Darcy Number Research Articles

Visualization of thermo-magnetic natural convective heat flow in a square enclosure partially filled with a porous medium using bejan heatlines and Hooman energy flux vectors: Hybrid fuel cell simulation

The aim of the article is to study the magneto-convective laminar incompressible flow within a differentially heated square enclosure partly filled with porous medium saturated with ionized helium gas in the presence of a transverse magnetic flux, as a model for emerging fuel cell designs. A Darcy model is utilized for porous medium bulk drag effects. Visualization of heat line and energy flux vectors is computed via Bejan's heat line approach and the Hooman energy flux vector method. The horizontal (i.e., bottom and top) wall boundaries are considered adiabatic and impermeable, while the side walls (cold and hot walls) are maintained at different but constant temperatures. The nonlinear coupled conservation equations under prescribed boundary conditions are solved with a finite difference-based vorticity-stream function approach. Appropriate convergence criteria factors are deployed. Furthermore, validation with earlier studies for the purely fluid, non-magnetic case i. e. in the absence of porous medium (Da) and Hartmann number (Ha) effects, is included. A parametric examination of the impact of Rayleigh number (Ra), Darcy number (Da) and Hartmann number (Ha) on temperature contours, heat lines, energy flux vectors and streamline patterns for ionized helium gas (Prandtl number, Pr = 0.71) is conducted. An increment in Hartmann magnetic number decreases the magnitudes of heat lines, suppresses heat transmission in the enclosure and curtails flow circulation. Enhancing the magnetic parameter and Rayleigh number however boosts the average heat transfer rate. Average Nusselt number at the left hot wall is elevated with increasing permeability of the porous medium and Rayleigh number. Mid-section velocities are enhanced in the porous layer but depleted in the non-porous layer of the enclosure with greater Rayleigh number. The simulations find provide some insight into applications in emerging hybrid electromagnetic fuel cells enabling better visualization of the thermal/fluid characteristics via the novel heat flow visualization heat-function and energy flux vectors analogy.

Linear Model for Two-Layer Porous Bed Suspended with Nano Sized Particles

Two immiscible fluids flows are materialized in science and technology; the combined convection of the two immiscible fluids in a square conduit is reviewed in this study. The nanofluid and pure viscous fluid which do not mix are discussed, and both layers saturated with a porous matrix have different permeabilities. The Dupuit–Forchheimer and Tiwari–Das models are applied to outline the permeability of the layer and nanofluids, respectively. The finite difference method is utilized to find the solutions of conservation equations along with suitable boundary and interface conditions. The boundary condition for the velocity is no slip at all the boundaries, while continuity of velocity and shear stress are used at the interface. The left and right walls are kept at constant but different temperatures, the top and bottom walls are isolated, and the continuity of temperature and heat flux is assumed at the interface. Grashof number, Brinkman number, Darcy number, inertia parameter, permeability ratio, solid volume fraction, thermal conductivity and viscosity ratios, different nanoparticles, and various base liquids of the two-layered fluids are engaged. The velocity is depleted by the inertia and viscosity ratio while it is accelerated with the Darcy and Grashof numbers. The energy distribution was not modulated significantly with any of the dimensionless numbers. Using copper nanoparticles doped in mineral oil and ethylene glycol produced the peak momentum. Diamond nanoparticles dropped in water catalysis showed the best heat transfer rate.

Unsteady buoyant convection of nanofluid in a porous annulus: impacts of size and location of thermal source–sink pairs

The current numerical investigation deals with natural convection in a nanofluid-saturated porous cylindrical annulus subjected to partial heating and cooling of side walls by adopting Brinkman-extended Darcy Model to govern the fluid flow in porous media. By choosing five different locations and four lengths of thermal source–sink pairs, the impact of discrete heating–cooling on fluid flow, thermal transport rates and thermal mixing in a porous annular enclosure has been addressed. From the vast range of numerical simulations, the results provide information on the proper size and location of source–sink combinations to dissipate maximum thermal transport along with better thermal mixing in the enclosure. The importance of porosity, nanoparticle concentration, Darcy and Rayleigh numbers on overall thermal dissipation rate has also been discussed. The results showed that identifying an optimum source–sink location along with an appropriate choice of other control parameters can lead to higher thermal transport enhancement and thermal mixing.

Enhancement of thermoelectric energy harvesting of thermal fluctuations with thermocapillary flows in phase-change materials embedded in metallic foams

We present a complete model of a thermoelectric micro-energy harvester to transform ambient thermal fluctuations into electricity that couples a thermoelectric generator (TEG) to a heat storage unit (HSU) filled by a phase-change material embedded in a metallic foam subjected to convective flows due to the Marangoni effect. Convective heat transfer at the HSU is simulated with the Darcy–Brinkman–Forchheimer bulk equations, a shear stress boundary depending on the porosity, and the phase change with the enthalpy-porosity method. The porous matrix weakens the Marangoni effect by decreasing the surface shear stress while increasing the effective thermal conductivity. Introducing the metallic foam multiplies the efficiency in the transformation between thermal spatial gradients and electric energy with respect to only Marangoni driving. For a heat storage unit of 8 cm times 1 cm, Darcy number Da =10^{-2} coupled to a TEG with Seebeck coefficient 0.027, we find that the harvested energy multiplies with respect to a base PCM driving by Marangoni: 3.2 times (23.6 J) for porosity epsilon =0.95, 3.9 times (28.4 J) for epsilon =0.9, 4.3 times (31 J) for epsilon =0.85. Decreasing the permeability augments the resistance to the convective flows and slightly reduces the electric energy generated.

LBM-MHD Data-Driven Approach to Predict Rayleigh–Bénard Convective Heat Transfer by Levenberg–Marquardt Algorithm

This study aims to consider lattice Boltzmann method (LBM)–magnetohydrodynamics (MHD) data to develop equations to predict the average rate of heat transfer quantitatively. The present approach considers a 2D rectangular cavity with adiabatic side walls, and the bottom wall is heated while the top wall is kept cold. Rayleigh–Bénard (RB) convection was considered a heat-transfer phenomenon within the cavity. The Hartmann (Ha) number, by varying the inclination angle (θ), was considered in developing the equations by considering the input parameters, namely, the Rayleigh (Ra) numbers, Darcy (Da) numbers, and porosity (ϵ) of the cavity in different segments. Each segment considers a data-driven approach to calibrate the Levenberg–Marquardt (LM) algorithm, which is highly linked with the artificial neural network (ANN) machine learning method. Separate validations have been conducted in corresponding sections to showcase the accuracy of the equations. Overall, coefficients of determination (R2) were found to be within 0.85 to 0.99. The significant findings of this study present mathematical equations to predict the average Nusselt number (Nu¯). The equations can be used to quantitatively predict the heat transfer without directly simulating LBM. In other words, the equations can be considered validations methods for any LBM-MHD model, which considers RB convection within the range of the parameters in each equation.

The Effect of a Magnetic Field on the Stability of Fluid Flow in a Porous Channel

In this article, we investigate the stability of flowing fluid in a horizontal cavity. The walls of which consists of a porous material under the influence of a magnetic field (MF) perpendicular to the cavity plane. We find that the real part of the wave velocity is negative for any value of the Hartmann number and the Ghratshof number is greater than zero. In addition, we convert the governing equations to non-dimensional formulas. So, we obtain that Reynolds number, wave number (k), Hartmann number , Ghratshof number, and Darcy number affect on the stability of the model through the graphs of these numbers.

Multi-objective hydrothermal optimization of nanoparticles shape in nanofluid flow through an annular pipe having metal foam fins

The objective of this contribution is twofold. First, a multi-objective hydrothermal optimization is performed to determine the optimum nanoparticles (NPs) shape in nanofluid flow through an annular pipe having metal foam fins, in the forced convection regime. Second, the suitability of the performance evaluation criterion (PEC), as a common method to analyze the hydrothermal performance of thermal systems, is examined by comparing the outcomes of the multi-objective optimization method with those obtained by maximizing PEC. It is found that, depending on the value of the Reynolds number, the cylindrical-shaped NPs or the blade-shaped NPs exhibit the highest heat transfer performance. The lowest value of the required pumping power belongs to the spherical-shaped NPs. The results indicate that PEC attains its maximum value in the spherical-shaped NPs, the NPs fraction of 0.01, the nondimensional fin length of 0.4, the Darcy number of 0.01, and the medium porosity of 0.75. However, the multi-objective algorithm provides several optimum circumstances, each for a certain criterion indicating the suitability of this strategy for hydrothermal performance optimization of the system.

Analysis of hydrothermal performance for laminar forced convective flow through a wavy channel with porous blocks

Purpose The purpose of this paper is to investigate computationally the hydrothermal characteristics for forced convective laminar flow of water through a channel with a top wavy wall and a flat bottom wall having metallic porous blocks. Design/methodology/approach The governing equations are solved computationally using a finite element method–based numerical solver COMSOL Multiphysics® for the following range of parameters: 10 ≤ Reynolds number (Re) ≤ 500 and 10–4 ≤ Darcy number (Da) ≤ 10–1. Findings The presence of porous blocks significantly influences the heat transfer rate, and the value of local Nusselt number increases with the increase in Da. The value of the average Nusselt number decreases with Da for the top wall and the same is enhanced for the bottom wall of the wavy channel with porous blocks (WCPB). The value of the average Nusselt number for WCPB is significantly higher than that of the wavy channel without porous block (WCWPB), plane channel without porous block (PCWPB) and plane channel with the porous block (PCPB) at higher Re. For PCPB, the performance factor (PF) is always higher than that of WCWPB and WCPB for Da = 10–4 and Da = 10–3. Also, PF for WCPB is higher than that of WCWPB for higher Re except for Da = 10–4. Further, the value of for WCPB is higher than that of PCPB at Da = 10–2 and 10–1 at Re = 500. Practical implications The current study is useful in designing efficient heat exchangers for process plants, solar collectors and aerospace applications. Originality/value The analysis of thermo-hydraulic characteristics for laminar flow through a channel with a top wavy wall and a flat bottom wall having metallic porous blocks have been analyzed for the first time. Further, a comparative assessment of the performance has been performed with a wavy channel without a porous block, a plane channel without a porous block and a plane channel with porous blocks.

Double-diffusive magneto-natural convection of nanofluid in an enclosure equipped with a wavy porous cylinder in the local thermal non-equilibrium situation

The emphasis of this study is the numerical analysis via finite volume method of magneto-natural convection induced by double-diffusion of a nanofluid within an enclosure equipped with a wavy porous cylinder. The porous cylinder which is under the local thermal non-equilibrium (LTNE) situation is modeled using Darcy-Brinkman-Forchheimer design as momentum equation. Results are interpreted and evaluated in terms of the dimensionless controlling variables such as Rayleigh number (Ra), Lewis number (Le), Hartman number (Ha), buoyancy ratio parameter (N), Darcy number, (Da), and medium's porosity (ε) at a given heat transfer coefficient for the solid/fluid inter-phase (H) and modified conductivity ratio (γ) of the porous wavy cylinder region. Corcione's correlations with 4% of Al2O3 nanoparticles is used to estimate nanofluid's thermal conductivity and viscosity. Findings show that the heat and mass transfers are improved by Ra, Da, and ε augmentation and Ha and Le diminution. In general, as Le increases, the mass transfer rate rises while the corresponding heat transfer rate drops. Also, we identified a value of N where Le-dependent heat and mass transfer rates are minimal. Moreover, the heat transfer rate is found to be more sensitive to the changes in the porous medium's properties of the wavy porous cylinder than the mass transfer rate. Furthermore, the LTNE situation everywhere exists in the porous wavy cylinder except for the center where the LTE is realized.

Analysis on the Effects of Different Receiver Structures and Porous Parameters on the Volumetric Effects and Heat Transfer Performance of Porous Volumetric Solar Receiver

The volumetric solar receiver is an important heat transfer component of the concentrated solar power (CSP) system. Moreover, in order to improve the absorption of concentrated solar radiation, the porous media are widely used in volumetric solar receiver. In recent years, many studies were concerned with the effects of porosity, pore number density and size, Reynolds number, and Darcy number on heat and flow performance in volumetric solar receiver. However, there are few studies on the effects of structure type and geometric parameters on the volumetric effects and the radiation characteristics of a porous volumetric solar receiver with nonuniform heat flux boundary condition. In this contribution, in order to analyze the effects of structures on the volumetric effects and heat transfer performance, we design six types of porous volumetric receiver structures, whereas the volume of different structures keeps consistent. Then, we apply the local non-thermal-equilibrium model and also consider the effect of structure shape on the concentrating solar radiation transport characteristic. Furthermore, we apply the Gaussian distribution model (GDM) to simulate the actual nonuniform heat flux boundary condition. The result shows that drum-type structure (RT-III) has the best heat transfer performance among the six structures; e.g., the outlet average temperature of fluid is up to 851 K when the pore size and porosity of porous media are 2 mm and 0.7, respectively, which is 41 K increased than the scaled cone-type structure. In addition, the pore diameter has less influence on the outlet temperature of fluid, but it has greater influence on the pressure drop. The contribution can provide a reference for this type of solar receiver design and reconstruction.

Computer simulation of Cu: AlOOH/water in a microchannel heat sink using a porous media technique and solved by numerical analysis AGM and FEM

Extensive improvements in small-scale thermal systems in electronic circuits, automotive industries, and microcomputers conduct the study of microsystems as essential. Flow and thermic field characteristics of the coherent nanofluid-guided microchannel heat sink are described in this perusal. The porous media approximate was used to search the heat distribution in the expanded sheet and Cu: γ - AlOOH/water. A hybrid blend of Boehme copper and aluminum nanoparticles is evaluated to have a cooling effect on the microchannel heat sink. By using Akbari Ganji and finite element methods, linear and non-linear differential equations as well as simple dimensionless equations have been analyzed. The purpose of this study is to investigate the fluid and thermal parameters of copper hybrid solution added to water, such as Nusselt number and Darcy number so that we can reach the best cooling of the fluid. Also, by installing a piece of fin on the wall of the heat sink, the coefficient of conductive heat transfer and displacement heat transfer with the surrounding air fluid increases, and the efficiency of the system increases. The overall results show that expanding values on the NP (series heat transfer fluid system maximizes performance with temperatures) volume division of copper, as well as boehmite alumina particles, lead to a decrease within the stream velocity of the Cu: AlOOH/water. Increasing the volume fraction of nanoparticles in the hybrid mixture decreases the temperature of the solid surface and the hybrid nanofluid. The Brownian movement improves as the volume percentage of nanoparticles in the hybrid mixture grows, spreading the heat across the environment. As a result, heat transmission rates rise. As the Darcy number increases, the thermal field for solid sections and Cu: AlOOH/water improves.

Significance of thermo-diffusion and chemical reaction on MHD Casson fluid flows conveying CNTs over a porous stretching sheet

The current work aims to study the magnetic field on mixed convection heat transfer with Casson fluid flow over a porous medium in the presence of the thermo-diffusion mechanisms with CNTs. CNTs have a wide range of industrial, biomedical, energy production, and space cooling applications because they can improve the thermal properties of the base material. A non-linear governing equation is mapped to a set of ordinary differential equations via a similarity transformation. The highlight of the work is the Casson fluid using CNTs over a Darcy porous media through momentum and temperature profile with the support of an analytical process. The various plots are also used to discuss the effects of physical parameters such as Casson fluid; mass transpiration, Darcy number, and Dufour-Soret number on the profiles, which are depicted through graphical representations. Modulating temperature distribution in different industrial applications, including solar collector design, electronic cooling, building ventilation, etc. The study reveals that the thermal radiation, Casson fluid; mass transpiration, Darcy number, Dufour-Soret number, Lewis number, and Prandtl number increase the values of Richardson number, also buoyancy ratio, magnetic parameter, and volume fraction decrease the values of buoyancy ratio, also controls the transfer of heat. Studies are done on boundary effect studies that have an impact on fluid flow. The behavior of physical factors, such as the Lewis, Prandtl, Dufour, and Soret numbers influencing temperature and concentration field, is determined and presented using graphs. Highlights The analytical method is used to executive equations are converted in the highly non-linear ODEs are obtained from PDEs. The importance of SWCNTs and MWCNTs in modifying MHD, Richardson number, Buoyancy ratio Casson nanofluid parameter and Heat source/sink, carbon nanofluid flow is examined. Stretching/shrinking sheet under the impact of Magnetic parameter, Prandtl number, and mass transpiration non-Newtonian heating is considered. Water is a base fluid mixed with SWCNTs and MWCNTs. The impact of magnetic parameter, Casson fluid, Darcy number, Lewis number, Dufour number, Soret number, mass transpiration, and nanoparticle volume fraction parameter studied.

On the Magnus effect of a rotating porous circular cylinder in uniform flow: A lattice Boltzmann study

In this paper, a multiple-relaxation-time–lattice Boltzmann method is used to simulate the steady fluid flow through and around a rotating porous circular cylinder in uniform flow. This study aims at investigating the effect of Darcy number (10−6≤Da≤10−2), velocity ratio (0≤VR≤2), and Reynolds number (Re = 20 and 40) on the Magnus lift as well as on the flow pattern and pressure coefficient inside and around the rotating porous cylinder. The results reveal that besides the enveloping and detached wakes reported in the literature for rotating solid cylinders, in this study, a new type of the wake called confined wake is observed within the rotating porous cylinders at high Darcy numbers and velocity ratios of less than one. It is seen that the Magnus lift increases almost linearly with the velocity ratio for Da≤10−3; however, through curve-fitting, the rate of increase is shown to decrease with Darcy number in a non-linear manner. For Darcy numbers higher than 10−3, the Magnus lift varies non-linearly with both the velocity ratio and Darcy number in such a way that, interestingly, for Re=40 and very high Darcy numbers of 7.5×10−3 and 10−2, the Magnus lift becomes negative showing a behavior called the inverse Magnus effect.

Entropy Study of Hybrid (Al2O3–Cu/H2O) Nano-Fluid in a Cylindrical Cavity with Wavy Sides Under the Effect of a Parallel Magnetic Field

The convection and entropy of a hybrid nanofluid were investigated in a cylindrical chamber. Inside the cylinder, we have added a rectangular fin with a temperature of Th. Tc applied on the right waving wall. Insulation installed on both the top and bottom walls. The induction of a steady magnetic field is included in this research. Governing equations are resolved by Galerkin finite element method (GFEM) and it’s utilized to treat the controlling equations obtained by giving different characteristics of fluid like The porosity, cylinder rayon and the size of the nano particles with Rayleigh, Hartmann and Darcy numbers. This information is crucial for controlling both fluid flow and the heat transfer rate for normal convection. The results of the solution demonstrate that Da influences the entropy and leads to a decrease in the generation of entropy. The Nusselt mean differs in a straight line with the dynamic. The domain of flows through the sublime modes while it acts contrary with the magnetic force. The use of a rectangular fin inside a cylindrical enclosure rather than traditional ones, as well as the evaluation of its optimal dimensions, was novel in this paper. Moreover, the novelty of this study is that it fills a research gap by looking into the effect of the specific shape of the walls of the porous chamber on heat flux.

Analysis of the laminar flow and enhanced heat transfer rate through a triangular array of cylinders embedded in a fluid-saturated porous media with mixed convection

An investigation of the laminar flow and enhanced heat transfer rate through a triangular array of cylinders embedded in a fluid-saturated porous media is considered. Mixed convection covering forced convection cases has been studied using a finite-volume approach considering the local thermal equilibrium model. Darcy–Brinkman–Forchheimer momentum equations are used to characterize the porous media in question. The study is constrained to the low and intermediate Peclet number (Re = 5 to 40, Pr = 50) and high range of Darcy number (Da = 10−3 to 10−1) for different cylinder spacings (0.7 ≤ φ ≤ 0.99). It is anticipated that porous media would enhance heat transfer, but it derives a multiple order in pressure drop indubitably not desirable in many heat applications. Investigation reveals that both mean drag coefficient (CD ) and Nusselt number (Nu) are strong functions of Da and φ; however, the influence of the buoyancy parameter is mainly witnessed in higher permeability levels and high Peclet numbers. The study also details the effect of governing parameters on mean gap velocities and pressure coefficients. Surprisingly, recirculation wakes exist for the lowest cylinder spacing (φ = 0.7) in high fluid momentum. Low φ and high permeability are desirable in a forced convection regime, whereas both φ and Da should be high in the case of mixed convection. This article also quantifies the combined effect of the Darcy number and buoyancy parameter on the heat-transfer enhancement ratio. A maximum of 112% increase in Nu for φ = 0.7 and 452% for φ = 0.99 are reported at Da = 10−1, but at the cost of higher pressure drop in the latter case.

Convective heat transfer and MHD flow through semi-porous cylindrical filters embedded in an impermeable region

• Magnetic field dominate the flow field and suppresses the flow rate. • Maximize the effect of Hartmann number leads to rise in temperature distribution for the flow. • Heat transfer in the flow is enhanced as the Darcy number increased. • A negative pressure gradient influences the flow immensely. In the present study, profound applications of agriculture, industries and engineering have motivated to report an exact solution for steady incompressible axisymmetric electrically conducting fluid flow through a semi-porous media surrounded by an impervious cylinder. The uniqueness of this paper is to propose the effect of Joule heating by incorporating convection terms which allows to improve the flow model. Further, the non-dimensional parameters applied on equations governing the flow, takes the form of non-linear differential equations. A Differential Transform Method (DTM) is employed directly to the non-linear differential equations. A succinct solution for velocity and temperature is obtained and its variation on the fluid flow is observed for different dimensionless parameters. This is an interesting phenomenon graphically revealed by Skin-friction and Nusselt number. A good agreement is found in the current paper with certain added features by making the equations compatible and well matched with the previous work of research investigators.

CFD Study of MHD and Elastic Wall Effects on the Nanofluid Convection Inside a Ventilated Cavity Including Perforated Porous Object

Cost-effective, lightweight design alternatives for the thermal management of heat transfer equipment are required. In this study, porous plate and perforated-porous plates are used for nanoliquid convection control in a flexible-walled vented cavity system under uniform magnetic field effects. The finite element technique is employed with the arbitrary Lagrangian–Eulerian (ALE) method. The numerical study is performed for different values of Reynolds number (200≤Re≤1000), Hartmann number (0≤Ha≤50), Cauchy number (10−8≤Ca≤10−4) and Darcy number (10−6≤Da≤0.1). At Re = 600, the average Nusselt number (Nu) is 6.3% higher by using a perforated porous plate in a cavity when compared to a cavity without a plate, and it is 11.2% lower at Re = 1000. At the highest magnetic field strength, increment amounts of Nu are in the range of 25.4–29.6% by considering the usage of plates. An elastic inclined wall provides higher Nu, while thermal performance improvements in the range of 3.6–6% are achieved when varying the elastic modulus of the wall. When using a perforated porous plate and increasing its permeability, 22.8% increments of average Nu are obtained. A vented cavity without a plate and elastic wall provides the highest thermal performance in the absence of a magnetic field, while using a porous plate with an elastic wall results in higher Nu when a magnetic field is used.

Flow around porous square cylinders with a periodic and scalable structure

This study experimentally explores the effect of permeability as an isolated control parameter on the downstream wake structure and the aerodynamic properties of the porous square cylinder. An addictive manufacturing technique was utilized to fabricate the porous cylinder, comprised of a simple cubic lattice structure. This unique manufacturing technique, coupled with a periodic and scalable lattice structure, allows us to decouple the permeability from the porosity. Based on permeability measurements in this study, Darcy number (Da) of the porous cylinder varies in a range of 2.44×10−5−4.07×10−4, while its porosity (Φ) is fixed at 0.7. Interesting features in the downstream wake are observed due to the mutual interaction between longitudinal and lateral bleeding flows. This mutual interplay prevents wake entrainment, relocating a recirculation bubble downstream and making its size smaller with increasing Da. When Da>2.0×10−4, in particular, a steady wake is observed in the near-wake despite the low degree of porosity. Finally, we proposed a new approach to defining the length scale of the flow adjustment that is suitable for the porous square cylinder and comparable to the porous cylinder with a circular cross-section.

Double-diffusive peristaltic MHD Sisko nanofluid flow through a porous medium in presence of non-linear thermal radiation, heat generation/absorption, and Joule heating

This article studied a numerical estimation of the double-diffusive peristaltic flow of a non-Newtonian Sisko nanofluid through a porous medium inside a horizontal symmetric flexible channel under the impact of Joule heating, nonlinear thermal radiation, viscous dissipation, and heat generation/absorption in presence of heat and mass convection, considering effects of the Brownian motion and the thermophoresis coefficients. On the other hand, the long wave approximation was used to transform the nonlinear system of partial differential equations into a nonlinear system of ordinary differential equations which were later solved numerically using the fourth-order Runge–Kutta method with shooting technique using MATLAB package program code. The effects of all physical parameters resulting from this study on the distributions of velocity, temperature, solutal concentration, and nanoparticles volume fraction inside the fluid were studied in addition to a study of the pressure gradients using the 2D and 3D graphs that were made for studying the impact of some parameters on the behavior of the streamlines graphically within the channel with a mention of their physical meaning. Finally, some of the results of this study showed that the effect of Darcy number Da and the magnetic field parameter M is opposite to the effect of the rotation parameter Omega on the velocity distribution whereas, the two parameters nonlinear thermal radiation R and the ratio temperature {theta }_{w} works on a decrease in the temperature distribution and an increase in both the solutal concentration distribution, and the nanoparticle's volume fraction. Finally, the impact of the rotation parameter Omega on the distribution of pressure gradients was positive, but the effect of both Darcy number Da and the magnetic field parameter M on the same distribution was negative. The results obtained have been compared with the previous results obtained that agreement if the new parameters were neglected and indicate the phenomenon's importance in diverse fields.

Investigation of the free convection of nanofluid flow in a wavy porous enclosure subjected to a magnetic field using the Galerkin finite element method

This paper provides a numerical simulation to explore natural convection and fluid flow in a three-dimensional wavy cubical enclosure subjected to a magnetic field along the z-axis. The main emphasis of this research is to discuss entropy generation inside a wavy porous cavity. Brinkman's extended non-Darcy flow model is utilized for studying the free convection in the permeable medium. The equations of the mathematical model were solved using Galerkin finite element method. Isotherms, streamlines, velocity profile, and variation of average Nusselt number are calculated for various values of Ra (103–106), Ha (0–100), and φ (0–0.08). The results show that the streamlines and isotherms increase with the increase of Darcy and Rayleigh numbers. Besides, entropy generation is sensitive to the values of Darcy number, Rayleigh number, Hartmann number, and the number of undulations. Besides, the variation of the Nusselt number is affected by the Rayleigh number and Darcy number. The highest Nusselt number can be obtained for a maximum number of undulations only at the small values of the Rayleigh number. Otherwise, this pattern will be changed in the opposite trend at large values of the Rayleigh number. At the highest values of Ra, increasing Ha from 0 to 100 decreased Nuavg by 23%, while increasing Da from 10−5 to 10−2 enhanced it by 177%. Finally, it was found that the values of Beavg are only being enhanced by only one parameter, namely the Hartmann number. The obtained results from the current non-Darcy flow model agree well with the available literature results.