Stability analysis of $Cu-C_6{H}_{9}Na{O}_7$ and $\mathrm{Ag}-C_6{H}_9\mathrm{Na}{O}_7$ nanofluids with effect of viscous dissipation over stretching and shrinking surfaces using a single phase model

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.