Darcy-Brinkman Equation Research Articles

Inertial Deposition of Aerosol Particles in a Periodic Row of Porous Cylinders

The aerosol flow through a periodic row of parallel porous cylinders is investigated. The air flow field outside the cylinders is described by the Navier–Stokes equations of viscous incompressible fluid. The extended Darcy–Brinkman equations are used to calculate the flow velocity inside a porous cylinder. The dependence of the efficiency of the deposition of aerosol particles by inertial impaction and interception on the Stokes number for various values of the Darcy number is studied. Comparison of the results obtained from the numerical model and an approximate analytical model is given. The combined approximate formula for the deposition efficiency of a cylindrical fiber in a parallel array proposed by Müller et al. (2014) is extended for the porous cylindrical fiber. The aerosol flow through the porous body composed by a random array of cylinders is calculated to estimate the interior deposition.Copyright 2015 American Association for Aerosol Research

Constant wall heat flux boundary condition in micro-channels filled with a porous medium with internal heat generation under local thermal non-equilibrium condition

Forced convection heat transfer in a micro-channel filled with a porous material saturated with rarefied gas with internal heat generation is studied analytically in this work. The study is performed by analysing the boundary conditions for constant wall heat flux under local thermal non-equilibrium (LTNE) conditions. Invoking the velocity slip and temperature jump, the thermal behaviour of the porous-fluid system is studied by considering thermally and hydrodynamically fully-developed conditions. The flow inside the porous material is modelled by the Darcy–Brinkman equation. Exact solutions are obtained for both the fluid and solid temperature distributions for two primary approaches models A and B using constant wall heat flux boundary conditions. The temperature distributions and Nusselt numbers for models A and B are compared, and the limiting cases resulting in the convergence or divergence of the two models are also discussed. The effects of pertinent parameters such as fluid to solid effective thermal conductivity ratio, Biot number, Darcy number, velocity slip and temperature jump coefficients, and fluid and solid internal heat generations are also discussed. The results indicate that the Nusselt number decreases with the increase of thermal conductivity ratio for both models. This contrasts results from previous studies which for model A reported that the Nusselt number increases with the increase of thermal conductivity ratio. The Biot number and thermal conductivity ratio are found to have substantial effects on the role of temperature jump coefficient in controlling the Nusselt number for models A and B. The Nusselt numbers calculated using model A change drastically with the variation of solid internal heat generation. In contrast, the Nusselt numbers obtained for model B show a weak dependency on the variation of internal heat generation. The velocity slip coefficient has no noticeable effect on the Nusselt numbers for both models. The difference between the Nusselt numbers calculated using the two models decreases with an increase of the temperature jump coefficient.

Theoretical analysis of high flow conductivity of a fracture induced in HiWay fracturing

HiWay (or channel) fracturing has been a new technology for development of unconventional oil and gas resources in recent years. It has been carried out more than 4000 times worldwide, and obtained good performance in oil and gas recovery. HiWay fracturing improves the flow conductivity of fractures by constructing inhomogeneous distributions of proppant and stable, open flow channel in hydraulic fractures. However, the mechanism and impact factors of high flow conductivity of HiWay fractures are not very clear. To the best of our knowledge, there are no relevant research reports available for such analysis. In this paper, it is first assumed that the fluid flow in proppant clusters follows the Darcy's law and the flow in the channels with proppant clusters is laminar viscous flow, which can be described using Stokes equation. However, the coupling of Darcy-Stokes equations is difficult, and some untrivial interface conditions at the interface between the porous and free-flow regions should be introduced, this will increase greater complexity in numerical computation. As an alternative approach, the Darcy-Brinkman equation is often used for this coupling flow problem, which provides a unified equation with continuous variable coefficients in the two different flow regions. Therefore, there is not necessary to introduce specific interface conditions any more. In this work, we first applied the Darcy-Brinkman equation to model the fluid flow in hydraulic fractures, and then the upscaling of Darcy-Brinkman equation is conducted to evaluate the equivalent permeability of a fracture by using homogenization theory and finite element numerical simulation. Finally, various impact factors of flow conductivity of a hydraulic fracture, such as the cluster shape, cluster distribution, cluster size, etc., are analyzed based on the equivalent permeability. Results show that the permeability of a hydraulic fracture is considerably greater than thst of proppant cluster when the free-flow region is well connected in the fracture, and the geometric properties of proppant clusters are also the key influencing factors for the flow conductivity. Therefore, in HiWay fracturing process, how to construct the well-connected free-flow region in hydraulic fractures is most important, and the flow conductivity of proppant cluster is not the keypoint. However, the surface roughness and stress sensitivity of the hydraulic fractures have not been considered in this work, it will be considered in the future work.

Finite Element Solution of Darcy–Brinkman Equation for Irregular Cross-Section Flow Channel Using Curved Triangular Elements

The finite element method of solution with optimal subparametric higher-order curved triangular elements is used to solve the 3-D fully developed Darcy–Brinkman flow equation through channel of irregular cross-section. Extensive numerical computation and numerical experimentation are done using the quadratic, cubic, quartic and quintic order triangular elements, which reveals that the parameters’ influence on the velocity distributions are qualitatively similar for all the cross-sections irrespective of whether they are of regular or irregular cross-sections. The quintic order curved triangular element yields the solution of a desired accuracy of 10-6. The method can be easily employed in any other irregular cross-section channels.

Convection–radiation heat transfer in solar heat exchangers filled with a porous medium: Homotopy perturbation method versus numerical analysis

The case of combined conduction–convection–radiation heat transfer usually occurred in solar thermal usages is the aim of the present study. This type of combined heat transfer in heat exchangers filled with a fluid saturated cellular porous medium is investigated. The flow is modeled by the Darcy–Brinkman equation. The steady state model of this combined heat transfer is solved semi-analytically based on the homotopy perturbation method (HPM) and numerically based on the finite difference method. No analytical solution has been previously proposed for the problem. Effects of porous medium shape parameter (s) and radiation parameters (Tr and λ) on the thermal performance are analyzed. Furthermore, a discussion on the accuracy and limitations of the HPM in this kind of problems is represented. This study shows that semi-analytical methods (like HPM, VIM, DTM, and HAM) can be used in simulation and prediction of thermal performance of solar energy harvesting systems.

Dynamics of a pre-lens tear film after a blink: Model, evolution, and rupture

A mathematical model is developed to investigate the dynamics and rupture of a pre-lens tear film on a contact lens. The contact lens is modeled as a saturated porous medium of constant, finite thickness and is described by the Darcy-Brinkman equations with stress-jump condition at the interface. The model incorporates the influence of capillarity, gravitational drainage, contact lens properties such as the permeability, the porosity, and the thickness of the contact lens on the evolution and rupture of a pre-lens tear film, when the eyelid has opened after a blink. Two models are derived for the evolution of a pre-lens tear film thickness using lubrication theory and are solved numerically; the first uses shear-free surface condition and the second, the tangentially immobile free surface condition. The results reveal that life span of a pre-lens tear film is longer on a thinner contact lens for all values of permeability and porosity parameter considered. An increase in permeability of contact lens, porosity or stress-jump parameter increases the rate of thinning of the film and advances the rupture time. The viscous-viscous interaction between the porous contact lens and the pre-lens tear film increases the resistance offered by the frictional forces to the rate of thinning of pre-lens tear film. This slows down the thinning process and hence delays the rupture of the film as compared to that predicted by the models of Nong and Anderson [SIAM. J. Appl. Math. 70, 2771–2795 (2010)] derived in the framework of Darcy model.

Darcy-Brinkman Flow Through a Bumpy Channel

The forced flow through a channel with bumpy walls which sandwich a porous medium is studied. The problem models micro-fluidics where, due to the small size of the channel width, the surface roughness of the walls is amplified. The Darcy-Brinkman equation is solved analytically through small perturbations on the ratio of bump amplitude to the half width of the channel. The first- order perturbation solutions give the three-dimensional velocity effects of the bumpiness and the second-order perturbation solutions give the increased resistance due to roughness. The problem depends heavily on the non-dimensional porous medium parameter \(k\) which represents the importance of length scale to the square root of permeability. Our solutions reduce to the clear fluid limit when \(k\) is zero and to the Darcy limit when \(k\) approaches infinity.

Finite element analysis of a projection-based stabilization method for the Darcy–Brinkman equations in double-diffusive convection

This paper presents a projection-based stabilization method of the double-diffusive convection in Darcy–Brinkman flow. In particular, it is concerned with the convergence analysis of the velocity, temperature and concentration in the time dependent case. Numerical experiments are presented to verify both the theory and the effectiveness of the method.

Viscous flow in a curved tube filled with a porous medium

Viscous flow in a tube is basic in fluid mechanics. However, there are cases where the tube is filled with a porous medium, such as those in filters, catalytic reactors or matrix-filled biological pores. In these cases the appropriate governing equation is the DarcyBrinkman equation, where both the resistance of the matrix and the bounding walls are taken into account [1, 2]. There are many articles on the flow in a straight tube filled with a porous medium, notably Refs. [3– 10]. This paper considers the flow in a curved tube filled with a porous medium. Previous literature include the works of Avramenko and Kuznetsov [11, 12] who solved analytically for the flow in a curved channel and a curved rectangular tube. Full numerical solution was also done for a helically coiled tube [13]. We shall introduce a Ritz method to treat the flow through a porous medium in a general curved tube, then apply the method to a curved tube of elliptic cross section. The Ritz method and its convergence have been known in elasticity (e.g. [14]), but its applications

Combined Non-Newtonian and Viscous Shear Effects on Porous Squeeze Film Behavior

On the basis of the Stokes microcontinuum theory, a numerical study of the combined effects of non-Newtonian couple stress and viscous shear forces on porous squeeze film behavior for parallel circular plates is presented in this article. A modified Reynolds equation governing the film pressure is derived using the Stokes constitutive equations to account for lubricant couple stress effects due to the presence of additives or suspended particles. The Darcy-Brinkman equations describing the fluid flow within the porous plate are used to account for viscous shearing forces. Numerical solutions for the squeeze film characteristics including film pressure, radial velocity, squeeze velocity, and response time are calculated for various values of the couple stress and viscous shear parameters. The simulation results indicate that the combined effects of couple stresses and viscous shearing forces are considerable.

An Analytical Study for Fluid Flow in Porous Media Imbedded Inside a Channel With Moving or Stationary Walls Subjected to Injection/Suction

This paper reports a new analytical solution for 2D Darcy-Brinkman equations in porous channels filled with porous media subjected to various boundary conditions at walls. The governing equations of fluid flow through porous medium are reduced to a nonlinear ordinary differential equation (ODE) based on physics of fluid flow. The obtained ODE is solved analytically using homotopy perturbation method (HPM). The analytical models for velocity profile and pressure distribution along the length of channel are validated with data available in the open literature and an independent numerical study using finite volume method (FVM). It was shown that there is an excellent agreement between the presented models and the results of the CFD and previous works. Finally, the effects of Reynolds (Re) and Darcy (Da), numbers suction or injection parameters (α,β) and wall axial velocity coefficients (λ and γ) on velocity profiles and pressure drop in different cases are investigated. The models are applicable to analyze flow in channels filled with and without porous media for both moving and stationary walls and can be used to predict flow in micro and macro channels and over stretching sheets in porous medium as well as study of vapor flow in evaporator section of flat plate heat pipes.

Effect of the Wavy permeable Interface on Double Diffusive Natural Convection in a Partially Porous Cavity

Two-dimensional, double diffusion, natural convection in a partially porous cavity satured with a binary fluid is investigated numerically. Multiple motions are driven by the external temperature and concentration differences imposed across vertical walls. The wavy interface between fluid and porous layer is horizontal. The equations which describe the fluid flow and heat and mass transfer are described by the Navier-Stokes equations (fluid region), Darcy-Brinkman equation (porous region) and energy and mass equations. The finite element method was applied to solve the governing equations. The fluid flow and heat and mass transfer has been investigated for different values of the amplitude and the wave number of the interface and the buoyancy ratio. The results obtained in the form of isotherms, stream lines, isoconcentrations and the Nusselt and Sherwood numbers; show that the wavy interface has a significant effect on the flow and heat and mass transfer.

A numerical simulation of viscous shear effects on porous squeeze-film using the Darcy-Brinkman model

In recent decades, problems related to the squeeze of fluid films in the presence of a porous medium draw attention of researchers and are the subject of many applied studies for industry and biomechanics. Our concerns in this paper are the numerical simulation of the viscous shear stresses effects on the fluid film characteristics between two discs with one porous. This study is based on the coupling, at the fluid film-porous disc interface, of the Darcy-Brinkman equations in the porous medium and the modified Reynolds equation describing the flow in the fluid film. The system of equations obtained is discretized by the means of finite differences method and solved numerically using the technique of Successive Over-Relaxation (SOR). The results show that the viscous shear effects increase the radial and the axial fluid film velocities as well as the squeeze film velocity but decrease the response time. Moreover, these effects are enlarged for smaller viscous shear parameter and for smaller fluid film thickness.

Equal-order finite elements with local projection stabilization for the Darcy–Brinkman equations

For the Darcy–Brinkman equations, which model porous media flow, we present an equal-order H 1-conforming finite element method for approximating velocity and pressure based on a local projection stabilization technique. The method is stable and accurate uniformly with respect to the coefficients of the viscosity and the zeroth order term in the momentum equation. We prove a priori error estimates in a mesh-dependent norm as well as in the L 2-norm for velocity and pressure. In particular, we obtain optimal order of convergence in L 2 for the pressure in the Darcy case with vanishing viscosity and for the velocity in the general case with a positive viscosity coefficient. Numerical results for different values of the coefficients in the Darcy–Brinkman model are presented which confirm the theoretical results and indicate nearly optimal order also in cases which are not covered by the theory.

Flow through a finned channel filled with a porous medium

Abstract Viscous flow through a channel filled with a porous medium and obstructed by periodic fins is studied analytically. The Darcy–Brinkman equation is solved by series expansion and boundary collocation. The problem is governed by both the porous medium parameter k and the fin geometry. There are no recirculating eddies for parallel flow along the fins, while for transverse flow recirculating cells may exist between the fins, but can be suppressed by increased k . The resistance is smaller (flow larger) if the pressure gradient is along the fins than that transverse to the fins.

Gas–liquid dynamics at low Reynolds numbers in pillared rectangular micro channels

Most heterogeneously catalyzed gas–liquid reactions in micro channels are chemically/kinetically limited because of the high gas–liquid and liquid–solid mass transfer rates that can be achieved. This motivates the design of systems with a larger surface area, which can be expected to offer higher reaction rates per unit volume of reactor. This increase in surface area can be realized by using structured micro channels. In this work, rectangular micro channels containing round pillars of 3 μm in diameter and 50 μm in height are studied. The flow regimes, gas hold-up, and pressure drop are determined for pillar pitches of 7, 12, 17, and 27 μm. Flow maps are presented and compared with flow maps of rectangular and round micro channels without pillars. The Armand correlation predicts the gas hold-up in the pillared micro channel within 3% error. Three models are derived which give the single-phase and the two-phase pressure drop as a function of the gas and liquid superficial velocities and the pillar pitches. For a pillar pitch of 27 μm, the Darcy-Brinkman equation predicts the single-phase pressure drop within 2% error. For pillar pitches of 7, 12, and 17 μm, the Blake-Kozeny equation predicts the single-phase pressure drop within 20%. The two-phase pressure drop model predicts the experimental data within 30% error for channels containing pillars with a pitch of 17 μm, whereas the Lockhart–Martinelli correlation is proven to be non-applicable for the system used in this work. The open structure and the higher production rate per unit of reactor volume make the pillared micro channel an efficient system for performing heterogeneously catalyzed gas–liquid reactions.

Poiseuille flow in a fluid overlying a highly porous material

This paper investigates the instability of Poiseuille flow in a fluid overlying a highly porous material. A two layer approach is adopted, where the Darcy–Brinkman equation is employed to describe the fluid flow in the porous medium, with a tangential stress jump boundary condition at the porous/fluid interface. The basic velocity profiles are continuous due to the interfacial boundary conditions. It is shown that for certain parameter ranges the neutral curves are no longer bimodal, such that the two modes of instability corresponding to the fluid and porous layers, respectively, are not distinct.

Unsteady Oscillatory Flow and Heat Transfer in a Horizontal Composite Porous Medium Channel

The problem of unsteady oscillatory flow and heat transfer in a horizontal composite porous medium is performed. The flow is modeled using the Darcy-Brinkman equation. The viscous and Darcian dissipation terms are also included in the energy equation. The partial differential equations governing the flow and heat transfer are solved analytically using two-term harmonic and non-harmonic functions in both regions of the channel. Effect of the physical parameters such as the porous medium parameter, ratio of viscosity, oscillation amplitude, conductivity ratio, Prandtl number and the Eckert number on the velocity and/or temperature fields are shown graphically. It is observed that both the velocity and temperature fields in the channel decrease as either of the porous medium parameter or the viscosity ratio increases while they increase with increases in the oscillation amplitude. Also, increasing the thermal conductivity ratio is found to suppress the temperature in both regions of the channel. The effects of the Prandtl and Eckert numbers are found to decrease the thermal state in the channel as well.

Drying Corn Grains Contained in a Vertical Channel with Constant Heat Flux on Walls

A previous report[ 1 ] developed a numerical model to study the influence of thermal and mass Rayleigh numbers as well as Darcy and Lewis numbers on heat and mass transfer during the drying by natural convection of coffee grains which are treated as a porous medium. The code uses the Darcy-Brinkman momentum equation and the classical natural convection equations. In this work the numerical computer code has been validated by comparing experimental results obtained during the drying of a corn bed.

Stability analysis of thin film flow along a heated porous wall

The time evolution of a thin liquid film flowing down a heated solid porous substrate is investigated. Using the Navier–Stokes and Darcy–Brinkman equations in the film and the porous layer, respectively, the problem is reduced to the study of the evolution equation for the free surface of the liquid film derived through a long-wave approximation. A linear stability analysis of the base flow is performed and the critical Reynolds and Marangoni numbers are obtained. A nonlinear analysis using continuation techniques shows that the base flow yields to stationary surface structures ranging from surface waves to large amplitude structures resembling sliding drops or ridges. It is also shown under what conditions the porous layer can be replaced by an effective slip boundary condition at the liquid-solid interface. Then, the corresponding slip length is calculated from the porous layer characteristics (thickness, porosity, and Darcy number).