ABSTRACT In this work, the effects of a magnetohydrodynamics (MHD) field on the heat and mass transport properties of a chemically reactive, radiating Casson ternary nanofluid (TNF) across an exponentially stretching surface in a porous medium are examined. Water‐based silicon dioxide (SiO 2 ), copper (Cu), and silver (Ag) nanoparticles compose the nanofluid. The research examines how velocity, temperature, and concentration profiles are simultaneously affected by permeability, thermophoresis, radiation, Brownian motion, and chemical reactions. To explore the effects of important parameters, the governing partial differential equations (PDEs) are converted into ordinary differential equations (ODEs) using a similarity transformation and solved numerically in MATLAB using the boundary value problem (BVP) fourth‐order code (bvp4c). The derived quantities are analyzed using the response surface methodology (RSM). The increased porosity, magnetic field intensity, and the Casson fluid parameter reduce velocity. Additionally, Brownian motion and radiation increase temperature dispersion. According to the results, the optimization study shows that the model is highly accurate in R 2 , with the reaching 96.07%, the reaching 99.68%, and the reaching a 99.65% level of significance. These findings provide novel nanofluid heat and mass transport technologies for industrial and technical applications.

This study investigates the magnetohydrodynamic (MHD) characteristics of hybrid nanofluid flow confined between concentric cylindrical surfaces. The governing momentum and energy equations are formulated and solved subject to appropriate boundary conditions to analyze fluid motion and heat transport mechanisms. Particular attention is given to the roles of the Darcy number and magnetic field strength in regulating flow behavior and thermal performance. Comparative assessments demonstrate the reliability of the results and reveal that hybrid nanofluids provide enhanced thermal effectiveness in restricted geometries. An increase in the Darcy number lowers resistance within the porous medium, leading to higher fluid velocities and improved heat transfer, which is beneficial for geothermal and subsurface energy applications. Variations in the shape factor influence the flow by reducing velocity while promoting a more uniform temperature distribution, thereby contributing to greater thermal stability in control and regulation systems. Intensifying the internal heat generation elevates temperature gradients and induces fluid expansion, strengthening heat transfer in systems such as cooling devices and nuclear reactors. Furthermore, higher magnetic field intensity suppresses fluid motion due to Lorentz forces. The increased wall shear stress caused by inertial factors causes the skin friction coefficient increase along with the Reynolds number. The thinner thermal boundary layer causes the Nusselt number to escalation with the Reynolds number. These findings offer valuable insights for the design and optimization of advanced thermal management systems, energy technologies, and industrial applications.

Oil and water-based immiscible nanofluids are being investigated for enhanced oil recovery in the petroleum industry. The removal of leftover oil from reservoir rocks can be enhanced by these nanofluids, which contain nanoparticles dissolved in either water or oil. The procedure of mixing several kinds of solid nanoparticles with different thermophysical properties such as bi/tetra-nanoparticles, into a nanoliquid based on kerosene and water, respectively is crucial. Two immiscible fluids’ flow characteristics in a horizontal semi-corrugated conduit are examined while both electric and magnetic fields are applied. Two regions make up the channel: the first is filled with a Casson bi-nanofluid that conducts electricity and contains two distinct kinds of nanoparticles, and the second is full of permeable matrix soaked in a conducting electrical Newtonian tetra-nanofluid having four distinct kinds of nanoparticles, which has been overlooked in prior published literature. Closed-form functions for the fluid velocity and temperature were identified by applying the perturbation method with appropriate conditions. For a range of problem parameter values, an analysis has been performed for velocity, temperature, shear stress, and heat transmission rate. The effects of pertinent fluid flow parameters on the Bejan and entropy generating numbers were evaluated in detail. The results show the behaviors of two immiscible nanofluids flowing in a channel are more affected by the electric fields, porous media, and the width ratio of the channel. Mathematical evaluation that is intended to be helpful in applications involving oil/water-based nanoflows is presented in the proposed study.

Enhancing heat transfer efficiency and understanding fluid flow behavior in convergent–divergent channels remain significant challenges in advanced thermal engineering applications. This study presents a novel numerical investigation of magnetohydrodynamic (MHD) Jeffrey–Hamel flow through a porous medium in convergent and divergent channels by simultaneously incorporating Joule heating, thermal radiation, activation energy, and stretching/shrinking wall effects in the presence of hybrid nanofluids—an aspect not previously addressed in a unified framework. Three different hybrid nanofluids are considered to evaluate their comparative thermal performance. The governing equations are solved using the bvp5c method. The influences of key physical parameters, including the magnetic parameter, inertia parameter, porosity parameter, stretching/shrinking parameter, and channel angle, on the velocity distribution are examined. Additionally, the effects of the magnetic parameter, Eckert number, thermal radiation, and heat source strength on the temperature field are analyzed, along with the role of activation energy on the concentration profile. The results reveal that the combined effects of magnetic field and inertia significantly enhance the skin-friction coefficient. Heat transfer is intensified with increasing magnetic parameter and heat source strength, as reflected by higher Nusselt numbers, while mass transfer is enhanced with increasing magnetic parameter and decreasing activation energy. Among the hybrid nanofluids studied, the MoS2-Al2O3 water hybrid nanofluid exhibits the highest heat transfer efficiency, highlighting its potential for advanced MHD-based thermal systems.

This study introduces novel numerical methods that employ spectral Galerkin and collocation techniques with shifted Schröder polynomials (SSPs) to solve linear and nonlinear second-order two-point boundary value problems (SOTBVPs). The proposed techniques are formulated through a reduced sequence of modified sets of SSPs. The unknown expansion coefficients are determined by using spectral Galerkin and collocation techniques. The resulting algebraic systems are efficiently solved using appropriate numerical solvers. Illustrative examples are provided to validate and demonstrate the accuracy, efficiency, and applicability of the proposed methodologies.

Reline, being a sub-class of ionic liquids consisting of hydrogen bond acceptor (HBA) and hydrogen bond donor (HBD), exhibits properties such as low melting points, high thermal stability, and tunable properties. It is one of the deep eutectic solvents with the composition Choline Chloride-Urea and is also known as an environmentally friendly solvent in various chemical processes. This attribute of the reline aligns well with sustainable practices in the chemical industry as an impressive alternative to conventional solvents. Its application extends to some of the major fields that mankind is dependent upon, such as pharmaceuticals, biotechnology, energy storage, etc. On the other hand, single-walled carbon nanotubes provide a higher thermal conductivity and specific heat capacity, making them useful in various heat exchangers, thermal storage systems, metallurgical applications, etc. Hence, in this study, the nanofluid is formed by considering reline as the base fluid with SWCNT suspensions. The simulation is performed to understand the various aspects of reline-based nanofluid flow inside elliptic ducts. The study explores the impact of the Reynolds number, the aspect ratio of the duct, and the thermophysical properties of SWCNT-reline nanofluid on the system's behaviour. Navier-Stokes' equation is utilised to perform the Mathematical analysis to understand the flow and heat transfer behaviour of SWCNT-reline nanofluid. From the results, it was clearly observed that the velocity at the narrow region decreased as the pressure rose, and Reynold's number profile indicated the presence of turbulent flow behaviour. (c) 2026 L&H Scientific Publishing, LLC. All rights reserved.

Abstract This study explores entropy generation in the magnetohydrodynamic flow of a tetra hybrid nanofluid. ( $${\text{Al}}_{2} {\text{O}}_{3} - {\text{Cu}} - {\text{SiO}}_{2} - {\text{TiO}}_{2} /{\text{H}}_{2} {\text{O}}$$ ) over a permeable surface embedded in a Darcy–Forchheimer porous medium, motivated by the need to enhance thermal system efficiency and minimise energy losses. The model incorporates key physical effects, including multiple dissipation mechanisms (Ohmic, viscous, Darcy, and Forchheimer), thermal radiation, internal heat generation, activation energy, and chemical reaction. The transformed nonlinear equations are solved numerically using the bvp4c technique. Results reveal that thermal radiation and heat source significantly elevate temperature and entropy generation, indicating increased irreversibility, while magnetic field and porous resistance strongly suppress fluid motion. Activation energy enhances species concentration, whereas a higher Schmidt number reduces mass diffusion. The Bejan number decreases under stronger magnetic and porous effects but increases with radiation and inertial resistance. The combined influence of these parameters provides deeper insight into controlling heat and mass transfer in advanced nanofluid systems. The findings are particularly relevant for applications in energy systems, cooling technologies, and thermal management, where minimising entropy generation is crucial for improving performance and efficiency.

The distinction between homogeneous and heterogeneous processes is essential for comprehending their influence on heat diffusion flow within the framework of Cattaneo–Christov heat flux (CCHF) models. Motivated by the limitations of classical Fourier heat flux model, the present work aims to examine the flow of circulation of liquid across stretched coaxial cylinders, utilizing the CCHF model and both homogeneous and heterogeneous reactions (HH). Furthermore, the momentum equation takes a magnetic field, and the temperature equation takes a heat source/sink (H-SS) effect. The two-dimensional flow equations are transformed into ordinary differential equations (ODEs) by comparable transformations. The converted ODEs are subsequently explored through numerical solutions adopting the Runge-Kutta-Fehlberg (RKF-45) method. To optimize the rate of thermal distribution in a system, Taguchi optimization technique was introduced. The results indicate a decrease in velocity for the magnetic parameter and a rise in velocity for the curvature parameter. Without a thermal relaxation parameter, the heat source/sink and magnetic parameters improve thermal dispersion. Both homogeneous and heterogeneous factors result in a decrease in mass transfer. Further, the Taguchi analysis reveal that magnetic parameter influence on rate of heat transfer with delta value of 21.238. The combined parametric values M = 1.5, γ=0.1,Q1=0.3&λ*=0.07 contributes majorly in the rate of heat transfer with SNR value 29.7100. The outcomes of the work are help in polymer extraction, catalytic reactors, thermal insulation operators and improved thermal management systems.

The present study involves the parametric analysis of natural convection driven by buoyancy within a square cavity subject to a spatially varying magnetic field, which results from a localized magnetic field source. The Marker and cell approach is employed to numerically solve the dimensionless mass, momentum, and energy equations. The present study addresses the effect of a localized magnetic field source and several partially active walls on the thermal-magnetic behavior within the square cavity, unlike previous researches, where most have focused on uniform magnetic fields. It is found that due to the presence of non-uniform magnetic fields, the buoyancy-driven natural convection is highly affected by the partially active wall configurations and results in the development of unique thermal boundary layers. The heat transfer rate becomes less since the fluid flow is restricted due to a change in velocity profile to linear one due to an increase in Hartmann number (Ha). Furthermore, average value of the Nusselt number is found to decrease by about 44.7

This study explores the wire coating process, a key technique used to enhance both the mechanical strength and environmental resistance of conductive wires by applying a molten polymer layer. To capture the complex behavior of radiative molten polymers during coating, a third‐grade fluid model is considered. The analysis considered the effects of porosity, thermal radiation, and temperature‐dependent viscosity, with viscosity characterized through the Reynolds and Vogel models. The governing nonlinear differential equations were solved using a hybrid approach that combined perturbation methods, the fourth‐order Runge–Kutta Method, and an ensemble‐learning algorithm. Graphical results demonstrated how changes in physical parameters affect both heat transfer and flow characteristics. The study found that increasing the permeability parameter leads to a clear reduction in flow velocity, attributed to higher resistance within the porous medium. Moreover, thermal radiation is shown to accelerate the coating process when porosity is present, highlighting the strong coupling between fluid properties and the porous structure.

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.

This paper introduces a collocation approach for treating the ordinary and fractional Newell–Whitehead–Segel equation (NWSE). The integer and fractional derivatives of shifted Schröder polynomials (SPs), as well as some new theoretical results of these polynomials, are presented and used in conjunction with the collocation method to convert the equation with its underlying conditions into a system of equations that can be treated using a suitable numerical solver. A thorough error analysis is performed to evaluate the accuracy and dependability of our suggested method. Some numerical examples show that our suggested strategy is effective and accurate. The numerical results demonstrate that the suggested collocation approach yields accurate solutions using shifted SPs as basis functions.

This work is dedicated to proposing a spectral tau solution for the fractional Bagley-Torvik equation (FBTE). The suggested solution is stated as a sum of basis functions, which are selected to be the Schr o der polynomials (SPs). By applying the proposed spectral tau approach, we derive a system of linear equations, and we solve it approximately by applying the Gauss elimination method. The error analysis is analyzed in detail. Additionally, numerical comparisons with other methods found in the literature are conducted. The numerical results validate the suggested method's accuracy, computational efficiency, and ease of use.

In various industries, for advanced cooling, electronic device thermal management, and solar thermal systems ethylene glycol (EG)-based nanofluids are presented as efficient heat transfer agents. The proposed oxide tri-hybrid nanofluids comprising multiple oxide nanoparticles, i.e., Al2O3, TiO2 and SiO2 have significantly enhanced thermal properties and stability compared to mono and binary nanofluids. The current investigation aims at the transportation of heat characteristics of an EG-based tri-hybrid nanofluid over a curved Riga surface filled with a porous substance. The study focuses on thermal radiation effects. Moreover, the Riga plate is a magnetic device with alternating electrodes and magnets that provide a significant electromagnetic forcing mechanism for flow behavior. The designed mathematical model with their dimensional form needs transformation to their corresponding dimensionless form with the utility of similarity rules. Further, a numerical technique based on shooting is employed for the solution of the model for the attainment of physical factors. The physical properties of these factors are presented briefly through graphs followed by a comparative analysis.

This work presents an in-depth analytical study of the flow and heat transfer characteristics of a nanotriple fluid system—comprising copper, alumina, and silver nanoparticles—over a permeable, elastic, and deformable surface, subject to magnetohydrodynamics (MHD) and velocity slip conditions. Unlike many emerging numerical treatments of water-based nanotriple fluids, the primary objective is to derive exact, closed-form solutions, providing a substantial contribution to the analytical understanding of such complex systems. A unique aspect of this investigation is the identification of multiple algebraic-type solutions for the stretching/shrinking sheet problem, yielding dual solutions under injection and a single solution under suction conditions. In addition, critical numbers are identified as thresholds delineating the boundaries for the existence or absence of solutions. It is found that the number of solutions increases as the magnetic force strength decreases. Dual solutions are observed for both skin friction and thermal gradient in the exponential and algebraic cases. These analytical findings are further reinforced by extensive numerical computations, which offer robust validation of the exact solutions derived. Additionally, stability analysis is carried out in order to determine the stability of solutions, where the first branch demonstrates stability, and the second branch is unstable, highlighting the distinct behaviors within the solution branches.

This article examines a novel magnetic dynamical system consisting of a connected linearly damped transverse tuned absorber to a linearly damped spring pendulum (SP), where its suspension point moving along a Lissajous trajectory. The motion is considered under the influence of a magnetic field, a harmonic moment at the suspension point, and a harmonic force on the spring’s radial direction. The controlling equations of motion (EOM) are derived using the second kind of Lagrange’s equations (LEs) and are analytically solved up to a higher order of approximation via the traditional perturbation approach known by multiple-scales method (MSM) to achieve new results. Solvability criteria are obtained in view of the removal of secular terms, and therefore the system’s modulation equations (ME) are achieved according to the examined resonance cases. The Routh-Hurwitz criterion (CRH) is employed as a well-established approach for assessing the stability of linearized systems through the analysis of the characteristic polynomial’s coefficients. This approach is particularly suited to the model under investigation, where the stability regions were identified and examined in relation to the solutions at the steady-state scenario. Graphical representations, including time histories plots, curves of resonance response, and stability regions, are provided to elucidate how various physical parameters affect the system’s behavior. Furthermore, phase portrait diagrams have been drawn which, are powerful tools for gaining deep insights into the behavior and properties of dynamical systems. The numerical solutions (NS) for the EOM are obtained using fourth-order Runge-Kutta algorithms (4RKA) and then compared with the achieved analytic approximate ones. This comparison highlights the strong consistency between them and confirms the accuracy of the applied perturbation technique. This work holds significance for its potential applications in both theoretical physics and engineering, including the analysis of flexible arm robotics control, vibrational dynamics of flexible arms, pump and compressor systems, rotor dynamics, transportation equipment, and shipboard crane operations.

This study investigates the magnetohydrodynamic flow of a dusty micropolar fluid along a deformable permeable elastic surface, accounting for the effects of wall mass flux, microrotation, and fluid-particle interactions. Through an analytical approach, the existence of unique, dual, and triple solutions for velocity, skin friction, and thermal profiles is established. The findings demonstrate that for a stretching surface, a unique solution is maintained for the suction case, whereas for a shrinking surface, dual or triple solutions emerge depending on the suction or injection parameter. Notably, a critical mass flux value is identified, below which no physical solution exists. The study further highlights that increasing the fluid-particle interaction parameter enhances velocity profiles for both stretching and shrinking cases. Moreover, strong suction leads to multiple temperature solution branches, reflecting the complex thermal behavior in micropolar dusty fluids. The influence of a magnetic field is found to enhance both velocity and temperature distributions. These findings provide critical insights for controlling particle deposition and heat transfer in advanced manufacturing processes like polymer sheet extrusion and MHD-based coating technologies. The identified multiple solutions and critical thresholds offer direct design principles for optimizing system performance and avoiding operational failure in industrial applications involving micropolar dusty fluids.

This article examines the mixed convection heat transfer analysis of magnetohydrodynamic (MHD) viscous fluid flow over a multilayer channel flanked between nanofluids comprising a porous medium. The research highlights the significance of multilayer nanofluid flow in practical applications such as petroleum filtration process, cooling of electronic systems, nuclear reactors, and solar thermal systems. Ethylene glycol (EG) serves as a coolant due to its excellent miscibility with copper (Cu), enhancing its corrosion resistance properties in the fluid flow system. The purpose of this study is to analyze the impact of magnetic fields on heat transfer in a multilayer flow of Cu-EG-based nanofluids with different nanoparticles. The comparative performance of copper, copper oxide, and silver nanoparticles in enhancing the heat transfer rate is investigated. The non-dimensional equations are highly coupled, and non-linear differential equations are resolved analytically by utilizing regular perturbation approaches to get a closed-form solution. A comparative analysis between analytical results and numerical methods was done, demonstrating excellent agreement for the fluid flow momentum profile. This study reveals the magnetic field reduces heat transfer in the fluid flow system EG as a base fluid, and silver nanoparticles outperform copper, and copper oxide nanoparticles in thermal conductivity. These findings give rise to optimizing nanofluid-based systems in engineering and industrial applications.

The multilayer flow model plays a crucial role in various industrial applications, including electronic cooling systems, filtration processes, chemical catalytic reactions, cooling system in nuclear reactors, solar collectors, and thermal storage systems. This study analyzed the impact of thermal radiation on the magnetohydrodynamics (MHD) flow of a viscous fluid between H2O-Cu nanofluids in the presence of porous medium and an internal heat source. The governing equations are formulated based on fundamental physical laws. These equations, which are nonlinear and coupled, are solved using the perturbation technique through a series expansion approach. The study explores heating and cooling scenarios in MHD flows, radiative heat generation in water-Cu nanofluids, and the influence of different nanoparticles in vertical multilayer channels. A comparative analysis between Bruggemann and Maxwell models of thermal conductivity and analytical, numerical techniques demonstrates excellent agreement in the results. The dominance of Lorentz force enhances the heat transfer rate, while radiation effects intensify the impact of internal heat source. Furthermore, velocity and temperature are observed to be higher in gold nanoparticles compared to Cu, and Al2O3, due to superior thermal performance.

In this work, steady two-dimensional boundary layer flow of an electrically conductive hybrid nanofluids past a shrinking sheet, along with magnetic field, thermal radiation, suction/injection, heat source/sink and velocity slip model, has been investigated. Particularly, eight mixed hybrid graphene nanofluids of copper, silver, aluminium oxide and titanium dioxide were studied and compared with the classical graphene-water nanofluid. Appropriate similarity transformations have been used to convert the governing PDEs into a set of non-linear ODEs, then, we deduced exact solutions of the flow and temperature. In addition, possibility of obtaining no, unique and dual solutions for these functions were introduced as critical values and regions via graphs. Moreover, validation of the present solutions with those in the literature has been tabulated. Furthermore, asymptotic expression and local extremum for the most important equation were studied. On comparing with those results in the literature at some special values of the included parameters, excellent agreements were gotten. It was mentioned that two conditions have to be simultaneously applied to result the temperature dual solution with restrictions on choosing values of specific parameters. Further, in the injection case, the temperature distributions are larger by about 66.7% on comparing with those in the suction case, for all almost the investigated parameters. To get the highest temperature, it was deduced that the copper is to be firstly added to the water and then mixed with the graphene to produce the hybrid nanofluid. Graphene-copper/water employs as a heater on increasing the Eckert number, shrinking parameter and second velocity slip (for N < 0 ). Furthermore, it acts as cooler with an increase of magnetic field, cheat source/sink parameterc, suction/injection parameter, solid volume fraction, first velocity slip and second velocity slip (for N > 0 ).