The current investigation is carried out on the thin film flow of Reiner-Philippoff fluid of boundary-layer type. We have analyzed the flow of thin films of Reiner-Philippoff fluid in the changeable heat transmission and radiation over a time-dependent stretching sheet in 2D. The time-dependent governing equations of Reiner-Philippoff fluid model are simplified with the help of transformation of similarity variables. To investigate the behavior of the Reiner-Philippoff fluid with variable stretching surface for different physical effects, we considered thermophoresis and Brownian motion parameters in the flow. The Homotopy Analysis Method is implemented in the reduced model to achieve a solution of the original problem. A numerical convergence of the implemented method is also analyzed. The behavior of temperature, velocity, and concentration profiles have been investigated with the variation of skin friction, Nusselt number, and Sherwood number. A comparative graphical survey is presented for the velocity gradient, under different parameters. An analytical analysis is presented for the time-dependent parameter over thin film flow. The results we obtained are better than the previously available results. For the survey, the physical representation of the embedded parameters, like, $\beta$ depends on the stretching parameter $\zeta$ , and the Reiner-Philippoff fluid parameter $ϵ$ are discussed in detail and plotted graphically. Prandtl number $Pr$ , Brownian motion parameter $Nb$ , thermophoretic number $Nt$ , and Schmidt number $Sc$ are presented by graphs and discussed in detail.

The 3D Carreau fluid flow through a porous and stretching (shrinking) sheet is examined analytically by taking into account the effects of mass transfer, thermal radiation, and Hall current. The model equations, which consist of coupled partial differential equations (PDEs), are simplified to ordinary differential equations (ODEs) through appropriate similarity relations. The analytical procedure of HAM (homotopy analysis method) is employed to solve the coupled set of ODEs. The functional dependence of the hydromagnetic 3D Carreau fluid flow on the pertinent parameters are displayed through various plots. It is found that the x-component of velocity gradient ($f^{{}^{\prime }}\left(\eta \right)$) enhances with the higher values of the Hall and shrinking parameters ($m,\varrho$), while it reduces with magnetic parameter and Weissenberg number ($M,We$). The y-component of fluid velocity ($g\left(\eta \right)$) rises with the augmenting values of m and M, while it drops with the augmenting viscous nature of the Carreau fluid associated with the varying Weissenberg number. The fluid temperature $\theta \left(\eta \right)$ enhances with the increasing values of radiation parameter ($Rd$) and Dufour number ($Du$), while it drops with the rising Prandtl number ($Pr$). The concentration field ($\varphi \left(\eta \right)$) augments with the rising Soret number ($Sr$) while drops with the augmenting Schmidt number ($Sc$). The variation of the skin friction coefficients ($C_{fx}$ and $C_{fz}$), Nusselt number ($N{u}_x$) and Sherwood number ($S{h}_x$) with changing values of these governing parameters are described through different tables. The present and previous published results agreement validates the applied analytical procedure.

The boundary-layer equations for mass and heat energy transfer with entropy generation are analyzed for the two-dimensional viscoelastic second-grade nanofluid thin film flow in the presence of a uniform magnetic field (MHD) over a vertical stretching sheet. Different factors, such as the thermophoresis effect, Brownian motion, and concentration gradients, are considered in the nanofluid model. The basic time-dependent equations of the nanofluid flow are modeled and transformed to the ordinary differential equations system by using similarity variables. Then the reduced system of equations is treated with the Homotopy Analysis Method to achieve the desire goal. The convergence of the method is prescribed by a numerical survey. The results obtained are more efficient than the available results for the boundary-layer equations, which is the beauty of the Homotopy Analysis Method, and shows the consistency, reliability, and accuracy of our obtained results. The effects of various parameters, such as Nusselt number, skin friction, and Sherwood number, on nanoliquid film flow are examined. Tables are displayed for skin friction, Sherwood number, and Nusselt number, which analyze the sheet surface in interaction with the nanofluid flow and other informative characteristics regarding this flow of the nanofluids. The behavior of the local Nusselt number and the entropy generation is examined numerically with the variations in the non-dimensional numbers. These results are shown with the help of graphs and briefly explained in the discussion. An analytical exploration is described for the unsteadiness parameter on the thin film. The larger values of the unsteadiness parameter increase the velocity profile. The nanofluid film velocity shows decline due the increasing values of the magnetic parameter. Moreover, a survey on the physical embedded parameters is given by graphs and discussed in detail.

The present study is about the investigation of different aspects of viscous and electrically conducting Titania nanofluids with different base fluids. A three dimensional geometry is assumed for the steady nanofluid flow over an inclined rotating disk by applying magnetic field. The time-independent partial differential equations are constructed from the demonstrated geometry for the continuity, momentum and energy balance. By using similarity variables transformation these equations are reduced to a system of nonlinear ordinary differential equations. A numerical technique is used to solve the reduced system of equations. State variables are depicted to investigate the effects of various parameters with their variation. The influence of different physical parameters, like magnetic parameter M, Hall parameter m, porosity parameter γ, radiation parameter Rd and thickness parameter δ are briefly discussed graphically. In addition, the Nusselt number and skin friction are discussed graphically. A comparison of the applied numerical approach with the homotopy analysis method is carried out in the tabular form. Tables show the reliability of our technique verses the homotopy analysis method. The convergence of the implemented technique is presented by graph for the number of iterations performed.

The dynamics of the 3-dimensional flow of magnetized Carreau fluid past a paraboloid surface of revolution is studied through thermal radiation and mass transfer analysis. The impacts of Brownian motion and chemical reaction rate are considered on the flow dynamics. The system of nonlinear PDEs are converted to coupled ODEs by employing suitable transformation relations. The developed ODEs are solved by applying the standard procedure of homotopy analysis method (HAM). The impacts of various interesting parameters on the state variables of the Carreau fluid (velocity components, temperature, concentration, and shear stress) are explained through various graphs and tables. It is found that the horizontal velocity components augment with the rising magnetic parameter and Grashof number values. The fluid temperature augments with the higher values of the pertinent parameters except Prandtl number. The Nusselet number and fluid concentration enhance with the augmenting Brownian motion parameter. The shear stress augments with the rising Grashof number. The agreement of the obtained and published results validate the accuracy of the employed technique.

This study is related to the heat energy transfer during 3D nanofluid (water-based) motion over a rotating surface by incorporating the combined impacts of thermal radiations and couple stress. The flow is modeled by a set of non-linear coupled PDEs, which is converted to a set of coupled non-linear ODEs by using suitable similarity transformations. The transformed equations are solved with the built-in NDSolve command. The effects of relevant interesting parameters on the nanofluid velocity components and temperature distribution are explained through various graphs. It is found that the velocity component $f\left(\eta \right)$ is increased with higher values of $\gamma$ and $A_0$ while it drops with an increasing rotation parameter and nanoparticle volume fraction. The fluid temperature increases with higher ${\alpha }_{nf},Rd,{ϵ}_2,{ϵ}_3,A_1$ and drops with increasing $Pr,{ϵ}_1$ and couple stress parameter $\left(A_0\right)$. The Nusselt number remains constant at a fixed $Pr$ and $Rd$, whereas it increases with increasing $Pr$ and is reduced with rising $Rd$. A comparison between the achieved results is carried out with the analytical results through different tables. An excellent agreement is observed between these results.