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This paper analyzes the thermodynamic second law analysis of mixed convection heat transfer of Carboxymethyl Cellulose (CMC)-water based viscoelastic hybrid nanofluid flow in a vertical parallel plate channel filled with porous medium. In mixed convection flow, where both buoyancy and viscous forces play significant roles in the flow behavior, the inclusion of viscous dissipation in the analysis is crucial. The governing equations of the problem are converted into a system of ordinary differential equations using appropriate similarity transformations, which are then solved by the Homotopy Analysis Method (HAM). The behavior of non-dimensional velocity, temperature, skin friction coefficient, Nusselt number, entropy generation and Bejan number profiles for a range of pertinent flow parameter values is displayed graphically and deliberated. Study reveals that a decrease in the dominance of viscous forces relative to inertial forces within the porous medium lowers entropy generation in the system. [ABSTRACT FROM AUTHOR]

In this analysis, the collective effects of rotation, viscous dissipation and vertical throughflow on the onset of convective movement in Jeffrey fluid saturated permeable layer is studied. The improved Darcy model is applied to depict the rheological performance of Jeffrey fluid flow in porous medium. The approximate analytical solution with overall error 0.4 % and numerical solution accurate to one decimal place are presented using the Galerkin process. The analysis reveals that the convective motion concentrates in the top layer if it occurred with sufficiently high value of the Darcy–Eckert number. The rotation factor and the Péclet number postponement the onset of convective drive while, the Gebhart number quicken it weakly. In the occurrence of rotation, the Jeffrey factor displays dual impact on the coming of convective movement. The magnitude of the convection cell declines with increasing the rotation factor, the Jeffrey factor and the Péclet number, while it decreases with enhancing the Gebhart number. It is also found that in the lack of rotation, the Jeffrey factor has no impression on the extent of the convective cell, whereas in the nonexistence of the Péclet number, the Gebhart number has no impact on the arrival of convective drive as well as on the magnitude of the convective cells. [ABSTRACT FROM AUTHOR]

This paper focuses on flow structures and thermal fields of the Carreau–Yasuda (CY) nanofluid model through a two-dimensional, wavy, complicated vertical asymmetrical conduit filled with porous elements. Formulations of the viscous dissipation in the case of CY nanofluids are derived and nonlinear radiation flux as well as joule heating are examined. Buongiorno's nanofluid approach, which involves Brownian motion and thermophoresis aspects is considered. The electrical conductivity of the suspension is considered as a variable where it depends upon the ambient temperature and concentration distributions and the Joule heating impacts are not neglected. The approach of solving the problem is contingent upon converting the system to dimensionless form then the lubrication approach with low magnetic Reynold numbers is applied. Numerical solutions are found for the resultant system, and wide ranges are considered for Weissenberg number We, non-Newtonian parameter n and Darcy number D a , namely, 0 ≤ We ≤ 2 , − 0. 5 ≤ n ≤ 1. 5 and 0 ≤ D a ≤ 1. 6 , respectively. The major results indicated that gradients of the pressure are higher in case of shear thickening (n > 1) comparing to in the instance of shear thinning (n < 1). Also, the velocity is enhanced, close to the channel's lowest portion, as the Weissenberg number is growing. The variable electrical conductivity gives a higher mass transfer rate compared to the constant property. [ABSTRACT FROM AUTHOR]

This paper aims at developing non-similar solutions for heat transfer augmentation in ternary hybrid nanofluids. Nanofluid is composed of three distinct (Silver, Copper, Aluminum oxide) nanosize particles while water is considered as a base fluid. Darcy–Forchheimer expression with variable porosity and permeability is adopted. Joule heating and viscous dissipations are also considered. Non-similar approach is utilized. Numerical solutions are computed by bvp4c solver of MATLAB. Graphical illustrations for flow and thermal fields behavior are provided. Comparative results are obtained for ternary hybrid nanofluid and nanoliquid. Physical quantities such as skin drag coefficient and Nusselt number are computed and interpreted. Our results reveal that rate of heat transfer augments substantially for Ag/water nanofluid in comparison to other classes of nanofluid. [ABSTRACT FROM AUTHOR]

In this analysis, we have considered heat transmission in two-dimensional steady laminar nanofluid flow past a wedge. Magnetohydrodynamic (MHD), Brownian motion, viscous dissipation and thermophoresis effects are considered over the porous surface. Similarity transformations have been used to change the governing partial differential equations (PDEs) into nonlinear higher-order ordinary differential equations (ODEs). Governing ODEs with boundary conditions are then converted to the system of first-order initial value problem. After that the modeled system is solved numerically by RK4 technique. Impact of the magnetic number, Eckert number, Prandtl number, Lewis number, Brownian motion, thermophoresis and permeability parameters on the flow domain is analyzed graphically as well as in tabular form. It is noted that magnitude of Nusselt number for the flow regime increases with the increase of nondimensional parameter P r , N b , N t while opposite behavior is observed in case of R. [ABSTRACT FROM AUTHOR]

Purpose: This investigation delves into the rationale behind the preferential applicability of the non-Newtonian nanofluid model over alternative frameworks, particularly those incorporating porous medium considerations. The study focuses on analyzing the mass and heat transfer characteristics inherent in the Williamson nanofluid's non-Newtonian flow over a stretched sheet, accounting for influences such as chemical reactions, viscous dissipation, magnetic field and slip velocity. Emphasis is placed on scenarios where the properties of the Williamson nanofluid, including thermal conductivity and viscosity, exhibit temperature-dependent variations. Design/methodology/approach: Following the use of the OHAM approach, an analytical resolution to the proposed issue is provided. The findings are elucidated through the construction of graphical representations, illustrating the impact of diverse physical parameters on temperature, velocity and concentration profiles. Findings: Remarkably, it is discerned that the magnetic field, viscous dissipation phenomena and slip velocity assumption significantly influence the heat and mass transmission processes. Numerical and theoretical outcomes exhibit a noteworthy level of qualitative concurrence, underscoring the robustness and reliability of the non-Newtonian nanofluid model in capturing the intricacies of the studied phenomena. Originality/value: Available studies show that no work on the Williamson model is conducted by considering viscous dissipation and the MHD effect past over an exponentially stretched porous sheet. This contribution fills this gap. [ABSTRACT FROM AUTHOR]

This work analyzes the impact of viscous dissipation on an unstable stretching surface in a porous medium with heat generation and thermal radiation—an important factor for numerous engineering applications like cooling baths and plastic sheets. Using MATLAB's Runge-Kutta fourth-order approach, the controlling partial differential equations are converted into highly nonlinear ordinary differential equations that can be solved numerically. The findings show that a decrease in the skin friction coefficient, temperature profiles, velocity, and Nusselt number occurs when the unsteadiness parameter is increased. In contrast to the Prandtl number, which rises with temperature profile and reduced Nusselt number, the Eckert number rises with a dimensionless temperature profile and reduced Nusselt number. Reduced Nusselt number and temperature profile affect the heat generation parameter; a decrease in skin friction coefficient and velocity profile correlate with the porosity parameter. Furthermore, the radiation parameter rises as the temperature distribution and Nusselt number decrease.

The current study investigates the three-dimensional radiative and convective Casson hybrid nanofluid flow and heat transfer with the Cattaneo–Christov heat flux model over an inclined spinning and extending disk subjected to an applied magnetic field. Additionally, the study considers the impacts of Joule’s heating and viscous dissipation. Mathematical modelling of the nanofluid flow problem containing Ag and multiwalled carbon nanotubes (MWCNT) nanoparticles with water as the base fluid in a Darcy medium is done using a cylindrical coordinate system. The simplified system of equations is subjected to the spectral quasilinearisation method (SQLM) approach for the graphical and tabular representations. Examining key parameters, such as magnetic field, Bejan number, angle of inclination, disk movement parameter and disk rotation reveals interesting results on velocity and temperature profiles. The research concludes that the Bejan number increases with higher values of temperature ratio, radiation and magnetic parameters, while it decreases with increasing Casson parameter and Brinkman number. Radial wall friction decreases with improved magnetic field, temperature ratio, stretching and porosity parameters, but tangential wall friction increases. The present results are compared with the one already existing in literature to validate the numerical scheme and the results are found to agree well with the previously published work. The application of hybrid nanofluid flow over rotating and stretching disks is widespread in various fields, including rotating machinery, electronic devices, patient treatment instruments, crystal growth method, etc. [ABSTRACT FROM AUTHOR]

The idea of a hybrid nanofluid (HNF) has sparked curiosity among many scientists because of its ability to enhance thermal characteristics, leading to elevated rates of heat transfer (HT). These HNFs are utilized in various engineering and industrial settings, such as electronics cooling, manufacturing, naval structures, biomedical applications, and drug delivery. The current study investigates the analysis of irreversibility in EMHD [Cu + TiO2/H2O]h flow over a stretching sheet with radiation and viscous dissipation. The governing PDEs are converted into ODEs using similarity variables. These ODEs are then solved using the RKF method along with a shooting technique. The effects of different physical parameters on the velocity and temperature distributions of the HNF, as well as on HT and surface drag force, are thoroughly examined and presented in graphs. The velocity of [TiO2/water]n flow declines as the magnetic field strength rises, but it rises with greater electric field values for [Cu + TiO2/water]h. The temperature of the [Cu + TiO2/water]h increases with elevated levels of radiation, Eckert number, and heat generation strength. Higher Reynolds and Brinkman numbers result in a rise in entropy generation for [Cu + TiO2/H2O]h, whereas the Bejan number decreases to the same extent. The HT rate in [Cu + TiO2/H2O]h increases by 3.05% as the Eckert number rises, while it drops by 4.01% when there is significant thermal radiation. Skin friction reduces by 3.21% in [TiO2/water]n as the electric field strength increases, whereas it decreases by 4.05% with an increase in magnetic field strength. These discoveries offer valuable perspectives on furthering the utilization of HNFs in engineering and industrial operations. [ABSTRACT FROM AUTHOR]

Entropy measures the disorderness and randomness in the thermal systems. It has significant influence over efficiency and performances of the thermal systems. The motive of the research paper is to present a comparative analysis of entropy generation of a heat dissipative Darcy–Forchheimer flow of copper (Cu/H2O)-based mono and (CuAl2O3/H2O)-based hybrid nanofluid under the influence of thermal dispersion. The mathematical model of the conceptualized flow problem is formulated using single phase nanofluid model along with Darcy–Forchheimer equation for non-Darcy porous medium flow. The system of dimensional Partial Differential Equation (PDE) depicting the flow problem is converted in the system of dimensionless Ordinary Differential Equation (ODE) using the suitable similarity variables and has been solved by MATLAB’s bvp4c package. The flow variables, engineering parameters like skin friction and Nusselt number along with entropy generation, have been analyzed for the active parameters inherited in the problem. The findings suggest that heat transfer rate on the surface enhances with the increment in thermal dispersion parameter. Further, it is reported that the hybrid nanofluid generates lesser entropy as compared to the mono-nanofluid. This research has potential to serve the real-life applications based on electronics and geothermal systems. [ABSTRACT FROM AUTHOR]

The present work examines the convective Casson fluid flow over a stretching sheet with viscous dissipation and porous medium. The impact of thermal and mass transport is examined in terms of Soret and Dufour effects. The flow behaviour and heat transfer are investigated by subjecting the fluid to an induced magnetic field and suction/injection. The governing equations describing the flow problem are formulated using partial differential equations (PDEs). Similarity transformations are utilized to transform the governing equations into a system of ordinary differential equations (ODEs). Following that, the MATLAB solver bvp5c is utilized to grasp the results. Graphs serve as a function to display the consequences of flow on attributed parameters. Furthermore, the engineering properties such as skin friction, Nusselt number, and Sherwood number are represented through tables and graphs. Besides, the results of the present study are validated with the existing results, whereas a fine correspondence has been noticed. The findings show that the velocity of the fluid decreases by 8%, when the parameter of the induced magnetic field rises from 0.1 to 0.2. Similarly, a 3% decrement in the velocity profile is observed, while increasing the suction/injection parameter from 1.2 to 1.4. The temperature of the fluid is escalated for the increasing values of the Eckert number. Our analysis indicates a positive correlation between the Nusselt number and increasing Dufour number values. Similarly, the Sherwood number exhibits enhancement with a rise in the chemical reaction parameter. This combination has some applications in designing cooling systems for microfluidic devices, optimizing drag reduction in magnetohydrodynamics (MHD) devices, and developing novel separation processes. [ABSTRACT FROM AUTHOR]

Our present study intends to examine the unsteady thin film flow and heat transfer of Ag - H 2 O nanofluid on a stretching sheet embedded in a porous medium. Radiative heat, Ohmic heating with slip, and convective boundary conditions are taken into account. The effect of various nanoparticle shape factors is also investigated. Nanofluid thermal conductivity depends on nanoparticle shape. The primary time-dependent equations are transformed using similarity transformation and solved using a prominent Runge–Kutta fourth-order and shooting iterative process. Graphs and numerical values of physical factors pertinent to fluid flow are obtained using MATLAB software. The skin friction coefficient and the heat transfer rate are also investigated and tabulated. Results reveal that magnetic, porosity, and unsteady parameters all have a diminishing impact on velocity profiles. Radiation parameter has a positive impact on temperature distribution, while an opposite trend is observed for magnetic fields. Findings indicate that magnetic field and porosity enhance the skin friction coefficient, whereas heat transfer rate increases with Biot number and slip parameter. Platelet-shaped nanoparticles are shown to be the most efficient in heat transfer. The outcomes were compared to previously reported results and found excellent agreement. [ABSTRACT FROM AUTHOR]

This work analyzes the impact of viscous dissipation on an unstable stretching surface in a porous medium with heat generation and thermal radiation-an important factor for numerous engineering applications like cooling baths and plastic sheets. Using MATLAB's Runge-Kutta fourth-order approach, the controlling partial differential equations are converted into highly nonlinear ordinary differential equations that can be solved numerically. The findings show that a decrease in the skin friction coefficient, temperature profiles, velocity, and Nusselt number occurs when the unsteadiness parameter is increased. In contrast to the Prandtl number, which rises with temperature profile and reduced Nusselt number, the Eckert number rises with a dimensionless temperature profile and reduced Nusselt number. Reduced Nusselt number and temperature profile affect the heat generation parameter; a decrease in skin friction coefficient and velocity profile correlate with the porosity parameter. Furthermore, the radiation parameter rises as the temperature distribution and Nusselt number decrease. [ABSTRACT FROM AUTHOR]

The influence of conjugate heat transfer based on radiation and magnetohydrodynamic utilization is an attractive research area, with progressive features; it has many applications in thermal engineering, heat exchangers, cooling phenomenon, magnetic cell separation, energy production, hyperthermia, etc. Following the motivating significances of the current research topic, this paper explores the influences of an electrically conducting and radiating fluid with internal friction, heat generation and thermal radiation embedded in a porous medium past a flat permeable plate. The boundary layer equations are dimensionally transformed and solved numerically using implicit finite difference technique called the Keller-box method. The effect of various fluid obeying parameters such as magnetic field, viscous dissipation and heat generation on the flow factors such as velocity and temperature are graphed and discussed in this paper. Growing heat source, energy inside the fluid boosts, thereby increasing the energy of the flow. Increase in generation of energy reduces the viscosity of the flow reduces thereby increasing the velocity of the flow particles. [ABSTRACT FROM AUTHOR]

This work inspects 3D magnetohydrodynamic hybrid nanofluid flow on a permeable elongating surface. The emphasis of this paper is on the study of hybrid nanofluid flow within a rotating frame, taking into account the simultaneous impact of both thermal and velocity slip boundary conditions. The chosen base fluid is water, and the hybrid nanofluid comprises two nanoparticles Cu\text{Cu} and Al2O3{\text{Al}}_{2}{\text{O}}_{3}. The effect of the magnetic and porosity parameters is taken into account in the momentum equation. The thermal radiation, Joule heating, and heat source are considered in the energy equation. Using a similarity system, we transform the PDEs of the proposed model into ODEs, which are then solved numerically by the bvp4c technique. The magnetic field shows a dual nature on primary and secondary velocities. Enrich magnetic field decreases the primary velocity and enhances the secondary velocity. The rotation parameter has an inverse relation with both velocities. The temperature profile amplified with the escalation in heat source, magnetic field, rotation factor, and Eckert numbers. The skin friction is boosted with magnetic parameters while the Nusselt number drops.

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.

This study encapsulated the research methodology utilized in the flow behaviors of Williamson nanofluid and analyzed the associated mass heat transfer. The study concentrated on examining the magnetohydrodynamic behavior of nanofluids in the presence of heat generation effects and the inclusion of dissipative energy on a vertical nonlinear stretching surface submerged within a Darcy porous medium. The rationale for including variable viscosity and variable conductivity in this research was to precisely evaluate the mechanisms of heat and mass transfer, particularly with regard to the fluctuations in fluid properties. The objective was to enhance the understanding of how these varying properties impact the overall heat and mass transfer processes. The initial formulation of the phenomenon, initially presented as partial differential equations, was transformed into ordinary differential equations by employing appropriate dimensionless variables. The ultimate streamlined version of the model was then numerically solved utilizing the shooting method. By employing the numerical shooting method, we portrayed nanofluid patterns in velocity, temperature, and concentration fields, alongside essential parameters such as skin friction coefficient, Sherwood number, and Nusselt number. The significant key findings highlighted that both the porous parameter and the magnetic number increasingly affected temperature and concentration distributions. Additionally, increasing the thermophoresis parameter resulted in higher concentration and corresponding temperature levels. Graphical presentation and physical explanations were used for analysis, and the study's outcomes were compared to existing literature, affirming a strong agreement that validated the solutions.

This study encapsulated the research methodology utilized in the flow behaviors of Williamson nanofluid and analyzed the associated mass heat transfer. The study concentrated on examining the magnetohydrodynamic behavior of nanofluids in the presence of heat generation effects and the inclusion of dissipative energy on a vertical nonlinear stretching surface submerged within a Darcy porous medium. The rationale for including variable viscosity and variable conductivity in this research was to precisely evaluate the mechanisms of heat and mass transfer, particularly with regard to the fluctuations in fluid properties. The objective was to enhance the understanding of how these varying properties impact the overall heat and mass transfer processes. The initial formulation of the phenomenon, initially presented as partial differential equations, was transformed into ordinary differential equations by employing appropriate dimensionless variables. The ultimate streamlined version of the model was then numerically solved utilizing the shooting method. By employing the numerical shooting method, we portrayed nanofluid patterns in velocity, temperature, and concentration fields, alongside essential parameters such as skin friction coefficient, Sherwood number, and Nusselt number. The significant key findings highlighted that both the porous parameter and the magnetic number increasingly affected temperature and concentration distributions. Additionally, increasing the thermophoresis parameter resulted in higher concentration and corresponding temperature levels. Graphical presentation and physical explanations were used for analysis, and the study's outcomes were compared to existing literature, affirming a strong agreement that validated the solutions. [ABSTRACT FROM AUTHOR]

This investigation explores the convective MHD stagnation point flow of Casson nanofluid over a stretching sheet in a porous medium with higher-order chemical reactions and multiple slips. The examination of heat transmission is carried out in the presence of radiation, Joule heating, viscous dissipation, and heat source. The system of governing equations was simplified by using appropriate transformations. The transformed equations are tackled numerically by a Runge-Kutta-based shooting technique with the bvp5c MATLAB package. The graphical and numerical results for different parametric values are discussed and presented through figures and tables. It is observed that an increase in the magnetic field, porosity, and Casson fluid parameter reduces the velocity whereas the opposite trend is seen in the case of the thermal Grashof number and the solutal Grashof number. Increasing values of radiation and Joule heating parameters, and the Eckert number, lead to an increase in the temperature. The chemical reaction, suction, and solutal slip were found to reduce the concentration. The friction factor was reduced due to the higher values of the thermal Grashof number, solutal Grashof number, and the velocity ratio parameter. Also, an increase in the Brownian motion and thermal slip parameters decreases the heat transfer rate. The computational results of the Nusselt number have been validated with published results in the literature and found good agreement. The results obtained in this investigation have applications to biomedical, engineering, and industrial sectors such as food processing, polymer manufacturing, glass, and fiber production, improving oil recovery, and material processing. [ABSTRACT FROM AUTHOR]

This present research is focused on unsteady MHD (Magnetohydrodynamics) nanofluid flow past a vertically accelerated plate through a porous medium in the existence of heat source and radiation impacts. All three ingredients are thought to be nanofluids based on water that can conduct electricity. The effects of electromagnetic fields and radiation are considered. The numerical explanations of the governing equations are achieved using the finite difference method. Graphical demonstrations of the temperature and velocity fields of Ag-water and $Ti{O_2}$TiO2-water are shown. Ag-water has higher velocities than $Ti{O_2}$TiO2-water. Ag-water has a higher temperature than $Ti{O_2}$TiO2-water. When the magnetic force is kept the same between the plate and the fluid, the velocity curves are found to be a lot higher than when it is kept the same between the plate and the fluid. [ABSTRACT FROM AUTHOR]

Due to dynamical applications of nanoparticles in various engineering and biomedical applications, our work focuses on examining and exploring the numerical study of bioconvected unsteady Williamson fluid flow at a heated permeable stretched sheet in a porous medium, along with the existence of particular Cattaneo–Christov heat and mass flux, viscous dissipation, activation energy, Brownian and thermophoresis motion. A mathematical model has been developed by considering the effects of the aforementioned terminologies on the flow field which are expected to be similar to the physical behavior and already published work. By using suitable similarity transformations, the system of the flow field is transformed from partial differential equations to ordinary differential equations. The developed differential system is then solved numerically through bvp4c in the computational software MATLAB. The tabular and graphical analysis have been presented against velocity, temperature, concentration, and density profiles that imply the impact of physical parameters. The skin friction coefficient, rate of heat, and mass transfer are also examined. The results coincide very well to previous published work for limiting cases that authenticate the validity of the current work. It was observed that upon increasing magnetic field (M) and porosity (Kp) parameters, the velocity profile decelerates but the opposite behavior was seen for motile density, concentration, and temperature profiles. The temperature and concentration of Williamson nanofluid reduced by enhancing the significant thermal and mass stratification. The concentration profile declines for rising values of Schmidt number (Sc), the chemical reaction rate (σ), and temperature difference (δ) parameters but accelerates for higher values of Activation energy (Ea). Also, by increasing Peclet number (Pe), bioconvection Lewis number (Lb), and microorganism concentration difference parameter Ω motile density profile decreases. Analyze the impacts of bioconvection and activation energy on Williamson nanofluid. We numerically imitate the outcomes of viscous dissipation, bioconvection, and activation energy at a heated permeable stretched sheet. We examine the effect of A, M, We, Kp, and S on − 1 2 C f (R e) 1 2 . We examine the effect of M, Kp, and We on S h x (R e x) − 1 2 . We examine the effect of the local density number with M, Kp, We, A, and injection and suction parameters. [ABSTRACT FROM AUTHOR]

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.

The aim of the present research is to discuss the numerical aspects of heat-mass transfer in power-law nanofluids on an stretched surface. In addition, the novelty in this research lies in its thorough exploration and incorporation of parameters such as viscous dissipation, slip velocity, and convective boundary conditions into the analysis. This distinguishes the study from previous work and underscores its originality. For non-Newtonian fluids, a power-law model is employed, while the nanofluid system associate the influences of thermophoresis and the Brownian motion. The fluid’s thermal conductivity is considered to change based on temperature, while the concentration of nanoparticles at the surface is maintained at a constant level. A heated fluid situated beneath the lower surface can act as a heat convection mechanism source. A process of similarity transformation is employed to simplify the equations related to the mass, momentum, thermal energy, and nanoparticle concentration into nonlinear ordinary differential equations. These equations are then treated numerically with the help of the shifted Chebyshev polynomials of the sixth order and the spectral collocation method. The proposed technique reduces the existing problem into a system of algebraic equations formulated as a constrained optimization challenge. Subsequently, the optimization technique is applied to determine the unknown coefficients of the series solution. Graphical representations depict the impacts of nanofluid parameters. A quantitative assessment is presented in a tabular format to illustrate a comparison with previously published results for specific scenarios, revealing a notable level of agreement.

An incompressible unsteady free convective viscous MHD rotating flow past a moving plate embedded in a porous medium is considered with the influence of viscous dissipation, heat source effects. It is assumed that the flow rotates with angular velocity which is normal to the plate and also that a transverse magnetic field is applied along the normal to the plate. Appropriate dimensionless quantities are applied to change the governing equations into dimensionless form. Then the equations are solved numerically using the Galerkin finite element method. Some important characteristics of the fluid are studied. The results are in good agreement with the available literature.

This article investigates the impact of time-dependent magnetohydrodynamics free convection flow of a nanofluid over a non-linear stretching sheet immersed in a porous medium. The combination of water as a base fluid and two different types of nanoparticles, namely aluminum oxide (Al2O3) and copper (Cu) is taken into account. The impacts of thermal radiation, viscous dissipation and heat source/sink are examined. The governing coupled non-linear partial differential equations are reduced to ordinary differential equations using suitable similarity transformations. The solutions of the principal equations are computed in closed form by applying the MATLAB bvp4c method. The velocity and temperature profiles, as well as the skin friction coefficient and Nusselt number, are discussed through graphs and tables for various flow parameters. The current simulations are suitable for the thermal flow processing of magnetic nanomaterials in the chemical engineering and metallurgy industries. From the results, it is noticed that the results of copper nanofluid have a better impact than those of aluminium nanofluid. [ABSTRACT FROM AUTHOR]

The main aim of the present study is the heat transfer analysis of MHD Eyring-Powell flow over a stratified stretching sheet immersed in a porous material. Mixed convection as well as viscous dissipation effects are considered in order to observe the heat transfer analysis. To strengthen the energy equation viscous dissipation effect is incorporated in this study. Here, temperature distribution is carefully examined to assess rate of heat transmission for the present consideration. Present consideration can have many applications in various engineering fields. Later, the most appropriate similarity transformations are utilized to transform the governing partial differential equations into a nonlinear set of ordinary differential equations. Thereafter, a powerful and convergent procedure namely Runge-Kutta-Fehlberg procedure together with shooting technique is applied to obtain numerical solution for the condensed problem. Graphs are drawn for various values of flow controlling parameters to observe clear insight of the present study through velocity, temperature and concentration profiles. Numerical solutions are tabulated for comparison purpose. The main observations indicate that the temperature and velocity of the fluid decrease over a stretched sheet as the Eyring-Powell fluid material parameter increases. Moreover, mixed convection exhibits a significant impact on the present study, as the parameter value increases. correspondingly, velocity increases, but temperature shows a reverse nature. [ABSTRACT FROM AUTHOR]

The studies of a first-order chemical reaction have been presented in various published investigations, while the nth chemical reaction should be examined. Also, optimizing the effect of process variables is important for a clear understanding of the heat transfer mechanism. Therefore, this paper presents an optimization and sensitivity analysis for the heat transfer and shear rate utilizing response surface methodology (RSM) for dissipative-magneto highly reactive Casson nanoliquid flow with an electric field. The irreversibility property is disclosed, and both active and passively controlled cases are assumed. The governing system is transformed into similar forms and solved using the Blottner technique together with the Finite Difference Method (FDM). The main attraction of the current investigation is to optimize the rate of heat transfer as well as the shear rate by utilizing robust statistical approaches. Further, sensitivity analysis for the significant factors is, also, demonstrated for the proposed responses. The major outcomes revealed that the skin fraction is enhanced as the magnetic parameter is altered, while the electrical coefficient causes a reduction in the velocity gradients. In the active-controlled case, the power index of the chemical reaction reduces the mass transfer rate. [ABSTRACT FROM AUTHOR]

This article describes the Al2O3/Cu/Ag-H2O nanofluid stagnation-point flow through a porous medium with emphasis on heat transfer as well as magnetohydrodynamic (MHD) behavior and entropy generation. The main motivation is to do the MHD flow of nanofluid and look into the repercussions of viscous dissipation, which has many applications in various industries, especially the extrusion process where the durability of the end product is of interest. The governing equations are solved using similarity transformations and DTM-Padé approximations, and the solutions are then compared to the outcomes of the numerical technique (shooting technique and Runge-Kutta 4th order method), which validates the accuracy of the research. The ranges of the parameters are taken as -2< Q < 3, 0 < Gr < 6, 0 < Gc < 6, 0 < Br < 3, 0< ? < 0.3. Some important findings are: entropy generation is to decrease for higher values of Br in a streamline manner within asymptotic pattern; viscous heating is dominant in the neighborhood of the plate for higher value of Br and for small values the distribution is smooth across the flow field; decrease in entropy generation for higher value of Br shows that there is a base value or residue left over energy in the process of entropy generation, which is important in its own revelation and application basis. [ABSTRACT FROM AUTHOR]

A numerical investigation of the natural convection magnetohydrodynamic (MHD) flow of a visco-elastic, incompressible, and electricity-conducting fluid in a medium with porosity is carried out. The novelty of the study is to examine the impact of visco-elasticity on MHD boundary layer flow, which plays a crucial role in engineering and industrial applications. The model differential equations obtained are in a coupled form which are highly non-linear. The non-dimensional partial differential equations are simplified numerically by employing Crank-Nicolson finite difference scheme. The impact of vital parameters of the flow are discussed and presented graphically. Also, the outcomes of the physical parameters, such as skin friction, heat transfer rate, and mass transfer rate, are derived and analyzed through tables. It is observed that the visco-elastic parameter enhances the velocity profiles in the whole flow region. Also, the velocity gradient growths with the higher value of the visco-elastic parameter and the enhanced value of the magnetic parameter diminishes the profile of velocity. [ABSTRACT FROM AUTHOR]

Through the investigation, in this work, we focused at the steady flow of a Casson-Williamson fluid due to an stretchable, impenetrable sheet with Ohmic dissipation. It is assumed that the impermeable stretched sheet is incorporated into a porous media and has a rough surface. The porous media through which the non-Newtonian fluid is flowing are supposed to obey Darcy's law. Magnetic and electric fields' impacts are considered. We investigate how the process of heat transfer is affected by viscous dissipation and varying thermal conductivity. On the basis of a little magnetic Reynolds number, the controlling basic equations are represented by a system of nonlinear ordinary differential equations. The shooting technique is used to get a numerical solution for this system, which controls both the temperature and velocity fields. Graphical representations of the impact of various parameters on the velocity and temperature profiles are shown. Regarding the significant results, we note that the local electric parameter tends to improve both the velocity and temperature fields, while the porous parameter, Casson parameter and slip velocity parameter decrease the velocity profiles. [ABSTRACT FROM AUTHOR]

The main aim of this numerical investigation is to describe the influence of viscous dissipation and magnetic Ohmic heating on the steady-state flow of Casson nanofluid through a porous medium over a stretching sheet under heat source/sink effects. The Brownian motion and thermophoretic effects are included in the energy and concentration equations to describe the nanofluid behavior within the boundary layer regime. The Soret and chemical reaction effects with concentration slip conditions are considered. A standard Casson model has been used to distinguish non-Newtonian flow behavior from those of Newtonian fluids. Motivated by the applications of Casson fluid, the present problem is developed based on the stretching sheet geometry. The nonlinear, coupled, two-dimensional partial differential equations are derived to describe the thermal and flow transport behaviour. Appropriate scaling transformations are used to reduce the dimensional complexity of the produced differential equations. A BVP4C numerical scheme is utilized to produce similarity solutions to the governing equations. Accordingly, it is noted that Casson nanofluid velocity is considerably decreased and the thermal field is increased with increasing values of magnetic number. Thermal transport increases owing to the enhanced Brownian motion parameter. Increasing Casson fluid and Eckert parameters increases the thermal distribution field. Increasing Soret and thermophoretic numbers increases the concentration diffusion field. Finally, the produced BVP4C solutions are validated with the available solutions and found to be in excellent agreement. [ABSTRACT FROM AUTHOR]

The proficiency of hybrid nanoparticles in increasing heat transfer has impressed many researchers to further analyze the working of those fluids. In the current study, the impact of entropy generation on EMHD hybrid nanofluid (copper-alumina) flow over a rotating disk in the presence of the porous medium, Darcy‐Forchheimer, heat generation, viscous dissipation, and thermal radiation. By applying the self-similarity variables, the partial differential equations are converted into ordinary differential equations. After that, the dimensionless equations are numerically solved by using the Runge-Kutta technique, and also the comparison is done between the numerical technique (R-K method) and the homotopy perturbation method (HPM) where HPM yields a more effective and dependable conclusion. To highlight their physical significance, unique characteristic graphs are shown for the profiles of velocity, temperature, and entropy generation, along with a suitable explanation. The hybrid nanofluid velocity decreases with larger values of the magnetic parameter, but the velocity profile increases with the higher electric field. It is observed that both skin friction and nusselt number are increasing function of magnetic parameter and electric field parameter. [ABSTRACT FROM AUTHOR]

A numerical study of the heat and mass transfer of a micropolar nanofluid flow over a stretching sheet embedded in a porous medium is carried out in this investigation. The main objective of this work is to investigate the influence of Arrhenius activation energy, heat source and viscous dissipation on the fluid velocity, microrotation, temperature, and concentration distribution. The equations governing the flow are transformed into ordinary differential equations using appropriate similarity transformations and solved numerically using bvp4c solver in MATLAB. Graphs are plotted to study the influences of important parameters such as magnetic parameter, porosity parameter, thermophoresis parameter, Brownian motion parameter, activation energy parameter and Lewis number on velocity, microrotation, temperature and concentration distribution. The graphical representation explores that the velocity of the liquid diminishes for increasing values of magnetic parameter, whereas the angular velocity increases with it. This study also reports that an enhancement of temperature and concentration distribution is observed for the higher values of activation energy parameter, whereas the Lewis number shows the opposite behavior. The effects of various pertinent parameters are exposed realistically on skin friction coefficient, Nusselt and Sherwood numbers via tables. A comparison with previous work is conducted, and the results show good agreement.

This study is of significant importance as it focuses on exploring the intricate dynamics of viscous dissipation over a layered stretching surface in the context of MHD free and forced convective flow influenced by Powell-Eyring fluid. The primary aim of this research is to analyze the impact of heat emission and immersion phenomena on the flow, and to achieve this, conformable transformations of similarity have been pragmatic to adapt the governing equations from a set of non-linear PDE's into ODE's. The research methodology employed in this study involves numerical analysis using a 4th order Runge-Kutta formulation, implemented through MATLAB. The study precisely examines the stimulus of different flow constraints on velocity and temperature profiles. The outcomes of this investigation are presented through graphical representations, providing a comprehensive view of the flow characteristics. Furthermore, this study goes beyond the basic analysis by presenting important engineering metrics, the Nusselt number and skin-friction coefficient, explicitly across dissimilar variables. In conclusion, the findings of this research reveal as the Eyring-Powell fluid velocity profile rises, the temperature profile decreases, while the temperature profile rises with rising radiation.

Present study is dedicated to analyze the closed form solution of nanofluid flow over a stretching/shrinking sheet with dual availability. Flow is developed through two-dimensional boundary layer theory. Appropriate tensor is used to generate the continuity, energy, and momentum equations. Converted governing partial differential equations (PDEs) into dimensionless non-linear ordinary differential equations (ODEs) by adoption of favorable similarity variables. The dimensionless ODEs of energy and momentum produced a dual nature solution in closed form under certain conditions. To deal with the nanofluid, the Koo-Kleinstreuer and Li (KKL) model is used, and the equations are solved using well-known software Maple. The effect of porosity Φ, suction/injection fw, stretching/shrinking λ, and magnetic effect M on skin friction, velocity, temperature, and streamlines are well explored and showcased. The results for the stable solutions have been showed that the upper branch's fluid velocity is increasing as the magnetic parameter M rises whereas the lower branch's fluid velocity is decreasing as M rises. Additionally, the CuO-nanofluid's velocity is impacted by the volume fraction of nanoparticles, with an increase in volume fraction causing a decrease in velocity. On both the lower and upper branches, the temperature profile is seen to improve as the Biot number increases. On the other hand, as the magnetic parameter varies and the magnetic field increases, the local Nusselt number against suction/injection decreases, as well as the rate of heat transfer in the upper branch decreases.

This study obtains the semi-analytical solutions and explores the thermal characteristics of a thermally developing flow in a parallel plate channel, partially filled with a porous medium at the core, while considering local thermal non-equilibrium conditions, and viscous dissipation for the first time. The developing temperature field, for a porous medium filling of volume fraction fixed at 0.9, indicates an increasing difference in temperature between the solid and fluid phases with the axial distance. Besides, heat flux bifurcation starts in the entrance region as more heat diffuses to the solid at the interface, more markedly with a decreasing Darcy number, Da. When the thermal transport is dominated by interstitial heat transfer between solid and fluid in the porous medium, the thermal entrant length is the shortest. An increasing amount of porous medium filling with such quality and an increasing Da tend to shorten the thermal entry length. Nonetheless, as a lower Da porous media convects more heat away from the interface, it expedites the thermal development in cases where interstitial heat transfer in porous medium is poor. With an increasing porous medium fraction, the impact of viscous dissipation on heat transfer magnifies at the entrance region, thereby reducing the effectiveness of porous medium insert.

In this study, we investigated the heat and mass transport properties of a non-Newtonian Casson-Williamson nanofluid flow. We explore the effect of viscous dissipation and the velocity slip boundary condition on the mechanism of heat and mass transfer due to a stretching sheet which embedded in a porous medium with heat generation under the influence of both thermal radiation and a uniform magnetic field. All physicochemical characteristics of Casson-Williamson nanofluid are considered to be constant. The nanofluid concentration is investigated under chemical repercussions as a result of the movement of the nanofluid particles. This study assumes that there is no suction (solid wall). A set of nonlinear partial differential equations with boundary conditions are used to mathematically model this physical problem. The numerical solution for the differential equations with the related boundary conditions was illuminated using the Runge–Kutta approach in conjunction with the shooting technique. The numerical examination is then pictorial displayed to show the impact of various governing factors on velocity, temperature, and concentration. The non-Newtonian nanofluid has a faster velocity in the absence of a magnetic field than in the presence of it, although the temperature field has the opposite trend. Further, the skin-friction coefficient increased as the porosity parameter increased, whereas the rate of heat transfer dropped.

Objective: The objective of current study is to discuss the effects of the Soret and Dufour with radiation absorption, applied heat source and viscous dissipation on an unsteady MHD mixed convective flow with velocity slip condition across a semi-infinite vertical permeable plate in porous medium. Method: A similarity transformation is used to turn the governing partial differential equations with proper boundary conditions into coupled, non-linear ordinary differential equations with variable coefficients. The inbuilt MATLAB solver bvp4c is used to generate numerical solutions. Findings: The effects on momentum, thermal and solutal boundary layers for various parametric values are graphically depicted. Skin friction, Nusselt number and Sherwood number are all tabulated and discussed in detail. An improvement in radiation absorption corresponds to enhancement of the heat transfer rate up to 59% while leading to a decline in mass transfer rate around 20%. The momentum, thermal and solutal boundary layers are all found to be boosted when the Soret effect is higher. For higher estimation of slip effect, the skin friction is found to decay around 23%. Also, as more time goes by the thermal and concentration boundary layers are enhanced. Novelty: Results obtained in this studied has also been compared and verified with available scientific literature and is found to be in good agreement, which establishes assurance in the numerical results reported in the study.

An incompressible unsteady free convective viscous MHD rotating flow past a moving plate embedded in a porous medium is considered with the influence of viscous dissipation, heat source effects. It is assumed that the flow rotates with angular velocity which is normal to the plate and also that a transverse magnetic field is applied along the normal to the plate. Appropriate dimensionless quantities are applied to change the governing equations into dimensionless form. Then the equations are solved numerically using the Galerkin finite element method. Some important characteristics of the fluid are studied. The results are in good agreement with the available literature. [ABSTRACT FROM AUTHOR]

This article discusses the impact of chemical reaction and radiation on an unstable two‐dimensional laminar flow around a viscous fluid over a semi‐infinite, vertical absorbent surface that moves progressively. The governing classification of partial differentiation was converted into an ordinary differentiation system in this case. To get numerical solutions, the Galerkin finite element technique is applied to nondimensional velocity, micro‐rotation, temperature, and concentration profiles. The consequences of skin friction, the combined pressure quantity, the mass, and heat assignments at the boundary are formed using different fluid properties and flow conditions. Physical quantities and their effects Graphs depict the radiation parameter R, thermal conductivity k, Eckert number Ec, and other velocities, micro‐rotation, temperature, and concentration factors. The main findings of this current problem is showing the chemical reaction effects on velocity and concentration. It is observed that both the velocity and concentration of the fluid decrease when Kr increases. [ABSTRACT FROM AUTHOR]

A theoretical study is conducted in order to scrutinize the thermodynamic first law of the MHD nanofluid flow with an inclined magnetic field, radiation, heat source/sink, viscous dissipation, Joule heating, concentration power-law exponent, and the chemical reaction on a porous stretching surface immersed within a permeable Darcian medium. Cobalt ferrite nanoparticles (CoFe2O4) have been combined with pure water to form a nanofluid called CoFe2O4/H2O. The controlling mathematical equations for MHD nanofluid flow are transformed through similarity transformation into non-dimensional equations. The exact solutions for the energy and mass transfer equations are provided in terms of confluent hypergeometric function. The effects of controlling parameters on the velocity, temperature, and concentration profiles are discussed and illustrated with figures and tables. According to the results, increasing the concentration power-law exponent yields a rise in the Sherwood number (PSC) magnitude and the wall concentration (PMF). Furthermore, the 3-D plots showed that the skin friction coefficient is directly related to the Hartmann number, suction parameter, and nanoparticle volume fraction parameter.

A numerical analysis of the incompressible two-dimensional flow of a non-Newtonian Williamson fluid is offered by expanding the sheet embedded in a porous medium and combining it with the Cattaneo-Christov model. Additionally, it is considered that the thermal conductivity and fluid viscosity both change as a linear function of temperature and an exponential function, respectively. The velocity, temperature and concentration field are all affected by thermal radiation, viscous dissipation, fluid variable properties, chemical reactions, and the slip velocity phenomenon. When the appropriate variables are employed, a system of non-linear, non-dimensional parameters emerges. The shooting method is used to numerically address this system. To better comprehend the impact of dimensionless parameters on dimensionless velocity, concentration, and temperature profiles, physical descriptions are prepared and justified using graphical representations. The values of the local skin-friction coefficient, the rate of heat transfer, and the rate of mass transfer are also investigated using tables. The behavior of changing fluid properties, on the other hand, establishes the link between Williamson fluid flow and the rate of heat mass transfer. According to the results, increasing the slip velocity and viscosity factors lowers both the Nusselt number and the Sherwood number. Also, due to an increase in Deborah number and the chemical reaction parameter, the temperature profiles decrease.

This computational work explores the heat and mass transfer of copper water nanofluid flowing along an inclined plate with varying surface temperature and concentration in the presence of a magnetic field and radiation through a permeable medium. The dimensionless governing equations are solved numerically using an efficient finite-difference technique, which is fast convergent and unconditionally stable. The findings are reviewed and illustrated through graphs for pertinent parameters.