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Present inquiry explores steady electro magneto hydrodynamic flow on porous exponentially vertical cone of Casson Williamson nanofluid. In this novel research work, we have successfully developed a multipurpose and adaptable penta-hybrid nanofluid by mixing nanoparticles of silver (Ag), gold (Au), magnesium oxide (MgO), copper (Cu), titanium dioxide (TiO2) with base fluid and are used in many innovative applications, including coatings, sensors, energy storage, water purification, enhanced heat transfer, biological purposes and lubricants. Numerous industries can benefit considerably from the combined features of these nanoparticles, which can improve the base fluid’s functioning and performance. Further, it aims to determine thermal and mass transportation of penta-hybrid nanofluid. The controlling PDEs can be turned into non-linear ODEs via similarity conversions, solved using fifth order Runge-Kutta-Fehlberg technique via shooting technique. Penta-hybrid nanofluid/ C2H6O2is excelled in heat transport to greater extent than other hybrid nanofluids with 62.6 %. Moreover, entropy generation rate is performed, and irreversibility demonstrates that, compared to previous efforts, total entropy generation rate minimize energy loss by 247.16 % in present study. Estimating impact of flow patterns and transmission rates are useful application with varied industrial and engineering applications, environmental restoration, and biomedical applications such as targeted medication administration.

One of the fundamental aspects of solving difficult and nonlinear mathematical ideas is the use of Artificial Neural Networks due to their exceptional efficiency in handling such problems. In many complex fields such as computational fluid system, biological computation, and biotechnology, a distinct computing structure is provided by Artificial Neural Networks, which is extremely valuable. The main purpose of this article is to dig out the abilities of the Levenberg-Marquardt technique using back-propagation artificial neural networks regarding the fluid mechanics and the heat transport assessment in nanoparticles. This interdisciplinary field explores heat and mass transfer through objects and fluids, and their impacts on temperature as well as concentration distributions. With the help of mathematical modelling and numerical solution methodologies, researchers can simulate and analyze these processes. The present analysis communicates the Eyring-Powell fluid flow caused by a rotating disk placed in the horizontal direction. The fluid flow over a rotating disk through non-linear partial differential equations is modeled. After converting the partial differential equations to ordinary ones, they are tackled numerically through shooting technique. The Levenberg-Marquardt algorithm using back-propagation artificial neural networks technique is used with reference datasets, having 70 % training, 15 % testing, and 15 % to validation. The method is validated with the help of mean squared error, error histogram and comprehensive regression analysis. These figures show the accuracy of the proposed method for solving nonlinear problems. Flow features such as velocity, temperature as well as the concentration profiles are exemplified quantitatively and have been graphically discussed. Velocity of the fluid decreases for porosity and increases for fluid parameter while temperature increases for thermophoresis and Brownian motion parameters. Consistency is shown by getting a minimum absolute error approaching zero, showing the strength of the proposed approach.

A computational study is conducted on the magnetohydrodynamic peristaltic circulation of Casson nanofluid within a non-uniform conduit when Joule heating, thermal radiation, and combined mass/heat transportation impacts are present and the porous medium is saturated. The preparation of nanofluid involves the suspension of copper oxide nanoparticles in blood, with blood serving as the base fluid in this instance. Basic flow equations are linearized mathematically by assuming a high wavelength and a low Reynolds number. For both the fluid and particle phases, analytical formulae for temperature, velocity, concentration profiles, and volumetric flow rate are provided. Numerical integration is applied for estimating the friction force and the parameters of the pumping rate. The impact of the model’s different parameters is shown graphically in detail using the Mathematica program. The skin friction coefficient behavior as well as the Sherwood and Nusselt numbers behavior have been graphically illustrated for the relevant parameters. Notably, raising the medium permeability, Casson parameter, and Hartmann number improve temperature fields, velocity, Sherwood number, and skin friction coefficient; however, they have a reverse effect on concentration profiles and Nusselt number in the range −1

A comprehensive study of the nature of micropolar fluid on an unsteady MHD-free convective transference through a perforated sheet is discussed considering the suction and Lorentz’s force effects. The corresponding similarity equations of temperature, momentum, concentration, and angular momentum equations have been modified into the non-dimensional similarity equations of the temperature, momentum, concentration, and angular momentum equations by utilizing the similarity technique. The modified equations have been resolved by exerting the shooting technique with the help of the finite difference method (FDM) through MATLAB. The fluid flow is inversely proportional to the changes in micro rotational effect (Δ). The concentration boundary layer gets thinner as the Schmidt number (Sc) advances and hence the concentration of the fluid decreases. Micropolar fluid helps reduce drag forces and also acts as a coolant. The heat transmission rate advances by around 391%, and 110% due to growing amounts of Pr from 0.71 to 7.0, and v0 from 0.6 to 3.0, respectively. The surface couple stress decays by about 33%, 45%, and 36% due to increasing values of M from 0.6 to 3.6, ∆ from 3.0 to 6.0, and λ from 0.1 to 1.2, respectively.

This study investigates the role of time-fractional derivatives in the entropy analysis of mixed convection in a reacting nanofluid within a vertical permeable channel saturated with a Darcy-Forchheimer porous medium. This is crucial for enhancing heat and mass transfer, incorporating memory effects, and addressing delayed responses in various engineering applications. Key phenomena such as thermophoresis, porous medium permeability, buoyancy forces, chemical reactions, viscous dissipation, Brownian motion, and velocity slip are considered. The study presents an advanced computational methodology that integrates the Euler wavelets collocation method with an implicit difference scheme to discretize the system of time-fractional partial differential equations. This advanced numerical framework is thoroughly validated, ensuring high accuracy in capturing the complex interactions between fluids and solids. The study reveals that a 20% increase in the Eckert number leads to a 15% rise in entropy generation, signifying greater energy dissipation within the system. Likewise, higher Reynolds numbers contribute to increased entropy generation, emphasizing the flow’s dissipative nature. On the other hand, a 10% increase in pressure gradient and Forchheimer parameters results in a 12% reduction in entropy generation, demonstrating their ability to control the system’s irreversibility. These findings pave the way for more optimized and energy-efficient designs in engineering systems involving porous media.

Recent research has demonstrated that non-Newtonian fluids exhibit superior heat transfer capabilities compared to conventional fluids. However, current methods for enhancing heat transfer in non-Newtonian fluids have their limitations. This study aims to address these limitations by investigating the influence of chemical reactions and activation energy on heat transfer within convergent/divergent channels, as described by the Buongiorno model. The study highlights several key outcomes like Higher convergence angles lead to pronounced concentration variations and gradients, whereas divergent channels result in more uniform concentration profiles. The parameter reflecting kinetic energy dominance over thermal energy shows that increasing this parameter decreases fluid temperature due to energy conversion to kinetic form. Additionally, a higher value enhances solute dispersion and energy transfer, while entropy generation increases with higher values, indicating greater irreversibility and energy losses. These insights are crucial for optimizing channel design in non-Newtonian fluid applications, aiding in the control of concentration gradients, management of temperature variations, and overall improvement in efficiency. Numerical computations of the proposed model were performed using the BVP4c scheme.

In a recent scenario, the flow of Casson fluid via porous media has practical applications in various engineering processes, such, as the design of cooling systems for electronic devices, oil recovery in porous reservoirs, and polymer extrusion processes, etc. The proposed investigation illustrates the thermal and solutal transfer rates in a conducting Casson fluid flow over an expanding surface. However, the emphasis goes to the behavior of the Ohmic heating and Darcy dissipation when considering the transverse magnetic field and porous matrix. The governing flow phenomena with their dimensional form are altered into a non-dimensional set of equations with the help of suitable similarity rules. Further, an adequate numerical simulation is adopted to solve the transformed equations using the in-house bvp4c function in MATLAB. The physical parameters involved in the flow problem and their behavior on the governing flow phenomena are presented graphically and described briefly. Prior to this investigation, the conformity of the current numerical output obtained for the heat transfer rate was validated with the earlier work with a good correlation. Moreover, the major outcomes are; the non-Newtonian Casson parameter retards the axial and transverse velocity profile and the shear rate also decreases significantly. The Eckert number caused by the inclusion of the dissipative heat encourages the fluid temperature throughout.

This study intends to understand the fundamental characteristics of hybrid nanofluids (HNF) in a porous cavity using computational modeling, which will aid in developing HNF within hydrodynamics and enable adaptability. The present research investigates the enhancement of the transmission of heat in a double lid-driven trapezoidal porous cavity containing a heated block, utilizing a HNF composed of aluminum oxide/copper-water (Al2O3/Cu-H2O) employing the Finite Element Method (FEM) based on Galerkin weighted residual (GWR). Additionally, the Darcy-Brinkman-Forchheimer comprehensive approach has been used to demonstrate fluid flow in the porous media. The governing parameters such as the Darcy number (10−4 ≤ Da ≤ 10−2), the porosity of the porous medium (0.6 ≤ ε ≤ 0.9), inclined angle (π/16 ≤ ϕ ≤ π/3), and solid volume fraction (0.001 ≤ δ ≤ 0.05) have been chosen to evaluate the impact within dimensionless time (0.1 ≤ τ ≤ 1). The simulation outcomes of flow and thermal fields are shown through streamlines, isotherms, average heat transfer rate at the hot surfaces, and average temperature inside the cavity. The outcomes of this study show that the HT rate rises by 41.02% when increasing the solid volume fraction (svf) from 0.1% to 5%. Also, it has been found that if the inclined angle is enhanced from π/16 to π/3 at non-dimensional time τ = 0.5, the average HT augmented by 63.16%.

Magnetic nanoliquids stand inimitable when compared with conventional liquids as their distinctive magnetic attributes can be regulated utilizing magnetic fields. For this reason, heat transference can be stimulated or regulated subjected to externally imposed magnetic fields. Magnetic nanoliquids are useful in comparison to orthodox or non-magnetic nanoliquids. These encompasses, energy conversion, electronics, hydraulics, thermal engineering and bioengineering. This study elaborates the magnetic dipole impact on convectively heated rheological nanomaterial confined by stretchy surface. Mathematical modeling is based on thermally radiative viscoplastic (Casson) model. Porous medium features are scrutinized through Darcian Forchheimer (DF) relation. Two-component Buongiorno nanomaterial model which captures Brownian diffusive together with thermophoretic diffusion is under consideration. Energy and solutal transportation expressions capture thermal source and chemical reaction effects. The dimensionalized nanomaterial flow model for stretching flow is obtained by deploying similarity variables. Numerical solutions are computed through Bvp4c scheme. The physical outcomes are elucidated graphically and arithmetically on dimensionless quantities (i.e., Nusselt number, temperature, skin-friction, velocity, Sherwood number and concentration streams). A benchmark is reported to authenticate the acquired numeric solutions. It is further visualized that nanomaterial temperature escalates subject to higher estimations of thermal Biot number, Curie temperature factor, radiation factor, thermophoresis parameter, heat source, ferrohydrodynamic interaction factor and Brownian diffusive parameter while it diminishes with Prandtl number and dissipation factor.

This study addresses the complex dynamics of heat transfer in magnetohydrodynamic (MHD) systems involving Jeffrey, Williamson, Maxwell, and Newtonian fluids, focusing on how chemical reactions, activation energy, porosity, and mixed convection impact fluid behavior. The problem is critical due to the significant influence these factors have on industrial processes and applications involving non-Newtonian fluids. The developed a mathematical model represented by partial differential equations (PDEs), which were solved using similarity transformations and the fourth order Runge-Kutta (R-K) method combined with shooting technique, with MATLAB software facilitating the solution process. The results reveal that variations in magnetic field strength, porosity, and buoyancy force significantly affect fluid velocities, while radiation, Brownian motion, and thermophoresis alter temperature profiles. Furthermore, chemical reaction rates, Schmidt number, relaxation constant, and activation energy influence fluid concentrations. Key findings include that increasing porosity and magnetic field strength generally decreases fluid velocity, while higher radiation and Prandtl numbers reduce temperature. Chemical reactions and activation energy decrease fluid concentrations, with non-Newtonian fluids showing more pronounced effects compared to Newtonian fluids. The novelty of this work lies in its comprehensive analysis of multiple interacting parameters and their combined effects on heat transfer in MHD systems, providing insights that extend beyond previous studies in the literature. This research offers valuable implications for optimizing fluid dynamics in various industrial applications, including food processing, ink formulation, and friction reduction.

Nanofluids have significant industrial applications and motivating heat transfer characteristics for various factors that contribute to the movement of nanofluids are essential significance. The aim of the present study focused on importance of heat transfer analysis on engine oil conveying copper nano particles embedded in a porous rotating disk with potential application in aerospace technology. In this model we considered magnetohydrodynamics, non-linear thermal radiation, thermophoresis and Brownian motion. The present investigation utilized copper nanoparticle and engine oil as a base fluid. The mathematical flow equations are transformed into ordinary differential equations (ODEs) by employing suitable self-similarity variables. The resultant ordinary differential equations are solved numerically by using the Midrich technique in the Maple software. The results are calculated to measure the impact of active parameters on velocity, temperature, concentration equations are presented graphically and in tabular form. Higher values of the magnetic field parameter the velocity profile decreased while the opposite tendency we noticed on energy profile. When increasing the radiation parameter values the energy profile increased. In both cases when increasing the magnetic field and radiation parameter values the Nusselt number profile increased. The research has important implications in a number of real-world situations. In particularly, the advancement of aircraft technology has presented manufacturers with new criteria and problems for the functioning of their devices. It is essential that, in order to guarantee the secure operation of aerospace machinery, the failure mechanisms be identified and the operational durability of critical structural components be improved as quickly as possible.

This manuscript merits at examining of an embedded orthogonal plate inside porous domain exposed to uniform heat influx to elaborate on how micro polar fluid factors and permeability characteristic affect the local skin friction, heating rate, and angular velocity inside boundary layer. The plate is subjected to forced convective micro polar fluid influx under steady, incompressible, and viscous circumstances. To facilitate dependable numerical solution, similarity approach is implemented to mutate set of coupled governing equations relevant to the adopted study into a constrained dimensionless differential equations. Computational analysis has been executed hiring Runge-Kutta scheme by Matlab function bvp4c to settle the governing equations. Study's results are highlighted graphically the impact of micro polar fluid factors on the local skin friction, heating rate, and angular velocity curves. High degree of acceptability of present findings compare with prior research results. It is found that the rising of Darcy parameter drives to decrease linearly both heating rate and local skin friction. Among the examined factors, the benchmark parameter of decreasing skin friction is the porosity. Additionally, it is found that an increasing of Prandtl number and micro rotation element lead to enhance Nusselt number. Once curves of micro rotation are interfered at certain distance from the plate due to increase in porosity, Darcy, Forchheimer's, and microelement rotation, the micro rotation curves are inverted.

This study investigates fluid flow and natural convection heat transfer in a parallelogram-shaped cavity with a porous medium under non-uniform temperature condition. The bottom wall is heated with a non-uniform temperature, the side walls are cooled, and the upper wall is adiabatic. The governing equations, along with corresponding boundary conditions, are formulated in dimensionless stream function and vorticity using the extended Darcy model for the porous medium. The equations are solved by employing the finite difference method. The study presents outcomes for isotherms, streamlines, and Nusselt numbers across range of parameters, including the Rayleigh number (104≤Ra≤108), Darcy number (10−1≤Da≤10−6), Prandtl numbers (Pr=0.71,1,7.1), fluid porosity (ϵ=0.8), and inclination angles (30∘≤ϕ≤90∘), about flow and heat transfer characteristics. The results indicate that an increase in Ra, Pr, and ϕ, along with a decrease in Da, leads to an enhancement in the strength of vortexes. It is observed that for constant Ra⁎=RaDa, Rate of heat transfer increases with an increase in Ra and a decrease in Da. The maximum value observed is 10.15. Also rate of Heat transfer increases with the increase in angle and maximum rate of heat transfer is observed at ϕ=90∘.

This study investigates the complex dynamics governing free convection flow in nanofluids under Magnetohydrodynamics (MHD) within a porous medium along an infinite vertical surface. We specifically focus on scenarios involving constant and uniform heat flux as well as heat sources. The governing equations, incorporating the Boussinesq approximation, continuity, and momentum equations, are formulated and solved using Caputo-Fabrizio derivatives and Laplace transforms. This comprehensive approach integrates the Boussinesq approximation and Caputo-Fabrizio fractional derivatives, while the use of Laplace transforms enables precise analytical solutions. Additionally, our investigation highlights the significant impact of nanometer-sized copper particles on fluid properties, transforming the behavior from Newtonian (water-like) to non-Newtonian. This transformation profoundly affects thermal conductivity and velocity behavior within the system. Overall, this research establishes a robust foundation for future studies and provides a framework for exploring non-Newtonian nanofluid systems in the context of MHD-driven free convection flow.

This study observes the numerical analysis of Powell-Eyring hybrid nanofluid along a stretching sheet. The movement of sheet in x-direction is taken to be the source of motion in the fluid. Nanoparticles are considered as agents for increasing thermal properties of fluids. This research emphasizes the amalgamation of copper oxide and carbon nanotubes with engine oil as a base fluid. A significant increase is observed by adding these agents. Porous medium is considered using Darcy law. Magnetics field is also taken for studying mass transfer analysis. For heat transfer mechanisms, thermophoresis and Brownian diffusions are taken into account. The whole procedure is shown through nonlinear differential equations. Appropriate transformations are used for simplification purposes. After transformations, partial differential equations are converted into ordinary differential equations and handled through numerical technique. Shooting numerical technique has been used to solve the governing system of equations, and the obtained results are thoroughly examined and discussed graphically. The thermal characteristics of the hybrid nanofluid are analyzed, in the presence of thermal radiation and thermophoresis effects. To validate our results, a comparison analysis is drawn with existing literature in some specific scenarios.

The current research examines the combined effects of solar energy and thermal buoyancy around a laminar jet placed in a porous medium. The governing boundary layer equations are dimensionalized by using appropriate dimensionless variables. The numerical solution of the dimensionless boundary layer equations is obtained using the finite difference method. The impact of physical parameters, which are Darcy number, dimensionless porous medium inertia coefficient, Prandtl number, radiation parameter, and dimensionless fluid's absorption parameter, on velocity and temperature profile, is shown graphically, while the influence of the above parameters on the heat transfer rate is presented in tabular form. It is keenly observed that for Darcy number velocity profile decreases while reverse behavior is noted for temperature distribution. For the dimensionless radiation parameter, both the velocity and temperature profile decrease. The main novelty of the current work is to improve the thermal performance of natural convection heat transfer system in the presence of thermal radiation placed in porous medium.

The flow problem discussed in this research is an investigation of compressible viscous fluid in an unstable 2D MHD laminar driven vortex and stable boundary layer flow, passing a heated stretched sheet at varying speeds via a porous medium exposed to a magnetic field with viscous dissipation, thermal diffusion, thermal radiation, chemical reaction and Dufour effects. With the aid of similarity transformations, PDEs that come out in governing equations of this formulations are converted into non-linear sets of ODEs. Well-defined bvp4c technique used to numerically solve these converted equations. With use of various graphs, the consequences of different factors on the distributions of velocity, temperature and concentration were examined numerically and graphically. Physical parameters for this issue are also discussed.

This paper uses three analytical methods to find the novel results in the two-dimensional time-dependent MHD oscillatory flow and heat transfer in an asymmetric wavy porous channel. The corresponding constitutive equations become complex by considering the pressure gradient as a complex function. This results in the existence of both real and imaginary solutions for fluid velocity and temperature. The changes in parameters like the Hartmann number, wavelength, Grashof number, and porous medium shape factor will affect the real fluid velocity. Still, only radiation parameter changes will affect the real fluid velocity and temperature. Increasing the Hartmann number, porous medium shape factor, and radiation parameter will drastically reduce real fluid velocity. The Reynolds number should theoretically affect the imaginary velocity, but according to the boundary condition definition, the imaginary velocity value becomes zero. Altering the frequency of the wavy walls and the Peclet number will only affect the imaginary temperature component, and increasing these parameters will increase the average imaginary temperature profile. The uniqueness of this study lies in the resolution of complex differential equations in a way that has not been attempted before. The results indicated that each set of three solutions was nearly identical, highlighting the outcomes' precision.

The analysis of melting heat transfer over an expansive sheet of variable thickness is of the utmost importance in various industrial and engineering sectors, such as injection molding of polymers and composites, cooling of nuclear reactors, hot rolling, etc. Thus, the current work examines the flow dynamics, melting heat transport, and entropy generation of a magneto-hybrid nanofluid over a stretching device with variable thickness in porous media. A water-based hybridized nanofluid is formed using copper (Cu) and aluminium oxide (Al2O3) nanoparticles in the presence of thermal radiation, viscous dissipation, and Joulean heating. The transport partial derivatives were transmuted to ordinary derivatives using appropriate similarity quantities. These equations were numerically solved through shooting techniques combined with the Runge–Kutta–Fehlberg (RKF) method. Diverse figures and tables are sketched to illustrate the outcomes of the various parameters involved in the study. The investigation reveals an enlargement in the heat-bounding surface with an escalation in the magnitude of volume fraction, melting heat transfer, and magnetic field terms. In contrast, the momentum-bounding surface depletes with these parameters. Furthermore, as thermal radiation and Eckert numbers rise, the thermal gradient increases. The entropy generation increases with higher Brikman number and porosity term, but the melting heat parameter causes a decline in the entropy profiles.

The Maxwell model of fluid flow in a rotating frame over a porous media is investigated in this paper. Binary chemical reactions and fluid movement under activation energy are both covered in this study. The impact of mass and heat transmission along the boundary layer is investigated in an equilibrium process. Using the method of similarity transformation, the controlling partial differential equations are changed into ordinary differential equations. The results are confirmed using the bvp4c Matlab built-in programme, and the altered equations are resolved utilizing a 4th order Runge Kutta based shooting method. Reversible and irreversible processes, activation energy, chemical reactions, Deborah numbers, and rotation parameters are some of the parameters for which the results are offered in tables and graphs. The prior objective of this study is to examine the impact of activation energy and chemical reactions on Maxwell fluid flow in an equilibrium setting. The concentration boundary layer for reversible flows is significantly finer than that of irreversible flows with the influence of activation energy, chemical reaction, and rotation factors. A reduced boundary layer thickness can improve the rates of heat and mass transmission in tremendous applications, such as exchangers of heat and chemical reactors. In this chemical process, sulphuric acid is utilized as a catalyst along with Maxwell fluid which affects the boundary layer reaction rate and selectivity. This is crucial for efficient catalytic process design. Controlling reaction rates using fluid elasticity and reaction kinetics is useful in operations that need accurate product production.

This study investigates the entropy generation analysis of natural convection flow within a diamond-shaped cavity filled with porous media. The research focuses on understanding the effects of a chamfered vertex and the introduction of one or two circular cold objects within the cavity. Using COMSOL Multiphysics software, the study examines various Rayleigh (Ra) and Darcy (Da) numbers to analyze the flow dynamics, heat transfer (HT), and entropy generation (Egen). Results indicate that the presence of cold objects significantly influences the flow patterns, enhances HT, and increases Egen due to fluid friction (FF). Higher Ra values lead to more vigorous convective currents, while lower Da values indicate a transition to conduction-dominated regimes. In addition, the heat transfer rate increases by 5 % by introducing a single cold cylinder inside the cavity for Da = 10−2 and Ra = 106. This enhancement rises to 16 % when two cold cylinders are introduced. The study provides valuable insights into optimizing cooling processes within diamond-shaped cavities by identifying key parameters that affect HT and irreversibility.

This research investigates the unsteady flow behavior of a magnetohydrodynamic Williamson nanofluid (WNF) over a permeable stretching sheet within a porous medium. It considers the influence of chemical reactions, Joule heating, heat generation/absorption, thermal radiation, and viscous and porous dissipation. The study on magnetohydrodynamic WNF over a permeable stretching sheet within a porous medium has gained much significance due to its numerous industrial and engineering applications. The study employs the Fibonacci wavelet series collocation method (FWSCM) for analysis. An appropriate similarity transformation transforms the governing partial differential equations (PDEs) into nonlinear coupled ordinary differential equations (ODEs). These ODEs are solved using FWSCM. The accuracy of the FWSCM is validated with some previous studies, the Mathematica NDSolve command, and the Haar wavelet collocation method (HWCM). The graphs and tables are used to discuss the physical parameter’s impact. The findings of this study reveal that the Nusselt number increases by 45.32 % with the Weissenberg number and 31.25 % with the unsteadiness parameter. In comparison, it decreases by 45.01 % with the heat generation/absorption parameter, 4.89 % with the porosity parameter, and 1.95 % with the chemical reaction parameter.

This study employs the Hybrid Analytical-Numerical (HAN) method to investigate steady two-dimensional magnetohydrodynamic (MHD) nanofluid flow over a permeable wedge. Analyzing hyperbolic tangent nanofluid flow, the governing time-independent partial differential equations (PDEs) for continuity, momentum, energy, and concentration transform into a set of nonlinear third-order coupled ordinary differential equations (ODEs) through similarity transformations. These ODEs encompass critical parameters such as Lewis and Prandtl numbers, Brownian diffusion, Weissenberg number, thermophoresis, Dufour and Soret numbers, magnetic field strength, thermal radiation, power law index, and medium permeability. The study explores how variations in these parameters impact the velocity field, skin friction coefficient, Nusselt, and Sherwood numbers. Noteworthy findings include the sensitivity of fluid velocity to parameters like Weissenberg number, power law index, wedge angle, magnetic field strength, permeability, and melting heat transfer. The skin friction coefficient experiences a significant increase with specific parameter changes, while Nusselt and Sherwood numbers remain relatively constant. The local Reynolds number significantly affects Nusselt and Sherwood numbers, with a less pronounced impact on the skin friction coefficient. The study's uniqueness lies in employing the analytical HAN method and extracting recent insights from the results.

This research introduces novel theoretical assumptions through the use of a non-Newtonian Sutterby model with variable properties, demonstrating significant advancements in thermal conductivity and diffusivity, which enhance heat and mass transfer in fluids, particularly under magnetic field exposure. The study is notable for its comprehensive examination of the effects of thermal radiation and viscous dissipation within a porous medium. Additionally, it explores the role of suction velocity to provide a deeper understanding of transport phenomena. Following the Darcy hypothesis, the fluid flow is modeled as resulting from the linear stretching of an elastic sheet in a saturated porous medium. The core physical model, comprising equations of mass, motion, concentration, and energy, is transformed into ordinary differential equations using appropriate similarity transformations. Employing the shooting technique, the numerical solution to the problem is obtained. The research uncovers and quantitatively analyzes intriguing physical parameters influencing velocity, concentration, and temperature fields. These parameters are further investigated both numerically and graphically, providing valuable insights into their effects. Quantitative outcomes include enhanced thermal and concentration fields by magnetic field parameter, the porous parameter and the viscosity parameter and improved the rate of mass transfer by both suction and thermophoretic parameters, highlighting the model’s efficacy in optimizing fluid dynamics especially under magnetic fields. The obtained outcomes were juxtaposed with previous studies, revealing a significant level of agreement.

Many external factors influence the properties of nanoparticles in the working fluid, which subsequently determines the efficacy attributes of the ensuing nanofluid. Incorporating these factors is essential for studying heat and mass transfer in hybrid nanofluids, as it provides a detailed representation of the mechanism. Such factors are porous medium, heat source, Thermophoresis, and Brownian motion. The rotation of a cylinder can alter the flow field and boundary layer around it, which could be used for flow management scenarios such as drag reduction or separation of boundary layers control, and the complex flow patterns generated through a rotating cylinder could be utilized for improved mixing and heat transfer in processes in industry. Thus, the current novel aspect of the study examines the impacts of thermophoresis and Brownian motion over a rotating cylinder in the presence of a porous media comprised of water-based TiO2 and Fe2O4. The physical circumstances are primarily represented as PDEs based on the considered flow conditions, which then get transformed into a system of ODEs using the relevant similarity variables. The numerical solution is obtained by using RKF-45th and the shooting procedure. The flow symmetry and the effects of the specific variables associated with the problem are examined and delivered graphically. The findings confirm that the drag coefficient improves with porous material, altering fluid dynamics. Heat transmission steadily drops as the thermophoretic parameter gets larger, but mass transfer displays a contrary trend for evolving Brownian motion parameters. Also, the current results are validated with existing past studies.

Engineering technology is rapidly changing, and activation and binary chemical reactions have many uses in the fields of chemical engineering, processing food, and mobility and Geothermal reservoir, etc. Energy activation stimulates reactants with the least energy whenever a chemical transition occurs. The thermophysical features of viscoelastic fluids are based on internal energy change, which has been discussed for years. With this significance the current proposal focuses on the heat transfer process in reversible esterification reactions. The reversible chemical reaction and the activation energy with hydromagnetic movement of Casson fluid are considered. A complex multivariate partial differential equation set can be reduced to ordinary differential equations using appropriate variables. A numerical method is utilized to determine the solutions. In order to examine the outcomes of the concentration boundary layer utilizing the R-K-based shooting technique, the study primarily considers the reversible esterification process, which involves an ethanol-based reversible chemical reaction. Also, the bvp4c solver was employed to validate the results and appraise the precision of the R-K methodology. The temperature field, the momentum field, and the volumetric concentration of the esterification process are analysed in relation to a variety of numerical values. Graphical analysis considers the pertinent physical ramifications for temperature, velocity, and concentration contours. Explications that furnish a comprehension of the values accompany the tabular depictions of the heat transfer rate, local Sherwood number and skin friction. Reversible and irreversible flows differ considerably in the assessment of Sherwood number, local Nusselt number and rate of shear stress, when the inertial parameter, temperature difference parameter and activation energy are measured.

Researchers and scientists became interested in the newly developed idea of hybrid nanofluids because of their exceptional thermal conductivities, which resulted in enhanced thermal performance. These fluids have a wide range of uses in the industrial and technical fields. This novel class of fluids is playing a vital role in solar energy, automotive industry, cooling of the machine and manufacturing, heating and ventilation systems, electronic cooling, nuclear systems cooling, biomedical fields, space, ships, and defense systems. In light of the physical significance of the above-mentioned class of fluids, the current investigations are carried to examine the thermal performance of the mixture of cylindrical shaped aluminum oxide Al2O3 and copper Cu nanoparticles suspended in ethylene glycol using third-grade non-Newtonian fluid flow model. The fluid flow is induced by the stretching of the permeable sheet in exponential manner that is embedded in a Dacry-Forchheimer porous medium. The effects of Lorentz force applied normal to the flow and suction impacts have been incorporated in the current proposed model. The solutions of the flow model transformed to ordinary differential equations are computed using MATLAB built-in solver bvp4c. The solutions are presented in graphical representations. Graphical solutions reflect that fluid velocity slows down by growing the volume fractions of the nanoparticles. Increasing values of viscoelastic parameters reflect that there is a reduction in velocity and intensification in profiles of temperature and concentration have been noted. Rising values of cross-diffusion parameter is leading to the reduction in magnitude of velocity, and augmenting behavior of temperature and concentration has been seen. Observations indicate rising third-grade fluid parameter and porosity parameter lead to reduction in velocity. Increasing Schmidt number gives reduced Sherwood number, and magnetic field parameter improves the rate of hear transfer (Nusselt number) and skin friction coefficient for both Al2O3/ethylene glycol nanofluid and Al3O3-Cu/ethylene glycol-based hybrid nanofluid. The suggested model's validity is confirmed through comparison with previously released findings. Additionally, a grid independence test (sensitivity analysis) has been performed to verify the accuracy and verification of the numerical model.

Mixed convective nanofluid flow has substantial importance in improvement of thermal performance, and thermal engineering to meet the global energy crisis. In this study, mixed convective nanofluid flow in a porous-wavy channel with an inner heated triangular obstacle under magnetic field effect is numerically examined. Nanofluid within the channel is heated and cooled from its bottom and top wavy-surfaces. A heated triangular cylinder is located at the centerline of the wavy-channel. Finite element method is utilized to solve the non-dimensional governing equations. The code is validated comparing present results with published numerical and experimental results. The response surface method is also implemented to analyze the obtained results and its sensitivity. The numerical results indicate that strength of flow velocity is accelerated with rising Reynolds number, Darcy numbers and inlet-outlet ports length but declined for Hartmann number and volume fraction. Heat transferring rate and heat transfer irreversibility are substantially increased for higher values of Reynolds number, inlet-outlet ports length, Darcy number and nanoparticle volume fraction but a reverse trend is occurred for magnetic field effect. The thermal performance is found significantly improved with simultaneous increment in Re, ϕ, Da and decrement in Ha. Positive sensitivity is achieved for input factors Re, ϕ, Da in computing Nuav while negative sensitivity to Ha. Heat transfer rate is found more sensitive to the impact of Re and ϕ compared to Da and Ha. 45.59 % more heat transmission potentiality is developed for using Al2O3–H2O nanofluid (vol.5 %) instead of using base fluid water. Heat transfer enhancement rate is decreased by 36.22 % due to impact of magnetic field strength. In addition, 84.12 % more heat transferring rate is recorded in presence of triangular obstacle. Moreover, irreversibility components are influenced significantly for the presence of heated triangular obstacle. Bejan number is also found declined for increasing physical parameters. The findings of this investigation may offer a guideline for finding experimental results to design high-performance convective heat exchangers.

The development on entropy generation in fluid flows has applications in many medical equipment such as cryogenic devices and therapeutic heat apparatus. This study looks at how porous medium, multi-slips, and entropy formation affect the pumping of Jeffrey nanofluid flow through an asymmetric channel containing motile microorganims. A lubrication theory is used to neglect the fluctuation effects in the flow. Numerical simulations are utilized to generate data for specific physical features of the problem utilizing the Shooting approach on Mathematica. Following a thorough research, it is appropriate to conclude that the porous medium's permeability reduces flow speed along the walls while increases at the center of the flow region. Graphical representation of the results further reveals that entropy production can be decreased by including high thermal slip and low viscous slip elements. It is also worth noting that the Brinkman number reduces the thermal distribution in the flow while it helps in increasing the flow speed.

The flow phenomena become more significant and varied when an extra heat source, a chemical reaction, and thermal and solutal buoyancy forces exist. Transport property optimization results from the current demand in several sectors dependent on the solutal reactant and the heat source given. Partial differential equations are transformed into ordinary equations using appropriate similarity transformations. Python was used as an efficient tool to solve equations in this research. According to the available facilities and libraries, Python can efficiently solve various physics and engineering problems. The homotopy perturbation method (HPM) and Akbari-Ganji Method (AGM) have been used to solve differential equations. The results calculated in Python indicate the high accuracy of calculations in Python. The effect of different parameters on the profile of velocity, temperature, and concentration has been investigated graphically. Also, the friction coefficient (Cf), Nusselt number (Nu), and Sherwood number (Sh) have also been investigated in this study. The results show that the velocity profile decreases with increased magnetic field intensity. Also, the porosity parameter has an inverse effect on the velocity and temperature profile, so the growth of the porosity parameter reduces the fluid velocity while improving the temperature profile.

This study delves into the intriguing interplay between activation energy, chemical reactions, no-slip effects, and radiation on the flow dynamics of a Maxwell fluid over a Riga plate nestled in a permeable medium. By initially formulating partial differential equations, which are then transformed into a set of ordinary differential equations using adept similarity transformations, we embark on a numerical journey aided by MATLAB. Unraveling the intricate tapestry of influences, we meticulously scrutinize the impact of activation energy, chemical reaction rates, no-slip parameters, and radiation on various flow characteristics. We explore the ramifications of velocity, energy distribution, skin friction, concentration profiles, and Nusselt number distributions through insightful plots. The essence of our findings reverberates throughout engineering and industrial domains, potentially reshaping practices and procedures. Our key revelations echo a narrative where heightened activation energy and adherence to no-slip conditions accentuate concentration profiles, contrasting the decrement induced by chemical reactions.

One of the most effective techniques for electronic-cooling devices involves the use of nanofluid, a substance known for its superior heat transfer capabilities. The main challenge in modern production engineering lies in the efficient cooling of electronic devices. Nanofluids have attracted widespread interest in various technical fields due to their exceptional characteristics, which enable effective cooling of electronic devices while improving energy efficiency. In the field of electronic systems, engineers are exploring the potential of nanofluids in a range of applications and phenomena. Choosing the right heat transfer fluid to dissipate heat is crucial in managing electronic component temperatures. This study examines the effects of heat sinks and thermal radiation on the flow of Darcy-Forchheimer nanofluid over a vertical plate, with the aim of enhancing the thermal durability of electronic components. The governing mathematical model has been refined to accurately represent the physical coefficients. To develop a relevant algorithm in the FORTRAN environment, primitive variables are employed in steady-state and oscillatory models, nanomaterial and heat sink. Outputs are presented numerically and graphically using Tecplot 360, revealing that an increase in heat sink capacity leads to a reduction in surface temperature due to the heat sink's ability to absorb excess thermal energy. The use of porous Darcy-Forchheimer material is shown to lower the fluid temperature. The study also shows that the heat transfer rate decreases with an increase in the heat sink coefficient, and the oscillatory frequency of heat transfer reduces with a decrease in the Prandtl number. Decreasing behavior of heat transfer is depicted for maximum thermophoretic factor due to heat sink impact.

In this paper, the study investigates the problem of the creeping flow of a non-Newtonian couple stress fluid through a linearly porous walled slit within a Darcy porous medium. The well-established approach, the Inverse Method approach, was employed to attain the precise solution of the governing equations. The inverse technique is utilized to transform the momentum equations into stream functions. The physical parameters are all computed: longitudinal velocity, transverse velocity, fractional reabsorption, leakage flux, axial pressure, volume flow rate, and mean pressure. The data was also graphed, indicating that the initial flow rate affects longitudinal velocity but not transverse velocity. The streamlines are greatly affected by the initial flow rate, resulting in straighter and more uniform patterns as the flow rate increases. Parameters like permeability and the couple stress parameter also have an impact on physical quantities. Because of the porosity parameter, backward flow occurs at the slit's end. The novelty of this research lies in the investigation of couple stress fluid flow within both linear channel walls and a porous medium. The present research focuses on how kidney disease affects fluid flow in renal tubules. It's critical because fibers, lipids, and waste particles can clog or disrupt these channels, lowering kidney performance. Understanding this helps in the management of kidney disease and provides insights for better biomedical engineering in the development of improved renal medicines.

We investigated two dimensional magnetohydrodynamic (MHD) stagnation point upper-convected Maxwell (UCM) fluid flow through a stretch sheet in a porous medium having the effects of heat generation/absorption and thermal stratification. Utilizing suitable similarity transformations, the governing partial differential equations (PDEs) of the fluid flow and heat transfer phenomena are converted to nonlinear dimensionless ordinary differential equations (ODEs). We numerically solved these nonlinear ODEs and compared our results of skin friction and Nusselt number with previous work which demonstrated accuracy of the presented solutions. We also illustrated the graphical behavior of dependent variables that is of the velocity and temperature fields versus the key parameters involved. It showed that velocity field decreases for Deborah number, porosity and magnetic parameters. The temperature of the fluid showed an increasing behavior for Deborah number, magnetic, porosity, and heat source/sink parameters while declining for thermal stratification and velocity ratio parameters.

To handle the continuous thermal power supply is very important in thermal systems and industries, so magnetized heat transport with non-Newtonian fluid is an appropriate platform for thermal power energy. In view of this usefulness, this paper provides a numerical assessment of electrically conducting Walter's B fluid flow through nonlinear permeable stretching sheet subject to mixed convective, porous medium, and Newtonian heating phenomena. Thermophoresis and Brownian motion effects have been included in energy and species diffusion equations to account for the impacts of nonlinear thermal radiation, joule heating, heat source and chemical reaction. The system of governing equations has been reduced to dimensionless nonlinear coupled equations using similarity transformations. The bvp4c technique has been used to obtain the solution for velocity, temperature, and concentration distributions. We consider both regions of assisting and opposing flow and provide various tables and graphs to examine the impact of varying values of emerging factors. Heat source and thermal radiation characteristics are found to enhance the thermal boundary layer thickness, which may be used to dissipate heat. The magnetic parameter augmented the fluid temperature and decelerated the velocity field. The chemical reaction parameter accelerated the concentration distribution. Further, the Sherwood number, Nusselt number, and skin friction coefficient have been determined and compared well to prior research that has been published.

The influence of heat radiation on HMT, MHD on a porous stretched sheet has been explained in this study. The governing PDEs are converted to ODEs via similarity transformations. The coefficient of skin friction, rate of HMT are calculated using the MATLAB software for various parameter values. The controlling boundary layer PDEs are turned into a system of coupled nonlinear ODEs via similarity techniques, which are numerically solved using the shooting technique in conjunction with the fourth order Runge Kutta method, and then ANN is applied to them. Validation of numerical results using ANN results. We discovered that using engineering standpoints, an ANN model may produce high-efficiency estimates for heat transfer rates. The achieved R squared values of 99% for predicting Skin friction, Nusselt number and Sherwood number coefficients highlight the remarkable effectiveness of neural network models in these predictions. This efficacy not only demonstrates the accuracy of the models but also results in a significant reduction in the computational time required compared to traditional numerical methods. Furthermore, when compared to alternative numerical approaches, the current ANN model stands out for its applicability to more complex mathematical models, because its efficiency in minimizing both time and processing capacity demands in solving such problems.

The increasing demand for thermal management due to the high heat density heat exchangers for different technological and industrial processes draw the attention of researchers in heat transfer analysis to investigate the simultaneous solution to this problem. Keeping this in mind, the current work addressed the utilization of ternary hybrid nanofluid, a mixture of graphene oxide with silver and copper, to examine the heat and mass transfer enhancement of MHD Maxwell fluid. In the current model, the author discussed heat and mass transfer analysis of Maxwell MHD ternary hybrid nanofluid flow across a porous medium over an extending sheet. The energy and concentration equations are expressed to include the effects of Brownian motion and thermophoretic diffusion. Additionally, the effects of thermal radiation, activation energy, chemical reactions, and convective boundary conditions are considered. The similarity transformations are used to convert the nonlinear PDEs into ODEs. The impact of embedded factors on velocity, temperature, and concentration profiles is perceived graphically and mathematically by using the HAM technique. The convergence of computed solutions is ensured by h-curves. The significant outcomes from the analysis expressed that the flow distribution increases with slip and suction/injection factors. The concentration of the ternary hybrid nanofluid increases with an increase in thermophoretic diffusion (0.1–0.5), thermal biot number, and solutal biot number (0.1–0.5), however, a declining behavior is perceived with rising quantity of Schmidt number (0.6–0.8) and Brownian motion (0.1–0.5) parameters. Nusselt number shows a dominant behaviour with a higher magnetic factor. Similarly, the skin friction increases with an increase in thermal biot number and thermophoresis factor. It is revealed that the skin friction along x− direction increases from (0.78138, 0.833702, 0.897022, and 0.972526) for ternary hybrid nanofluid. Nusselt number increases from 2 to 5% (0.208936, 0.214288, 0.219799, 0.22555) when the volume fraction varies from 0.01 to 0.04 respectively.

This study explores using hybrid nanofluids to improve water quality by enhancing heat transfer and substance decomposition. Nanofluids effectively remove pollutants, optimise heat transfer, control pollution sources, and regulate fluid dynamics, which can lead to efficient pollution management in water systems. Thus, the present research examines the flow of an unsteady hybrid Al2O3–Cu/water nanofluid near the stagnation region in a porous medium, considering the discharge concentration and convective boundary conditions. Governing equations in ordinary differential equations are obtained using similarity transformations. The BVP4C solver in MATLAB is employed to expose dual solutions. The volume fraction of copper φa2, the suction/injection parameter (S), and the unsteadiness parameter (A), collectively contribute to the delay of the boundary layer separation. Increasing the values of φa2,A, and S enhances convective heat transfer. When the sheet shrunk between the range of −16.2 and −13, hybrid nanofluid has higher convective thermal transfer than nanofluid. Moreover, an increment in φa2 and S raises the skin friction coefficients and mass diffusion rates. Stability analysis reveals that the first solution is stable while the second one is unstable.

The paper aims to investigate the unsteady magnetohydrodynamics (MHD) flow with Darcy-Forchheimer effect and heat transportation. The system incorporates a Casson nanofluid (CNF) with heat transfer during melting and slip velocity, influenced by heat source/sink and thermal radiation. The motivation of investigating the current topic comes because of the squeeze flow is of practical physics. The mathematical process involves converting nonlinear partial differential equations (PDEs) into nonlinear ordinary differential equations (ODEs). The nonlinear ODEs are analytically solved via the Homotopy perturbation method (HPM), while considering the appropriate boundary conditions (BCs). Through a non-dimensional procedure, many dimensionless physical quantities are achieved. The primary results of velocity, temperature profiles, local skin-friction, and the local Nusselt number are shown and analyzed based on several non-dimensional parameters. It is found that the increasing of the radiation factor values, leads to a drop in the velocity and temperature of the CNF, as well as the local skin-friction and Nusselt number. Additionally, the increase of the slip velocity factor results in higher velocity, temperature, and local Nusselt number. Meanwhile, it reduces the local skin-friction.

The primary goal of current study is to formulate a novel mathematical model employing the Prabhakar fractional operator, aimed at examining the dynamic behaviour of Jeffrey fluid flow and heat transfer phenomena under ramped wall temperature conditions. Our investigation entails a meticulous investigation of magnetohydrodynamic (MHD) natural convective flow within Jeffrey fluids, where we derive precise analytical solutions utilizing the Prabhakar fractional derivative, characterized by a non-singular type kernel. Furthermore, our study integrates fundamental principles such as Fick’s and Fourier’s laws into the model, leveraging a multi-parameter Mittag-Leffler kernel associated with the fractional operator. We delve into the intricacies of fluid flow near an infinitely vertical plate, considering characteristics such as velocity u0. To address the complexities of the problem, we express it in context the of partial differential equations along side generalized boundary conditions, employing a set of appropriate variables to transform these equations into a dimensionless form. Utilizing Laplace transform, we analyse the equations of fractional system, presenting outcomes both in the form of series and through specialized functions. We systematically explore the influence of key parameters α, Pr, β, Gm, Sc, γ, Gr on fluid flow dynamics, unveiling significant insights. Our comparative analysis reveals the superior performance of the Prabhakar-like non-integer approach over existing operators, substantiated by graphical representations of the results. Additionally, we extend our investigation to various limiting cases, including Newtonian fluids and second-grade, in both fractionalized and classical forms, thus highlighting the versatility and applicability of our proposed model within fluid dynamics research.

The current investigation delves into the convective heat and mass transfer characteristics of third-grade radiative nanofluid flow within a porous medium over a Riga plate configuration. The Riga plate structure incorporates magnets and electrodes strategically arranged on a plate surface. To enhance the accuracy of energy and concentration expressions within the third-grade fluid flow, the Cattano Christov Double Diffusion model is employed. Entropy generation analysis is conducted by applying the second law of thermodynamics, and Darcy's model is employed to characterize the behavior of a porous medium. Appropriate similarity transformations have been used to convert the partial differential equations monitoring the fluid flow model into dimensionless ordinary differential equations. The Galerkin weighted residual method is employed to resolve these equations numerically. The findings contain detailed explanations of how relevant factors affect the temperature field, concentration field, velocity field, entropy generation, and Bejan number, in addition to graphic representations of the results. The findings indicate that the medium's porosity and Brinkman number promote entropy generation. The Bejan number and entropy production is affected by the thermal radiation parameter, which first rises and then declines after a certain distance.

This study investigates the complex interplay between radiation and magneto-flow on a porous, perpendicular plate immersed in a viscoelastic Maxwell fluid. The significance lies in understanding how the Lorentz force shapes dynamics and energy transfer in such systems, which is crucial for various industrial applications. Utilizing a comprehensive model that accounts for heat generation and absorption, we transform partial differential equations into ordinary ones through flow similarity analysis, providing valuable insights into the system's behavior. Employing the bvp4c solver in MATLAB, we evaluate velocity and energy profiles in Maxwell fluid flow. Graphical representations allow for a thorough examination of the influence of thermophysical parameters. Our findings reveal an intensified velocity profile attributed to prolonged material relaxation times.Furthermore, temperature profiles progressively increase with higher radiation and porous parameters. These results hold significant implications for industrial applications, particularly in Maxwell fluids' heat transport studies. Insights from this research can inform the design and optimization of systems such as ovens, boilers, cross-flow heat exchangers, and solar panels, enhancing efficiency and performance. Overall, this study contributes novel insights that advance understanding beyond previous efforts in the literature, paving the way for further exploration in this field. The primary finding of this study indicates that as the Rayleigh number increases, the velocity profile experiences an increase, whereas the temperature profile shows a decrease.

The significance of the study lies in illuminating the consequences of thermophoresis and Brownian motion on heat and mass transfer, offering valuable insights crucial for practical applications. The purpose of this study is to look into how Casson–Williamson and Maxwell nanofluids flow over a stretching sheet in order to understand how heat and mass move in that situation. The mathematical structure comprises a set of partial differential equations (PDEs) converted into ordinary differential equations (ODEs) through a similarity transformation. Subsequently, the well-established MATLAB BVP4C method, integral to the finite difference approach, is applied to solve these ODEs. For the assurance of the method’s reliability and precision, the acquired outcomes are cross-referenced with existing literature, which is a crucial step in validation. The numerical results are depicted visually and in tables, specifically highlighting the impulse of different influencing elements on the Nusselt number, friction factor, and Sherwood number. Notably, an enhancement within the Brownian motion parameter (Nb) and the thermophoresis parameter (Nt) is found to correspond to higher local Nusselt numbers, indicating an enhancement in heat transfer. Conversely, elevated quantities of the Brownian motion parameter (Nb) result in a reduction in local Sherwood numbers. Physically, a higher Brownian motion parameter causes significant nanofluid particle displacement, enhancing their kinetic energy and intensifying heat generation in the boundary layer. The magnetic parameter (M) influences speed profile decline caused by the Lorentz force, which hinders fluid motion.

Hybrid nanoparticles, which are nanoparticles composed of multiple materials or phases, find applications in various fields due to their unique properties resulting from the combination of different materials. In such fluids, viscosity changes significantly in response to alterations in temperature or pressure. For example, these fluids become less viscous as they are heated and more viscous as they are cooled. Similarly, in certain situations, the application of a temperature variation can cause a change in viscosity. This property is commonly observed in materials like polymers and some oils. The article aims to investigate the heat transfer and flow behavior of fluid from an elastic sheet of hybrid nanoparticles, considering the temperature-dependent variable viscosity and thermal conductivity. The obtained equations for the mathematical model are partial differential equations converted to ODEs by applying a suitable similarity transformation. The findings show that the magnetic field opposes fluid motion. Our main findings are that adding nanoparticles to the base fluid significantly increased its heat conductivity. This improvement has great potential for uses requiring effective heat transmission, especially in engineering systems where heat dissipation plays a crucial role. The study also reveals the complex relationships that influence thermal conductivity, including slip boundary effects, viscosity fluctuations, magnetohydrodynamic effects, and porous media dynamics. Understanding these interactions is critical for optimizing heat transport processes in porous media applications. Comprehending these interplays is essential for refining heat transport mechanisms in applications involving porous media. The graphical representations are used to explain the physical behavior of various model parameters. Previous outcomes are also contrasted with the current ones. The findings show that the magnetic field opposes fluid motion.We range the subsequent list of values for parameters in every graph unless indicated accordingly.ϵ=0.1,Pr=2,M=0.5,R=0.2,K=5,fw=0.1,s=0.1 ,λ=1,φ1=0.02,φ2=0.04.

The consequence of thermal performance and mass distribution of endothermic/exothermic chemical reaction and pollutant concentration on the nanoliquid stream via cylinder/plate in the presence of permeable media is explored in the present study. The advancement of efficient waste disposal and pollution prevention techniques might result from studies on the fluid flow and dispersion of contaminants in cylinders and plates. Further, TiO2 nanoparticle offers improved thermal conductivity, helps in wide range of technological and industrial applications. The governing partial differential equations (PDEs) of the fluid problem are modeled and converted to ordinary differential equations (ODEs) utilizing similarity variables. The resultant ODEs are numerically solved using Runge Kutta Fehlberg’s fourth-fifth order (RKF-45) scheme. The influence of various non-dimensional parameters on the velocity, thermal, and concentration profiles are illustrated with a graphical representation. The comparison between cylinder and plate geometry is also displayed in graphs. The novel outcomes show that the augmentation of the chemical reaction parameter reduces the thermal profile of the endothermic case and elevates the thermal profile of the exothermic case. Elevating the local pollutant external source parameter leads to a rise in the concentration profile. It is around 10%–12% in Cf, 4%–6% in Nu and 8%–12% Sh observed in cylinder geometry than plate. The mass transfer rate reduces for improved values of solid fraction and activation energy. In all the modes, the cylinder geometry performs better than plate geometry.

A computational exploration is presented to emphasis on optimization and sensitivity inspection of the unsteady free convective heat transfer flow of hybrid nanofluid in a square cavity under multidirectional periodic magnetic field with aluminum foam porous medium. The vertical walls of the cavity are maintained at constant low temperature whereas the constant as well as the variable thermal conditions on the bottom wall is considered and the top wall is considered as adiabatic. The finite element technique has been applied to solve the governing equations after changing them into dimensionless form. In the numerical simulations, the Ag-MgO/water hybrid nanofluid has been engaged to gain insight into the flow and thermal arenas. The obtained outcomes are exhibited in terms of streamlines, isotherms and average Nusselt number as well as plotting the response surface and contours. The average Nu is displayed in line graphs, response surface and contours graphs as well as tabular form for different flow parameters. Response surface methodology is applied to achieve an optimization procedure to search for the best situation in which the highest heat transfer rate occurs when the distinct model parameters namely Darcy number, porosity and hybrid nanoparticles volume fraction are treated as the sensitivity of the parameters. A correlation equation between the output response and input variables has been derived with the help of response surface methodology. The magnetic field and its path regime the flow decoration of hybrid nanofluid suggestively. The Ag and Ag-MgO nanomaterial with water flow arenas are compared. While considering Ag-H2O nanofluid, the average Nu increases 23.97 % at Ra = 107 with ϕhnp = 0.02 whereas the corresponding increases is 25.68 % if the Ag-MgO/H2O hybrid nanofluid is taken into account. The average Nu upsurges suggestively, as hybrid nanoparticles volume fraction, period number, magnetic field leaning angle, Darcy number and Rayleigh number rises but porosity and Hartmann number have inverse effect. The outcomes showed that the average Nu is more sensitive to the hybrid nanoparticles volume fraction rather than the porosity or Darcy number. Also, it is discovered that the optimum condition arises at Da = 0.001, ε = 0.2 and ϕhnp = 0.02 with the final Nuave = 2358.21.

This study aims to explore the characteristics of tangent hyperbolic fluid flow along a stretching sheet. The sheet has suction or injection influences and is located inside a porous medium. The research inspects the flow and heat transfer (FHT) properties, taking into account the presence of a velocity slip condition. The flow of non-Newtonian magnetohydrodynamic fluid caused by a porous stretching sheet, taking into account thermal radiation and heat generation, has a wide range of engineering applications. These applications involve chemical reactors, energy distribution, storage of solar energy, and filtration processes. Mathematically, the flow problem is expressed as a collection of nonlinear partial differential equations. To numerically solve the resulting ODEs, the finite difference approach (FDM) is successfully used. Tables and graphs are used to display the various output values related to the hyperbolic tangent fluid. Among the different output values that appear are velocity and temperature. Significant observations from the study indicate that an increase in the power-law index, slip velocity parameter, porosity parameter, and magnetic number results in a decrease in the fluid's velocity and an increase in temperature. The completed comparison analysis shows a sizable degree of agreement with the earlier investigation.

The aggregation of nanoparticles is a major phenomenon having broad consequences in many fields. In order to fully utilize the capabilities of nanoparticles in a variety of applications and to evaluate the effects that they will have on the environment and biological systems, it is crucial to comprehend and manage aggregation. To develop and improve nanoparticle-based technology, researchers are still learning more about the aggregation processes. Thus, the present study investigates the combined effects of velocity slip and nanoparticle aggregation on Heat transfer (HT) analysis of MHD nanofluid (i.e., TiO2-C2H6O2) flow between porous convergent, divergent channel. The modified Krieger–Dougarty and Maxwell–Bruggeman models were utilized for nanoparticle aggregation. The modeling is based on nonlinear PDEs such as continuity, momentum, and heat equations. These equations are transformed into a system of nonlinear ODEs using similarity transformations and then solved numerically and analytically. The analytical solution has been constructed using the ADM method. The present results in particular cases are compared to results obtained by the HAM- package and by the Runge- Kutta Fehlberg 4th–5th order (RKF-45) for validation. The effects of active parameters on the velocity, temperature, concentration, skin friction, and Nusselt numbers are investigated. It is found that nanoparticle aggregation can limit fluid velocity in converging channels by increasing flow resistance through aggregate formation. Individual nanoparticles generate friction and lower velocity, while aggregated nanoparticles boost fluid density and velocity. In addition, it is found that the magnetic field lowers skin friction and increases HT due to Lorentz force, while porosity increases friction and HT. Nanoparticle concentration inversely affects friction, increasing friction without aggregation and decreasing friction with aggregation, with the HT rate rising with increased nanoparticle concentrations.

It is vitally important to comprehend non-Newtonian fluids flow behaviour from an industrial perspective. Non-Newtonian fluids are needed for a number of industrial and technical processes, including the production of paper, photographic films, and polymer sheet extrusion. There are several industrial applications for heat and mass transfer. However, thermal and solute relaxation time phenomena can't be predicted by classical heat and mass transfer laws (Fourier and Fick laws). This work aims to apply the modified Ohm law to thermal and mass transfer which are based on Fick's and generalized Fourier principles, respectively. The current study examines the three-dimensional Darcy-Forchheimer flow of non-Newtonian fluid “Casson fluid” via porous medium with ion slip and Hall effects across a stretched sheet. Thermophoresis and Brownian motion aspects are also considered. The diffusion phenomena are modelled using the Boungrino model. The modified Buongiorno model (MBM) for nanofluids is used to investigate the enhancement of heat transport. Together with practical nanofluid properties, it covers the mechanics of thermo-migration and random motion of nanoparticles. By adding the appropriate similarity transformations, this collection of (PDEs) that reflect the mathematical model is transformed into an (ODEs) system, which it is subsequently resolved utilizing the Lobatto IIIA technique's powerful computing capabilities via Matlab software. Data visualisations and numerical examples are presented considering many physical restrictions affect the fluctuation of velocities, temperatures, concentration, dimensionless shear stress, Nusselt number, and Sherwood number. According to certain findings, the existence of Hall and ion slip effects, and other factors contribute to an increase in horizontal velocity distribution. Furthermore, Fourier's law produces a wider temperature distribution than the Cattaneo-Christov heat flux model.

Our study focuses on numerically investigating natural convection within a three-dimensional trapezoidal cavity featuring a corrugated hot bottom wall, with a particular emphasis on enhancing heat exchange using a water-based hybrid nanofluid. Employing a steady-state regime assumption, the study considers laminar and incompressible three-dimensional flow. The Darcy-Forchheimer model is incorporated to account for inertial advection effects within the porous layer. Predictions of the thermal and hydrodynamic behaviors of the system are achieved by solving dimensionless equations utilizing the Galerkin Finite Element Method (GFEM). Notably, a range of influential parameters, including the volume fraction of nanoparticles (φ), Darcy number (Da) spanning from 10−5 to 10−2, Hartman number (Ha) across the range of 0–100, φ within the range of 0–0.08, porosity (ε) ranging from 0.2 to 0.9, and Rayleigh numbers (Ra) varying from 103 to 106 was explored. The investigation also evaluates the geometrical impact of the enclosure by considering undulation numbers (N) of the warm bottom ranging from 1 to 4. Our results provide novel quantitative insights. Specifically, we observe an inverse relationship between undulation number (N), Hartman number (Ha), and porosity (ε) with large Rayleigh (Ra) flows, significantly influencing effective heat transfer. Notably, this influence diminishes at lower Ra flows. At the highest Ra, we find that increasing Da, ε, and φ enhances the Nusselt number (Nuavg) by 26 %, 17 %, and 23.5 %, respectively. Conversely, increasing Ha and N reduces Nuavg by 13 % and 40 %, respectively. These findings highlight the nuanced interplay of parameters and offer valuable quantitative data for optimizing heat transfer processes in similar systems.