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The unsteady hydromagnetic free convection flow of rotating incompressible viscous fluids across an infinite moving plate with fractional thermal transportation is explored in the presence of heat source, Hall current and slip velocity effects. The Laplace transform method is used to convert the controlling PDEs corresponding to the temperature and velocity profiles into linear ODEs. To get the results, the classical model is modified to a fractional-order model using constitutive relations of the generalized Fourier’s law for heat flux. After making the equations dimensionless, solutions to the energy and velocity equations may be found. Graphs are plotted to check the insight of physical characteristics. Moreover, the Mittag-Leffler kernel is most affected. The memory of temperature improved by using generalized Mittag-Leffler kernel in comparison of exponential kernel. Some graphical representations of the temperature and velocity were created using the software application Mathcad. As a result, it is found that with the application of a generalized Mittag-Leffler kernel, the thermal and momentum boundary layers as well as the memory of fluid properties can be enhanced for larger values of the fractional parameter.

Technologies regarding solar heating, ventilation and air-conditioning (S-HVAC) are aimed at making modern 3D numerical forms that address the Sutterby flow ternary nanofluids circulating onward the convective heating and extendable seats. Heat transport includes joule heating, heat source or sink along with thermal radiation. Using fitting modifications, mathematically conveyed partial differential equations of energy, fixation and strength may decrease into ordinary differential equations (ODEs). They determine ODEs beyond dimension, and for this, the mathematical process is utilized. Copper–silver–aluminum alloys/sodium alginate (Cu–Ag–AA7075/C₆H₉NaO₇) was used to address the behavior of this research work. The natural attributes, for example, heat movement and surface drag coefficients, are numerically prepared and shown in figures and tables when there is an alteration in distant factors. The field of temperature was raised to develop the Biot number. This heat transport rate was hiked to 34.0839% although the shear stress rate was hiked to 32.8043% in the single nanoparticle case compared to the triple nanoparticle case. To validate the analysis, a comparison between the presented and existing is reported under certain assumptions on the flow parameters. It is found that the results are reliable and in line with the existing ones.

This research discusses the investigation of heat and mass transfer in a magnetohydrodynamic (MHD) Casson fluid (CF) flow over an exponentially porous stretching sheet. The analysis takes into account the existence of thermal radiation, viscous dissipation, chemical reactions, and the influence of velocity and thermal slips. A recognized Casson model is taken into account in order to distinguish the characteristics of Casson fluid from those of Newtonian fluids. Using the geometry under consideration, the current physical problem is modeled. Appropriate similarity conversions are implemented to reduce the resultant set of coupled nonlinear PDEs to a set of nonlinear ODEs. By implementing the Keller-Box technique, numerical solutions to these reduced non-dimensional governing flow field equations are obtained. Tables and diagrams are utilized to illustrate the physical behavior of various control parameters. The temperature profile is enhanced and velocity profile diminished as the CF parameter value increased, according to this study. An increase in the velocity slip factor resulted in a diminution in the velocity field, while a gain in the thermal and concentration contours. With growing amounts of the chemical reaction factor, the concentration profile exhibited a decline. Indeed, the similar outcomes elucidated in this paper exhibit a remarkable correspondence with solutions that have been previously documented in the academic literature. This research may be motivated by a desire to improve the comprehension of fluid flow in different engineering and environmental situations, where these conditions are common, such as geothermal energy extraction, thermal management, chemical processing industries, and environmental control technologies.

This study investigates the impact of variable permeability as well as chemical reactions on the oscillatory free convective flow that passes parallel porous flat plates with fluctuating temperature and concentration in the presence of a magnetic field. A vertical channel is assumed to be rotating at an angular velocity $\mathrm{\Omega }$. Periodic free stream velocity causes oscillations in one plate, while periodic suction velocity causes oscillations in the other plate. Complex variable notations are used to solve the governing equations. The perturbation technique is used to derive analytical expressions for the temperature, concentration, and velocity fields. In this study, various parameters were investigated in relation to mean velocity, mean temperature, mean concentration, amplitude, and phase difference. The study also examines the impact on secondary velocity, primary velocity, temperature, concentration, and heat transfer rate during transients. The outcomes are presented graphically for the physical parameters of the problem. The findings contribute to optimizing systems and improving efficiency in heat transfer, fluid dynamics, and environmental remediation.

The principal aim of the work is to investigate the flow of Williamson fluid on a power law extended lubrication surface with partial sliding under the Magnetohydrodynamic issue on account of alterable thickness lubricated film. Accounting for the influence of microbes, assumption of activation energy, Cattaneo–Christov mass and heat flux took place in the equations of concentration and also in the temperature. Our paper will provide remarkable help to the medical and industrial areas because of the inspection under electro osmosis along with MHD effects in the Williamson fluid flow. With the influence of boundary conditions, built-in equations were studied. The BVP4C technique adopted to solve numerically the transformed ordinary differential equations from nonlinear partial differential equations using innumerable variables. The significant outlines of microorganisms, temperature, concentration and velocity were discussed. Decreases in the velocity distribution were observed with magnetic parameters. Also, an increase in friction coefficient was noticed as 3.1069% for rising magnetic field strength.

Swirling flows are important in rheological devices, spin coatings and lubrication, so we set out to investigate what makes chemically reactive non-Newtonian spinning flows across a disk with a radially applied magnetic field so interesting. Nanofluids are thermally enhanced working fluids with many interesting physical properties. This study takes its inspiration from rotating disk oxidations used in the medical techno industry and builds a mathematical model of a continuous convective von Kármán swirling flow including Jeffrey, magnetic, Joule/ohmic and chemical reactions. The wall anisotropy slips and the concentration-induced blowing effects are included. By using the bvp4c approach, the transformed boundary conditions (BCs) are addressed. Graphical representations of the effects of involved parameters on the density distribution of motile microorganisms, concentration, temperature and dimensionless velocity components are shown. Supporting evidence from prior research is included. Novel bioreactors, membrane oxygenators, bio-chromatography and food processing should take note of the study’s findings. As Jeffrey’s parameter upsurges, there is a decrease in radial velocity. As the Jeffrey parameter increases, there is a decrease in the circumferential velocity. Radial flow is significantly enhanced near the wall as the radial slip parameter ($\delta u)$ increases. As the Eckert number grows, the quantity of temperature increases. Concentration distribution closer to the disk to grow as Le increases. The concentration and diffusivity of microorganisms drop as the number of motile microorganisms thickens.

Development of thin Casson liquid (CL) film on a heated nonlinear flat stretching surface is examined under influences of thermal radiation and transverse magnetic field. The velocity and temperature at any point of the stretching surface are assumed as the generalized nonlinear functions of the distance of that point. The analytical expressions for velocity components and temperature are obtained using long-wave approximation technique. The numerical solution for nonlinear film evolution equation is incurred by the Newton–Kantorovich method. It is found that initial non-uniform film thickness becomes flat with due course of time. It is further observed that the film thinning rate enhances for larger values of the Marangoni number and radiation parameter. It is also discovered that the rate of film thinning diminishes for the larger Hartmann number and Casson parameter.

The objective of this study is to determine the irreversible losses and associated entropy generation within a fluid system, considering the combined effects of magnetic field, convective boundaries, and porous media. It accomplishes this objective by a thorough investigation into the second law analysis and entropy generation of a magnetohydrodynamic (MHD) Eyring–Powell fluid flowing through a symmetric porous medium. To achieve this, the governing equations for the Eyring–Powell fluid are formulated using the conservation laws of mass, momentum, and energy, while incorporating the magnetic field’s effects. In order to account for the porous character of the medium, the equations are coupled with the Darcy model. Using appropriate computational techniques, the resulting system of partial differential equations is numerically solved. The local irreversibility ratio calculates the system’s entropy generation number, revealing its distribution. The Hartmann number and Eyring–Powell fluid parameters are also studied. The primary findings indicate that $A^{\ast }$ enhances velocity and diminishes temperature and entropy, while $B^{\ast }$ has the opposite effect. Entropy is also increased by Hartmann and Brinkman numbers, which are a result of the enhanced heat transfer and stronger magnetic fields. The findings emphasize the need and importance of studying irreversible losses and improving fluid system energy efficiency.

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.

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.