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This research utilizes the bvp4c method to conduct a detailed numerical analysis of the hydrothermal behavior of magnetized hybrid nanofluids flowing across a permeable curved surface. The study explores the impact of crucial parameters such as curvature, magnetic field strength, viscosity and suction/injection, alongside the heat absorption coefficient, on the transport properties of copper (Cu) and ferric oxide (Fe₃O₄) nanomaterial’s suspended in water. Results reveal that as the curvature parameter increases, velocity profiles exhibit a decrease under suction conditions and an increase under injection conditions for both conventional and hybrid nanofluids. Furthermore, higher magnetic parameters are found to decrease velocities in general. Hybrid nanofluids display enhanced velocity and thermal performance compared to conventional nanofluids, manifesting higher skin friction and heat transfer rates. Temperature profiles exhibit a complex interplay with curvature, magnetic parameters and the injection/suction scenario, where injection conditions intensify thermal effects. The incorporation of the heat absorption coefficient further amplifies the thermal efficiency of hybrid nanofluids. These findings, supported by previous research, offer valuable insights for optimizing industrial processes, especially in sectors like ceramics, plastics and polymers, where efficient heat management is paramount.

This paper investigates the influence of magneto-tangent hyperbolic nanofluid on the flow of a tri-hybrid nanoliquid consisting of ${\mathrm{\text{MoS}}}_2,\mathrm{\text{Si}}{\mathrm{\text{O}}}_2$, and $\mathrm{\text{GO}}$ particles suspended in $\mathrm{\text{EG}}$. The entropy production is encountered in this analysis. The fluid flows over a stretch sheet is considered. In addition, the energy equation also assumes the existence of a uniform heat source or sink and thermal radiation. Furthermore, the concentration equation emphasizes the chemical reaction. The current proposed model yields a set of nonlinear governing equations. The modeled formulation is transformed into a dimensionless system through the application of a suitable alteration. The complex nonlinear equation system was solved using the bvp4c through numerical methods. The main motive of this exploration is to emphasize the rate of heat and mass transfer in a flow of ${\mathrm{\text{MoS}}}_2,\mathrm{\text{Si}}{\mathrm{\text{O}}}_2$, and $\mathrm{\text{GO}}/\mathrm{\text{EG}}$-based hybrid nanofluid across a stretch sheet. The graphical study illustrates that Weissenberg number and magnetic field enhancement result in decreasing the velocity. But thermal layer, entropy production, and Bejan number are enhanced with larger values of Weissenberg number and magnetic field. This study focuses on different profiles with various flow parameters. Furthermore, we have compared the tri-hybrid nanofluid with the hybrid and mono nanofluid in all the figures and tabular format. Additionally, we have compared tri-hybrid, hybrid, and mono nanofluid using graphs for velocity, temperature, concentration, entropy production, and Bejan number.

The aim of this study is to analyze heat transfer over two horizontal concentric cylinders in the influence of MHD, internal heat source containing porous nanofluids and thermal radiation are considered. The novelty of this work is internal heat source and porous media of H₂O–Cu nanofluids with the Lorentz effect are investigated and its applications are cooling systems, and heat exchangers. In addition, transformation for the momentum and energy equation is applied to obtain a set of ODEs for governing equations in the heat transfer flows. Further, the numerical technique BVP4C is used to solve the resulting system of nonlinear, coupled equations with boundary conditions. The influence of Hartmann number, volume fraction, radiation parameter, internal heat source parameter, Darcy number and different nanoparticles are examined in velocity and temperature profiles. The results show good agreement with the existing work of velocity and temperature graphs. Moreover, they reveal that thermal radiation significantly influences temperature distribution within the annulus, leading to a higher heat transfer rate. Furthermore, the presence of a porous medium and internal heat source modulates the flow patterns. This study provides optimizing MHD nanofluid systems for engineering applications such as thermal management systems, hyperthermia treatment in cancer therapy, food processing, rotating machinery and cooling systems.

We compute general-relativistic polytropic models of magnetized rotating neutron stars, assuming that magnetic field and rotation can be treated as decoupled perturbations acting on the nondistorted configuration. Concerning the magnetic field, we develop and apply a numerical method for solving the relativistic Grad–Shafranov equation as a nonhomogeneous Sturm–Liouville problem with nonstandard boundary conditions. We present significant geometrical and physical characteristics of six models, four of which are models of maximum mass. We find negative ellipticities owing to a magnetic field with both toroidal and poloidal components; thus the corresponding configurations have prolate shape. We also compute models of magnetized rotating neutron stars with almost spherical shape due to the counterbalancing of the rotational effect (tending to yield oblate configurations) and the magnetic effect (tending in turn to derive prolate configurations). In this work such models are simply called "equalizers." We emphasize on numerical results related to magnetars, i.e. ultramagnetized neutron stars with relatively long rotation periods.

Magnetohydrodynamic flow of nanofluids and heat transfer between two horizontal plates in a rotating system have been examined numerically. In order to do this, the group method of data handling (GMDH)-type neural networks is used to calculate Nusselt number formulation. Results indicate that GMDH-type NN in comparison with fourth-order Runge–Kutta integration scheme provides an effective means of efficiently recognizing the patterns in data and accurately predicting a performance. Single-phase model is used in this study. Similar solution is used in order to obtain ordinary differential equation. The effects of nanoparticle volume fraction, magnetic parameter, wall injection/suction parameter and Reynolds number on Nusselt number are studied by sensitivity analyses. The results show that Nusselt number is an increasing function of Reynolds number and volume fraction of nanoparticles but it is a decreasing function of magnetic parameter. Also, it can be found that wall injection/suction parameter has no significant effect on rate of heat transfer.

Lattice Boltzmann method (LBM) was used to simulate two-dimensional MHD Al₂O₃/water nanofluid flow and heat transfer in an enclosure with a semicircular wall and a triangular heating obstacle. The effects of nanoparticle volume fraction ($0\le \varphi \le 0.05$), Rayleigh number $(10^4\le Ra\le 10^6)$, Hartmann number $(0\le Ha\le 60)$ and heating obstacle position (Cases 1–7) on flow pattern, temperature distribution and rate of heat transfer were investigated. The results show that with the enhancing Rayleigh number, the increasing nanoparticle volume fraction and the reducing Hartmann number, an enhancement in the average Nusselt number and the heat transfer appeared. The effect of Ha on the average Nu increases by increasing the Ra. It can also be found that the action of changing the heating obstacle position on the convection heat transfer is more important than that on the conduction heat transfer. The higher obstacle position in Cases 6 and 7 leads to the small value of the average Nusselt number. Moreover, the effect of Ha on average Nu in Case 1 at $Ra=10^6$ is more significant than other cases because the flow pattern in Case 1 is changed as increasing Ha.

This research attempts to draw a fuller understanding of the magnetohydrodynamic flow of Williamson fluid owing to a continuously moving permeable surface. A detailed illustration of the importance of the effects of viscous dissipation and the variable fluid properties phenomena are also given. Meanwhile, this work has also revived the interest of the slip velocity phenomenon. The success is met after formulating the thorough equations which describe this model, thus it is no longer difficult to get into the numerical solution for this model via the shooting method. By analogy, this interesting subject also stimulates us to pursue further details which can be obtained from both the drag force or the skin-friction coefficient and the rate of heat transfer. Based on our apparent results, the presence of magnetic field, suction phenomena and slip velocity was the reason to reduce the surface velocity. Also, the same parameters can serve as a helpful guide for governing the rate of heat transfer.

This work investigates the time-dependent squeezed and extruded flow of viscous incompressible and electrically conducting fluid between two convective parallel plates under an impressed transversely applied magnetic field. The governing nonlinear partial differential equations are reduced to dimensionless form using some selected dimensionless parameters and solved numerically in Matlab. A detail analysis illustrating the influence of various governing parameters like extrusion, squeezing, Eckert, Hartmann number, dimensionless time and Biot number on fluid velocity, normal fluid velocity, temperature, rate of heat transfer and local coefficient of skin friction are highlighted and discussed. The results reveal that rate of heat transfer and coefficient of local skin friction can be optimized by increasing/decreasing the squeezing parameter. Also, the result also reveal that increasing the extrusion parameter and the Hartmann number reduces the rate of heat transfer and enhances the coefficient of local skin efriction.

Current continuation describes the computational study concerning with the unsteady flow of Eyring–Powell magneto nanoliquid over a bidirectionally deformable surface. Transference of activation energy is used in the improvement of binary chemical reaction. Nonlinear significance of thermal radiation is also incorporated in the energy equation. Investigation has been carried out through convective Nield’s boundary restrictions. Firstly, useful combination of variables has been implemented to alter the governing PDEs into ODEs. Later on, Keller-Box approach has been adopted to obtain the numerical solution of the physical problem. Physical interpretations of obtained results are also described for temperature and mass concentration distributions through various graphs. Rate of heat transportation has been explained through tabular data for acceptable ranges of involved engineering parameters. It is detected that escalating amount of Brownian constraint provides a constant temperature distribution. It is also inspected through present investigation that escalating amounts of activation energy factor, thermophoresis parameter, radiation parameter, Biot number and temperature ratio parameter improve the concentration field. Moreover, the amount of heat transport has considerably improved by increasing the amounts of temperature controlling indices and Biot number. Convergence analysis and error estimations of the numerical solution are also presented through various mesh refinement levels of the computational domain. Finally, comparison benchmarks with the restricted cases have been presented for the validation of the results obtained through the present parametric investigation.

Current investigation deals with influence of inclusion of nanoparticles within the permeable medium within a tank with circular outer wall. The inner surface is hot and the radiative term has been imposed in temperature equation. Vorticity formula helps us to remove the pressure terms from equations and CVFEM was incorporated to calculate the amount of scalars in each node. With correlating the current data from previous paper, verification procedure was done which demonstrates good accuracy. Permeability has crucial role and greater values of Da results in stronger thermal penetration and isotherms become more disturbed. Intensity of cell augments with rise of Da about 70% in absence of Ha. Impose of Rd cannot affect the isotherms too much while it can change the Nu regarding the definition of this factor. When $\text{Da}=0.01$, growth of Ha can decline the strength of eddy about 35%. Given $\text{Ha}=20$, as Da increases, Nu enhances about 10.24% and 0.25% when $\text{Ra}=1$e5 and 1e3, respectively. Replacing platelet with sphere shape can augment the Nu about 0.38% and 0.6% when $\text{Ha}=20$ and 0, respectively.

In this paper, porous chamber with considering nanomaterial as operating fluid has been scrutinized. The transportation of nanopowder was controlled by magnetic force and insert of porous media boosts the cooling rate. Such zone needs special model to involve the impact of porous media and in this paper, non-Darcy technique was utilized. Low fraction of hybrid nanomaterial leads to good accuracy of homogeneous model and empirical correlations have been employed to forecast the features of operating fluid. Entropy generation was studied to find the influence of each term on irreversibility of unit. Also, two significant functions were calculated, namely, Be and Nu. Influences of Ra, Da and Ha on contours plots were reported in outputs. As Ra augments, the convection becomes stronger and augmentation of $\mathrm{\Psi }$ proves this fact. Also, temperature of elliptic surface declines about 48% with intensifying Ra. Temperature of elliptic surface augments about 34.6% with augment of Ha while it declines about 25.7% considering greater permeability. Nu augments about 238%, 11.49% with rise of Ra, Da but it declines about 28.3% with rise of Ha. Be intensifies with rise of Ha about 7.76% while it reduces about 75.2% with augmentation of Ra.

This communication is to analyze the Marangoni convection MHD flow of nanofluid. Marangoni convection is very useful physical phenomena in presence of microgravity conditions which is generated by gradient of surface tension at interface. We have also studied the swimming of migratory gyrotactic microorganisms in nanofluid. Flow is due to rotation of disk. Heat and mass transfer equations are examined in detail in the presence of heat source sink and Joule heating. Nonlinear mixed convection effect is inserted in momentum equation. Appropriate transformations are applied to find system of equation. HAM technique is used for convergence of equations. Radial and axial velocities, concentration, temperature, motile microorganism profile, Nusselt number and Sherwood number are sketched against important parameters. Marangoni ratio parameter and Marangoni number are increasing functions of axial and radial velocities. Temperature rises for Marangoni number and heat source sink parameter. Activation energy and chemical reaction rate parameter have opposite impact on concentration profile. Motile density profile decays via Peclet number and Schmidt number. Magnitude of Nusselt number enhances via Marangoni ratio parameter.

To detect the influence of MHD on migration of hybrid nanopowders, CVFEM has been employed in this paper. Mixture of Fe₃O₄ and MWCNT was added in water and for calculating properties, experimental formulas were utilized. To increase the convective mode, porous media has been used and tank was experienced in the horizontal magnetic field. In governing equations, there exist two new terms, one for permeable media and the other for MHD effect. Such complex physics needs special numerical approach and CVFEM has been utilized for this goal. Final formulation of PDEs did not have pressure terms and the stream function scalar was introduced. As permeability of zone enhances, nanopowders can transfer faster and the interaction of them with wall enhances, so, Nusselt number augments as well as stream function. Besides, employing higher Ha, the force against the buoyancy force increases and the velocity of operating fluid declines which provides lower convective flow. With rise of Ha, stream function declines about 48% and Nu declines about 31.92% when $\mathrm{\text{Da}}=100$. As Da rises, Nu rises about 33.43% when $\mathrm{\text{Ha}}=0$. Augment of Rd leads to augmentation of Nu.

Many studies have been performed on non-Newtonian fluid flow through the nonuniform stretching sheet. In most cases, the problem is assumed to be two-dimensional fluid flow and a similarity solution is exploited. In this paper, we consider the effect of viscous dissipation on the MHD non-Newtonian fluid flow and heat mass transfer due to slendering stretching sheet with thermal radiation. Both Williamson and Casson models are opted here while the mathematical modeling is used for the principle flow equations. By dimensionless transformation, the governing equations are transformed to identically coupled three equations along with three common boundary conditions imposed. They are then solved numerically by using Chebyshev spectral method for different values of the physical parameters involved. These governing parameters are graphically shown to have a considerable influence on the fluid flow and heat mass transfer characteristics of this model. Local heat transfer rate is found to depend on both magnetic parameter and Casson parameter in addition to the Eckert number dependence. Likewise, due to increase of Casson parameter, the sheet temperature and the thermal boundary layer thickness enhances and also the same behavior is observed for increasing the radiation parameter.

In this study, the numerous solutions to Falkner–Skan flow of a Maxwell fluid with nanoparticles are investigated, considering the nonlinear radiation and magnetic domain. The flow described above can be expressed in accordance with PDEs that are transformed into ODEs by choosing suitable variables of similarity. The fourth- and fifth-order Runge–Kutta–Fehlberg method can be utilized to solve these reduced ODEs by applying the shooting approach. The graphs were drawn to explain the effects of different parameters on different fluid profiles for both the lower- and upper-branch solutions. This study shows that the velocity outlines improve both solutions by increasing local Deborah numbers slightly. Besides, an increase in radiation reduces the thermal gradient for both solutions, thereby reducing the concentration gradient for both solutions contributing to raised Brownian motion and Lewis numbers.

In this investigation, numerical modeling for the behavior of nanomaterial inside a porous zone with imposing Lorentz force has been illustrated. The working fluid is a mixture of H₂O and CuO and due to concentration of 0.04, it is reasonable to use the homogeneous model. Two-temperature model for porous zone was employed in which new scalar for calculating temperature of solid region was defined. CVFEM has been applied to model this complex physics. Radiation terms were considered and their influence on Nu has also been considered. Verification with benchmark proves greater accuracy. Dispersing nanopowders helps the fluid to increase velocity and reduce the temperature of inner wall. Rise of Ra results in three strong eddies inside the zone which creates two thermal plumes and it reduces the temperature of square surface about 68%. With rise of Nhs, the power of counter-clockwise vortex reduces about 61.6% and inner wall becomes warmer about 33.3%. Raising the Ha makes thermal plume to vanish and cooling rate decreases about 46.6%. Augment of Nhs makes Nu to reduce about 5.08% while augment of Ra makes it to augment about 35.64%. Also, augmenting Ha makes Nu to decline about 56.45%.

In this paper, He’s homotopy perturbation method (HPM) is used, which is an approximate analytical method for solving numerically the problem of Newtonian fluid flow past a porous exponentially stretching sheet with Joule heating and convective boundary condition. The major feature of HPM is that it does not need the small parameters in the equations and hence the determination of classical perturbation can be discarded. Due to the complete efficiency of the HPM, it becomes practically well suited for use in this field of study. Also, the obtained solutions for both the velocity and temperature field are graphically sketched. The results reveal that the proposed method is very effective, convenient, and quite accurate to systems of nonlinear differential equations. Results of this study shed light on the accuracy and efficiency of the HPM in solving these types of nonlinear boundary layer equations.

In this work, an approximate analytical solution for the problem of non-Newtonian Casson fluid flow past a porous exponentially stretching sheet with Joule heating and convective boundary condition is obtained using a relatively new technique; He’s homotopy perturbation (HPM). The major feature of HPM is that it does not need the small parameters in the equations, and hence the determination of classical perturbation can be discarded. Due to the complete efficiency of the HPM, it becomes practically well suited for use in this field of study. Also, the obtained solutions for both the velocity and temperature field are graphically sketched. The results reveal that the method is very effective, convenient and quite accurate to systems of nonlinear equations.

In this paper, an efficient numerical treatment for magnetohydrodynamic flow and heat transfer calculations is presented, based on the differential transformation method (DTM), and the finite element method (FEM). This numerical treatment is obtained for the ordinary differential equation (ODE) which describes physically the boundary layer flow due to a permeable shrinking sheet. The DTM and FEM are utilized in this study because of their capacity to solve linear and nonlinear systems of ODEs, as well as their accuracy and convenience of use. These methods are approximate analytical that can usually get the solution in a series form. These numerical procedures are effective for this type of physical problem of varying degrees of complexity. The numerical calculations yield that the dimensionless velocity enhances when the wall mass suction for the flow is increased after the magnetic field is imposed. Also, the dimensionless velocity was found to increase for the large value of the magnetic parameter.

In this study, the combined effect of Joule heating and Soret number on magnetohydrodynamics (MHD) braking of incompressible stratified liquid metal packed between a confined rotating annulus has been investigated. Both axial sinusoidal temperature and as well as sinusoidal concentration have been imposed in this braking system. However, radial cooling is enforced on the rotating disc of the braking system. The Soret effect on concentration gradient has been observed when the stratified liquid is exposed to a temperature gradient along with an axial magnetic field at different buoyancy ratios.

A significant difference in convection heat and mass flow can be seen. The average shear wood decreased from 8.97 to 2.36 and the average Nusselt number decreased from 8.6 to 2.8 between the outer rotating disk and inner stationary disks at positive buoyancy ratios. This huge difference in heat and mass convection near the inner disk with respect to the outer brake disk confirms the MHD braking effect. Moreover, comparative studies of mass convection with respect to convective heat transfer have been observed based on the average magnitude of Sherwood and Nusselt numbers at both positive and negative buoyancy ratios.