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The impact of the double‐diffusive natural convection of water in a square enclosure with diffusion‐thermo and thermo‐diffusion around the non‐Darcy porous medium is studied numerically. The top and bottom side walls of the cavity are maintained at constant temperatures, while the vertical left and right walls are considered to be cold. Water is considered the working fluid. The resulting nondimensional partial differential equation is solved by the Marker and Cell (MAC) method by using a staggered grid system. The Dufour and Soret effects processes will be a significantly essential role in the double‐diffusive velocity flow processes. For different values of the relevant parameters, the fluid flow velocity, temperature, and concentration are presented for various Rayleigh numbers, Prandtl numbers, non‐Darcy, thermo‐diffusion, and diffusion‐thermo. The streamlines, isotherms, and iso‐concentration contours are obtained using the MATLAB software. Finally, it is observed that the Nusselt numbers increased with non‐Darcy, thermo‐diffusion, and Rayleigh numbers and decreased with diffusion‐thermo effects. [ABSTRACT FROM AUTHOR]

Present study investigates influence of Soret-Dufour effects on MHD unsteady flow of a tetra-hybrid nanofluid (Al2O3, Cu, SiO2 and TiO2 with base fluid water) within non-Darcy porous stretching cylinder. Additionally, chemical reaction, activation energy, and heat generation are considered. This research contributes to the understanding of how these nanofluids can optimize heat and mass transfer process in applications such as advanced cooling systems, solar collectors, biomedical devices, and chemical reactors. Tetra-hybrid nanofluids are selected as per novel aspects for their exceptional ability to adapt their properties for diverse applications, including advanced thermal management systems and scenarios requiring high thermal and electrical conductivity. The comparison between hybrid, tri-hybrid, and tetra-hybrid nanofluids serves to evaluate how increasing complexity and diversity in nanoparticle combinations impact thermal and flow characteristics. The prevailing PDE's undergo transformation into nonlinear ODE's through the utilization of similarity variables and numerically solved using fifth order Runge-Kutta Fehlberg method with shooting method. It is established that rising unsteady parameter values result in increasing velocity profile and rising shape factor parameter result in higher heat transfer. Specifically, the Nusselt number increases by 24 % in the tri-hybrid and 11 % in the tetra-hybrid with a higher Soret number, whereas the Sherwood number decreases by 38 % in the tri-hybrid and 26 % in the tetra-hybrid nanofluid. Employing sensitivity analysis, this study also aims to investigate impact of output responses such as local Nusselt number and local Sherwood number on input parameter Dufour number, Soret number and chemical reaction parameter for tri-hybrid and tetra-hybrid nanofluid. It is found out that Dufour number in tetra-hybrid nanofluid has the more significant impact on the Nusselt number, whereas the Soret number predominantly affects the Nusselt number in tri-hybrid nanofluid. These findings underscore the potential of tetra-hybrid nanofluid in enhancing the performance of various industrial and environmental processes.

Analysis of nanofluid flow phenomena through a stretching sheet in a non-Darcy porous medium has great importance in enhancing transport processes in energy systems like hybrid fuel cell technology, glass blowing, etc. The current study looks at two-dimensional flow over a convectively heated linear stretching sheet in presence of a magnetic field. The boundary layer flow caused by a sheet that is linearly stretched has been investigated numerically. Employing the appropriate similarity transformation, the governing partial differential equations, which define the flow regime, are converted into a set of ordinary differential equations. The mathematical calculations are carried out using a finite difference algorithm with the aid of Newton's linearization method, which allows us to handle non-linear terms very smoothly. The outcomes of this work have been demonstrated are based on eight parameters, such as Prandtl number [Formula: see text], Lewis number [Formula: see text], Brownian motion parameter [Formula: see text], thermophoresis parameter [Formula: see text], magnetic parameter [Formula: see text], porosity parameter [Formula: see text], Frochemier number [Formula: see text], and convection Biot number [Formula: see text]. The impacts of aforementioned parameters on thermal and concentration boundary layers are depicted graphically. It has been found that the heat transfer in the boundary layer rises as [Formula: see text] enhances, and deposits aggravating particle away from the fluid region and increases the volume percentage of nanoparticles. Moreover, it has been found that as the [Formula: see text] number rises, the concentration profiles become steeper and the species border layer becomes thinner. Furthermore, it has been encountered that the reduced Sherwoord number rises as Le number increases for [Formula: see text] but decreases as [Formula: see text] number increases for [Formula: see text].

Abstract The present research work interprets the entropy generation in the magnetohydrodynamic multiple slip flow of Casson nanofluid initiated due to stretching and coaxially rotating surfaces of two disks inside a non-Darcy porous medium under the dominance of the diacritic Hall current and heat generation. The energy field is explored by incorporating the consequences of distinctive thermal radiation, viscous dissipation, and Joule heating. The present study overcomes the barrier of heat and mass transport by annexing Cattaneo–Christov double diffusion effects. A substantive mathematical problem is modeled by assigning nonlinear partial differential equations together with multiple slip boundary conditions. A compatible similarity transformation comprised in the current study is exerted to produce a set of nonlinear ordinary differential equations with competent boundary conditions. The resulting mathematical model is numerically solved via dispensing the successive linearization method. The present article deals with an in-depth exploration of diagnostic flow parameters’ attributes against the flow field and efficient physical quantities with the help of distinctive graphs and tables. As per regression analysis, the maximum relative error for the regression model on the skin friction coefficient in the radial direction ranging from .0097236 to .017225% is less than that of the other physical quantities. Besides, augmenting the thermophoretic diffusion boosts diluting the concentration of nanoparticles. [ABSTRACT FROM AUTHOR]

This study investigates the phenomena of fluid flows, general heat transfer and entropy generation on transient natural convection in a square cavity occupied with a saturated non-Darcy porous medium with dissipation effects and thermal radiation. The governing equations for the fluid flows and heat transfer, with corresponding boundary and initial conditions in the porous system are solved with the use of the alternate direct implicit method (ADI). The impact of the dimensionless parameters on the flow and thermal effect including entropy generation, with the fluid structure interaction cases are discussed. The following ranges have been used for simulations for the relevant dimensionless parameters: Grashof Number (103≤Gr≤107), Rayleigh number (103≤Ra≤106), Eckert number (0≤Ec≤5×10-5), the Forschheimer resistance (0≤Γ,γ≤10), radiation parameter (0≤Rd≤5) and Prandtl number (0≤Pr≤10). The present numerical work, compared with previously published work are made, and excellent agreement is discovered. According to the simulation results, the top right wall and base of the heated wall are where entropy is being generated most. Furthermore, as the Forschheimer resistance increases the flow rate, the heat transfer, and entropy production reduce. As Pr decreases at Ra=105 the buoyancy force takes control of the flow fields when the porous medium’s resistance weakens. As a result, the fluid flow becomes stronger.

This paper studies the effect of thermo-diffusion, electrical field, and nonlinear thermal radiation. Thermal radiant heat transfer has several industrial applications, and the analysis of radiation heat transfer in non-Newtonian fluids under different conditions has been widely studied. To better understand the problem of thermal nonlinear radiation heat transfer flow in non-Darcy Casson fluid on stretched surfaces, this research examines such a phenomenon. The Hybrid Analytical and Numerical Method (The HAN Method) is used to precisely analyze the steady flow of incompressible Newtonian electrically conducting non-Darcy Casson fluid on a vertical permeable stretchable plate with the presence of the magnetic field. The governing equations of this problem are reduced from the system of nonlinear partial differential equations (PDEs) into a system of nonlinear ordinary differential equations (ODEs) using similarity transformations. The results showed that Casson's parameter has a small and indirect effect on temperature and mass transfer, but it affects the velocity significantly and directly. Also, the magnetic field has a direct and significant effect on temperature and velocity but an indirect and significant effect on concentration. The electric field has a direct and significant effect on temperature and velocity but an indirect and significant effect on concentration. Also, the effects of parameters such as thermal buoyancy, inertial, porous permeability, Eckert number, Prandtl number, chemical reaction, Schmidt and Soret numbers, and porosity on velocity, temperature, and concentration have been studied.

Analysis of nanofluid flow phenomena through a stretching sheet in a non-Darcy porous medium has great importance in enhancing transport processes in energy systems like hybrid fuel cell technology, glass blowing, etc. The current study looks at two-dimensional flow over a convectively heated linear stretching sheet in presence of a magnetic field. The boundary layer flow caused by a sheet that is linearly stretched has been investigated numerically. Employing the appropriate similarity transformation, the governing partial differential equations, which define the flow regime, are converted into a set of ordinary differential equations. The mathematical calculations are carried out using a finite difference algorithm with the aid of Newton's linearization method, which allows us to handle non-linear terms very smoothly. The outcomes of this work have been demonstrated are based on eight parameters, such as Prandtl number (P r) , Lewis number (L e) , Brownian motion parameter ( N b ) , thermophoresis parameter ( N t ) , magnetic parameter (M) , porosity parameter (K) , Frochemier number (F r 1 ) , and convection Biot number (B i). The impacts of aforementioned parameters on thermal and concentration boundary layers are depicted graphically. It has been found that the heat transfer in the boundary layer rises as N t enhances, and deposits aggravating particle away from the fluid region and increases the volume percentage of nanoparticles. Moreover, it has been found that as the L e number rises, the concentration profiles become steeper and the species border layer becomes thinner. Furthermore, it has been encountered that the reduced Sherwoord number rises as Le number increases for N t < 0.1 but decreases as L e number increases for N t > 0.1. [ABSTRACT FROM AUTHOR]

A computational analysis on transient magnetohydrodynamics micropolar fluid flow through a non-Darcy porous medium has been investigated. A computer code based on finite difference technique was developed in Fortran programming language with the objective to analyze the behavior of fluid flow in a porous medium for the non-Darcy case. Two-dimensional mathematical model of the problem has been utilized to obtain the nonsimilar solutions. The finite difference technique was used to explicitly solve the dimensionless equations of fluid velocity, temperature, angular velocity, and concentration to study the behavior of the system for various parameters. In addition, the stability of the system has been performed and obtained that the system converged for the values of Prandtl and Schmidt number greater than and equal to 0.03. The behavior of the system was studied for non-dimensional time ranging from 0 to 80 for a small-time step of 0.005. The most significant influence of various parameters on fluid velocity, temperature, angular velocity, and concentration profiles within the boundary layer have represented graphically and discussed qualitatively. It was found that the system reached to a steady state at the time greater than or equal to 75. [ABSTRACT FROM AUTHOR]

The topic of two-dimensional steady laminar MHD boundary layer flow across a wedge with non-Newtonian hybrid nanoliquid (CuO-TiO2/C2H6O2) with viscous dissipation and radiation is taken into consideration. The controlling partial differential equations have been converted to non-linear higher-order ordinary differential equations using the appropriate similarity transformations. It is demonstrated that a number of thermo-physical characteristics govern the transmuted model. The issue is then mathematically resolved. When the method’s accuracy is compared to results that have already been published, an excellent agreement is found. While the thermal distribution increases with an increase in Eckert number, radiation and porosity parameters, the velocity distribution decreases as porosity increases.

The purpose of this research is to investigate the MHD flow of Williamson fluid through a non-Darcy porous medium with micro-rotation. The governing equations are first converted into a coupled system of ODEs with initial and boundary conditions. The shooting procedure with the Runge-Kutta method is used to solve the ODEs, and determine the impact of various parameters for heat and momentum transfer of boundary layer flow. Finally, all findings are graphed and compared with earlier results to ensure that the current outputs are perfectly valid. Furthermore, it can be seen from the characteristic graph that the system parameters, magnetic field M and non-Darcy F tend to reduce the fluid velocity. Also, a comparison between the two fluid models, Williamson fluid and micropolar fluid have been obtained via the effects of various parameters on skin friction and Nusselt number.

This work explores the second law analysis in the pseudoplastic, Newtonian and dilatant fluid flows over a truncated cone situated in a Forchheimer type of non-Darcy porous medium. The influence of viscous dissipation and thermal dispersion parameters on this free convective flow is studied in the presence of nonlinear Boussinesq approximation. The primary focus of this work is to have a clear idea about entropy generation in the flow of pseudoplastic, Newtonian and dilatant fluids over a truncated cone. Due to a complex nature of flow governing equations, the combination of local non-similarity approach and spectral local linearization method is found to be more accurate in comparison with other spectral methods; hence, this combination is proposed to discuss the solution of system of equations. An error analysis is employed to check the relevance of utilizing the spectral method for this kind of flow governing equations. In addition to this, the comparisons with the existing results, in particular cases are also inserted to show correctness and validity of the results. The study of physical quantities reveals that Nusselt number, entropy generation rate and Bejan number are higher for the dilatant fluid in comparison with the pseudoplastic and Newtonian fluids with or without above-mentioned effects. Increments in heat transfer and entropy generation rates are noticed with streamwise coordinate (ξ ) for all the fluids in the presence and absence of these effects which imply that heat transfer and entropy generation rates for the flow over a truncated cone lie between the flow over a full cone and a vertical plate. [ABSTRACT FROM AUTHOR]

The aim of this article is to analyze the mixed convective Ostwald–de Waele power-law nanofluid flow over vertical frustum of a cone in a non-Darcy porous medium using an efficient numerical technique. The involved power-law nanofluid model utilizes water as the base fluid, and Ti-alloy (Ti6Al4V) and multi-wall carbon nanotubes (MWCNTs) as the nanoparticles. The solution of resultant non-similarity equations subjected to boundary conditions is described using the local non-similarity technique along with an efficient spectral local linearization method. The error estimation is provided to show the efficiency of above-mentioned solution procedure. A detailed explanation about the impact of nanoparticle volume fraction on the dimensionless velocity and temperature profiles along with heat transfer rate and skin friction coefficient is also provided for both the opposing and aiding flow cases. On comparison of the present results in particular cases with the relevant published data, it is assured that this method gives highly accurate outcomes for this kind of very complex fluid flow problems. The domination of dilatant nanofluid over pseudo-plastic nanofluid in both the aiding and opposing flow cases is noticed for velocity profiles, and the velocity is decreased with an increment in the nanoparticle volume fraction. Also, the variation in profiles with a streamwise coordinate ξ shows non-similar nature of the problem. The use of Ti-alloy and MWCNTs in this work makes it very profitable in various important sectors like aerospace and medical sector. [ABSTRACT FROM AUTHOR]

The present investigation analyzes the influence of Cattaneo–Christov heat and mass fluxes on peristaltic transport of an incompressible flow. The fluid is obeying Bingham alumina nanofluid. The fluid flows between two co-axial vertical tubes. The system is expressed by a varying radially magnetic field with respect to the space. Soret effect and non-Darcy porous medium are taken into account. The governing system of equations is tackled by utilizing the approximations of long wave length with low Reynolds number and with the help of homotopy perturbation method (HPM). It is noticed that the axial velocity magnifies with an increase in the value of Bingham parameter. Meanwhile, the value of the axial velocity reduces with the elevation in the value of the magnetic field parameter. On the other hand, the elevation in the value of thermal relaxation time leads to a reduction in the value of fluid temperature. Furthermore, increasing in the value of mass relaxation time parameter makes an enhancement in the value of nanoparticles concentration. It is noticed also that the size of the trapped bolus enhances with the increment in the value of Bingham parameter. The current study has many accomplishments in several scientific areas like medical industry, medicine, and others. Therefore, it represents the depiction of the gastric juice motion in the small intestine when an endoscope is inserted through it. [ABSTRACT FROM AUTHOR]

Present study investigates influence of Soret-Dufour effects on MHD unsteady flow of a tetra-hybrid nanofluid (Al 2 O 3 , Cu, SiO 2 and TiO 2 with base fluid water) within non-Darcy porous stretching cylinder. Additionally, chemical reaction, activation energy, and heat generation are considered. This research contributes to the understanding of how these nanofluids can optimize heat and mass transfer process in applications such as advanced cooling systems, solar collectors, biomedical devices, and chemical reactors. Tetra-hybrid nanofluids are selected as per novel aspects for their exceptional ability to adapt their properties for diverse applications, including advanced thermal management systems and scenarios requiring high thermal and electrical conductivity. The comparison between hybrid, tri-hybrid, and tetra-hybrid nanofluids serves to evaluate how increasing complexity and diversity in nanoparticle combinations impact thermal and flow characteristics. The prevailing PDE's undergo transformation into nonlinear ODE's through the utilization of similarity variables and numerically solved using fifth order Runge-Kutta Fehlberg method with shooting method. It is established that rising unsteady parameter values result in increasing velocity profile and rising shape factor parameter result in higher heat transfer. Specifically, the Nusselt number increases by 24 % in the tri-hybrid and 11 % in the tetra-hybrid with a higher Soret number, whereas the Sherwood number decreases by 38 % in the tri-hybrid and 26 % in the tetra-hybrid nanofluid. Employing sensitivity analysis, this study also aims to investigate impact of output responses such as local Nusselt number and local Sherwood number on input parameter Dufour number, Soret number and chemical reaction parameter for tri-hybrid and tetra-hybrid nanofluid. It is found out that Dufour number in tetra-hybrid nanofluid has the more significant impact on the Nusselt number, whereas the Soret number predominantly affects the Nusselt number in tri-hybrid nanofluid. These findings underscore the potential of tetra-hybrid nanofluid in enhancing the performance of various industrial and environmental processes., Competing Interests: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (© 2024 The Authors.)

The purpose of this paper is to investogate the ectromagnetic and micropolar properties on biviscosity fluid flow with heat and mass transfer through a non-Darcy porous medium. Morever, The heat source, viscous dissipation, thermal diffusion and chemical reaction are taken into consideration. The system of non linear equations which govern the motion is transformed into ordinary differential equations by using a suitable similarity transformations. These equations are solved by making use of Rung–Kutta–Merson method in a shooting and matching technique. The numerical solutions of the velocity, microtation velocity, temperature and concentration are obtained as a functions of the physical parameters of the problem. Moreover the effects of these parameters on these solutions are discussed numerically and depicted graphically. It is found that the microtation velocity increases or deceases as the electric parameter, Hartman parameter and the microrotation parameter increase. Morever, the temperature increases as Forschheimer number, Eckert number increase. [ABSTRACT FROM AUTHOR]

In the present investigation, a problem of natural convective flow of a non-Newtonian power-law fluid over an inclined plate saturated in a non-Darcy porous medium is considered. Also, the nonlinear Boussinesq approximation and convective thermal boundary condition are taken into account to address heat and mass transfer phenomena of thermal systems which are operated at moderate and very high temperatures. The steady-state boundary layer equations are non-dimensionalized into non-similar form and then solved numerically by the local non-similarity method with successive linearisation method (SLM). The effects of various physical parameters on the fluid flow, heat and mass transfer characteristics are depicted graphically and analysed in detail.

The present problem shares out the influences of convective heating and thermal dispersion on thermally stable stratified power-law fluid flow over the vertical frustum of a cone situated in a non-Darcy porous medium. For the first time, it is considered in both aiding and opposing flow cases as it is readily applicable in realistic, practical situations. The local nonsimilarity approach, along with the spectral local linearization method, is used to analyze the fluid behavior and heat transfer within the boundary layer region. In addition, error analysis and comparison with the existing results are also included to validate the obtained results wherever feasible. It is significant to perceive the existence of flow separation in this study as more gain of the velocity of a power-law fluid is seen for aiding flow case in comparison with the opposing flow case. Also, it is evident from these results that the separation of flow is less in the presence of thermal stratification, but it is more in the absence of thermal stratification. Finally, thermal stratification significantly affects the heat transfer rate of power-law fluids besides delaying the boundary layer separation. [ABSTRACT FROM AUTHOR]

In this work, we studied the peristaltic motion of steady non-Newtonian nanofluid flow with heat transfer through a non-uniform inclined channel. The flow in this discussion obeys the power law model through a non-Darcy porous medium. Moreover, the effects of thermal radiation, heat generation, Ohmic dissipation and a uniform external magnetic field are taken in consideration. The governing equations that describe the velocity, temperature and nanoparticles concentration are simplified under the assumptions of long wave length and low-Reynolds number. These equations have been solved numerically by using Runge–Kutta–Merson method with the help of shooting and matching technique. The solutions are obtained as functions of the physical parameters entering the problem. The effects of these parameters on the obtained solutions are discussed and illustrated graphically through a set of figures. It is found that as Brownian motion parameter increases, the axial velocity decreases, whereas the nanoparticles concentration increases and it has a dual effect on the temperature distribution. Moreover, the axial velocity and temperature increase as Prandtl number increases, while the nanoparticles decrease. [ABSTRACT FROM AUTHOR]

This paper emphasizes the thermo-diffusion and viscous dissipation effects on double diffusive natural convection heat and mass transfer characteristics of non-Newtonian power-law fluid over a vertical cone embedded in a non-Darcy porous medium with variable heat and mass flux conditions. The Ostwald–de Waele power-law model is employed to describe the behavior of non-Newtonian fluid. Local non-similarity procedure is applied to transform the set of non-dimensional partial differential equations into set of ordinary differential equations and then the resulting system of equations are solved numerically by Runge-Kutta fourth order method together with a shooting technique. The influence of pertinent parameters on temperature and concentration, heat and mass transfer rates are analyzed in opposing and aiding buoyancy cases through graphical representation and explored in detail.

In this paper, time fractional flow of a Newtonian fluid through a uniform cylindrical tube with a non-Darcy porous medium in the presence of dust particles under the application of a uniform magnetic field along the meridian axis is discussed. The implication of time fractional order differential equations in flow problems and some benefits of fractional order differential equations are highlighted. The Laplace Decomposition Method (LDM) is used to obtain an approximate solution to the proposed problem. The impact of fractional order and integer order of the differential equations and also the effects of some important parameters on the flow system are shown in the forms of graphs and a table. The convergence test of the solution is done. It has been observed that the fractional order differential equation reveals more things like the decrease in dust particle velocity due to the increase in magnetic field for fractional order derivatives, whereas, no noticeable change in dust particle velocity due to the change in magnetic field for integer order derivatives are observed. Also, it is observed that an increase in a fractional order derivative decrease the fluid as well as the dust particle velocities. The skin friction at the walls of the tube are also highlighted. [ABSTRACT FROM AUTHOR]

Electrokinetically modulated flow through a hydrophobic microtube embedded in a Darcy–Forchheimer porous medium is investigated in this paper. The steric effect has been taken into account in the electrical double layer (EDL) region. Heat transfer is analysed in the case of Darcy–Forchheimer–Brinkman flow subject to Joule heating. The flow is supposed to take place under the combined influence of electroosmosis and imposed pressure gradient. The governing nonlinear partial differential equations for electric potential, fluid flow and heat transfer are solved numerically by developing an iterative finite difference method that has second order accuracy. The thermal efficiency is discussed under the purview of the second law of thermodynamics. Influences/impact of different physical parameters on velocity, temperature and entropy are investigated and demonstrated graphically. The paper shows that with an increase in the steric effect, the electrokinetic velocity diminishes and that thermal irreversibility is very high in the electrical double layer region, but it reduces drastically in the neighbourhood of the central region of the microtube. Results of the study are likely to be of profuse interest in the design and development of microfluidic devices that deal with critical types of fluid transport mechanism in non-Darcian porous media. [ABSTRACT FROM AUTHOR]

In this paper, a numerical analysis is performed to investigate the effects of double dispersion and convective boundary condition on natural convection flow over vertical frustum of a cone in a nanofluid saturated non-Darcy porous medium. In addition, Brownian motion and thermophoresis effects have taken into consideration, and the uniform wall nanoparticle condition is replaced with the zero nanoparticle mass flux boundary condition to execute physically applicable results. For this complex problem, the similarity solution does not exist and hence suitable non-similarity transformations are used to transform the governing equations along with the boundary conditions into non-dimensional form. The Bivariate Pseudo-Spectral Local Linearisation Method (BPSLLM) is used to solve the reduced non-similar, coupled partial differential equations. To test the accuracy of proposed method, the error analysis and convergence tests are conducted. The effect of flow influenced parameters on non-dimensional velocity, temperature, nanoparticle volume fraction, regular concentration field as well as on the surface drag, heat transfer, nanoparticle and regular mass transfer rates are analyzed.

The objective of the present study is to investigate the effect of nonlinear variation of density with temperature and concentration on the mixed convective flow of a micropolar fluid over an inclined flat plate in a non-Darcy porous medium in the presence of the convective boundary condition. In order to analyze all the essential features, the governing non-dimensional partial differential equations are transformed into a system of ordinary differential equations using a local non-similarity procedure and then the resulting boundary value problem is solved using a successive linearisation method (SLM). By insisting the comparison between vertical, horizontal and inclined plates, the physical quantities of the flow and its characteristics are exhibited graphically and quantitatively with various parameters. An increase in the micropolar parameter and non-Darcy parameter tend to increase the skin friction and the reverse change is observed in wall couple stress, mass and heat transfer rates. The influence of the nonlinear concentration parameter is more prominent on all the physical characteristics of the present model, compared with that of nonlinear temperature parameter.

A mathematical model for thermosolutal convection flow in an open two-dimensional vertical channel containing a porous medium saturated with reactive Newtonian fluid is developed and studied. Robin boundary conditions are prescribed, and a first-order homogenous chemical reaction is considered. The Darcy–Forchheimer model is used to simulate both the first- and second-order porous mediums' drag effects. For the general non-Darcy-case, a numerical solution is presented using the Runge–Kutta quadrature and a shooting method. The influences of thermal (0 ≤ λ 1 ≤ 15) and solute Grashof numbers (0 ≤ λ 2 ≤ 20) , Biot numbers (1 ≤ Bi 1 ≤ 10 , Bi 2 = 10) , Brinkman number (0 ≤ Br ≤ 0.5) , first-order chemical reaction parameter (2 ≤ α ≤ 8) , porous medium parameter (2 ≤ σ ≤ 8) and Forchheimer (inertial drag) parameter (0 ≤ I ≤ 12) on the evolutions of velocity, temperature and concentration (species) distributions are visualized graphically. Nusselt number and skin friction at the walls are also computed for specific values of selected parameters. The study is relevant to the analysis of geothermal energy systems with chemical reaction. [ABSTRACT FROM AUTHOR]

In current paper, it is aimed to investigate the entropy generation of electroosmotic flow aggravated by peristaltic pumping across a non-Darcy porous medium. We have implemented the Darcy Forchheimer model to interpret the permeability of porous media. The electro-magneto-hydrodynamic flow is considered in a symmetric channel. We have analyzed the flow characteristics, heat transfer and entropy generation for various values of joule heating parameter γ , Hartmann number H m , Darcy number Ω 2 , Forchheimer number c F and electroosmotic parameter m. It is found that entropy generation increases for increasing values of Darcy number Ω 2 and Forchheimer number c F . [ABSTRACT FROM AUTHOR]

This study deals with the transfer of mass and heat of nanofluid flow over three different geometries of the non‐Darcy permeable vertical cone/wedge/vertical plate. Influence of the Brownian motion and thermophoresis takes place due to the nanofluid. Boundary condition on the temperature is introduced at the surface where the thermal conductivity of the fluid obeys a linear relation with the temperature. The local linearization method is introduced for solving the governing equations, and is based on spectral discretization. To verify the numerical scheme, we compared our results with those in the existing literature. The impact of the governing parameters on the fluid velocity, temperature distribution, and concentration distribution of nanoparticles along with the Nusselt number and Sherwood number is discussed. Some important outcomes of the present study are that the Nusselt number is higher for the plane plate than that for the vertical cone and it significantly decreases with introduction of the radiation parameter. The nanofluid Lewis number decreases the diffusivity of mass of the nanofluid, and as a result it helps enhance the Sherwood number. [ABSTRACT FROM AUTHOR]

Here nonlinear radiation influence in slip flow of viscous nanomaterial is modeled and elaborated. Rate of entropy generation is evaluated employing second thermodynamics relation. Energy equation is modeled via heat generation and thermal stratification aspects. Non-Darcy porous medium is considered. Prandtl number has an essential contribution in momentum and energy equations to control the boundary layer. For this purpose, effective and without effective Prandtl fluid models are utilized in forced convective nanomaterial flow over a stretched surface. Appropriate similarity transformations give the nonlinear systems. Influence of pertinent variables versus Bejan number and entropy rate are elaborated through graphs. Engineering quantities such as Nusselt number and coefficient of skin friction are examined graphically. The main conclusions are presented. [ABSTRACT FROM AUTHOR]

Abstract Nanomaterials belong to a new class of materials with increased thermo-physical characteristics and heat transport performance. A comprehensive spectrum of applications in science and mechanical engineering could possibly help from them. Therefore keeping such effectiveness in mind we pointed to explore the characteristics of nanomaterial in non-Darcy mixed convective flow of viscous nanofluid through effective and without effective Prandtl number towards a stretched surface. Through implementation of second thermodynamics law total entropy rate is obtained. Stagnation point flow is pointed out. Shooting technique is implemented to tackle the nonlinear equations for numerical results. The variations in entropy generation, velocity, Bejan number and temperature fields for different estimations of flow parameters like mixed convection parameter (0.0–0.4), inverse Darcy number (0.0–0.4), Inertia parameter (1.0–5.8), velocity ratio parameter (0.0–0.4) and nanoparticle volume fraction (0.00–0.8) are discussed and the results are graphically presented. Skin friction and Nusselt number of considered flow are discussed via graphically sketch. Moreover, it is examined that velocity of fluid particles increases for both effective and without effective Prandtl number with rising estimations of nanoparticles volume fractions. It is also examined that entropy rate shows dual behavior verses nanoparticles volume fraction. Entropy rate enhances in case of effective Prandtl number, while decays in case without effective Prandtl number. Highlights • Total entropy generation rate is calculated. • Thermodynamic second law is implemented in modeling. • Nanoparticles comprise (γAl₂O₃-H₂O and γAl₂O₃-C₂H₆O₂) particles. • Non-Darcy porous medium is used. [ABSTRACT FROM AUTHOR]

Melting heat transfer and non-Darcy porous medium effects in MHD stagnation point flow toward a stretching surface of variable thickness are addressed. Brownian motion and thermophoresis in nanofluid modeling are retained. Zero mass flux condition for concentration at surface is imposed. The problem of ordinary differential system are analyzed numerically through shooting technique. Graphically results of various physical variables on the velocity, temperature and concentration are studied. Skin friction coefficient local Nusselt number and Sherwood number are also addressed through tabulated values. The results described here illustrate that the velocity field is higher via larger melting parameter. However reverse situation is examined for Hartman number. Moreover the influence of thermophoresis parameter on temperature and concentration is noted similar. [ABSTRACT FROM AUTHOR]

This study investigates the phenomena of fluid flows, general heat transfer and entropy generation on transient natural convection in a square cavity occupied with a saturated non-Darcy porous medium with dissipation effects and thermal radiation. The governing equations for the fluid flows and heat transfer, with corresponding boundary and initial conditions in the porous system are solved with the use of the alternate direct implicit method (ADI). The impact of the dimensionless parameters on the flow and thermal effect including entropy generation, with the fluid structure interaction cases are discussed. The following ranges have been used for simulations for the relevant dimensionless parameters: Grashof Number ( 10 3 ≤ G r ≤ 10 7), Rayleigh number ( 10 3 ≤ R a ≤ 10 6), Eckert number (0 ≤ E c ≤ 5 × 10 - 5 ), the Forschheimer resistance (0 ≤ Γ , γ ≤ 10), radiation parameter (0 ≤ R d ≤ 5) and Prandtl number (0 ≤ P r ≤ 10). The present numerical work, compared with previously published work are made, and excellent agreement is discovered. According to the simulation results, the top right wall and base of the heated wall are where entropy is being generated most. Furthermore, as the Forschheimer resistance increases the flow rate, the heat transfer, and entropy production reduce. As Pr decreases at Ra = 10 5 the buoyancy force takes control of the flow fields when the porous medium's resistance weakens. As a result, the fluid flow becomes stronger. [ABSTRACT FROM AUTHOR]

A mathematical model is presented for the steady, two-dimensional magneto-convection heat transfer of a two-phase, electrically-conducting, particle-suspension in a channel containing a non-Darcian porous medium intercalated between two parallel plates, in the presence of a transverse magnetic field. The channel walls are assumed to be isothermal but at different temperatures. The governing equations for the one-dimensional steady flow are formulated following Marble (1970) and extended to include the influence of Darcian porous drag, Forcheimmer quadratic drag, buoyancy effects, Lorentz body force (hydromagnetic retardation force) and particle-phase viscous stresses. Special boundary conditions for the particle-phase wall conditions are implemented. The governing coupled, non-linear differential equations are reduced from an (x,y) coordinate system to a one-dimensional (y) coordinate system. A series of transformations is then employed to non-dimensionalize the model in terms of a single independent variable, , yielding a quartet of coupled ordinary differential equations which are solved numerically using the finite element method, under appropriate transformed boundary conditions. The influence of for example Grashof free convection number (Gr), Hartmann hydromagnetic number (Ha), inverse Stokes number (Skm), Darcy number (Da), Forcheimmer number (Fs),particle loading parameter (PL), buoyancy parameter (B) on the fluid-phase velocity and particle-phase velocity are presented graphically. A number of special cases of the transformed model are also studied. The mathematical model finds applications in solar collector devices, electronic fabrication, jet nozzle flows, industrial materials processing transport phenomena, MHD energy generator systems etc.

This investigation describes the nanofluid flow in a non-Darcy porous medium between two stretching and rotating disks. A nanofluid comprises of nanoparticles of silver and copper. Water is used as a base fluid. Heat is being transferred with thermal radiation and the Joule heating. A system of ordinary differential equations is obtained by appropriate transformations. Convergent series solutions are obtained. Effects of various parameters are analyzed for the velocity and temperature. Numerical values of the skin friction coefficient and the Nusselt number are tabulated and examined. It can be seen that the radial velocity is affected in the same manner with both porous and local inertial parameters. A skin friction coefficient depicts the same impact on both disks for both nanofluids with larger stretching parameters. [ABSTRACT FROM AUTHOR]

The present work addresses flow in a non-Darcy porous medium. Flow is due to nonlinear stretching surface with variable thickness. Formulations consists of salient features of nonlinear thermal radiation, viscous dissipation, Joule heating and homogeneous-heterogeneous reactions. In whole analysis equal diffusions for reactants and autocatalyst are invoked. Resulting problems are numerically solved using built-in-shooting technique. Outcome of different thermophysical variables i.e., ratio of mass diffusion coefficient, radiation parameter, local inertia coefficient, inverse Darcy number, shape parameter, surface thickness parameter, Hartman number, Prandtl number, Biot number, strength of homogeneous reaction parameter, Eckert number, Schmidt number and strength of heterogeneous reaction parameter on the momentum, thermal and concentration distributions is investigated. Velocity is found to decay for higher estimation of Hartman number while thermal and concentration fields are enhanced for larger Eckert and Biot numbers. [ABSTRACT FROM AUTHOR]

In this contribution, the MHD pulsatile mechanism with heat transfer of an incompressible non-Newtonian nanofluid between two permeable parallel vertical plates is investigated. The effects of non-Darcy porous medium, radiation, Ohmic and viscous dissipation are included. By using the homotopy perturbation method, a solution is obtained, which is in a good agreement with the momentum, energy and nanoparticles equations. Numerical results for the velocity, temperature and nanoparticles distributions are obtained. [ABSTRACT FROM AUTHOR]

Comparative analysis for flow of CNTs nanofluids is discoursed in the presence of non-Darcy porous medium. The consequences of homogeneous/heterogeneous process and heat transfer through convection are employed. The flow induced is due to non-linear stretching sheet of variable thickness. The bottom of the variable thickness sheet is heated by convective processes from a heated fluid. The velocity, temperature and concentration functions are formulated for the stretched flow problem. Convergence control variables and square residual errors for series solutions are obtained through OHAM (Optimal Homotopy Analysis Method). Biot number corresponds to larger temperature distribution in case of MWCNT than SWCNT . Comparison of nanoparicles SWCNT and MWCNT for the CNTs nanofluid fluids is highlighted. Water and engine oil CNTs fluids have higher magnitude of Nusselt number when compared with kerosene oil CNT fluid. The heat transfer rate in the presence of MWCNT is higher than SWCNT . Comparison of present study with previous published data is made. The outcomes are found in favorable agreement. [ABSTRACT FROM AUTHOR]

The objective here is to analyze the influence of homogeneous and heterogeneous reactions in flow induced by convectively heated sheet with nonlinear velocity and variable thickness. Porous medium effect is characterized by Darcy-Forchheimer consideration. A simple isothermal model of homogeneous-heterogeneous reactions is used to regulate the temperature of stretched surface. Thermodynamics processes of homogeneous-heterogeneous reactions analyze the effect of temperature phase changes. Resulting problems are computed for the convergent solutions of velocity, temperature and concentration. Analysis for the influential variables on the physical quantities is graphically examined. Our computed results interpret that velocity field decays for larger magnetic parameter while temperature field enhances for higher estimation of Biot number. Let us examine two-dimensional flow of incompressible viscous fluid bounded by nonlinear stretching sheet with variable thickness. Fluid obeying non-Darcy features fills the porous space . The convectively heated sheet at has velocity . Here and are the positive constants. Note that the surface is of same thickness when . [ABSTRACT FROM AUTHOR]

This paper analyzes the flow, heat, and mass transfer characteristics of the mixed convection on a vertical plate with uniform and constant heat and mass fluxes in a doubly stratified non-Darcy porous medium saturated with couple stress fluid. The nonlinear governing equations and their associated boundary conditions are initially cast into dimensionless forms by pseudo-similarity variables. The resulting system of equations is then solved numerically using the Keller-box method. The numerical results are compared and found to be in good agreement with previously published results on special cases of the problem. The obtained results are displayed graphically to illustrate the effect of the couple stress, mixed convection, stratification parameters, and Forchheimer number on the dimensionless velocity, wall temperature, and wall concentration. The numerical values of heat and mass transfer rates for different values of governing parameters are also tabulated. [ABSTRACT FROM AUTHOR]

The present research work examines the entropy generation in the magnetohydrodynamic second-order slip flow of Casson nanofluid surpassing a horizontal stretching sheet inside a non-Darcy porous medium under the dominance of Hall current and nonlinear thermal radiation. The present model is made more realistic by taking second-order velocity slip flow. The energy field is pursued by incorporating the consequences of distinctive viscous dissipation and Joule heating. The chemical reaction incited by activation energy is comprised in the current exploration. A substantive mathematical problem is modeled by assigning nonlinear partial differential equations together with second-order velocity slip and convective boundary conditions. A compatible similarity transformation comprised is exerted to produce a set of nonlinear ordinary differential equations with competent boundary conditions. The resulting mathematical model is numerically solved via the spectral quasi-linearization method. The present article deals with an in-depth exploration of the characteristics of diagnostic flow parameters against the-flow field and other efficient physical quantities with the help of distinctive graphs and tables. As per the regression analysis, the maximum relative error for the reduced Nusselt number ranging from. 000090231 % to. 00015936 % is less than that of the other physical quantities. Magnetic force, thermophoresis, and Brownian motion assist in lessening the heat transport rate, but it gets enriched under the Hall current effects. For the increasing Casson parameter, fluid movement tends to rise near the sheet's surface and gets decelerated afterwards. The intense Hall current accelerates the Casson fluid's motion. But the velocity components in x and z -directions become abated across the flow region due to increasing the first-order velocity slip parameter. Besides, the enhancement in the magnitude of the second-order velocity slip parameter undermines the velocity components in x and z -directions. [Display omitted] The present study incoporates • influences of non-Darcy porous medium, second-order slip and Hall current, • heat transfer phenomenon with activation energy and dissipation effects, • the entropy generation analysis is performed to analyse the loss of heat, • regression analysis to predict the behaviour of significant quantities. [ABSTRACT FROM AUTHOR]

• The analysis of radiation heat transfer in non-Newtonian fluids under different conditions has been accomplished. • The Hybrid Analytical and Numerical Method has been applied. • The results showed that Casson's parameter has a small and indirect effect on temperature and mass transfer. • The electric field has a direct and significant effect on temperature and velocity. This paper studies the effect of thermo-diffusion, electrical field, and nonlinear thermal radiation. Thermal radiant heat transfer has several industrial applications, and the analysis of radiation heat transfer in non-Newtonian fluids under different conditions has been widely studied. To better understand the problem of thermal nonlinear radiation heat transfer flow in non-Darcy Casson fluid on stretched surfaces, this research examines such a phenomenon. The Hybrid Analytical and Numerical Method (The HAN Method) is used to precisely analyze the steady flow of incompressible Newtonian electrically conducting non-Darcy Casson fluid on a vertical permeable stretchable plate with the presence of the magnetic field. The governing equations of this problem are reduced from the system of nonlinear partial differential equations (PDEs) into a system of nonlinear ordinary differential equations (ODEs) using similarity transformations. The results showed that Casson's parameter has a small and indirect effect on temperature and mass transfer, but it affects the velocity significantly and directly. Also, the magnetic field has a direct and significant effect on temperature and velocity but an indirect and significant effect on concentration. The electric field has a direct and significant effect on temperature and velocity but an indirect and significant effect on concentration. Also, the effects of parameters such as thermal buoyancy, inertial, porous permeability, Eckert number, Prandtl number, chemical reaction, Schmidt and Soret numbers, and porosity on velocity, temperature, and concentration have been studied. [ABSTRACT FROM AUTHOR]

The significance of the Soret effect on the boundary-layer stagnation-point flow past a stretching/shrinking sheet in a nanofluid-saturated non-Darcy porous medium is investigated in this study. The nanofluid-saturated porous medium is considered by incorporating the Brownian motion and thermophoresis effects. A similarity transformation is used to reduce the governing fluid flow equations into a set of differential equations and then solved numerically by an accurate implicit finite-difference method. The flow; temperature; concentration and nanoparticle concentration fields; skin friction coefficient; and heat, mass, and nanoparticle mass transfer rates are affected by the complex interactions among the various physical parameters involved in the analysis. These profiles are illustrated graphically in order to reveal interesting phenomena. [ABSTRACT FROM AUTHOR]