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A numerical formulation for solving homogeneous anisotropic heat conduction problems based on the use of an isotropic fundamental solution is presented in detail. The analysis is carried out assuming a generic position of the coordinate axes, which may not coincide with the principal directions of orthotropy of the material. The two primary integral equations of the method are derived from the governing differential equation of the problem. Then, the numerical procedure is developed by rewriting the internal degrees of freedom that arise from the domain discretization in terms of the boundary nodes and solving the resulting system of linear equations for the boundary unknowns only. Special attention is given to the differentiation of singular integrals which yields additional terms as well as to the evaluation of the resulting Cauchy principal value integral. The main feature of the proposed formulation is its generality, which makes possible its direct extension to solve the problem of three‐dimensional heat conduction in anisotropic media and, foremost, to three‐dimensional orthotropic and anisotropic elasticity or elastoplasticity.

The effects of anisotropy and radiation cannot be considered negligible while investigating the stability of the fluid in convection. Hence, the purpose of this paper is to analyze how these effects could affect the system while considering a couple-stress dielectric fluid. Therefore, the study establishes the effect of thermal radiation in a couple-stress dielectric fluid with an anisotropic porous medium using Goody's approach (Goody, 1956).

To analyze the effect of radiation on the onset of convection, the Milne–Eddington approximation is employed to convert radiative heat flux to thermal heat flux. The equations are further developed to approximate for transparent and opaque medium. Stability of the quiescent state within the framework of linear theory is performed. The principle of exchange of stabilities is shown to be valid by means of single-term Galerkin method. Large values of conduction–radiation and absorptivity parameters are avoided as fluid is considered as liquid rather than gas.

The radiative heat transfer effect on a couple-stress dielectric fluid saturated anisotropic porous medium is examined in terms of Milne–Eddington approximation. The effect of couple-stress, dielectric, anisotropy and radiation parameters are analyzed graphically for both transparent and opaque medium. It is observed that the conduction–radiation parameter stabilizes the system; in addition, the critical Darcy–Rayleigh number also shows a stabilizing effect in the absence of couple-stress, dielectric and anisotropy parameters, for both transparent and opaque medium. Furthermore, the absorptivity parameter stabilizes the system in the transparent medium, whereas it exhibits a dual effect in the case of an opaque medium. It was also found that an increase in thermal and mechanical anisotropy parameters shows an increase in the cell size, whereas the increase in Darcy–Roberts number and conduction–radiation parameter decreases the cell size. The validity of principle of exchange of stability is performed and concluded that marginal stability is the preferred mode than oscillatory.

The effects of anisotropy and radiation on Rayleigh–Bénard convection by considering a couple-stress dielectric fluid has been analyzed for the first time.

The Galerkin finite element method (FEM) based on the characteristic-based split (CBS) scheme is applied to simulate the nanofluid flow and thermal fields inside an inclined geometry filled by a heat-generating hydrodynamically and thermally anisotropic non-Darcy porous medium using the local thermal non-equilibrium model (LTNEM). Property of the hydrodynamic anisotropy is taken in both the Forchheimer coefficient and permeability and these tools are considered as functions of inclination of the principal axes. Also, the thermal conductivity for the porous phase is assumed to be anisotropic.

The Galerkin FEM based on the CBS scheme is applied to solve the partial differential equations governing the flow and thermal fields.

It is noted that the net rate of the heat transfer between the nanofluid and solid phases are influenced by variations of the anisotropic properties. Also, the system is reached to the thermal equilibrium state at H > 100. Further, the maximum nanofluid temperature is reduced by 12.27% when the nanoparticles volume fraction is varied from 0% to 4%.

This paper aims to study the nanofluid flow and heat transfer characteristics inside an inclined enclosure filled with a heat-generating, hydrodynamically and thermally anisotropic porous medium using the CBS scheme. The LTNEM is considered between the nanofluid and porous phases while the local thermal equilibrium model (LTEM) between the base fluid (water) and the nanoparticles (alumina) is taken into account. The Galerkin FEM is introduced to discretize the governing system of equations. Also, examine influences of the anisotropic properties (permeability, Forchheimer terms and thermal conductivity of the porous medium), inclination angle and nanoparticles volume fraction on the net rate of the heat transfer between the nanofluid and porous phases and on the local thermal non-equilibrium state is one of the concerns of this paper.

Thermal conduction anisotropy, which is defined by the dependency of thermal conductivity on direction, is an important parameter in many engineering and research studies such as the design of nuclear waste depositional sites. In this context, the authors aim to investigate the effect of grain shape in thermal conduction anisotropy using pore scale modeling that utilizes real shapes of grains, pores and throats to characterize petrophysical properties of a porous medium.

The authors generalize the swelling circle approach to generate porous media composed of randomly arranged but regularly oriented elliptical grains at various grain ratios and porosities. Unlike previous studies that use fitting parameters to capture the effect of grain–grain thermal contact resistance, the authors apply roughness to grains’ surface. The authors utilize Lattice Boltzmann method to solve steady state heat conduction through medium.

Based on the results, when the temperature field is not parallel to either major or minor axes of grains, the overall heat flux vector makes a “deviation angle” with the temperature field. Deviation angle increases by augmenting the ratio of thermal conductivities of solid to fluid and the aspect ratios of grains. In addition, the authors show that porosity and surface roughness can considerably change the anisotropic properties of a porous medium whose grains are elliptical in shape.

The authors developed an algorithm for generation of non-circular-based porous medium with a novel approach to include grain surface roughness. In previous studies, the effect of grain contacts has been simulated using fitting parameters, whereas in this work, the authors impose the roughness based on the its fractal geometry.

Natural convection from a semi‐infinite vertical plate embedded in a fluid saturated porous medium is studied both analytically and numerically. The plate is assumed to be heated isothermally or by a constant heat flux. The porous medium, modeled according to Darcy’s law, is anisotropic in permeability with its principal axes oriented in a direction that is oblique to the gravity vector. In the large Rayleigh number limit, the governing boundary‐layer equations are solved in closed form, using a similarity transformation. Comparisons between the numerical solution of the full equations and analytical solutions are presented for a wide range of the governing parameters. The effects of the anisotropic permeability ratio K^*, of the orientation angle of the principal axes θ, and of the Rayleigh number R_H on the flow and heat transfer are investigated. Results indicate that the anisotropic properties of the porous medium considerably modify the heat transfer, velocity and temperature profiles from that expected under isotropic conditions.

This study aims to evaluate the melting process of the phase-change RT-35 material in a shell and tube heat exchanger saturated with a porous medium. Titanium porous media with isotropic and inhomogeneous structures are studied. The considered tubes in the shell and tube exchanger are made of copper with specific thicknesses. The phase-change material has a non-Newtonian behavior and follows the endorsed Carreau–Yasuda Model.

The enthalpy–porosity method is used for modeling of the melting process. The governing equations were transferred to their dimensionless forms. Finally, the equations are solved by applying the Galerkin finite element method.

The findings for different values of the relative permeability (K*) and permeability deviation angle (λ) are represented in the forms of charts, streamlines and constant temperature contours. The considerable effects of the relative permeability (K*) and deviation angle (λ) on the flow line patterns of the melting phase-change material are some of the significant achievements of this works.

This study was conducted using data from relevant research articles provided by reputable academic sources. The data included in this manuscript have not been published previously and are not under consideration by any other journal.

A study on heat and mass transfer behavior for an anisotropic porous medium embedded in square cavity/annulus is conducted using incompressible smoothed particle hydrodynamics (ISPH) method. In the case of square cavity, the left wall has hot temperature T_h and mass C_h and the right wall have cool temperature T_c and mass C_c and both of the top and bottom walls are adiabatic. While in the case of square annulus, the inner surface wall is considered to have a cool temperature T_c and mass C_c while the outer surface is exposed to a hot temperature T_h and mass C_h. The paper aims to discuss these issues.

The governing partial differential equations are transformed to non-dimensional governing equations and are solved using ISPH method. The results present the influences of the Dufour and Soret effects on the fluid flow and heat and mass transfer.

The effects of various physical parameters such as Darcy parameter, permeability ratio, inclination angle of permeability and Rayleigh numbers on the temperature and concentration profiles together with the local Nusselt and Sherwood numbers are presented graphically. The results from the current ISPH method are well-validated and have favorable comparisons with previously published results and solutions by the finite volume method.

A study on heat and mass transfer behavior on an anisotropic porous medium embedded in square cavity/annulus is conducted using Incompressible Smoothed Particle Hydrodynamics (ISPH) method. In the ISPH algorithm, a semi-implicit velocity correction procedure is utilized, and the pressure is implicitly evaluated by solving pressure Poisson equation (PPE). The evaluated pressure has been improved by relaxing the density invariance condition to formulate a modified PPE.

The purpose of this paper is to calculate near field and far field scattering of SH waves by multiple multilayered anisotropic circular inclusions using parallel volume integral equation method (PVIEM) quantitatively.

The PVIEM is applied for the analysis of elastic wave scattering problems in an unbounded solid containing multiple multilayered anisotropic circular inclusions. It should be noted that this numerical method does not require the use of the Green’s function for the inclusion – only the Green’s function for the unbounded isotropic matrix is needed. This method can also be applied to solve general elastodynamic problems involving inhomogeneous and/or anisotropic inclusions whose shape and number are arbitrary.

A detailed analysis of the SH wave scattering problem is presented for multiple multilayered orthotropic circular inclusions. Numerical results are presented for the displacement fields at the interfaces and the far field scattering patterns for square and hexagonal packing arrays of multilayered circular inclusions in a broad frequency range of practical interest.

To the best of the authors’ knowledge, the solution for scattering of SH waves by multiple multilayered anisotropic circular inclusions in an unbounded isotropic matrix is not currently available in the literature. However, in this paper, calculation of displacements on interfaces and far field scattering patterns of multiple multilayered anisotropic circular inclusions using PVIEM as a pioneer of numerical modeling enables us to investigate the effects of single/multiple scattering, fiber packing type, fiber volume fraction, single/multiple layer(s), the multilayer’s geometry, isotropy/anisotropy and softness/hardness.

This study aims to present a new numerical model for the simulation of water flow through porous media of anisotropic character, based on the network simulation method and with the use of the free code Ngspice.

For its design, it starts directly from the flow conservation equation, which presents several advantages in relation to the numerical simulation of the governing equation in terms of the potential head. The model provides very precise solutions of streamlines and potential patterns in all cases, with relatively small meshes and acceptable calculation times, both essential characteristics when developing a computational tool for engineering purposes. The model has been successfully verified with analytical results for non-penetrating dams in isotropic media.

Applications of the model are presented for the construction of the flow nets, calculation of uplift pressures, infiltrated flow and average exit gradient in anisotropic scenarios with penetrating dams with and without sheet piles, being all this output information part of the decision process in ground engineering problems involving these retaining structures.

This study presents, for the first time, a numerical network model for seepage problems that is not obtained from the Laplace's governing equation, but from the water flow conservation continuity equation.

This study aims to carry out an analysis for flow and heat transfer of a new hybrid nanofluid over a vertical flat surface embedded in a saturated porous medium with anisotropic permeability at high Rayleigh number. Here the hybrid nanofluid is considered as the working fluid, with different kinds of small particles in nanoscale being suspended.

The generalized homogenous model is introduced to describe the behaviors of hybrid nanofluid. Within the framework of the boundary layer approximations, the governing equations embodying the conservation equations of total mass, momentum and thermal energy are reduced to a set of fully coupled ordinary differential equations via relevant scaling transformations. A flow stability analysis is performed to examine the behavior of convective heat energy. Accurate solutions are obtained by means of a very efficient homotopy-based package BVPh 2.0.

Results show that the linear correlations of physical quantities among the base fluid and its suspended nanoparticles are adequate to give accurate results for simulation of behaviors of hybrid nanofluids. Heat enhancement can be also fulfilled by hybrid nanofluids. A flow stability analysis suggests the heat-related power index m > −1/3 for satisfying the increasing behavior of convective heat energy.

Free convection of a hybrid nanofluid near a vertical flat surface embedded in a saturated porous medium with anisotropic permeability is investigated for the first time. The simplified hybrid nanofluid model is proposed for describing nanofluid behaviors. The results of this proposed approach agree well with those given by the traditional hybrid nanofluid model and experiment. It is expected that, by using different combinations of various kinds of nanoparticles, the new generation of heat transfer fluids can be fabricated, which possess similar thermal-physical properties as regular nanofluids but with lower cost.