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The use of nanoparticles in heat transmission is an interesting research subject and many scholars have shown an interest in this topic. In accordance with the motivating relevance of nanomaterials, the current research advocates for an evaluation of Casson nanofluid using porous medium thermal analysis in conjunction with vibrant implementations of mixed convection through a chemical reaction. Additionally, a nonlinear suction/injection phenomenon is taken into consideration. A consistent flow pattern is created by the stretched porous structure. The extended Casson nanofluid model is used to identify the heat production of non-Newtonian fluids. The essential incentives for choosing the Casson nanofluid model are validated as it simultaneously achieves Casson fluid, nanofluid and porous medium outcomes. The convective transport of nanofluid has been addressed using convective temperature boundary conditions and convective flow. A dimensionless form with similarity variables is used to simplify the fundamental equations for the provided flow model. The fourth-order Runge–Kutta (RK) strategy is deployed to arrive at a numerical solution for simulated flow. The physical conveyance of flow variables is accessed visually. It is calculated from engineering quantities (i.e., skin friction, Nusselt number and Sherwood number) to generate the numerical values. The velocity declines with increasing porosity and material factors while opposite outcomes are witnessed for buoyancy factors (mixed convection and buoyancy ratio) and injection factor. [ABSTRACT FROM AUTHOR]

Viscoelastic liquids are of great interest for industrial and engineering applications due to their unique properties. This research presents a comparative examination of tri-nanomaterial Darcy-Forchheimer flow, involving Oldroyd-B, Maxwell, and Jeffrey (OMJ) nanofluids, over a permeable stretching sheet. The analysis integrates the influences of quadratic thermal radiation, activation energy, magnetic field, and heat source. Additionally, Buongiorno's model has been utilized along with convective thermal boundary conditions. By leveraging local non-similar approach (LNSA), the modelled highly nonlinear partial differential equations (PDEs) are transformed into a set of ordinary differential equations (ODEs) which are further solved using the finite-difference-based bvp5c solver. It is determined that a rise in the Forchheimer parameter reduces the velocity field. The least and highest temperature profile is observed for Maxwell and Oldroyd-B nanofluids, respectively. It is further noted that per-unit increase in the Forchheimer parameter declines the drag coefficient by 15.43%, 23.87%, and 14.49% for Oldroyd-B, Maxwell, and Jeffrey nanofluid, respectively. As the temperature ratio raises, the heat transfer rate and mass transfer rate increase. Additionally, it is seen that the Oldroyd-B has the highest transfer rates followed by Jeffrey and Maxwell nanofluid. [ABSTRACT FROM AUTHOR]

In this paper, we consider an adaptive hybridizable discontinuous Galerkin (HDG) method based on the discrete fracture model for approximation of a single-phase flow in fractured porous media. We are interested in the case that the flow rate in fracture is large enough to justify the use of Forchheimer's law for modeling flow within the fracture, while Darcy's law is applied to the surrounding matrix. The HDG method could be designed to simulate the flow in porous media with reduced fractures which consist of many straight lines or planes. More specifically, we use piecewise polynomials of degree k to approximate the velocity and pressure in fracture and surrounding porous media. The existence and uniqueness of discrete solutions are proved by the Brouwer fixed point theorem, and an efficient and reliable a posteriori error estimator is obtained with respect to an energy norm. Moreover, the HDG scheme, the existence and uniqueness of discrete solutions, and the a posteriori error estimates are also extended to the problem with non-planar, embedded, and intersecting fractures. Finally, several numerical examples are provided to validate the performance of the obtained a posteriori error estimator. [ABSTRACT FROM AUTHOR]

This research work presents the numerical solution of the Darcy–Forchheimer flow of Casson nanofluidic model (DFFCNFM) on stretching sheets with Newtonian heating by utilizing a combination of nonlinear input–output neural networks with backpropagation of Levenberg–Marquardt computational approach. The presented study investigates the impact of electroosmosis forces on the boundary layer of the Casson nanofluid, focusing on viscous and Joule dissipations. A dataset for DFFCNFM is generated for the different events with backward differentiation formula (BDF) by varying Casson fluid parameter (β) , permeability parameter (Da), electric parameter ( E 1) , Reynolds number (Re) relative to stretching velocity, magnetic field (M) , Eckert number (Ec), and Prandtl number (Pr). The artificial intelligence-inspired technique via nonlinear input–output neural networks with backpropagation of Levenberg–Marquardt is utilized from the generated dataset for DFFCNFM to find the approximate solutions. The satisfactory performance levels, as indicated by the mean square error (MSE), have been attained consistently with magnitude around 10 − 1 2 –10 − 1 4 for all scenarios of DFFCNFM. The precision and performance validation is effectively established by the negligible MSE, close proximity to the unit value of regression metric, and the distribution of instances in error-histograms. [ABSTRACT FROM AUTHOR]

Fluid flow across a rotating disk has significant technical and industrial applications, including rotors, turbines, fans, centrifugal pumps, spinning disks, viscometers, etc. The impact of different-shaped nanoparticles immersed in the fluid controlled the thermophysical characteristics of the fluid, which were utilized in several sectors to accelerate thermal advancement. In the present problem, the tri-hybrid nanofluid flows over a rotating disk with three different shapes, namely spherical, cylindrical, and platelets, respectively, for Al2O3, multi-layered carbon nanotubes, and graphene nanoparticles immersed in the base fluid water. Under convective conditions, the tri nanofluid's thermal expansion is more significant when combined with Joule heating, Cattaneo-Christov heat flux, and nonlinear thermal radiation. The Galerkin Finite Element Method is used to solve the simplified form of PDEs after a similarity transformation is introduced to convert them into ODEs. The skin friction coefficient and the heat transfer rate are subjected to a quadratic regression analysis; the results are shown in tables. Compared to the base fluid, the Nusselt number reveals an improvement of around 5.72% for nanofluid, 7.35% for hybrid nanofluid, and 17.18% for tri-hybrid nanofluid when the strength of radiation parameter and Brinkman number is raised. Platelet-shaped nanoparticles observed a significant tendency to enhance the rate of heat transfer, which is more prominent for the tri-hybrid nanofluid than the hybrid and mono nanofluids. Each graph features a comparison of ternary hybrid, hybrid, and mono nanofluid with other significant physical parameters. It was noted that the entropy of the system significantly intensified with Reynolds number and temperature ratio, while it was controlled by radiation parameters. The uses of ternary nanofluids include energy storage devices, adsorbents, sensors, imaging, catalysts, therapeutic activity, and more. [ABSTRACT FROM AUTHOR]

This paper intends to discuss the hybrid nanofluid flow through a cone in a Darcy–Forchheimer porous media, containing base fluid Methanol (CH3OH), nanoparticles of titanium oxide (TiO2) and copper (Cu). We have examined the flow under the influences of mixed convection, viscous dissipation, chemical reaction, and nonlinear heat source. In hybrid nanofluid flow, viscous dissipation and mixed convection play a crucial role in heat exchangers. By accounting for both mixed convection and viscous dissipation, the addition of nanoparticles into a base fluid improves the transfer of heat and optimizes the performance of the cooling system. Using the laws of conservation, governing equations have been constructed. The flow model is converted using the appropriate similarity transformations from partial differential equations to ordinary differential equations. The homotopy analysis method is used to solve the updated system of equations. When Forchheimer inertial drag parameter is increased, the velocity profile decreases, but it rises when the Darcy number is increased. As the value of exponential heat source parameter rises, the temperature profile increases as well. The result exposes that with an increment in nanoparticle volume fraction, temperature profile also rises but velocity profile decreases. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to describe the stead MHD mixed convection flow over a permeable vertical flat plate embedded in a Darcy–Forchheimer porous medium. Using appropriate similarity variables, the partial differential equations are transformed into ordinary (similar) differential equations, which are numerically solved using the bvp4c function in MATLAB. The numerical results are used to present graphically and in tables, illustrations of the reduced skin friction, reduced Nusselt number, velocity, and temperature profiles. Dual (upper and lower branch) solutions are discovered in this exciting analysis. Although numerous studies on the mixed convection past a vertical plate embedded in a fluid-saturated porous medium exist, none of the researchers have focused on the Darcy–Forchheimer flow with asymptotic solutions. The behavior of the flow and heat transfer has been thoroughly analyzed with the variations in governing parameters, such as Darcy–Forchheimer G , suction/injection S , MHD M , and mixed convection λ parameters. [ABSTRACT FROM AUTHOR]

In this work, the Cattaneo–Christov double diffusion model is used to analyze the three-dimensional boundary layer flow of an upper-convected Maxwell fluid flowing over a bi-directional stretching surface. The model assumes that the fluid's diffusivity is concentration-dependent, whereas dynamic viscosity and thermal conductivity are temperature-dependent; and the model takes into account the influence of the magnetic field of uniform strength. The problem is formulated using the conservation rules of mass, momentum, and energy as well as the boundary layer approximation. The resulting nonlinear system is then numerically solved using the bvp4c procedure in MATLAB. The obtained results are set forth graphically to illustrate the variations of different parameters. It is contemplated that the increase in thermal relaxation parameter results in a drop in fluid temperature while there is an enhancement in the concentration profile. Also, the temperature field has a direct relationship with the temperature-dependent viscosity. This research has proven its utility in industries where there is a need to analyze fluid flow over surfaces that can stretch in two directions. [ABSTRACT FROM AUTHOR]

The flow of carbon nanotubes (CNTs) via the Riga plate has a substantial influence, and it is used in the industrial sector, like fluid stirring, thermal reactors, semiconductors, tissue regeneration, gene delivery to organs, etc. However, the 3D Darcy–Forchheimer flow (DFF) of CNTs through the Riga plate with nonlinear thermal radiation and a non-uniform heat sink/source has not been investigated. As a consequence of this, the objective of this paper is to explore the impact of the nonlinear radiative DFF of water-based CNTs past a heated Riga plate with a non-uniform heat sink/source and homogeneous and heterogeneous reactions. In addition, the energy equation is constructed using the Cattaneo–Christov heat flux concept. By making use of appropriate variables, the governing flow models are mutated into a system of ordinary differential equations. These reduced equations are analytically and numerically computed by homotopy analysis method (HAM) and the bvp4c scheme. The brief discussion and visual depiction of the dominance of different factors on x- and y-direction velocities, thermal, nanoparticle concentration (NPC), surface drag coefficients, and local Nusselt number are presented in tables and figures. The growth of the Forchheimer number leads to a noticeable enrichment in the velocity profile in both directions. When the nanoparticle’s volume fraction and space-dependent heat source/sink parameters are increased, the thermal profile gets better. The NPC profile exhibits suppression as the intensity of both heterogeneous and homogeneous responses rises. [ABSTRACT FROM AUTHOR]

Ferromagnetic hybrid nanofluids can be employed in electronics and microelectronics cooling applications to minimise the accumulation of heat and effectively eliminate excess heat. By increasing the heat transfer rate, these nanofluids serve to maintain suitable operating temperatures and avoid device overheating. This study examines the influence of convective heating on the fluid flow of a three-dimensional ferromagnetic Casson hybrid nanofluid (composed of Mn-ZnFe2O4/CoFe2O4 nanoparticles) over a radiative Riga sensor device. The investigation takes place within a permeable medium characterised by Darcy–Forchheimer dynamics. Additionally, the analysis incorporates the assessment of the interaction of viscous dissipation. To establish a standardised set of governing partial differential equations along with their associated boundary circumstances, suitable similarity transformations are implemented. Following this, the resultant transformed ordinary differential equations are efficiently solved using the bvp5c solver. The solution process employs the shooting technique facilitated by MATLAB software. The impact of these influencing factors was carefully observed and thoroughly analysed using graphical representations. Specifically, the effects of pertinent factors on shear stress and heat transfer rates are concisely depicted in tabular formats.

AbstractIn this article, an effort is made to analyze the 3−D flow of hybrid nanofluid over a rotating stretching sheet. The hybrid nanofluid contains engine oil as a base fluid amalgamated with nano-particles MoS2 and ZnO. The flow is assumed to pass through a Darcy–Forchheimer medium subjected to the influence of an inclined magnetic field. In addition, heat radiation, Joule heating, viscous dissipation, nonlinear heat source, and quartic autocatalytic chemical reaction are applied to discourse heat and mass transfer features. The PDEs representing the physical problem are converted to ODEs with the introduction of similarity transformations. The solution of the problem is determined numerically by using MATLAB built-in bvp4c method. The proposed mathematical model’s validation is ascertained by comparing it with an earlier study in limiting case. In this regard, an excellent consensus is accomplished. The current investigation reveals that the flow along axial direction gets retarded by enhancing the values of rotational parameter, inertial parameter, and magnetic parameter, whereas the flow along radial direction gets accelerated. The increasing value of angle of inclination admirably resists the motion of nanofluid and hybrid nanofluid. The concentration profile gets decreased under the increasing influence of inclination angle and Schmidt number while it enhances for homogeneous reaction parameter. The enhancing values of rotational parameter, solid volume fraction and temperature, and exponential dependent heat generation parameters enhance the value of Nusselt number. [ABSTRACT FROM AUTHOR]

Purpose: This paper demonstrates the way tangent hyperbolic nanofluid flow through a vertical cone is influenced by varying viscosity and varying thermal conductivity. This study also seeks to illustrate the impact of convective boundary conditions on a fluid. The mathematical modeling also takes the Darcy–Forchheimer effect into account.Methodology: Using the appropriate similarity transformation, the fluid problem is reduced into a set of nonlinear ordinary differential equations. These systems of differential equations are evaluated numerically by applying the Optimal Homotopy Asymptotic Method.Findings: The nature of emergent parameters is examined in relation to the temperature distribution, nanoparticle concentration profile, and velocity profile. An increase in variable viscosity corresponds to a decrease in fluid velocity, while enhanced thermal conductivity results in elevated fluid temperature. The skin friction coefficient, Sherwood, and Nusselt numbers are numerically examined for active concerned parameters. These findings can be put into practice in a variety of fields such as polymer cooling systems and medication.Originality: Existing literature has yet to explore the combination of tangent hyperbolic nanofluids with varying viscosity and thermal conductivity under convective boundary conditions. [ABSTRACT FROM AUTHOR]

The theme of this work is to investigate the heat transfer assessment of three-dimensional flow of magnetized Casson hybrid nanofluid through a rotating permeable Riga plate using Darcy–Forchheimer model. The comparative analysis between thermal conductivity model of Hamilton-Crosses, Xue and Yamada-Ota model is also presented. The hybrid nanofluid is the combination of nanoparticles single-walled carbon nanotubes and multi-walled carbon nanotubes which are suspended in base fluid sodium alginate. The flow and heat transfer behavior of working fluid are influenced by the slip velocity, convective heating, viscous dissipation. The solution of existing nonlinear partial differential equations of three-dimensional flow of Casson hybrid nanofluid is obtained by employing the bvp4c command of MATLAB. For the validation of our code and numerical results, a comparison is made with existing results. From the numerical outcomes, it is observed that increasing axial slip parameter, the thermal boundary layer decreases which reveal that the thermal conductivity model of Yamada-Ota hybrid nanofluid is superior to that of Hamilton Crosser and Xue model. Moreover, Hybrid nanofluids also show an improvement in the axial and transverse velocity components as the magnitude of buoyancy parameter and Hartmann number increase. These findings may provide the valuable insights for the fields of geothermal engineering, energy conservation, and nuclear waste disposal. [ABSTRACT FROM AUTHOR]

The present study is to highlight the significance of viscous dissipation, Joule heating and magnetic field on the stagnation point Darcy-Forchheimer flow of THNF (trihybrid nanofluid) around a spinning sphere containing Oxytactic and gyrotactic microorganisms. The analysis also includes the effects of heat generation and higher-order chemical reaction. A trihybrid nanofluid consisting of water (H2O) as the base fluid and TiO2,Cu and Fe3O4 nanoparticles is used. The Hamilton–Crosser model of trihybrid nanofluid is used in this work to compare their respective performance. This model is applicable to the optimization and design of sophisticated cooling systems where effective heat transfer is necessary, like electronic cooling, nuclear reactors, and high-performance heat exchangers. This model is used in biomedical engineering to comprehend drug delivery systems where exact control over reactions and fluid dynamics is required. Oxytactic microorganisms are important because they help break down contaminants in environmental engineering and wastewater treatment procedures. The fundamental system of the PDEs (partial differential equations) and the pertinent BCs (boundary conditions) have been transformed for computation using a few appropriate transformations. The numerical estimations for the appropriate system of differential equations are derived by employing the shooting approach (Bvp4c). Extensive studies are conducted to examine the effects of pertinent limitations on the rates of heat transfer, mass transfer, density of Oxytactic and gyrotactic microorganisms for both hybrid and trihybrid nanofluid. Among the various findings of this investigation is the observation that a higher Oxytactic and gyrotactic microorganisms Schmidt numbers result in a slower trihybrid nanofluid and hybrid nanofluid Oxytactic and gyrotactic microorganism's distributions. By increasing the value of Peclet number up to 7% the values of local density gyrotactic microorganisms enhance up to 9% .

The hybrid nanofluids are crucial for enhancing heat transfer efficiency in different technological and industrial operations like nuclear reactor cooling, fuel cells, drug delivery systems, etc. Taking this factor into account, the current investigation focused on the MHD Darcy–Forchheimer flow of water-based hybrid nanofluid (Cu-Al2O3) and mono nanofluid (Cu) past a stretched surface with injection/suction. Additionally, the investigation focuses on analyzing the consequences of nonlinear thermal radiation and heat consumption/generation. The governing higher-order PDEs are rehabilitated into ODEs through a suitable transformation process. Finally, we solve these equations using the quantitative approach of the Bvp4c algorithm in MATLAB and visualize the results in tables and graphs. Our study revealed that the intensified magnetic field, porosity, and injection/suction parameters suppress the fluid velocity. The fluid heat is enhanced when intensifying the radiation, and nanoparticles volume fraction parameters. Augmenting the magnetic field parameter results in a reduction in the skin friction coefficient. It is also found that the highest dwindling percentage of the skin friction coefficient is 24% (HNF), 33.38% (NF), and 25.70% (VF) when the magnetic field parameter changes from 0 to 1. The largest growing percentage of the local Nusselt number is 34.02% (HNF), 36.75% (NF) and 39.52 (VF), which occurs when the suction/injection parameter is modified from −0.3 to 0.

The increasing demand for thermal devices in industry necessitates enhanced heat transfer efficiency. This study examines the steady, two-dimensional, incompressible laminar MHD boundary layer flow of a nanofluid in water. A system of boundary value problems is formulated and addressed using similarity variables and a novel iterative method based on the operational matrix technique. The effectiveness of the numerical method is demonstrated by computing the local truncation error. The numerical method exhibits rapid convergence and low computational cost. It is both a direct and iterative approach. The study explores the impact of various parameters on concentration, temperature, and velocity profiles. Findings indicate that the porosity parameter and Prandtl number significantly influence temperature and concentration distribution, while the inertia coefficient has a comparatively minor effect. The analysis presents promising results with potential for further improvement in future research.

The current communication deliberates the consequences of the Darcy–Forchheimer flow of Eyring–Powell nanofluid past a slippery surface containing activation energy and motile microorganisms. The flow is influenced by the consequences of Brownian motion, thermal radiation, the Cattaneo–Christov heat-mass flux theory, and thermophoresis. The framed flow models are transformed into ordinary derivative equations by adopting appropriate conversion variables. The transformed equations are numerically tackled by using the bvp4c scheme in MATLAB. The study is remarkable for its comprehensive analysis of the interplay of several flow factors, such as the Forchheimer number, Richardson number, bioconvection Rayleigh number, radiation, thermophoresis, Brownian motion, thermal and mass relaxation time parameters. The outcomes are visualized through tables and diagrams, which provide significant insights into the intricate physical mechanisms involved in this multifaceted subject. Evidently, the velocity profile declines when there is a rise in the buoyancy ratio parameter and the opposite trend is obtained for the Richardson number. The temperature grows when there is a larger magnitude of the thermophoresis parameter and it reduces for greater values of the time relaxation parameter. The activation energy and mass relaxation parameters enhance the concentration profile. The microbe density increases when enhancing the quantity of Peclet number and it declines for bioconvection Lewis number.

The Casson fluid flow over a stretching sheet is studied in this work in the presence of porous media with melting heat transfer and chemical reaction, combining nanoparticles of Ti4Al6V,AA7072 and AA7075 with a base fluid of sodium alginate (NaAlg). The porous medium Darcy Forchheimer effect is included in the momentum equation. Through the effects of melting, the process of heat transfer is described. In order to explore the features of heat transmission in the absenteeism of LTECs (local thermal equilibrium conditions), the current study employs a mathematical model that has been simplified. For both the solid and liquid phases, the LTNE model yields two different fundamental thermal gradients. This proposed model compares the YOM (Yamada-Ota model) and XM (Xue model), two well-known trihybrid nanofluid models, in terms of performance. After applying the proper transformations, modulated non-linear PDEs (partial differential equations) are simplified in ODEs (ordinary differential equations) and the bvp4c method is used to solve them numerically. A graphic discussion of the relevant factors' significance on the pertinent fields has been presented. It is observed that the Casson ternary hybrid nanofluid temperature and velocity distributions predominate at higher melting parameter. The solid phase's heat transport rate rises with an upsurge in the interphase heat transfer parameter.

This study offers a numerical assessment of unsteady laminar flow, heat and mass transfer of conventional hybrid nanofluid considering activation energy which is noticed in chemical processes. Further, non linear radiation and bioconvection flow with first order slip is also accounted. Nanoparticles as Cu and Al2O3 in the base fluid H2O are included in this analysis. Similarity transformation is applied to the governing model equations to reduce dimensionless form and ''bvp4c'' a MATLAB solver, is applied to get numerical outputs of flow problem. Validation of numerical outcomes is done by an analytical approach optimal auxiliary function method (OAFM). This analysis reports better thermal characteristics when volume fractions 0.0 ≤ φ1 ≤ 0.5 and 0.0 ≤ φ2 ≤ 0.5 of both nanoparticles Cu and Al2O3 are enhanced. Results for present analysis are interpreted numerically and graphically for distinct enhanced values of dimensionless parameters and physical quantities. The high resistance offered by velocity slip (0 ≤ Sv1 ≤ 1.0) and Forchheimer parameter (0 ≤ Fr ≤ 10) cause a drop in velocity profiles. It is noticed that bioconvection Peclet number (0 ≤ Pe ≤ 0.7) leading to a drop (χ(η) = 0.139972 to χ(η) = 0.0817063, at η = 1) in density distribution of motile microorganisms.

Abstract In this paper, we carried out a numerical analysis of the fluid dynamics and heat transfer occurring between two parallel disks. The study accounts for the impact of temperature-dependent fluid viscosity and thermal conductivity. We systematically investigated various parameters, including viscosity, thermal conductivity, rotational behavior (rotation or counter-rotation), and the presence of stretching, aiming to comprehend their effects on fluid velocity, temperature profiles, and pressure distributions. Our research constructs a mathematical model that intricately couples fluid heat transfer and pressure distribution within the rotating system. To solve this model, we employed the 'Particle Swarm Optimization' method in tandem with the finite difference approach. The results are presented through visual representations of fluid flow profiles, temperature, and pressure distributions along the rotational axis. The findings revealed that the change in Casson factor from 2.5 to 1.5 resulted in a reduction of skin friction by up to 65%, while the change in local Nusselt number was minimal. Furthermore, both the viscosity variation parameter and thermal conductivity parameters were found to play significant roles in regulating both skin friction and local Nusselt number. These findings will have practical relevance to scientists and engineers working in fields related to heat management, such as those involved in rotating gas turbines, computer storage devices, medical equipment, space vehicles, and various other applications.

The paper deals with the combined effect of non-Newtonian saturating fluid and horizontal flow rate on the thermal convection in a highly permeable, porous plane layer saturated with a power-law model. Asymmetric boundary conditions are assumed, with a cooled free surface at the top and a heated, impermeable, rigid wall at the bottom. The generalised Forchheimer equation is employed to model the power-law fluid movement. Convection cells emerge in the power-law fluid because of vertical temperature gradient imposed by the thermal boundaries. The onset of this scenario can be studied using linear stability theory, which leads to an eigenvalue problem. The latter is solved either numerically, employing shooting schemes, or analytically, using one-order Galerkin approach. The present study is considered an extension of the classical Prats problem. When the Peclet number, which defines the flow rate, is negligible, the configuration switches to the special case of Darcy-Rayielgh instability. The results show that the form drag exhibits a stronger stabilizing influence in shear-thinning fluids compared to shear-thickening and Newtonian ones since the saturating fluid is described by the power-law model. This scenario appears in the specific range of the Peclet number. In general, this investigation can be used to understand the heat transfer process in subsurface hydrocarbon reservoirs where the fluid may exhibit non-Newtonian behaviour.

Saving energy, shortening processing times, maximizing thermal efficiency, and lengthening the life of industrial equipment are all possible outcomes of heating and cooling optimization. In recent years, there has been a rise in interest regarding the development of high-efficiency thermal systems for the purpose of enhancing heat and mass movement. This study presents an investigation on the non-linear flow of a hybrid nanofluid comprising of Multi Walled Carbon Nanotubes (MWCNTs) and Single Walled Carbon Nanotubes (SWCNTs) over an extended surface, considering the effects of Magnetohydrodynamics (MHD) and porosity, with engine oil serving as the base fluid. Also, radiation and Darcy-Forchheimer flow is considered. The problem of regulating flow is transformed into ordinary differential equations (ODEs) by employing similarity variables. The Midrich Scheme is then used to implement a numerical solution to these equations in the program Maple. Through visual representations of fluid velocities and temperatures, the inquiry addresses several important factors, including magnetic parameters, porosity parameters, radiation parameters, Eckert numbers, inertia coefficients, and Biot numbers. The research has important implications in a number of real-world contexts. Due to its exceptional characteristics, such as reduced erosion, reduced compression drops difficulties, and greatly increased heat transfer rates, hybrid nanofluids are frequently used in heat exchangers. For instance, various cooling devices such as electromagnetic cooling systems, as well as heat exchangers including condensers, boilers, chillers, air conditioners, evaporators, coil preheaters, and radiators. Furthermore, it has the potential to be employed in pharmaceutical businesses and the field of biomedical nanoscience.

Carbon nanotubes (CNTs) are gaining popularity due to their expanding uses in industrial and technical processes, such as geothermal reservoirs, water and air filters, coatings, solar collection, ceramic material reinforcement, electrostatic dissipation, etc. In addition, the CNTs have superior electrical conductivity and biocompatibility. Based on the aforementioned applications, the current work examines the time-dependent and Darcy–Forchheimer flow of water/glycerin-based Casson hybrid nanofluid formed by single-walled CNTs and multi-walled CNTs over a Riga plate under velocity slip. The energy expression is modeled through nonlinear thermal radiation and viscous dissipation impacts. The incorporation of convective boundary condition into the current model improves its realism. By employing suitable variables, the governing models are re-framed into ordinary differential equations. The bvp4c and the homotopy analysis method are used to find the computational results of the re-framed equations and boundary conditions. The novel characteristics of a variety of physical parameters on velocity, temperature, skin friction coefficient (SFC), and local Nusselt number (LNN) are discussed via graphs, charts, and tables. It is found that the fluid velocity decays when enriching the Forchheimer number, unsteady and porosity parameters. The radiation parameter plays an opposite role in convective heating and cooling cases. The modified Hartmann number enhances the surface drag force, and the Forchheimer number declines the SFC. The unsteady parameter develops the heat transfer rate, and the Forchheimer number suppresses the LNN. The simulated flow problem has many applications in engineering sectors, including ceramic manufacture, heating and cooling systems, energy storage units, thermodynamic processes, and other fields.

The paper deals with the combined effect of non-Newtonian saturating fluid and horizontal flow rate on the thermal convection in a highly permeable, porous plane layer saturated with a power-law model. Asymmetric boundary conditions are assumed, with a cooled free surface at the top and a heated, impermeable, rigid wall at the bottom. The generalised Forchheimer equation is employed to model the power-law fluid movement. Convection cells emerge in the power-law fluid because of vertical temperature gradient imposed by the thermal boundaries. The onset of this scenario can be studied using linear stability theory, which leads to an eigenvalue problem. The latter is solved either numerically, employing shooting schemes, or analytically, using one-order Galerkin approach. The present study is considered an extension of the classical Prats problem. When the Peclet number, which defines the flow rate, is negligible, the configuration switches to the special case of Darcy-Rayielgh instability. The results show that the form drag exhibits a stronger stabilizing influence in shear-thinning fluids compared to shear-thickening and Newtonian ones since the saturating fluid is described by the power-law model. This scenario appears in the specific range of the Peclet number. In general, this investigation can be used to understand the heat transfer process in subsurface hydrocarbon reservoirs where the fluid may exhibit non-Newtonian behaviour. [ABSTRACT FROM AUTHOR]

Darcy-Forchheimer model has been used to consider the mathematical and statistical aspects of Prandtl nanofluid flow on a stretched curvy geometry, with homogenic-heterogenic reactions, nonlinear radiation, exponential heat, Joule heating, velocity slip, and convective heat conditions. An account of entropy significance has been given to boost the applicability of the study. The 4-5th ordered numerical tool, Runge-Kutta-Fehlberg, has been employed to establish the plots for the considered flow. ANOVA and Taguchi optimisation technique is used to obtain the optimal condition in enhancing the heat transfer rate for modelled mathematical problem. Here, the study reveals that the increasing homo-heterogenic strength parameters foster the concentration profile. The study also found that the thermal curves are positively affected by the radiation parameter and the temperature differential parameter. In addition to this, graphical portraits of isotherms and streamlines have been given to characterise the flow and heat pattern. Taguchi method reveal that first level of Prandtl number, magnetic parameter, Weissenberg number, heat source parameter and third level of curvature parameter, produce maximum Nusselt number. Heat source parameter has large contribution of about 49.45% among the other parameters and Prandtl number has the least contribution of about 1.4% for optimisation. [ABSTRACT FROM AUTHOR]

The concept of heat transport over a flat stretching surface for the inclusion of two unlike types of metal and oxide nanoparticles in a solvent water to investigate the role of hybrid nanofluid. The inclusion of Darcy–Forchheimer inertial drag over an elongating surface embedding in a porous medium enriches the flow behavior. However, the main attraction of the proposed investigation is the implementation of the Mintsa model conductivity combined with convective boundary condition that boosts up the thermophysical model. The suitable choice of similarity rules is useful for the transformation of governing equations into its non-dimensional form, and further traditional Runge–Kutta fourth-order method associated with shooting is employed to solve the flow profiles. The characteristics of controlling parameters on the flow phenomena along with the rate coefficients are deployed via graphs and tables. This investigation gives comparative results with earlier studies in particular cases. The important outcomes of the proposed investigation are: The augmented particle concentration favors in enhancing the fluid temperature in the permeable medium, whereas drag opposes the velocity profile significantly. [ABSTRACT FROM AUTHOR]

The incompressible miscible displacement of two-dimensional Darcy–Forchheimer flow is discussed in this paper, and the mathematical model is formulated by two partial differential equations, a Darcy–Forchheimer flow equation for the pressure and a convection–diffusion equation for the concentration. The model is discretized using a fully mixed virtual element method (VEM), which employs mixed VEMs to solve both the Darcy–Forchheimer flow and concentration equations by introducing an auxiliary flux variable to ensure full mass conservation. By using fixed point theory, we proved the stability, existence and uniqueness of the associated mixed VEM solution under smallness data assumption. Furthermore, we obtain optimal error estimates for concentration and auxiliary flux variables in the |$\texttt {L}^{2}$| - and |$\textbf {L}^{2}$| -norms, as well as for the velocity in the |$\textbf {L}^{2}$| -norm. Finally, several numerical experiments are presented to support the theoretical analysis and to illustrate the applicability for solving actual problems. [ABSTRACT FROM AUTHOR]

Industrial and business environments may benefit profoundly from the heat transfers provided by magnetohydrodynamic nanofluid flow via cone, wedge, and plate. Conical geometries are frequently employed in the creation of fluid system nozzles and diffusers. Conical nozzles promote fluid flow whereas conical diffusers diminish it. These parts are employed in a variety of applications, including jet engines and HVAC systems. In industrial systems, wedges are employed in relief valves that release pressure to regulate the discharge of pressurised fluids or gases. Motivated by this, this study aims to simulate the flow and heat transfer attributes of a magnetised Casson hybrid nanofluid (GO-MoS2/sodium alginate) subject to quadratic thermal radiation and nonlinear convection through cone-, wedge-, and plate-shaped geometries. The Darcy-Forchheimer and Cattaneo-Christov models were utilised to investigate the flow across porous media and heat transfer attributes, respectively. The solution to the governing equations is attained using bvp4c solver, and the effects of parameters like thermal relaxation parameter, Casson factor, Grashof number, magnetic parameter, Eckert number, and wall temperature on flow velocity and temperature are illustrated through graphs and tables, as are their consequences on physical quantities. The study reports that fluid velocity decreases with an increase in magnetic parameter, wall temperature, and thermal relaxation parameter, and fluid temperature decreases with an increase in magnetic parameter, Grashof number, and Eckert number. Further, the wall temperature and thermal relaxation parameters increase the heat transfer rate and reduce the skin friction coefficient. Moreover, a comparison with earlier published work is also done to verify the accuracy of the applied method and is reported to be consistent with previous studies. [ABSTRACT FROM AUTHOR]

The purpose of the presented article is to develop some new global regularity criteria for a magnetohydrodynamic (MHD) fluid flowing in a saturated porous medium. The effect of the porous medium, over the fluid flow, is characterized by a Darcy–Forchheimer law. The fluid, under study, is considered as one-dimensional and flowing in the x– direction with velocity component u. In addition, such a component is assumed to vary with the y– direction, i.e. u (y). Then, given the vorticity function w = - ∂ u ∂ y , such that ‖ w ‖ BMO 2 is sufficiently small, we develop the regularity criteria under the scope of the L 2 space. We extend our results to the spaces Ls , where s > 2. Afterward, we prove the Liouville-type theorem for the MHD Darcy–Forchheimer flow equation. Eventually, we obtain some characterization about the asymptotic behaviour of solutions, particularly, the nonuniform convergence in L 2 for t → ∞. [ABSTRACT FROM AUTHOR]

The impact of heat generation and activation energy on the Darcy-Forchheimer flow of magnetized fluid with nanoparticles in the condition of blood flowing arteries is being investigated by researchers from all over the world in an effort to understand the heat transmission component for the treatment of numerous diseases. The assessment of heat transmission has been made with the enclosure of influence like the viscous dissipation, heat source, Joules heating, and thermal radiation. Utilized is a tetra hybrid nanofluid made up of silver (Ag), gold (Au), titanium oxide (TiO2), coper (Cu), and Blood as the base fluid. The main justification for the use of gold, copper, and silver nanoparticles as drug delivery nanomaterials is their potential for drug transport and imaging. The purpose of the presented model is to assess the performance of the well-known tetra hybrid nanofluid models developed by Yamada-Ota and Xue. The main PDE are transformed into ODE via similarity renovations, and these ODE are then numerically solved using the MATLAB built-in bvp4c scheme. The arteries' internal temperature rises as a result of thermal radiation and viscous dissipation. The velocity and thermal distributions show an increasing behavior with greater variations in the flow parameter. The Yamada-Ota model outperforms the Xue tetra hybrid nanofluid model in terms of heat transmission effectiveness.

The Marangoni convective phenomena have a unique impact on industries and medical tools. These phenomena are more prominent in the presence of dual nanoparticles (NPs) over base fluids such as blood that are surrounded by a thin interfacial nanolayer, an important feature to control the physical and thermal properties of the NP. In this problem, we have analysed the thermo-solutal Marangoni convective Darcy-Forchheimer flow of nanomaterials with the impact of the interfacial nanolayer. The results of the system of an exponential heat source, non-linear radiation, joule heating, and activation energy are discussed. An appropriate transition is applied to rationalise the substantially paired and nonlinear governing equations and then processed by the Galerkin finite element method (G-FEM). The impression of different governing parameters on the governing systems in conjunction with entropy and Bejan number is demonstrated through graphical and tabular form. Graphs are drawn with an evaluation of general and hybrid nanofluids (HNFs) and different nanolayer thicknesses of NPs. Activation energy and chemical reaction parameters restrict the Sherwood number, and the same is observed for the Nusselt number with an increase in the Brinkman and Eckert numbers. The thickness of the interfacial nanolayer of the NPs restricts the entropy generation of the system, while the entropy is higher for the HNF than the nanofluid. An opposite feature was observed for the Bejan number.

This article presents a comprehensive study on the thermal behavior of a three dimensional viscoelastic micropolar nanofluid under the inspiration of Cattaneo-Christov heat conduction model. The nanofluid is subject to an exponentially stretchable sheet and the flow is characterized by Darcy-Forchheimer formulation. Similarity variables are used to convert viscoelastic nanofluids PDEs into ODEs. MATLAB platform with bvp4c command is used to solve resulting ODE. The impact of involving parameters such that velocity ratio, Schmidt number, Prandtl number, Peclet number, Forchheimer number, buoyancy ratio parameter, Brownian motion, Heat source, Magnetic parameter on velocity profiles, rotation, temperature concentration and density profiles via graphs and tables. Temperature curve is improved up for magnetic parameters, temperature exponent parameter and thermopherosis parameter while significance loss is noted for increasing value of Prandtl number Pr and relaxation time λT. This research bears significance in applications related to advanced heat exchangers, microfluidic systems, and other emerging technologies where the interplay of nanofluids, viscoelasticity, and non-local heat conduction is of paramount importance.

This research aims to propose a numerical scheme for solving boundary value problems. It is a two-stage, third-order accurate scheme known as a predictorcorrector scheme. The two main results are finding the region of the scheme where it is stable and determining the stability criterion for a set of linearized first-order differential equations. In addition, a mathematical model for heat and mass transfer of Darcy-Forchheimer flow of non-Newtonian nanofluid over the sheet is presented. The similarity transformations reduce PDE into a system of ODE for easier manipulation. The results are compared with the past research and those obtained by MATLAB SOLVER BVP4C. The results show that the velocity profile slightly decays by enhancing the Weisenberg number. [ABSTRACT FROM AUTHOR]

Nanolubricants fall among the newly evolved and emerged technologies used in lubrication and heat transfer systems due to their unique thermo-physical properties. They are crucial to the industry because they enhance surface performance, reduce maintenance and fuel costs and boost engine efficiency. They are also used in mechanical systems to efficiently minimize friction and surplus heat. The present investigation focuses to analyze Darcy–Forchheimer flow of magnetized ZnO-SAE50 nanolubricant across Riga plate. Riga plate is poised of an electromagnetic equator that consists of an everlasting magnet and a span-wise collection of irregular electrodes mounted over a planner surface and is utilized in achieving proficient flow. The flow takes place in the presence of viscous dissipation, heat source–sink and chemical reaction of higher order. Newtonian heating and thermophoretic particle deposition effects are also investigated in this study. The role of viscosity is quite important and dynamical in various engineering and industrial fields. The viscosity variation highly fluctuates the thermal systems. The present work also highlights the role of variable viscosity, owing to its importance. A novel micro-nano-convection model called the Patel model is invoked in perspective of the enhancement in thermal conductivity of nanolubricant. The use of ZnO-SAE50 nanolubricant (a brand-new form of nanolubricant) over Riga plate, variable viscosity effects and the application of Patel model are novel features of this study. The application of selected transformations causes the modeled PDEs to change into ODEs. The modeled problem is executed numerically with MATLAB bvp-4c solver. The alterations in the concerned profiles corresponding to various emerging parameters of interest are presented graphically in order to develop better understanding and make analysis. The outcomes reveal that larger modified Hartmann number significantly favors the velocity of nanolubricant ZnO-SAE50, while solid volume fraction, magnetic parameter and variable viscosity parameter halt it. The heat source–sink parameter, Eckert number and solid volume fraction are found to be helpful agents in improving temperature of nanolubricant ZnO-SAE50. In comparison with Newtonian heating (NH), common wall temperature condition (CWT) yields better thermal enhancement. The enhancing chemical reaction order ameliorates the concentration profile but it gets minified for chemical reaction parameter and thermophoresis parameter. The increasing values of magnetic parameter, variable viscosity parameter and inertial parameter tend to enhance the skin friction, while the modified Hartman number reduces it. The enhancing values of Eckert number, magnetic parameter and heat source–sink parameter increase the value of Nusselt number. [ABSTRACT FROM AUTHOR]

This research paper examines the Darcy–Forchheimer flow of second-grade hybrid nanofluid with thermophoretic particle deposition on a solar collector plate in a porous media. This study performs an extensive exploration of entropy generation. Solar collector plates play a crucial role in energy storage in solar power plants. They help to store and regulate energy at extreme temperatures. This work analyzes the performance of a solar collector plate when the conventional fluid of Ethylene Glycol (EG) is reciprocated by nanoparticles of zirconium dioxide and copper. The ramifications of Magntohydrodynamic (MHD) and Cattaneo–Christov heat and mass flux are also investigated. The expressions of mass and energy are generated by using the Cattaneo–Christov model of heat and mass flux. The Homotopy analysis method (HAM) is utilized to achieve the results of differential equations against various dimensionless parameters. The fluctuating behavior of velocity, concentration and temperature profiles is discussed graphically in this paper. Furthermore, tables are included for the numerical values of skin friction, Sherwood number and Nusselt number for several parameters. As the value of the Darcy parameter raises, the fluid's velocity distribution continuously reduces. The temperature distribution reduces along with the greater values of the thermal relaxation parameter. The concentration profile has shown decreasing impact due to the increasing value of the concentration relaxation parameter. [ABSTRACT FROM AUTHOR]

Abstract The present analysis aims to compute the non-similar solution of a chemically reactive Prandtl nanofluid across an elongated surface in a permeable medium, with a uniform magnetic field applied perpendicular to the surface. The Buongiorno model is used to study the random motion and thermophoresis of the nanoliquid by considering factors such as viscous dissipation, Joule heating, and multiple slips. By using appropriate transformations and a non-similarity approach, the governing flow equations are scaled down to the second level and numerically analyzed using the MATLAB bvp4c algorithm. The profound effects of the dimensionless parameters are illustrated graphically for the velocity, temperature, and solutal profiles. The impact of drag force, thermal, and solutal transmission at the interface of the deformable sheet are inspected in tabulated form. The results of this study revealed that higher Prandtl and elastic parameters lead to an increase in fluid velocity but a rise in thermal slip parameter results in the opposite behavior. An increase in Hartmann number and porosity parameters upsurges drag force, whereas an increase in the slip parameter reduces it. The research is validated by comparing it to previous studies, and a strong correlation is observed. It is examined that the percentage (%) error with both comparative papers is between −0.0017181 and 0.0006.377 which is almost negligible. Thus, confirming the reliability and precision of the formulated problem. [ABSTRACT FROM AUTHOR]

Stagnation point flow defines the fluid motion near the stagnation point of a solid surface and has utilization in the process of polymer extrusion, electronic gadgets cooling, nuclear reactor, etc. Furthermore, the main theme of this work is to analyze the heat transfer characteristics of stagnation point flow of Casson nanofluid over stretching sheet using induced magnetic field and Cattaneo-Christov model. The Casson fluid is a non-Newtonian fluid model; and in the current work the working fluid consists of nanoparticles of Au (Gold) and blood (Casson fluid). In the existing model, temperature of the surface depends on the distance from the stagnation point. Therefore, Darcy-Forchheimer porous model is incorporated. The solution of coupled nonlinear PDEs is obtained by bvp4c. The heat transfer and flow characteristics of Au-blood based Casson nanofluid are discussed graphically. The dual behavior is observed in the profiles of induced magnetic field with increase in the value of reciprocal magnetic Prandtl number, however the velocity outlines of Au-blood nanofluid declined. A comparison is presented with previous article and is found to be in good agreement. [ABSTRACT FROM AUTHOR]