occurrence of the multiple solutions of MHD Casson fluid in a porous channel. Governing partial differential equation of the proposed problem converted into nonlinear ordinary differential equations by using similarity transformation. Numerical technique known as shooting method is used to investigate the existence of the multiple solutions for the variations of different parameters. Effects of physical parameters on velocity profile, temperature, concentration, and skin friction are presented in pictorial and tabulation representation.