The magnetohydrodynamic thin film nanofluid sprayed on a stretching cylinder with heat transfer is explored. The spray rate is a function of film size. Constant reference temperature is used for the motion past an expanding cylinder. The sundry behavior of the magnetic nano liquid thin film is carefully noticed which results in to bring changes in the flow pattern and heat transfer. Water-based nanofluids like Al ${}_2$ O ${}_3$ -H ${}_2$ O and CuO-H ${}_2$ O are investigated under the consideration of thin film. The basic constitutive equations for the motion and transfer of heat of the nanofluid with the boundary conditions have been converted to nonlinear coupled differential equations with physical conditions by employing appropriate similarity transformations. The modeled equations have been computed by using HAM (Homotopy Analysis Method) and lead to detailed expressions for the velocity profile and temperature distribution. The pressure distribution and spray rate are also calculated. The comparison of HAM solution predicts the close agreement with the numerical method solution. The residual errors show the authentication of the present work. The CuO-H ${}_2$ O nanofluid results from this study are compared with the experimental results reported in the literature showing high accuracy especially, in investigating skin friction coefficient and Nusselt number. The present work discusses the salient features of all the indispensable parameters of spray rate, velocity profile, temperature and pressure distributions which have been displayed graphically and illustrated.

Buongiorno’s nanofluid model is followed to study the bioconvection in two stretchable rotating disks with entropy generation. Similarity transformations are used to handle the problem equations for non-dimensionality. For the simulation of the modeled equations, Homotopy Analysis Method is applied. The biothermal system is explored for all the embedded parameters whose effects are shown through different graphs. There exists interesting results due to the effects of different parameters on different profiles. Radial velocity decreases with increasing stretching and magnetic field parameters. Temperature increases with Brownian motion and thermophoresis parameters. Nanoparticles concentration decreases on increasing Lewis number and thermophoresis parameter while motile gyrotactic microorganisms profile increases with increasing Lewis and Peclet numbers. Convergence of the solution is found and good agreement is obtained when the results are compared with published work.

A time dependent symmetric 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 number. The unsteadiness parameter β has profound effect on the nanobioconvection flow within the plates. Optimal homotopy asymptotic method (OHAM) is followed to evaluate the transformed systems. 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.

The Arrhenius activation energy and binary chemical reaction are taken into account to consider the magnetohydrodynamic mixed convection second grade nanofluid flow through a porous medium in the presence of thermal radiation, heat absorption/generation, buoyancy effects and entropy generation. The items composing of the governing systems are degenerated to nonlinear ordinary differential equations by adopting the appropriate similarity transformations which are computed through Runge-Kutta-Fehlberg (RKF) numerical technique along with Shooting method. The solution is manifested through graphs which provides a detailed explanations of each profile in terms of involved parameters effects. The compared results maintain outstanding approach to the previous papers.