The unsteady flow and heat transfer characteristics of electrically conducting water based thin liquid film non-Newtonian (Casson and Williamson) nanofluids dispensed with graphene nanoparticles past a stretching sheet are considered in the presence of transverse magnetic field and non-uniform heat source/sink. Embedding the graphene nanoparticles effectively amplifies the thermal conductivity of Casson and Williamson nanofluids. Ordinary differential equations together with the boundary conditions are obtained through similarity variables from the governing equations of the prob lem, which are solved by the HAM (Homotopy Analysis Method). The solution is expressed through graphs and illustrated which show the influences of all the param eters. The convergence of the HAM solution for the linear operators is obtained. Favorable comparison with previously published research paper is performed to show the correlation for the present work. Skin friction coefficient and Nusselt number are presented through Tables and graphs which show the validation for the achieved results demonstrating that the thin liquid films results from this study are in close agreement with the results reported in the literature. Results achieved by HAM and residual errors are evaluated numerically, given in Tables and also depicted graphi cally which show the accuracy of the present work. In another study the steady two dimensional magnetohydrodynamic second grade nanofluid flow containing nanoparticles and gyrotactic microorganisms is considered using passively controlled nanofluid model boundary conditions. For the biofluid xxii xxiii the thermal boundary layer convective boundary conditions have been handled. The study has been restricted to gyrotactic microorganisms where compensating torques generated by shear and gravity effects manifest in gyrotaxis which controls the ori entation of upswimming microorganisms through rotary motions. By using the ap propriate similarity transformation for the velocity, temperature, nanoparticle volume fraction and motile microorganism density, the governing partial differential conserva tion equations under prescribed boundary conditions are transformed to the ordinary differential equations which are solved analytically by the HAM (Homotopy Anal ysis Method). Graphical solutions are presented to show the influences of all the parameters. Skin friction, wall heat transfer rate, nanoparticle mass transfer rate and microorganism transfer rate are evaluated in Table. Motile microorganism density function enhances with an increase in momentum slip. Applying optimal homotopy asymptotic method (OHAM), a time dependent symmet ric flow with heat transmission of a second-grade fluid containing nanoparticles and gyrotactic microorganisms between two parallel plates in two dimensions is explored. Partial differential equations furnish the nonlinear ordinary differential equations due to the usage of relevant similarity transformations. Motion declines due to second grade fluid, energy elevates due to thermophoresis, concentration enhances due to Brownian motion and gyrotactic microorganisms profile elevates due to Peclet num ber. The unsteadiness parameter has a profound effect on the nanobioconvection flow within the plates. Consistency and smoothness between the first and second orders of the optimal homotopy asymptotic method are revealed through graphs. Also, graphs are provided to manifest the impacts of each parameter. In last problem, the simultaneous flow and heat transfer of two nanoliquids in the environment of gyrotactic microorganisms and cubic autocatalysis chemical reaction through a porous medium is treated under the potentiality of buoyancy. Weissenberg parameter keeps direct relation and the Casson parameter have inverse relationship to xxiv the velocity in the existence of opposing features of dominant agent and space keep ing pores. Energy keeps low position to the peak position of slippery environment. The dual behavior on concentration profile for the ascending range of strength of homogeneous reaction parameter can be observed. Small organisms are treated pos itively through homogeneous reaction while porous medium, magnet existence and heterogeneous reaction keeps opposite behaviors. Required substitutions are adopted to receive different order differential equations from the original equations which are treated via an efficient scheme. The potentialities of all the representatives are put into graphs and are elucidated.