Brinkman Momentum Equation Research Articles

Magnetohydrodynamics of immiscible Newtonian fluids in porous regions of different variable permeability functions

This work is an analysis of the flow model often occurs in crude oil extraction, blood flow in the arteries, filtration of underground different fluids flowing together. In this analysis, a model of three-layered porous horizontal channel is considered. This problem is significant because of the different permeability functions used for each porous layer of the channel and flow of different Newtonian fluids takes place in these porous regions. The problem is solved for the general case which can be reduced into several particular cases. The flows inside the channel and in every porous layer is governed by the Brinkman's momentum equation for the porous medium. The momentum equations in each region of the present model are the Airy's inhomogeneous differential equations respectively. An analytical closed form solution of the flow in each region have been obtained. Authors have discussed various results of velocity profile, flow rate and wall shear stresses graphically. Our results agree with the previous published results.

A vorticity-based fully-mixed formulation for the 3D Brinkman–Darcy problem

We propose and analyze a fully-mixed finite element method to numerically approximate the flow patterns of a viscous fluid within a highly permeable medium (an array of low concentration fixed particles), described by Brinkman equations, and its interaction with non-viscous flow within classical porous media governed by Darcy’s law. The system is formulated in terms of velocity and pressure in the porous medium, together with vorticity, velocity and pressure of the viscous fluid. In addition, and for sake of the analysis, the tangential component of the vorticity is supposed to vanish on the whole boundary of the Brinkman domain, whereas null normal components of both velocities are assumed on the respective boundaries, except on the interface where suitable transmission conditions are considered. In this way, the derivation of the corresponding mixed variational formulation leads to a Lagrange multiplier enforcing the pressure continuity across the interface, whereas mass balance results from essential boundary conditions on each domain. As a consequence, a typical saddle-point operator equation is obtained, and hence the classical Babuška–Brezzi theory is applied to establish the well-posedness of the continuous and discrete schemes. In particular, we remark that the continuous and discrete inf–sup conditions of the main bilinear form are proved by using suitably chosen injective operators to get lower bounds of the corresponding suprema, which constitutes a previously known technique, recently denominated T-coercivity. In turn, and consistently with the above, the stability of the Galerkin scheme requires that the curl of the finite element subspace approximating the vorticity be contained in the space where the discrete velocity of the fluid lives, which yields Raviart–Thomas and Nédélec finite element subspaces as feasible choices. Then we show that the aforementioned constraint can be avoided by augmenting the mixed formulation with a residual arising from the Brinkman momentum equation. Finally, several numerical examples illustrating the satisfactory performance of the methods and confirming the theoretical rates of convergence are reported.

Comments on two analyses of thermal non-equilibrium Darcy–Brinkman convection in cylindrical porous media

The problem of forced convection heat transfer in porous media using the non-thermal equilibrium two-equation model is revisited analytically. Darcy-extended Brinkman momentum equation is used to describe the velocity profile inside the porous medium. The solution steps for the volume-averaged equations are outlined concisely to allow easy comparison to existing solutions in the literature. Using a case of air flow in open-cell metal foam, the fluid temperature is obtained by the current solution and by a solution existing in the literature. The pre-existing solution produced a physically-unexplained behavior. What seemed to be an error in the pre-existing solution was identified and described in detail. Robust solutions are useful in parametric studies, initial engineering design and for calibration of numerical solutions of more complex cases.

Fully Developed Forced Convection Heat Transfer in a Porous Channel with Asymmetric Heat Flux Boundary Conditions

An analytical study is performed on steady, laminar, and fully developed forced convection heat transfer in a parallel plate channel with asymmetric uniform heat flux boundary conditions. The channel is filled with a saturated porous medium, and the lower and upper walls are subjected to different uniform heat fluxes. The dimensionless form of the Darcy–Brinkman momentum equation is solved to determine the dimensionless velocity profile, while the dimensionless energy equation is solved to obtain temperature profile for a hydrodynamically and thermally fully developed flow in the channel. Nusselt numbers for the lower and upper walls and an overall Nusselt number are defined. Analytical expressions for determination of the Nusselt numbers and critical heat flux ratio, at which singularities are observed for individual Nusselt numbers, are obtained. Based on the values of critical heat flux ratio and Darcy number, a diagram is provided to determine the direction of heat transfer between the lower or upper walls while the fluid is flowing in the channel.

Effect of the Darcy number on the energy flow and operating conditions of a thermoacoustic porous-medium system

In this paper, a simplified porous medium thermoacoustic system is modeled to observe its energy interaction characteristics and identify its operating conditions mainly as a function of porous medium Darcy number. The governing Darcy–Brinkman momentum equation and energy equation are simplified and linearized by using a first order perturbation analysis. Similar perturbation analysis is usually used to solve the linear thermoacoustic problem in the low Mach number limit. Simplified momentum and energy equations are solved, in the frequency domain, in order to obtain the expressions of the fluctuating velocity ( u 1) and temperature ( T 1). Time averaged and space averaged heat fluxes and work fluxes are calculated using the expressions of fluctuating velocity and temperature. The effects of the drive ratio (DR), Darcy number ( Da δ ), temperature gradient (∇ T m ), and frequency ( f) on the heat flux, work flux, and operating conditions are discussed and graphically presented.

Rotating Brinkman–Lapwood Convection with Modulation

The purpose of this article is to analyze, theoretically, the effect of modulation on rotating Brinkman–Lapwood convection, i.e., buoyancy-driven convection in a sparse porous medium subjected to rotation. Darcy–Brinkman momentum equation with Coriolis term has been used to describe the flow. The system is considered rotating about an axis with non-uniform rotation speed. In particular, we assume that the rotation speed is varying sinusoidally with time. A linear stability analysis has been performed to find the critical Rayleigh number in modulated case. The effect of modulated rotation speed is found to have a stabilizing effect on the onset of convection for different values of modulation frequency and the other physical parameters involved.

Entropy generation for forced convection in a porous saturated circular tube with uniform wall temperature

A numerical study is reported to investigate both the First and the Second Law of Thermodynamics for thermally developing forced convection in a circular tube filled by a saturated porous medium, with uniform wall temperature, and with the effects of viscous dissipation included. A theoretical analysis is also presented to study the problem for the asymptotic region applying the perturbation solution of the Brinkman momentum equation reported by (K. Hooman, K., A.A. Ranjbar-Kani, A perturbation based analysis to investigate forced convection in a porous saturated tube, Journal of Computational and Applied Mathematics 162 (2) (2004) 411–419.). Expressions are reported for the temperature profile, the Nusselt number, the Bejan number, and the dimensionless entropy generation rate in the asymptotic region. Numerical results are found to be in good agreement with theoretical counterparts.

Analysis of electroosmotic flow in a microchannel packed with microspheres

The electroosmotic flow in a microchannel packed with microspheres under both direct and alternating electric fields is analyzed. In the case of the steady DC electroosmosis in a packed microchannel, the so-called “capillary model” is used, in which it is assumed that a porous medium is equivalent to a series of intertwined tubules. The interstitial tubular velocity is obtained by analytically solving the Navier–Stokes equation and the complete Poisson–Boltzmann equation. Then, using the volume-averaging method, the solution for the electroosmotic flow in a single charged cylindrical tubule is applied to estimate the electroosmosis in the overall porous media by introducing the porosity and tortuosity. Assuming uniform porosity, an exact solution accounting for the electrokinetic wall effect is obtained by solving the modified Brinkman momentum equation. For the electroosmotic flow under alternating electric fields in a cylindrical microchannel packed with microspheres of uniform size, two different conditions regarding the openness of the channel ends are considered. Based on the capillary model, the time-periodic oscillating electroosmotic flow in an open-ended microchannel in response to the application of an alternating electric field is obtained using the Green’s function approach to the Navier–Stokes equation. When the two ends of the channel are closed, a backpressure is induced to generate a counter flow, resulting in a new zero flow rate. Such induced backpressure associated with the counter flow in a closed-end microchannel is obtained analytically by solving the transient modified Brinkman momentum equation.

Conjugate heat transfer inside a porous channel

Analytical and numerical analyses have been performed for fully developed forced convection in a fluid-saturated porous medium channel bounded by two parallel plates. The channel walls are assumed to be finite in thickness. Conduction heat transfer inside the channel wall is also accounted and the full problem is treated as a conjugate heat transfer problem. The flow in the porous material is described by the Darcy–Brinkman momentum equation. The outer surfaces of the solid walls are treated as isothermal. A temperature dependent volumetric heat generation is considered inside the solid wall only. Analytical expressions for velocity, temperature, and Nusselt number are obtained after simplifying and solving the governing differential equations with reasonable approximations. Subsequent results obtained by numerical calculations show an excellent agreement with the analytical results.

Flow, thermal, and entropy generation characteristics inside a porous channel with viscous dissipation

We present both analytical and numerical analyses of fully developed forced convection and entropy generation in a fluid-saturated porous medium channel bounded by two parallel plates. The flow in the porous material is described by the Darcy–Brinkman momentum equation. Differentially heated isothermal walls are selected as thermal boundary conditions. Analytical expressions for velocity, temperature, Nusselt number, entropy generation rate, and heat transfer irreversibility are obtained after simplifying and solving the governing differential equations with reasonable approximations. Subsequent results obtained by numerical calculations show excellent agreement with analytical results.

AC electroosmosis in microchannels packed with a porous medium

This paper presents a theoretical study on ac-driven electroosmotic flow in both open-end and closed-end microchannels packed with uniform charged spherical microparticles. The time-periodic oscillating electroosmotic flow in an open-end capillary in response to the application of an alternating (ac) electric field is obtained using the Green function approach. The analysis is based on the Carman–Kozeny theory. The backpressure associated with the counter-flow in a closed-end capillary is obtained by analytically solving the modified Brinkman momentum equation. It is demonstrated that in a microchannel with its two ends connected to reservoirs and subject to ambient pressure, the oscillating Darcy velocity profile depends on both the pore size and the excitation frequency; such effects are coupled through an important aspect ratio of the tubule radius to the Stokes penetration depth. For a fixed pore size, the magnitude of the ac electroosmotic flow decreases with increasing frequency. With increasing pore size, however, the magnitude of the maximum velocity shows two different trends with respect to the excitation frequency: it gets higher in the low frequency domain, and gets lower in the high frequency domain. In a microchannel with closed ends, for a fixed excitation frequency, use of smaller packing particles can generate higher backpressure. For a fixed pore size, the backpressure magnitude shows two different trends changing with the excitation frequency. When the excitation frequency is lower than the system characteristic frequency, the backpressure decreases with increasing excitation frequency. When the excitation frequency is higher than the system characteristic frequency, the backpressure increases with increasing excitation frequency.

A perturbation based analysis to investigate forced convection in a porous saturated tube

Fully developed laminar steady forced convection inside a circular tube filled with saturated porous medium and with uniform heat flux at the wall is investigated on the basis of a Brinkman model which is appropriate for hyperporous materials which are of current practical importance. The “WKB method” is applied for small values of the Darcy number. For the case of large Darcy number, the solution for the Brinkman momentum equation is found in terms of an asymptotic expansion. With the velocity distribution determined, the energy equation is solved using the same asymptotic technique. The results for limiting cases were found to be in good agreement with those available in the open literature.

Forced convection in a fluid-saturated porous-medium tube with isoflux wall

Two solutions are presented to investigate fully developed laminar steady forced convection inside a circular tube filled with saturated porous medium and with uniform heat flux at the wall on the basis of a Brinkman model. The WKB method is applied for small values of the Darcy number. For the case of large Darcy number the solution for the Brinkman momentum equation is found in terms of an asymptotic expansion. Having known the velocity distribution, the energy equation is solved using the same asymptotic suggestions. The present solution is accurate for all limiting values of the Darcy number. A numerical solution is utilized for the problem also. This latter solution is performed to check the accuracy of the former one. It was seen that in the limiting regions -as we expected- the results of both methods are in complete agreement with each other and also with those previously shown in the open literature. The results found hear are appropriate for hyperporous materials which are of current practical importance.