The current problem is concerned with the study of magnetohydrodynamic ternary hybrid nanofluid flow over two distinct geometries i.e., cone and wedge. The ternary hybrid nanoliquid with MHD has a lot of engineering and industrial applications. In polymer data processing, cone and wedge geometries are frequently utilized. Therefore, the present problem is designed to the flow of ternary hybrid nanoliquid over multiple geometries. Hybrid nanoliquids performed well in the heat transport rate as compared to the nanoliquid and conventional liquid. Here in this study, the idea of ternary hybrid nanoliquid is introduced to improve the energy and mass transmissions which show more satisfactory results in the thermal and mass transmission performance. The impacts of chemical reaction and thermal radiation are also executed in this model. The formulation of the present study is performed in the form of PDEs which are then transformed into the ODEs by using suitable similarity transformations. The homotopic analysis scheme is implemented for the semi-analytical solution of the existing model. Some major results that materialize from the present simplification are that; the tri-hybrid nanoliquid velocity is greater for the rising nanoparticles volume fractions. The enlargement in radiation parameter enlarged the tri-hybrid nanoliquid thermal profile. The mass transfer rate of the ternary hybrid nanoliquid is lesser for the Schmidt number and chemical reaction. Intensification in nanoparticles volume fractions and radiation parameter has increased the ternary hybrid nanofluid heat rate transfer for both cone and wedge geometries.

In 2012, Rivaie et al. introduced RMIL conjugate gradient (CG) method which is globally convergent under the exact line search. Later, Dai (2016) pointed out abnormality in the convergence result and thus, imposed certain restricted RMIL CG parameter as a remedy. In this paper, we suggest an efficient RMIL spectral CG method. The remarkable feature of this method is that, the convergence result is free from additional condition usually imposed on RMIL. Subsequently, the search direction is sufficiently descent independent of any line search technique. Thus, numerical experiments on some set of benchmark problems indicate that the method is promising and efficient. Furthermore, the efficiency of the proposed method is demonstrated on applications arising from arm robotic model and image restoration problems.

In the current research community, certain new fractional derivative ideas have been successfully applied to examine several sorts of mathematical models. The fractal fractional derivative is a novel concept that has been proposed in recent years. In the presence of heat generation, however, it is not employed for the free convection Couttee flow of the Casson fluid model. The core interest of the present analysis is to examine the Casson fluid under the influence of heat generation and magnetic field. The flow of the Casson fluid has been considered in between two vertical parallel plates. The distance between the plates is taken as $l$ . The linear coupled governing equation has been developed in terms of classical PDEs and then generalized by employing the operator of the fractal-fractional derivative with an exponential kernel. The numerical solution of the proposed problem has been found employing the finite-difference technique presented by Crank–Nicolson. The Crank–Nicolson finite difference scheme has the advantage of being unconditionally stable and can be applied directly to the PDEs without any transformation to ODEs. This technique in sense of exponential memory has been revealed to be unreported in the literature for such a proposed problem. For graphical analysis, the graphs of velocity profile and thermal field have been plotted in response to several rooted parameters. For comparative analysis, the graphs for the parameter of fractal-fractional, fractional, and classical order have also been plotted. From the analysis, it has been found that the fractal-fractional order model has a large memory effect than the fractional-order and classical model due to the fractal order parameter.

The flow of fluid over a spinning disk has a broad scope of numerous applications. It is employed in various things, including medical equipment, the braking system of cars, gas turbines, plastic films, and glass production. As a result of these applications, we considered the phenomena of Darcy Forchheimer's three-dimensional flow on $Ti{O}_2-{Fe}_3{O}_4$ nanoparticles suspended in based CMC-water fluid. The influence of thermal radiation and convective conditions is studied. Moreover, the Buongiorno model is utilized to compute the Brownian motion and the thermophoretic effect. To generate the non-dimensionalized governing equations, suitable alterations are put into use. These equations are then utilized with Matlab BVP4c. Graphs are used to analyze the behavior of velocity distributions, and thermal and concentration profiles at different parameter values. In addition, the solutions to the flow problem have been analyzed in terms of several other physical variables on velocity, temperature, concentration, drag force, heat, and mass transfer. According to the findings, it is clear that an escalates in the value of the rotation parameter leads to an increase in the radial velocity and axial velocity. In contrast, an opposite pattern is followed in the Forchheimer number. Finally, some engineering quantities are evaluated numerically and presented in tabular forms.

This article presents the magnetohydrodynamic (MHD) flow of a nanoliquid due to a rotating sphere at a stagnation point. The flow is considered to be influenced by the magnetic field, dissipative, thermally radiative, and chemically reactive. Also, the thermophoretic and Brownian motion influences are taken into consideration. Some restrictions in the present analysis are taken: like there is no-slip and convective conditions, joule heating, Hall effects and buoyancy-driven. The solution of the present analysis is derived through the homotopy analysis method (HAM). The significance of several physical parameters on velocities, thermal and concentration profiles are shown with the help of Figures. Also, the significance of different physical factors on skin frictions, local Nusselt number and Sherwood number are demonstrated with the help of Tables. The outcomes show that the Nusselt number is lower for the larger Brownian motion parameter, Eckert number, and thermophoretic parameter, while the increment in the thermal radiation parameter augmented the Nusselt number. It is established that the increasing rotation, magnetic and positive constant parameters have increased the velocity profiles along the x-direction while reducing the velocity profiles along the z-direction of the nanoliquid flow. The increasing positive constant parameter reduces the thermal graph of the nanoliquid flow. Furthermore, the intensifying Eckert number, thermophoresis, Brownian motion, and thermal radiation factor have escalated the thermal profiles of the nanoliquid flow.