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This research presents an analytical study of magnetohydrodynamics (MHD)‐free convective heat and mass transfer flow of a nanofluid bounded by a semi‐infinite flat plate. A magnetic field of strength (B0)$( {{B}_0} )$ is applied throughout the fluid region. The plate is moving with a constant velocity (U0)$( {{U}_0} )$, temperature, and the concentration are assumed to be fluctuating with time harmonically from a constant mean at the plate. The frontier equations are assumed to be of an oscillatory nature and cracked analytically using the perturbation technique. The novelty of the present work is to examine the heat and mass transfer MHD flow for Cu‐water and TiO2‐water nanofluids in the presence of thermal radiation. The influence of physical parameters on the flow domain is described in the discussions by graphically and in tabular form. It was found that the fluid temperature and skin friction were reduced with the increased values of the radiation parameters for Cu‐water and TiO2‐water nanofluids. Also, it is noticed that the concentration boundary layer thickness decreases with an increase in chemical reaction parameters. [ABSTRACT FROM AUTHOR]

Nanofluids are emerging as a revolutionary replacement for traditional cooling agents. In addition to this, the motile microorganism helps to stabilize the nanofluid. In this direction, the present investigation addressed Casson fluid flow immersed with gyrotactic microorganisms due to an inclined surface. This flow is controlled by incorporating magnetic forces and restricted to passing through porous media. The influence of the chemical activities and heat convection is examined to complete the mathematical description of the flow situation. The total energy involvement is obtained by the addition of resistive heating and viscous dissipation terms in the energy equation. The current flow situation of equations of said physical condition is renovated into the ordinary differential equations (ODEs) with the implementation of proper similarity variables. The numerical solution is developed with the Runge-Kutta scheme. Graphical illustration for respective physical parameters is obtained via MATLAB. The numerical discussion of the present work is validated with previously published literature. The flow motion is controlled by strengthening the magnetic strength. Boost in the Schmidt number and bioconvection Schmidt number reduces the boundary layer thickness. [ABSTRACT FROM AUTHOR]

Ferrofluids are colloidal suspensions made of nanoscale ferromagnetic particles suspended in a base fluid. It has various medical applications like cell separation, drug targeting, magnetic resonance imaging, etc. Due to the wide range of uses for ferrofluids, this investigation looked into the way that how the fluid flows and transfers heat in the presence of thermal radiation, a porous medium, transverse magnetic field, and heat sink or source impacts. The current study examines a unique corner-heated enclosure problem with a quadratic Boussinesq approximation fluid flow and heat transfer under natural convection circumstances. The square cavity of length (L) is considered with hot and cold slits at the opposite–opposite corners, and it is filled with cobalt ferrite CoFe2O4–water H2O ferrofluid. Middle portion of the all the walls is adiabatic. The implications of magnetohydrodynamic are examined employing CoFe2O4 nanoparticles. The Marker-And-Cell method with finite difference technique is employed for the solution of the dimensionless constitutive equations. In this regard, physics of the problem is well explored for the various parameters such as Rayleigh number (Ra), Darcy number (Da), Hartmann numbers (Ha), solid volume fractions (φ) , thermal radiation parameter (Rd), nonlinear Boussinesq approximation parameter (δ) , and inclination angle of the cavity (ω) and are graphically represented. Suspending the CoFe2O4 ferroparticles in the base liquid water increases heat transmission by 14.5 % . A heat source or sink enhances the fluid velocity and boosts heat transmission by 91.93 % . [ABSTRACT FROM AUTHOR]

Non-Newtonian fluids within heterogeneous porous media may give rise to complex spatial energy and mass distributions owing to non-local mechanisms, the modeling of which remains unclear. This study investigates the natural convection heat and mass transfer of non-Newtonian fluids in porous media, considering the Soret and Dufour effects. A strongly coupled model is developed to quantify the coupled transport of energy and reactive pollutants with the non-Newtonian fluid. The constitutive equation for the non-Newtonian fluid is described by a two-sided Caputo type space fractional velocity gradient. The governing equation, with a symmetric diffusion term, is effectively solved using a stable and convergent shifted Grünwald–Letnikov formula. The influences of three important parameters, which are the average skin friction coefficient, the average Nusselt number, and the Sherwood number, on fluid heat and mass transfer are calculated and analyzed. Numerical results reveal a significant interaction between the fractional derivative and the buoyancy ratio number, both of which affect the average skin friction coefficient. Furthermore, the average Nusselt number increases with the Dufour number while decreasing with the average Sherwood number. These findings enhance our understandings of the dynamics of energy and mass co-transport in non-Newtonian fluids, particularly in relation to their constitutive equation featuring spatial non-local properties. [ABSTRACT FROM AUTHOR]

Chemical reactions, mineral diffusion, and deposition are pivotal in understanding the mechanisms of mineral deposition and the formation of seafloor sulfides in the hydrothermal circulation process. To understand the formation process of anhydrite in submarine hydrothermal systems, a computational model that combined component transport and chemical reactions was established and simulated using the mass transport model. The deposition rate of calcium sulfate was defined, and the effects of factors such as porosity, ion concentration, and inflow velocity on the temperature field in the reaction zone were thoroughly investigated. The distribution of temperature, porosity, and velocity during the reaction process was obtained, allowing for the identification of the chemical reaction patterns of certain ions in the early stages of hydrothermal activity. The simulation revealed the occurrence of biochemical reactions between two types of ions, leading to their deposition on the solid framework of a porous medium. With the increase in inflow velocity and solute concentration, the average porosities of the porous medium decreased by 0.495% and 0.468%, respectively, which consequently altered the structure of the rock. Such findings contribute to the inference of formation and extinction mechanisms of seafloor crusts and hydrothermal chimneys. [ABSTRACT FROM AUTHOR]

The large amount of energy consumed is produced as a result of the combustion of fossil fuels. This is due to the obvious pollution of the environment, the high cost and the possibility of running out of these sources that make alternative energy sources more important. Sunlight is a popular renewable energy source. Among the various alternatives, harnessing solar energy has become a promising option in all over the world being less expensive, environmentally friendly and impermeable. Parabolic collectors are an effective way to produce steam in thermoelectric power plants to convert radiant energy into thermal energy. However, these complexes can be improved by improving their design and modifying parameters related to thermal behavior. Heated receiver tubes are key components of a PTC device. These types of improvements can be achieved through use of Nano fluids and insert fins, or porous materials with in receiver tube to enhance the PTC performance. The current review papers seek to increase heat transfer in the receiver tube and reduce heat loss to improve convection fluid heat transfer. [ABSTRACT FROM AUTHOR]

Corn that are not properly dried can result in a lot of waste. Therefore, in this study, the corn pile was used as porous medium for hot air drying, and the coupling method of multiple physical fields was used to simulate and analyze the corn air‐drying silo with different initial conditions and different sizes. The results show that increasing the temperature of hot air could increase the temperature gradient inside corns and promote the evaporation of water. The higher the wind speed, the better the convective heat transfer effect between corn and air, thus improving the drying efficiency. The smaller the ratio of height to diameter of air‐drying silo, the more corn dried in the direction of hot air flow, resulting in poor drying effect. Increasing the inner diameter of the air‐drying silo will reduce the heat transfer efficiency. The lower the bed of the air‐drying silo, the closer the corns are to the hot air inlet, the faster the heating rate and the better the drying effect. The model has good performance and can be used as a mathematical tool to predict the change of maize wet heat field. Practical applications: In order to solve the integral problem of corn drying, the model uses corn grain pile as porous medium for hot air drying. The model investigated the changes of temperature field and humidity field during the drying process of grain pile in corn air‐drying silo. In addition, the characteristics of the model introduced in this study may qualify it for automatic control of corn air‐drying silo and online prediction of drying times to achieve the desired drying effect. [ABSTRACT FROM AUTHOR]

A numerical model was used to simulate natural convection heat transfer in a two-dimensional square enclosure filled with nanofluid and saturated with porous media. The enclosure contained three heated tubes with Rayleigh numbers (Ra) ranging from 102 to 3 × 104. The governing equations with Boussinesq approximation for natural convection were solved iteratively using the finite volume technique through ANSYS FLUENT-CFD commercial package. The Darcy–Forchheimer–Brinkman and Local Thermal Equilibrium models were applied for water and nanofluid flow across the porous zone. The numerical analysis was executed systematically for significant parameters that have a substantial impact on natural convection heat transfer, namely, aspect ratio (AR = 1.25 − 3.75), nanoparticle volume fraction (Φ = 0.25%, 0.5%, 0.7% and 1%), and the porosity of metal foam (ε = 0.3, 0.4, 0.5, 0.6 and 0.7). The aspect ratio was found to have a considerable impact on the averaged Nusselt number (Nuave). For AR = 3.75, the value of Nuave for all Rayleigh numbers increased by approximately 5.22 times compared to AR = 1.25. Moreover, Nuave remained approximately constant as the concentration of alumina-water nanofluid increased from 0.25% to 1%. The novelty of this study lies in the multi-objective optimal design approach adopted by adopting DX-12 with the outcomes of ANSYS FLUENT-CFD. The averaged Nusselt number and convection heat transfer rate reached their optimal values by nearly 13 and 9 times, respectively, at the lowest possible porosity (ε = 0.3), the lowest possible nanofluid concentration (Φ = 0), and the highest possible aspect ratio value (AR = 3.75). [ABSTRACT FROM AUTHOR]

This study aims to investigate the flow characteristics of mixed convection in a parallel plate channel filled with bidisperse porous medium (BDPM). The objective is to understand the effects of momentum slip and convective boundary conditions on the channel walls. A two-velocity two-temperature model is employed to analyze the flow and temperature distributions in both the fluid and solid phases. The governing equations are made dimensionless using appropriate similarity transformations. The Homotopy Analysis Method (HAM) is utilized to solve the resulting equations and analyze the velocity and temperature profiles. The influence of thermal radiation on temperature regulation is explored by controlling its intensity. The impact of the Biot number on enhancing the Nusselt number, a measure of convective heat transfer, is identified. Additionally, the study investigates the effects of various parameters on velocity, temperature, Nusselt number and skin friction. Graphical representations of these relationships are presented, and the findings are discussed comprehensively. [ABSTRACT FROM AUTHOR]

This research focuses on studying the effects of heat and mass transfer convective flow passing through an infinite vertical plate embedded in porous media under radiation and chemical reaction with constant heat and mass flux. A magnetic field of strength is functional throughout the fluid region. The novelty of the present work is to examine the heat and mass transfer magnetohydrodynamics flow in the presence of thermal radiation. The equations governing the flow, heat and mass transfer are solved analytically using the perturbation technique. Expressions for velocity, temperature, concentration, skin‐friction, Nusselt, and Sherwood numbers are obtained. The influence of physical parameters on the flow domain is described graphically and in tabular form. It is found that increase in radiation parameter reduces the velocity and temperature. Moreover, internal friction of the plate decreased with increasing values of radiation parameter. [ABSTRACT FROM AUTHOR]

Aiming to efficiently simulate the transient process of transpiration cooling with phase change and reveal the convection mechanism between fluid and porous media particles in a continuum scale, a new two-phase mixture model is developed by incorporating the local thermal non-equilibrium effect. Considering the low-pressure and high overload working conditions of hypersonic flying, the heat and mass transfer induced by capillary and inertial body forces are analyzed for sub-cooled, saturated and super-heated states of water coolant under varying saturation pressures. After the validation of the model, transient simulations for different external factors, including spatially-varied heat flux, coolant mass flux, time-dependent external pressure and aircraft acceleration are conducted. The results show that the vapor blockage patterns at the outlet are highly dependent on the injection mass flux value and the external pressure, and the reduced saturation temperature at low external pressure leads to early boiling off and vapor blockage. The motion of flying has a large influence on the cooling effect, as the inertial force could change the flow pattern of the fluid inside significantly. The comparison of the results from 2-D and 3-D simulations suggests that 3-D simulation shall be conducted for practical application of transpiration cooling, as the thermal protection efficiency may be overestimated by the 2-D results due to the assumption of an infinite width length of the porous plate. [ABSTRACT FROM AUTHOR]

This study examines forced convective heat transfer via an anisotropic porous channel in a couple stress flow. The flow field is assumed to be fully developed and governed by the Darcy Brinkman Forchheimer equation. The thermal field is assumed to be developing. The channel walls are subjected to constant heat flux. Since the momentum equation is non-linear and the thermal energy equation is linear, coupled equations are solved numerically using the finite difference method. The variation in the bulk mean temperature is linear with the axial distance for all values of the couple stress parameter and Darcy number. In the absence of axial conduction and heat sources or sinks in the flow field, it is easy to see that the energy gained by the fluid up to an axial distance is twice the axial distance. The parameters, anisotropic permeability ratio, and anisotropic angle enhance the heat transfer. The couple stress parameter lessens the enhancement in heat transfer. Anisotropy is critical in heat transmission for Darcy number, D a H ≤ 0.8. The heat transfer rate decreases by more than 40% due to couple stress fluid and anisotropic effects in the channel, as opposed to the Newtonian isotropic situation. This investigation's findings have been compared with previous experimental and numerical research. [ABSTRACT FROM AUTHOR]

We investigate the effective heat transfer in complex systems involving porous media and surrounding fluid layers in the context of mathematical homogenization. We differentiate between two fundamentally different cases: Case (a), where the solid part of the porous media consists of disconnected inclusions, and Case (b), where the solid matrix is connected. For both scenarios, we consider a heat equation with convection where a small scale parameter ε>0 characterizes the heterogeneity of the porous medium and conducts a limit process ε → 0via two-scale convergence for the solutions of the ε-problems. In Case (a), we arrive at a one-temperature problem exhibiting a memory term and in Case (b) at a two-phase mixture model. We compare and discuss these two limit models with several simulation studies both with and without convection. [ABSTRACT FROM AUTHOR]

The primary aim of the study is to offer a precise analysis of the impact of the radiation parameter and Soret number on the magneto hydrodynamic (MHD) free convective flow of an incompressible, electrically conducting viscous fluid (IECVF) over an inclined plate immersed in a porous material (IPIPM). The impulsively started plate with changing mass diffusion and temperature is taken into account. Perturbation technique is applied to solve the non-dimensional momentum equation along with the energy diffusion and mass diffusion equations. Equations are found for the fields of velocity (u), temperature (θ) and concentration (φ). Expressions are obtained for skin friction (SF), Nusselt number (Nu) and Sherwood number (Sh). The graphical representations of the outcomes of various parameters on 'u, 'θ', and 'φ' and on SF, Nu and Sh are drawn, to highlight their effects. [ABSTRACT FROM AUTHOR]

Forced convection heat transfer was studied in a horizontally heated circular pipe with constant heat flux. Porous medium was created using 1 and 3 mm stainless-steel balls (porosity: 0.3690 and 0.3912). Reynolds numbers ranged from 3,200 to 6,500 based on pipe diameter, with heat flux rates of 6,250 and 12,500 W m−2. ANSYS Fluent simulated a 51.4 mm diameter, 5 mm thick, 304 mm long stainless-steel pipe. Results showed increased turbulence and eddy formation. Analysis revealed higher convective heat transfer coefficient, pressure drop, and Nusselt number with increasing Reynolds number. Nusselt number also increased with 1–3 mm ball diameter. 6% porosity increase reduced pressure drop by 84.4%. Nusselt number rose by 46.7% (Reynolds 3,200–6,500) and 4.36% (heat flux 6,250–12,500 W m−2). [ABSTRACT FROM AUTHOR]

Unsteady MHD free convention isothermal vertical plate in a porous medium with radiation effect has been studied numerically. Implicit finite different technique is use as the main tool for the numerical approach. Numerical solutions for the effects of radiation on a MHD convective heat transfer past a semi-infinite porous plate with a magnetic field are obtained. It is assumed that the porous plate moves with a constant velocity in the direction of fluid flow, and the free stream velocity follows the exponentially increasing small numerical law. The magnetic field acts perpendicular to the porous surface which absorbs the fluid with a suction velocity varying with time. The governing equations for the flow are transformed into a system of nonlinear ordinary differential equations by implicit finite difference technique and then are solved numerically by using the numerical method. "A representative set of computational results for the velocity, temperature and concentration profiles as well as Local skin-friction coefficient, Local Nusselt number and Local Sherwood number are presented graphically for various governing parameters such as Magnetic parameter, Reynolds number, Ecart, Schmidt number, and Kinamatics. In the present analysis various comparisons with previously published work are performed and the results are found to be in a good agreement. [ABSTRACT FROM AUTHOR]

The research aims to examine the distribution of porosity and the combined heat and moisture movement while grains are being dried. This research concerns the porosity and flow of soybeans with different particle size ratios and the drying of soybeans with varying particle temperatures. Due to the similarity in shape between soybeans and balls, this article adopts a ball shape to study the heat and moisture transfer of soybean particles, which can also be used for the study of grains with similar shapes, such as mung beans and red beans. Random models of soybeans with varying proportions were created using modeling software Edem and UG. UDF programming was added to the preprocessing software Fluent to analyze the porosity, airstream allocation, and the interaction of temperature and moisture transfer in packed beds with various cylinder-to-particle size ratios and particle temperatures. A packed bed of soybeans was created, and the study examined the impact of cylinder-to-particle size ratios of 4.44, 5.6, and 6.25 on porosity. The results show that the radial porosity in the packed bed displays a fluctuating profile, with partial porosity increasing as the cylinder-to-particle size ratio increases. Increasing the ratio of cylinder size to particle size exacerbated the tortuosity of the flow paths within the packed bed. Simultaneously, the particle temperature increases, leading to a rise in the instantaneous heat transfer during the drying process, strengthening the ratio of moisture transfer within the packed bed. The method effectively models during convective heat and mass transfer in the liquid facies, as well as thermal and mass spread in the solid facies. The results of this study have been validated on physical models. The air temperature of 273 K is considered during the simulation process [ABSTRACT FROM AUTHOR]

This paper addresses the unsteady hydrodynamic convective heat and mass transfer of three fluids namely air, water, and electrolyte solution past an impulsively started vertical surface with Newtonian heating in a porous medium under the influences of magnetic field and chemical reaction. Suitable dimensionless parameters are used to transform the flow equations and the approximate analytic method employed to solve the flow problem. The results are illustrated graphically for the velocity, temperature, and concentration profiles. Though, low Prandtl numbers produce high-thermal boundary layer thickness, however, as a novelty, the presence of the magnetic field delayed the convection motion hence, the thermal boundary layer thickness is greater for water with high Pr = 7.0 as compared to air with low Pr = 0.71 and electrolyte solution with low Pr = 1.0. Practically, water with a high-Prandtl number can effectively absorb and release heat. This makes water useful in applications such as geothermal heat pumps and solar thermal collectors, industrial processes such as chemical reactions, distillation, and drying, and in oceanography in predicting the movement and behavior of ocean currents, which in turn can impact weather patterns and climate. Another major observation from the study is that the rate of cooling associated with air, water, or electrolyte impacts differently on the product being cooled. [ABSTRACT FROM AUTHOR]

This work introduces a new approach of analyzing convective heat transfer in porous medium by considering the foam structure as a type of fin. It provides the resulting heat transfer characteristics for the design of a longitudinally flowed tube bundle reformer used for the Micro Gas Turbine Solid Oxide Fuel Cell (MGT-SOFC) hybrid process. Owing to a limited experimental database available in literature for the above-mentioned situation, a physical model is initially introduced for a channel flow configuration between two large flat plates using a commercial PDE solver. This model is then validated with experimental results available in literature. A comparison with theoretical solutions is also conducted. Later, this model is modified/adapted for a pipe flow configuration. The physical model for a channel with representative cross-section shape of a longitudinally flowed tube bundle is more complex and is therefore built in a commercial CFD-Solver. A comparative study of the heat transfer behavior in channels of different cross-sections is performed based on a new dimensionless correlation, whose physical coherence with fin efficiency is explained and mathematically proved. The applicability of the heat transfer correlation from one cross-sectional shape to the other are discussed. The proposed new treatment of the porous medium as a fin structure considerably simplifies the heat transfer analysis in porous medium by the clear physical meaning behind fin efficiency and Biot number. This relationship contributes to a better understanding of the heat heat transfer characteristics in porous media in contrast to the correlation between Nusselt number and Reynolds number. Furthermore, this correlation enables a direct comparison between foam structures of different parameters because the fin efficiency is always between 0 and 1. The strong physical background of new correlations also enhances the reliability and plausibility at characterizing and designing the metal foam for heat transfer enhancement. [ABSTRACT FROM AUTHOR]

MHD mixed convection heat transfer of an ionized gas in a vertical microchannel filled with a porous medium is simulated and discussed in this study. The considered flow is hydrodynamically and thermally developing with Local Thermal Non – Equilibrium (LTNE) between the gas and the solid matrix. The Darcy – Brinkman – Forchheimer model is utilized to describe the flow filed in the porous medium. Moreover, both velocity – slip and temperature – jump boundary conditions are applied to the gas at the walls. The governing equations are solved by the finite – volume method. Results are presented and discussed in terms of the developed profiles of velocity and temperature of the constituents as well as the variations of the Nusselt number through the microchannel, the numerical values of the hydrodynamic and thermal entry lengths, and the fully – developed Nusselt number for different conditions. It is found that direct relations exist between the fully – developed Nusselt number and the Richardson number, the Reynolds number, the Hartmann number, the Biot number, the thermal conductivity ratio, and the Forchheimer number. With rise in the Knudsen number or the Darcy number, however, the Nusselt number deteriorates. The results indicate that the Knudsen number, the Hartmann number, the Biot number, and the thermal conductivity ratio are the most influential parameters on the fully – developed Nusselt number. It is envisaged that a tenfold increase in the Hartmann number and a hundredfold elevation in the Knudsen number are accompanied by 14% rise and 42% reduction in the fully – developed Nusselt number, respectively. [ABSTRACT FROM AUTHOR]

This study examines the complex dynamics of an incompressible, electrically conducting third-grade fluid (TGF) flow through a vertical micro-channel that is filled with porous media. The research utilizes asymmetrical convective cooling, a transverse magnetic field, and exothermic reactions to introduce a comprehensive model. The viscosity of the fluid, which is determined by the Naham-type law, changes significantly with temperature. Additionally, the convective heat transfer rates, which govern the asymmetric convective cooling at the surfaces of the micro-channel, are modeled using Newton's law of cooling, and Modified Darcy law is used to address the porous medium effect. To solve the complex system of nonlinear partial differential equations, we employ computational solutions derived from reliable and efficient finite difference methods (FDM). The method is tested for different time-steps and mesh sizes and it is found that the algorithms reliably produce the same results. Our investigation includes an evaluation of both spatial and temporal convergence of the FDM scheme. The study present qualitative discussions of graphical results, that highlight the impact of various flow parameters integrated into the system. The velocity and temperature profiles increases for higher values of thermal Grashoff number Gr and Reynolds number Re while opposite effect is noticed for increasing values of Hartman number M. [ABSTRACT FROM AUTHOR]

A multiphase transport model was formulated to explore the adaptability of local thermal and mass equilibrium/non-equilibrium (LTE/LTNE and LME/LMNE) assumptions in simulating freeze-drying of porous media. Saturated and unsaturated samples with the same initial mass and moisture content but different initial saturations were used in freeze-drying experiments and numerical simulations. Four pairwise combinations between LTE/LTNE and LME/LMNE were tested using the commercial software of COMSOL Multiphysics based on the finite element method. Results showed that the LTE-LMNE assumption achieved excellent agreements between numerical and experimental drying curves with RMSEs being 1.81% for the saturated sample and all less than 1.00% for the unsaturated samples. Based on distributions of saturation and temperature, and variations of effective mass diffusivity and thermal conductivity, the rate-controlling factor was determined to be mass transfer for the saturated material and heat transfer for the unsaturated one. Convective heat transfer played an unimportant role in freeze-drying. The LTE-LMNE based model provided satisfactory predictive capabilities under different conditions. Appropriately increasing the operating temperature could boost the drying rate. Changing the chamber pressure had insignificant effects with drying time differences only below 3%. The smaller the initial saturation, the shorter the drying time. The initially unsaturated frozen material with preformed pores significantly enhanced the freeze-drying process and improved the process economy. [ABSTRACT FROM AUTHOR]

Due to the complexity of the microstructure of porous media, it is of great significance to explore the heat transport mechanism in porous media in many engineering applications. In this study, an expression for effective thermal conductivity (ETC) of porous media embedded with randomly distributed damaged tree-like bifurcation networks is derived based on the theory of thermodynamics and fractal features of tree-like bifurcation networks. We investigate the effect of heat conduction and heat convection in porous media embedded with randomly distributed damaged tree-like bifurcation networks on the ETC of the porous media. It is found that our fractal model has good consistency with the existing available experimental data. In addition, the influence of the microstructural parameters of the model on heat transfer in the porous media is analyzed in detail. The research results can provide significant theoretical guidance for the development and design of heat transfer systems. [ABSTRACT FROM AUTHOR]

We used linear stability theory based on the normal mode decomposition technique and nonlinear stability theory based on the minimum representation of double Fourier series to study the criterion of appearance of the stationary convection and the oscillatory convection; and the rate of heat and mass transfer in a binary viscoelastic fluid mixture in a rotating porous medium under the effect of helical force. We have determined the analytical expression of the Rayleigh number of the system as a function of the dimensionless parameters. Expressions for heat and mass transfer rates are determined as a function of Nusselt and Sherwood number, respectively. The transient behaviors of the Nusselt number and the Sherwood number are studied by solving the finite amplitude equations using the Runge‐Kutta method. Then, the effect of each dimensionless parameter on the system is studied pointed out interesting results. According to the analysis of the different results obtained, it appears that in a porous medium saturated by a binary mixture of viscoelastic fluid in rotation, the Taylor‐Darcy number TaD$Ta_D$, the delay time parameter λ2 and the mass Rayleigh‐Darcy number RaCD$Ra_{CD}$ delay the onset of stationary and oscillatory convection. On the other hand, the helical force SD$S_{D}$, the relaxation time parameter λ1, the ratio of diffusivities τ and the number of Vadasz Va$Va$ accelerate the onset of stationary and oscillatory convection. In the case of the nonlinear stability study, at steady state, the helical force SD$S_{D}$ accelerates the rate of heat and mass transfer. On the other hand, the Taylor‐Darcy number TaD$Ta_D$, the diffusivity ratio τ and the mass Rayleigh‐Darcy number RaCD$Ra_{CD}$ retard the rate of heat and mass transfer. In the unsteady state, the helical force SD$S_{D}$, the relaxation time parameter λ1 and the mass Rayleigh‐Darcy number RaCD$Ra_{CD}$ accelerate the rate of heat and mass transfer. This is not the case for the Taylor‐Darcy number TaD$Ta_D$ and the delay time parameter λ2. [ABSTRACT FROM AUTHOR]

Due to enhanced heat transfer rate, the nanofluid and hybrid nanofluids have significant industrial uses. The principal objective of this exploration is to investigate how thermal radiation influences the velocity and temperature profile. A water-based rotational nanofluid flow with constant angular speed Ω is considered for this comparative study. A similarity conversion is applied to change the appearing equations into ODEs. Three different nanoparticles i.e., copper, aluminum, and titanium oxide are used to prepare different nanofluids for comparison. The numerical and graphical outputs are gained by employing the bvp-4c procedure in MATLAB. The results for different constraints are represented through graphs and tables. Higher heat transmission rate and minimized skin friction are noted for triple nanoparticle nanofluid. Skin coefficients in the x-direction and y-direction have reduced by 50% in trihybrid nanofluid by keeping mixed convection levels between the range 3 < ϵ ≤ 11 . The heat transmission coefficient with raising the levels of thermal radiation between 0.5 < π ≤ 0.9 and Prandlt number 7 < Pr ≤ 11 has shown a 60% increase. [ABSTRACT FROM AUTHOR]

A brush seal is a type of contact sealing technology that generates a great amount of heat during operations. The heat can affect the seal's performance and lifespan. To study the brush seals' temperature distribution, a new model considering the anisotropic heat transfer effect is established in this paper. The friction heat effect at the bristles' tip is studied. The temperature field and leakage rates are obtained by using combined finite element analysis (FEA)-computational fluid dynamics (CFD) analysis and the anisotropic heat transfer theory. The influence of operating and structural parameters on the temperature field and the sealing properties of the brush seal are investigated. It is shown that the value of the rotation rate and the interference can cause the temperature of the brush seal to increase. The pressure difference enhances the convective heat transfer from the brush seals. While the temperature at the bristles' tip increases, the radial average temperature of the bristles decreases significantly. In the case of a small pressure difference, the fence's height can increase the windward area, leading to stiff bristles and resulting in a temperature increase at the bristles' tip; however, the effective flow area increases, resulting in an acceleration of the radial temperature's decrease. To summarize, the porous medium model of anisotropic heat transfer provides a new method for studying brush seals, and it can reflect the temperature distribution and leakage performance of brush seals. [ABSTRACT FROM AUTHOR]

In this work, we investigate the two‐dimensional unsteady natural convective fluid flow problem in a porous‐corrugated enclosure with a fixed sinusoidal heated upper wall. The corrugations of the enclosure are discretely heated while vertical walls are maintained isothermally cold. Subject to where the heat sources are located, five different cases are taken into consideration. The vorticity–streamfunction equations are discretized using a transformation‐free higher order compact approach, and the hybrid BiCGSTAB technique is used to solve the system of algebraic equations that derives from the numerical discretization. To validate our findings, we first compare them to previously published numerical and experimental data. The numerically simulated outcomes are then examined over a variety of essential parameters, such as the Darcy (10−5 ≤ Da ≤ 10−1), Rayleigh (103 ≤ Ra ≤ 106), and Prandtl (0.1 ≤ Pr ≤ 10) numbers. Symmetric and asymmetric fluid flow phenomena are observed. Asymmetric flow phenomenon can be caused by miscible or non‐miscible movements of lighter fluids by heavier fluids, or almost exclusively by nonuniform buoyancy‐driven forces caused by density variations that have arisen because of variations in fluid temperature. The averaged Nusselt value for Case 1 and Case 5 exhibits the highest percentage ratio. The thermal boundary layer is strongly affected by compression, dispersion, suppression, the zone of stratification, and the outweighing of isotherms. The simulated results are visualized by stream functions, isotherms, local and averaged Nusselt number plots. [ABSTRACT FROM AUTHOR]

In this study, several methods to analyze convective heat transfer in a porous medium are presented and discussed. First, the method of Fourier was used to obtain solutions for reduced temperatures θ s and θ f . The results showed an exponentially decaying propagating temperature front. Then, we discuss the method of integration that was presented earlier by Schumann. This method makes use of a transformation of variables. Thirdly, the system of partial differential equations was directly solved with the Finite Difference method, of which the result showed good agreement with the Fourier solutions. For the chosen Δ τ and Δ ξ , the maximum error for θ f = 3.7 % . The maximum error for θ s for the first ξ and first τ is large (36%) but decays rapidly. The problem was extended by adding a linear heat source term to the solid. Again, making use of the change in variables, analytical solutions were derived for the solid and fluid phases, and corrections to the previous literature were suggested. Finally, results obtained from a numerical model were compared to the analytical solutions, which again showed good agreement (maximum error of 6%). [ABSTRACT FROM AUTHOR]

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In the current paper, the effect of a magnetic field on the fully developed forced convective flow and heat transfer is studied. An exact solution is extracted when the flow in the porous medium is governed by the Brinkman–Forchheimer Extended Darcy model. First, the problem formulation is explained to obtain a new system of mathematical formulation. Then, by utilizing the properties which are imposed into the problem, the exact closed‐form analytical solution of the problem is explored. Finally, the main results are illustrated to show the impact of the porous media‐shaped parameter, magnetic parameter, Forchheimer number, and viscosity ratio. It should be mentioned that the asymptotic results achieved in this study were compared with the exact results and it is found that they are in good agreement. [ABSTRACT FROM AUTHOR]

This article investigates the fluid dynamics and heat transfer properties in a trapezoidal enclosure containing a heated cylindrical object. It involves the interaction of multiple physical processes such as the magnetic field, thermal radiation, porous materials, and aqueous copper oxide nanoparticles. The governing partial differential equations are analyzed numerically through the continuous Galerkin finite element algorithm. The analysis takes into account various physical parameter factors, including the Richardson number (0 – 5) , the Hartmann number (5 − 40) , the Darcy number (0.001 − 0.1) , thermal radiation parameter (0.5 − 2) , and nanoparticle volume concentration (0.01 − 0.1). The physical mechanism of thermal and mass transfer in the enclosure caused by various factors is fully explored. In addition, the multiple expression programming (MEP) technique is implemented to report a comparative analysis of flow profiles and thermal distribution. The findings demonstrated that at low Ri, the primary flow within the cavity is driven by the shear friction generated by the moving walls. The growing importance of radiative heat transfer reduces the effectiveness of convective heat transfer, resulting in a decline in the average Nusselt number with R. The heat transfer rate rises up to 27.7% as ϕ augments; however, its value declines by 9.37% against Ha. The expected results obtained by the MEP approach are very consistent with the numerical ones. There is no doubt that the new MEP concept provides a valuable tool for researchers to predict the heat transfer behavior of any data set in cavities of different shapes. It is expected to provide new idea for the development of efficient cooling systems and the improvement of energy efficiency in various engineering applications. [ABSTRACT FROM AUTHOR]

In the present study, forced convection heat transfer on eight inline cylinders immersed in a 10 × 10 × 30 cm packed bed of porous medium is numerically explored with a range of Reynolds numbers from 1100 to 2250. The airflow passes through eight cylinders with an inline arrangement, each having a diameter of 15 mm and a length of 10 cm, immersed in a horizontal porous channel at a constant heat flux of 2000W/m². The commercial program ANSYS Fluent R.19 simulates the changes in pressure drop and temperature distribution by changing the Reynolds number and porosity. The dimensions of each porous pack are 10x10 cm in crosssection, 5 cm in length, and 5 cm in spacing from the next porous pack. The porosity values are (0.4001, 0.39112, and 0.3822). The general shape of all temperature contours shows that the high porosity near the cylinder wall enhances heat transfer from the heated cylinder surface. After that, the air temperature gradually decreases when going away from the cylinder surface. It can also be seen that the pressure drop decreases as particle diameter increases. [ABSTRACT FROM AUTHOR]

Combustion in a porous medium can be beneficial for enhancing reaction rate and temperature uniformity. Therefore, considering the combination with oxy-fuel combustion can address some shortcomings in oxy-fuel burners, a cylindrical two-layer porous burner model is established based on OpenFOAM in this paper. A two-temperature equation model is adopted for the simulation of the heat transfer process. The CH4 skeletal kinetic mechanism is adopted for complex chemistry integration based on OpenSMOKE++. Corresponding experimental methods were used for complementary studies. The walls of the burner are wrapped with three types of thermal insulation materials to present different levels of heat loss. The results show that considering the convection and radiative heat loss of the burner wall, the temperature near the wall is reduced by more than 300 K compared to the adiabatic condition. As a result, the flame propagation speed and CO oxidation rate slowed down. The stable range will be destructively narrowed by more than 50%, and CO emissions will increase by more than 10 times. These defects will be aggravated by increasing the diameter of the burner. It is observed that when the diameter of the burner increases from the initial 5 cm to 10 cm, the effect of heat loss on the stable range is insignificant. [ABSTRACT FROM AUTHOR]

The goal of this research is to examine how a magnetic field affects the unsteady flow of a hybrid nanofluid over a spinning disk that is inclined and stretched while the flow is surrounded by a non-Darcy porous medium. Furthermore, for heat transmission mechanisms, Joule heating and viscous dissipation are considered. The current article is made more realistic by imposing thermal radiation to enhance the heat transmission system under the effects of convection. Moreover, thermal and velocity slip conditions have also been incorporated into the current study. The equations that administer the flow problem along with constraints at the boundaries are converted to dimension-free form by employing a set of appropriate similarity transformations, which are then solved by the numerical technique Runge-Kutta method of order four (RK-4). The new and advanced trend for the convergence of the obtained results is validated through a neural networking approach. The temperature of hybrid nanofluid is augmented by an increase in the porosity parameter, the unsteadiness factor, the Eckert number, the magnetic field, and the Forchheimmer number, while for the values of the radiation factor, the thermal heat is decreasing near the disk and increasing away from the disk. The precision of the obtained results has been ensured by comparing them with established results, with good agreement among these results. [ABSTRACT FROM AUTHOR]

A new photopyroelectric detection configuration is proposed in order to measure the thermal effusivity of porous solids. Compared with the previously reported detection scheme this configuration makes use of a transparent window in front of the pyroelectric sensor. In such a way, the heat losses by convection at the sensor's irradiated surface are eliminated, and consequently, the conduction remains the only process responsible for the heat propagation in the whole detection cell. In the paper, the mathematical model for this new configuration is developed, with the main conclusion that the sample's thermal effusivity can be finally obtained via a fitting procedure with only two fitting parameters (instead of three as previously reported); in such a way, the possible degeneracy of the results is eliminated. The suitability of the method is demonstrated with application on some porous building materials and cellulose-based pressed powders. [ABSTRACT FROM AUTHOR]

Building a computational formula which is expressed by a system of nonlinear partial differential equations in two components was done in this job. The matter of energy transfer by free convection of a dissipative liquid running in a horizontal channel with a porous medium was examined in this study, and we describe the behavior of fluid flow through the channel and the temperature distribution within it. Numerical methods were used as the ODE45 method, one of the Rang-Kutta methods for processing the resulting differential equations. The effect of the Prandtl number, the Schmidt multitude, and the Gratshof amount were also studied. [ABSTRACT FROM AUTHOR]

In the present paper, recent advances in the application of nanofluids in heat transfer in porous materials are reviewed. Efforts have been made to take a positive step in this field by scrutinizing the top papers published between 2018 and 2020. For that purpose, the various analytical methods used to describe the flow and heat transfer in different types of porous media are first thoroughly reviewed. In addition, the various models used to model nanofluids are described in detail. After reviewing these analysis methods, papers concerned with the natural convection heat transfer of nanofluids in porous media are evaluated first, followed by papers on the subject of forced convection heat transfer. Finally, we discuss articles related to mixed convection. Statistical results from the reviewed research regarding the representation of various parameters, such as the nanofluid type and the flow domain geometry, are analyzed, and directions for future research are finally suggested. The results reveal some precious facts. For instance, a change in the height of the solid and porous medium results in a change in the flow regime within the chamber; as a dimensionless permeability, the effect of Darcy's number on heat transfer is direct; and the effect of the porosity coefficient has a direct relationship with heat transfer: when the porosity coefficient is increased or decreased, the heat transfer will also increase or decrease. Additionally, a comprehensive review of nanofluid heat transfer in porous media and the relevant statical analysis are presented for the first time. The results show that Al2O3 nanoparticles in a base fluid of water with a proportion of 33.9% have the highest representation in the papers. Regarding the geometries studied, a square geometry accounted for 54% of the studies. [ABSTRACT FROM AUTHOR]

We study the stagnation point boundary layer flow in a three-dimensional space with forced convection heat transfer in a porous medium. The local thermal non-equilibrium (LTNE) model is considered due to a hot (cold) fluid flowing in cold (hot) porous medium for which we take two different but coupled heat transport equations. The governing equations that describe the physical mechanism are solved numerically using the Chebyshev collocation method, and the results are qualitatively confirmed by predicting their far-field asymptotic analysis. In the asymptotic analysis, the governing equations are linearized about the edge of boundary layers and using the computational linear algebraic approach, and the solutions are expressed using the confluent hypergeometric functions of first kind. The results show that the thicknesses of both momentum and thermal boundary layers are found to be thinning for the three-dimensionality and porous parameters. The temperature of fluid and solid medium is identical at pore level when the interface heat transfer rate and porosity scaled are held large, and for an opposite case, the LTNE influences are rather strong. Due to lack of comparison of the present results, the stability analysis is performed on LTNE three-dimensional boundary layer solutions to show all the obtained solutions are practically feasible. The physical dynamics behind these interesting mechanisms are discussed in detail. [ABSTRACT FROM AUTHOR]

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 Ω on the velocity distribution whereas, the two parameters nonlinear thermal radiation R and the ratio temperature θ 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 Ω 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. [ABSTRACT FROM AUTHOR]

In the study of heat transfer in tree-like branching network, neither the heat convection caused by fluid flow in the tree-like branching network nor the asymmetric structure of the tree-like branching network can be ignored. In this work, we assume the porous media is embedded with a tree-like branching network that are characterized by damaged pipes. We investigated the effects of surface roughness on heat conduction and heat convection in the porous media embedded with the damaged tree-like branching network based on the fractal features of tree-like branching networks and the basic theory of thermodynamics. The proposed model for thermal conductivity can be expressed as a function of micro-structural parameters of the composite, such as the relative roughness, the ratio of thermal conductivity of the wall to that of the fluid in the micro-channel, the diameter ratio, the length ratio, the branching level, the number of damaged channels, the total number of branching levels, and the main tube porosity of the porous media. The effects of the micro-structural parameters of the model on its effective thermal conductivity have been analyzed in detail. It is believed that the joint expression of heat conduction and heat convection could enrich and develop the physical study of heat transport in porous media. [ABSTRACT FROM AUTHOR]

This study aims to obtain the local heat penetration and local heat convection characteristics in a system where a flow of heated vapour cedes thermal energy to a horizontal heatsink plate. The heat penetrates a bed of porous materials with variations in the thermal conductivity inside a sudden-enlargement-contraction-channel. The two non-dimensional parameters of interest are the local Nusselt and Metais-Eckert numbers. The solid particles used for the porous bed are copper, carbon steel, and ceramics. The study is conducted numerically via the software package Ansys-Fluent to solve the Navier-Stokes equation for the conservation of mass, momentum, and energy to obtain the profiles of local temperature and velocity, both in the porous bed and in the vapour stream. Results of this study show that the overall effective-thermal-conductivity of porous materials filled with vapour mainly affects the local Nusselt number. The local Nusselt number increases with increasing overall effective thermal conductivity of porous materials. The sudden enlargement-contraction of the channel affects the local Metais-Eckert number. This finding is proven by the average Metais-Eckert number's variation along the duct and by its nonproportionality to the overall-effective-thermal-conductivity of each porous material. Together with the local Reynolds number, the local Metais-Eckert number describes the flow regimes that locally occur in the system, namely, transitional-combined-convection, laminar-combinedconvection, and laminar-forced-convection. Additionally, based on the local Nusselts number, this study locates the critical point of change from enlargement-affected-zone to contraction-affectedzone at 80 mm along the duct's axis, which means that the switchover-point is not at the centre of the channel. [ABSTRACT FROM AUTHOR]

The goal of this research is to investigate numerically mixed convection flow and heat transfer characteristics from two rotating cylinders in a trapezoidal enclosure filled with porous material for a variety of factors including rotation speed and direction, Richardson number, and the location of the cylinders. The results of this presentation were checked against accessible data and determined to be very similar. The heat transmission properties within the trapezoid were found to be substantially dependent on the Richardson number, speed, and direction of the spinning cylinders, as well as the location of the cylinders according to the findings of this study. Furthermore, it was discovered that the CW-CW casing produced more heat transfer as compared to the CCW-CW scenario for various Richardson numbers when the cylinders placed vertically. Furthermore, this research revealed that, for higher Richardson values, the rotation speed has no effect on the heat transfer increase inside the trapezoidal enclosure. The effect of the cylinders' location on the average Nusselt number was also studied in this investigation. It was found the placing the cylinders in the vertical direction produced higher heat transfer compared with horizontal location for a trapezoid heated from below. [ABSTRACT FROM AUTHOR]

In this study, the laminar forced convection heat transfer and fluid flow characteristics of an annular pipe partly occupied with porous media are numerically investigated. The original sample considered here is the concentric clear fluid region with an aspect ratio (Ri) of 0.2. Three objectives are considered for developing the normal design, namely using eccentric clear fluid region, replacing the single clear fluid circular region by multi-circular ones having the same cross-sectional flow area, and using helical clear fluid region instead of the straight ones. The governing equations of the clear fluid zone and porous matrix zone are solved using the finite volume method. Constant wall heat flux is adopted on the outer pipe surface, and the Reynolds number is ranged from 100 to 1500. The concrete findings show that the heated wall temperature is reduced when the fluid region eccentricity increases and moves toward the wall. The maximum temperature gradient obtained is 8.6 K when one concentric fluid region is replaced by six small fluid regions distributed circumferentially having the same cross-sectional area of the original one. Moreover, around 9.1 K temperature gradient is observed when one straight fluid region is replaced by one helical fluid region having three helixes. [ABSTRACT FROM AUTHOR]

GeoChemFoam is an open-source OpenFOAM-based numerical modelling toolbox that includes a range of custom packages to solve complex flow processes including multiphase transport with interface transfer, single-phase flow in multiscale porous media, and reactive transport with mineral dissolution. In this paper, we present GeoChemFoam's novel numerical model for simulation of conjugate heat transfer in micro-CT images of porous media. GeoChemFoam uses the micro-continuum approach to describe the fluid-solid interface using the volume fraction of fluid and solid in each computational cell. The velocity field is solved using Brinkman's equation with permeability calculated using the Kozeny-Carman equation which results in a near-zero permeability in the solid phase. Conjugate heat transfer is then solved with heat convection where the velocity is non-zero, and the thermal conductivity is calculated as the harmonic average of phase conductivity weighted by the phase volume fraction. Our model is validated by comparison with the standard two-medium approach for a simple 2D geometry. We then simulate conjugate heat transfer and calculate heat transfer coefficients for different flow regimes and injected fluid analogous to injection into a geothermal reservoir in a micro-CT image of Bentheimer sandstone and perform a sensitivity analysis in a porous heat exchanger with a random sphere packing. [ABSTRACT FROM AUTHOR]

Various simulations and numerical studies have been carried out for free convection heat transfer of a square enclosure filled with porous matrix for sinusoidal and uniform heating conditions also varying Rayleigh values from 1 to 500 by using a mathematical finite element simulation. The porous media chosen for this research is saturated water medium (H2O). The enclosure properties have been predefined as follows: The top and base walls of the enclosure are made adiabatic, whereas the right wall is kept at a steady cold temperature. The hot left wall is treated at varying temperatures. The Nusselt values as well as mean Nusselt values have been quantified for the aforementioned values. It has been observed that the mean Nusselt values rise with the surge in Rayleigh values. For the Rayleigh value of 500, we found that uniform heating condition showed larger values of streamline function when compared to sinusoidal heating conditions. These simulations were performed by considering a solar fresnel as a real-world example. We hope that scholars and engineers collectively working on enhancing or aspiring to numerically model the thermal exchange through a square enclosure filled porous matrix will benefit from the information provided in this research article. [ABSTRACT FROM AUTHOR]

This study examines the effects of varying permeability and sinusoidal wall temperature fluctuations on the temporal heat transfer flow driven by natural convection inside a rectangular enclosure filled with a glass bead porous medium under local thermal nonequilibrium conditions for both the working fluid and the porous medium. The fluid’s thermal conductivity is assumed to be variable, and a Darcy–Brinkman–Forchheimer model is used to describe the fluid flow. The Galerkin-type finite element method simulates the constitutive equations governing the flow and heat transfer. The simulation explores the influence of different model parameters on the flow and thermal fields. The results indicate that at a steady state (τ=1.0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\tau = 1.0$$\end{document}), when glass bead diameter Dp increased from 0.01 to 1.0, the values of Nuf¯\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\overline{{{\text{Nu}}_{{\text{f}}} }}$$\end{document} and Nus¯\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\overline{{{\text{Nu}}_{{\text{s}}} }}$$\end{document} decreased by 48.8% and 26%, respectively. Besides, the value of the Nusselt number for the fluid increased by 280.61%, whereas the Nusselt number for the solid increased by 266.55% with the increase of the wave frequency n from 1 to 4. Furthermore, these physical quantities increased by 629.71% and 91.405% when the wave amplitude B rose from 0.1 to 1. [ABSTRACT FROM AUTHOR]