Bidirectional Williamson nanofluid flow towards stretchable surface with modified Darcy's law

Bidirectional flows of Williamson nanomaterial in porous space is discussed. Nonlinear model contains thermophoresis and Brownian motion. Bidirectional non-linear and time-dependent stretching sheet velocity is considered. Nanoparticles zero mass flux condition is accounted. Modified Darcy's law is invoked. Resulting nonlinear systems are computed by optimal homotopy analysis method (OHAM). Numerical values of velocity gradient coefficients and Nusselt and Sherwood numbers via involved pertinent variables are computed and addressed. Plots and tabulated values lead to physical interpretation. By increasing Williamson parameter and porosity parameter, bidirectional velcities slow down. While temperature is an increasing function of thermophoresis parameter and space and temperature dependent heat sources. Skin friction coefficients and Nusselt number are increasing functions of porosity parameter and Prandtl number while Sherwood number decreases for Brownian motion.