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 free convective hybrid nanofluid (Fe₃O₄+MWCNT/H₂O) magnetized non-Darcy flow over a porous cylinder is examined by considering the effects constant heat source and uniform ambient magnetic field. The developed coupled PDEs (partial differential equations) are numerically solved using the innovative computational technique of control volume finite element method (CVFEM). The impact of increasing strength of medium porousness and Lorentz forces on the hybrid nanofluid flow are presented through contour plots. The variation of the average Nusselt number (Nu_ave) with the growing medium porosity, buoyancy forces, radiation parameter, and the magnetic field strength is presented through 3-D plots. It is concluded that the enhancing medium porosity, buoyancy forces and radiation parameter augmented the free convective thermal energy flow. The rising magnetic field rises the temperature of the inner wall more drastically at a smaller Darcy number. An analytical expression for Nusselt number (Nu_ave) is obtained which shows its functional dependence on the pertinent physical parameters. The augmenting Lorentz forces due to the higher estimations of Hartmann retard the hybrid nanoliquid flow and hence enhance the conduction.

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 thermal energy transfer characteristics during hybrid nanofluid migration are studied in the presence of a variable magnetic field, heat source, and radiation. The flow is governed by the conservation laws of mass, momentum, and energy, whereas it is modeled by the coupled set of nonlinear partial differential equations (PDEs). Suitable similarity transformations are employed to convert the developed set of PDEs to a nonlinear system of coupled ordinary differential equations (ODEs). The simplified system of ODEs is solved by using the well-established analytical procedure of homotopy analysis method (HAM). The effects of varying the strength of the physical parameters on the thermal energy transfer during hybrid nanofluid motion between two plates in which one of the plate is porous, rotating, as well as stretching are investigated through tables and two-dimensional graphs. The porosity is modeled through the Koo–Kleinstreuer model (KKL) correlation. The analysis reveals that the skin friction and Nusselt number augment with the increasing strength of the magnetic field and nanomaterials’ concentrations. The gradient in the fluid velocity has a dual dependence on the strength of the applied magnetic field and Grashof number and drops with the higher values of the unsteadiness parameter. The fluid velocity constricts with the enhancing magnetic field due to higher Lorentz forces, and it also drops with the increasing rotation rate. The enhancing buoyancy associated with higher Grashof number values augments the fluid velocity. The fluid’s temperature rises with the augmenting nanomaterial concentrations, Eckert number, nonsteadiness, heat source strength, and radiation parameter, while it drops with the higher Grashof number and Prandtl number. The applied technique of the HAM shows good convergence over a wide range of the convergent parameter. This work has potential applications in the development of efficient thermal energy transfer systems.

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.