The present work explores the behavior of three-dimensional incompressible viscous fluid flow and heat transfer over the surface of a non-flat stretchable rotating disk. A variable thickness fluid is subjected under the influence of an external variable magnetic field and heat transfer. Navier–Stokes equation is coupled with Maxwell equations to examine the hydrothermal properties of fluid. The basic governing equations of motion are diminished to a system of nonlinear ordinary differential equations using appropriate similarity framework, which are further treated with numerical scheme known as parametric continuation method. The parametric continuation method has combined interesting characteristics of both shooting and implicit finite difference methods. For validity of the present numerical scheme, a comparison with the published work is performed and it is found that the results are in excellent agreement with each other. Numerical and graphical results for the velocity, temperature, and magnetic strength profiles as well as skin fractions and Nusselt number are presented and discussed in detail for various physical parameters. The heat transfer process is reduced with positive increment of no-flatness parameter $\zeta$ , while Prandtl number increases the heat transfer rate at the surface of the disk.

The steady magnetohydrodynamic ternary hybrid nanofluid flow over a slender surface under the effects of activation energy, Hall current, chemical reactions, and a heat source has been reported. A numerical model is developed to increase the rate of energy transfer and boost the efficiency and outcome of heat energy dissemination for a diverse range of biological applications and commercial uses. The rheological properties and thermal conductivity of the base fluids are improved by framing an accurate combination of nanoparticles (NPs). The ternary hybrid nanofluid has been prepared, in the current analysis, by the dispersion of magnesium oxide, titanium dioxide (TiO₂), and cobalt ferrite (CoFe₂O₄) NPs in the base fluid. The physical phenomena have been expressed in the form of a system of nonlinear PDEs, which are degraded to a dimensionless system of ODEs through the similarity replacement and numerically solved by employing the MATLAB software package bvp4c. The graphical and tabular results are estimated for velocity, mass, and energy curves vs distinct physical factors. It has been noticed that the variation in the magnetic effect enhances the energy profile while the increasing number of ternary nanocomposites (MgO, TiO₂, and CoFe₂O₄) in water lowers the energy curve. Furthermore, the effect of both Lewis and Peclet numbers weakens the motile microbe’s profile.

The current article illustrates a three dimensional hydrodynamics nanoliquid flow between two parallel rotating plates. The ambition of the study is to inquire the combined effects of electric and magnetic field on nanoliquid flow with mass and heat transfer characteristics. The base fluid of water is synthesized with copper oxide CuO nanoparticles. The subsequent arrangement of flow model is reduced to a dimensionless set of equations through similarity approach. For the numerical solution, the transformed equations are further tackled with the MATLAB package boundary value solver (bvp4c). To ensure the accuracy, the outcomes are also simulated by Runge Kutta fourth order method. The nature of velocity, heat, and mass transfer rate versus Reynolds number, electrical parameter, molecular diffusion, magnetic parameter and rotation parameter are sketched and discussed. The growing credit of Dufour and Soret number effectively enhances the mass transfer rate, because due to molecular diffusion, the net flux of molecules from a region of higher concentration to lower concentration increases, as a result mass transfer ratio enhances. It has been concluded that CuO nanoparticles positively effects the molecular energy transmission and can be used for cooling purposes.

The recent work provides the numerical investigation of an unsteady viscous nanofluid flow between two porous plates under the effect of variable magnetic field and suction/injection. Navier Stokes equations are modeled to study the hydrothermal properties of four different nanoparticles copper $(\mathrm{Cu})$ , silver $(\mathrm{Ag})$ , aluminum oxide $(\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3)$ , and titanium oxide $(\mathrm{Ti}{\mathrm{O}}_2)$ . The resultant nonlinear partial differential equations, governing the viscous fluid flow, are solved numerically using Crank–Nicolson scheme. The effect of important physical parameters such as volume fraction, magnetic strength, and porosity parameter are shown both graphically and in tabular form. It is found that due to the greatest thermal diffusivity for nanofluid $\mathrm{Ag}$ , comparatively the velocity increases more rapidly with the increasing value of volume fraction. Due to this effect, it is preferred to use nanofluid $\mathrm{Ag}$ for transportation purposes.

The steady incompressible slip flow with convective heat transport under the impact of a variable magnetic field has been taken into an account over a revolving disk. The temperature dependent viscosity, density, and thermal conductivity has been scrutinized. The obtained system of nonlinear differential equations governing the induced magnetic field, steady flow, and heat transmission has put down in polar cylindrical coordinates. The subsequent arrangement of nonlinear PDEs are subside into dimensionless system of ordinary equations, while making use of similarity abstraction. The modeled equations are tackled through Homotopy Analysis Method (HAM). The skin fraction coefficient, heat transmission rate, and Nusselt number (skin effects coefficient) are deliberated. From the results, It can be perceived that the slip factor effectively controls the heat and the flow characteristics. The influence of dimensionless numbers such as Batcheler number $\mathrm{Bt}$ and magnetic strength $R_3$ and $R_4$ are explored and shown graphically. Further the out-turn of Prandtl number, relative temperature difference, suction parameter, and slip factor on the temperature fields and velocity profile are discussed.

The aim of this article is to provide an analytical and numerical investigation to the viscous fluid flow, heat and mass transfer under the influence of a variable magnetic field. The governing system of partial differential equations are transformed by means of similarity transformations to a system of ordinary differential equations which are solved by Homotopy Analysis Method (HAM) and BVP4c. The effects of involved physical parameters are illustrated for the velocity components, magnetic field components, heat and mass transfers. Authentification of HAM results for various involved physical parameters are supported by comparison with numerical results obtained by BVP4c. It is observed that increasing distance between discs increase pressure on lower disc and torque on upper disc. It is also observed that increase in axial component of magnetic field increase fluid's axial velocity and increase in magnetic Reynold's number decrease magnetic flux it lower disc. Heat flux from lower to upper disc is increased by increase in Dufour number.