The purpose of the current study is to inquire the unsteady transverse magneto hydrodynamic three-dimensional flow induced by a bidirectional permeable stretching surface with variable concentration and temperatures fields. The flow of nano-material due to time dependent stretching surface through a porous media with prescribed surface temperature and concentration of nanoparticles are taken into accounts. The simulation applied for the nanomaterial contains the significance of Brownian movement and thermophoresis effects. Analytical treatment is made by employing homotopy analysis method (HAM) while the Keller box method is employed for numerical solution. The obtained solutions from both the methods are compared and are in good agreement with each other. The influences of involving parameters on concentration and temperature profiles are presented through tables and graphs. It is detected that the unsteady stretching parameter (S), temperature and concentration variation along x-direction (r), temperature and concentration variation along y-direction (s), Prandtl number (Pr) and Schmidt number (Sc) are displaying the identical behavior on concentration and temperature profiles. It is also founded that the large values of Brownian movement parameter (N_b ) and thermophoresis parameter (N_t) increase the width of thermal and concentration boundary layer.

This paper scrutinizes the consequences of radiation and heat consumption of MHD convective flow of nanofluid on a heated stretchy plate with injection/suction and convective heating/cooling conditions. The nanofluid encompasses with and nanoparticles. We enforce the suited transformation to remodel the governing mathematical models to ODE models. The HAM (homotopy analysis method) idea is applied to derive the series solutions. The divergence of fluid velocity, temperature, skin friction coefficient, local Nusselt number, entropy generation, and Bejan number on disparate governing parameters is exhibited via graphs and tables. It is seen that the fluid velocity in both directions is subsided when elevating the magnetic field and Forchheimer number. Also, the nanoparticles possess hefty speed compared to nanoparticles because the density of nanoparticles is high compared to that of nanoparticles. The fluid temperature upturns when enlarging the heat generation and radiation parameters. The skin friction coefficients and local Nusselt number are high in nanoparticles than in nanoparticles.

Investigation of expanding sheet flow in the existence of tiny particles is an interesting field of research and this type of mathematical model is usually expressed in the form of partial differential equations. Present exploration has been made to describe the unsteady 3D dynamics of MHD tangent-hyperbolic fluid impinging over an expanded sheet with the characteristics of haphazard movement and thermo migration of nanoparticles. Moreover, securitization has been accomplished in the environment of prescribed temperature and prescribed concentration restrictions at the stretchable surface. Novel combination of scaling transformations has been accustomed to exchange the governing equations into complicated system of ordinary differential equation and then solved by an innovative analytical scheme, namely, homotopy analysis method. The significance of involved scientific factors on temperature phase, concentration phase, condensed Nusslet number and condensed Sherwood number has been explained through various contours. It has been noticed that increasing choices of random movement $0\le N_b\le 1.0$ and thermo-migration of nanoparticles $0\le N_t\le 3.0$ reduce the estimate of heat transaction along with the estimate of mass transaction. Moreover, temperature and concentration circulations have improved with the advanced choices of Weissenberg number $0\le {We}_x\le 10.0$ and flow behavior index $0\le n\le 2.1.$ Graphical abstract