Forced convection gaseous slip flow in circular porous micro-channels

Laminar forced convection of gaseous slip flow in a circular micro-channel filled with porous media under local thermal equilibrium condition is studied numerically using the finite difference technique. Hydrodynamically fully developed flow is considered and the Darcy–Brinkman–Forchheimer model is used to model the flow inside the porous domain. The present study reports the effect of several operating parameters (Knudsen number (Kn), Darcy number (Da), Forchhiemer number (Γ), and modified Reynolds number ) on the velocity slip and temperature jump at the wall. Results are given in terms of the velocity distribution, temperature distribution, skin friction , and the Nusselt number (Nu). It is found that the skin friction is increased by (1) decreasing Knudsen number, (2) increasing Darcy number, and (3) decreasing Forchheimer number. Heat transfer is found to (1) decrease as the Knudsen number, or Forchheimer number increase, (2) increase as the Peclet number or Darcy number increase.     

Slip coefficients and macroparameter jumps of a diatomic gas with rotational degrees of freedom on a weakly curved spherical surface

Using the half-space moment method, the problem of the slip of a diatomic gas along a rigid spherical surface is solved within the framework of a model kinetic equation previously proposed which takes into account the rotational degrees of freedom of the gas. Second-order slip coefficients (correctionsC _m ^′ , β _R ^′ , and β _R to the isothermal and thermal slip which are linear with respect to the Knudsen number Kn) are obtained. The gas macroparameter jump coefficientsC _v andC _q, which are of the second order in the Knudsen number and characterize the discontinuity of the normal mass and heat fluxes on the gas-rigid phase interface, are calculated. These coefficients are given as functions of the tangential momentum accommodation coefficient, the translational and rotational energy accommodation coefficients, and the Prandtl number. The coefficients are calculated for certain diatomic gases. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 163–173, January–February, 2000.     

Mixed Convection Heat Transfer in the Annulus Between Two Concentric Vertical Cylinders Using Porous Layers

A numerical investigation of the steady-state, laminar, axi-symmetric, mixed convection heat transfer in the annulus between two concentric vertical cylinders using porous inserts is carried out. The inner cylinder is subjected to constant heat flux and the outer cylinder is insulated. A finite volume code is used to numerically solve the sets of governing equations. The Darcy–Brinkman–Forchheimer model along with Boussinesq approximation is used to solve the flow in the porous region. The Navier–Stokes equation is used to describe the flow in the clear flow region. The dependence of the average Nusselt number on several flow and geometric parameters is investigated. These include: convective parameter, λ, Darcy number, Da, thermal conductivity ratio, K _r, and porous-insert thickness to gap ratio (H/D). It is found that, in general, the heat transfer enhances by the presence of porous layers of high thermal conductivity ratios. It is also found that there is a critical thermal conductivity ratio on which if the values of Kr are higher than the critical value the average Nusselt number starts to decrease. Also, it found that at low thermal conductivity ratio (K _r ≈ 1) and for all values of λ the porous material acts as thermal insulation.     

The Use of Porous Fins for Heat Transfer Augmentation in Parallel-Plate Channels

In this study, a steady, fully developed laminar forced convection heat augmentation via porous fins in isothermal parallel-plate duct is numerically investigated. High-thermal conductivity porous fins are attached to the inner walls of two parallel-plate channels to enhance the heat transfer characteristics of the flow under consideration. The Darcy–Brinkman–Forchheimer model is used to model the flow inside the porous fins. This study reports the effect of several operating parameters on the flow hydrodynamics and thermal characteristics. This study demonstrates, mainly, the effects of porous fin thickness, Darcy number, thermal conductivity ratio, Reynolds number, and microscopic inertial coefficient on the thermal performance of the present flow. It is found that the highest Nusselt number is achieved at fully filled porous duct which requires the highest pumping pressure. The results show that using porous fins requires less pumping pressure with comparable high heat augmentation weight against fully filled porous duct. It is found that higher Nusselt numbers are achieved by increasing the microscopic inertial coefficient (A), the Reynolds number (Re), and the thermal conductivity of the porous substrate k ₂. The results show that heat transfer can be enhanced (1) with the use of high thermal conductivity fins, (2) by decreasing the Darcy number, and (3) by increasing microscopic inertial coefficient.     

Computational modeling of biomagnetic micropolar blood flow and heat transfer in a two-dimensional non-Darcian porous medium

We study theoretically and computationally the incompressible, non-conducting, micropolar, biomagnetic (blood) flow and heat transfer through a two-dimensional square porous medium in an (x,y) coordinate system, bound by impermeable walls. The magnetic field acting on the fluid is generated by an electrical current flowing normal to the x–y plane, at a distance l beneath the base side of the square. The flow regime is affected by the magnetization B ₀ and a linear relation is used to define the relationship between magnetization and magnetic field intensity. The steady governing equations for x-direction translational (linear) momentum, y-direction translational (linear) momentum, angular momentum (micro-rotation) and energy (heat) conservation are presented. The energy equation incorporates a special term designating the thermal power per unit volume due to the magnetocaloric effect. The governing equations are non-dimensionalized into a dimensionless (ξ,η) coordinate system using a set of similarity transformations. The resulting two point boundary value problem is shown to be represented by five dependent non-dimensional variables, f _ξ  (velocity), f _η (velocity), g (micro-rotation), E (magnetic field intensity) and θ (temperature) with appropriate boundary conditions at the walls. The thermophysical parameters controlling the flow are the micropolar parameter (R), biomagnetic parameter (N _H ), Darcy number (Da), Forchheimer (Fs), magnetic field strength parameter (Mn), Eckert number (Ec) and Prandtl number (Pr). Numerical solutions are obtained using the finite element method and also the finite difference method for Ec=2.476×10⁻⁶ and Prandtl number Pr=20, which represent realistic biomagnetic hemodynamic and heat transfer scenarios. Temperatures are shown to be considerably increased with Mn values but depressed by a rise in biomagnetic parameter (N _H ) and also a rise in micropolarity (R). Translational velocity components are found to decrease substantially with micropolarity (R), a trend consistent with Newtonian blood flows. Micro-rotation values are shown to increase considerably with a rise in R values but are reduced with a rise in biomagnetic parameter (N _H ). Both translational velocities are boosted with a rise in Darcy number as is micro-rotation. Forchheimer number is also shown to decrease translational velocities but increase micro-rotation. Excellent agreement is demonstrated between both numerical solutions. The mathematical model finds applications in blood flow control devices, hemodynamics in porous biomaterials and also biomagnetic flows in highly perfused skeletal tissue. Dedicated to Professor Y.C. Fung (1919-), Emeritus Professor of Biomechanics, Bioengineering Department, University of California at San Diego, USA for his seminal contributions to biomechanics and physiological fluid mechanics over four decades and his excellent encouragement to the authors, in particular OAB, with computational biofluid dynamics research.     

The Onset of Darcy–Brinkman Electroconvection in a Dielectric Fluid Saturated Porous Layer

The combined effect of a vertical AC electric field and the boundaries on the onset of Darcy–Brinkman convection in a dielectric fluid saturated porous layer heated either from below or above is investigated using linear stability theory. The isothermal bounding surfaces of the porous layer are considered to be either rigid or free. It is established that the principle of exchange of stability is valid irrespective of the nature of velocity boundary conditions. The eigenvalue problem is solved exactly for free–free (F/F) boundaries and numerically using the Galerkin technique for rigid–rigid (R/R) and lower-rigid and upper-free (F/R) boundaries. It is observed that all the boundaries exhibit qualitatively similar results. The presence of electric field is emphasized on the stability of the system and it is shown that increasing the AC electric Rayleigh number R _ea is to facilitate the transfer of heat more effectively and to hasten the onset of Darcy–Brinkman convection. Whereas, increase in the ratio of viscosities Λ and the inverse Darcy number Da ⁻¹ is to delay the onset of Darcy–Brinkman electroconvection. Besides, increasing R _ea and Da ⁻¹ as well as decreasing Λ are to reduce the size of convection cells.     

Effects of Knudsen number and geometry on gaseous flow and heat transfer in a constricted microchannel

A flow and heat transfer numerical simulation is performed for a 2D laminar incompressible gas flow through a constricted microchannel in the slip regime with constant wall temperature. The effects of rarefaction, creeping flow, first order slip boundary conditions and hydrodynamically/thermally developing flow are assumed. The effects of Knudsen number and geometry on thermal and hydrodynamic characteristics of flow in a constricted microchannel are explored. SIMPLE algorithm in curvilinear coordinate is used to solve the governing equations including continuity, energy and momentum with the temperature jump and velocity slip conditions at the solid walls in discretized form. The resulting velocity and temperature profiles are then utilized to obtain the microchannel C _f Re and Nusselt number as a function of Knudsen number and geometry. The results show that Knudsen number has declining effect on the C _f Re and Nusselt number in the constricted microchannel. In addition, the temperature jump on wall and slip velocity increase with increasing Knudsen number. Moreover, by decreasing the throttle area, the fluid flow characteristics experience more intense variations in the constricted region. To verify the code a comparison is carried out with available results and good agreement is achieved.     

Numerical simulation of the flow around a porous covering square cylinder in a channel via lattice Boltzmann method

In this paper, a detailed investigation on the flow past a porous covering cylinder is presented through the lattice Boltzmann method. The Brinkman‐Forchheimer‐extended Darcy model is adopted for the entire flow field with the solid, fluid, and porous medium. The effects of several parameters, such as porous layer thickness, Darcy number, porosity, and Reynolds number on flow field are discussed. Compared with the case of a solid cylinder, the present work shows that the porous layer may play an important role on the flow, the lift and drag force exerted on the cylinder. The numerical results indicate that the maximal drag coefficient C_d and maximal amplitude of lift coefficient C_l exist at certain Darcy number which is in the range of 10^?6–10^?2. Copyright © 2010 John Wiley & Sons, Ltd.     

Convection in a porous medium with velocity slip and temperature jump boundary conditions

The onset of convection in a rarefield gas saturating a horizontal layer of a porous medium has been investigated using both Darcy and Brinkman models. It is assumed that due to rarefaction both velocity slip and temperature jump exist at the boundaries. The results show that (i) when the degree of rarefaction increases the critical Rayleigh number as well as the critical wave number for the onset of convection increases, (ii) stabilizing effect of temperature jump is more than that of velocity slip, (iii) Darcy model is seen to be the most stable one when compared to Brinkman model or the pure gaseous layer (i.e. in the absence of porous medium).     

The onset of buoyancy-driven convection in a ferromagnetic fluid saturated porous medium

The onset of buoyancy-driven convection in an initially quiescent ferrofluid saturated horizontal porous layer in the presence of a uniform vertical magnetic field is investigated. The Brinkman-Lapwood extended Darcy equation with fluid viscosity different from effective viscosity is used to describe the flow in the porous medium. The lower boundary of the porous layer is assumed to be rigid-paramagnetic, while the upper paramagnetic boundary is considered to be either rigid or stress-free. The thermal conditions include fixed heat flux at the lower boundary, and a general convective–radiative exchange at the upper boundary, which encompasses fixed temperature and fixed heat flux as particular cases. The resulting eigenvalue problem is solved numerically using the Galerkin technique. It is found that increase in the Biot number Bi, porous parameter σ, viscosity ratio Λ, magnetic susceptibility χ, and decrease in the magnetic number M ₁ and non-linearity of magnetization M ₃ is to delay the onset of ferroconvection in a porous medium. Further, increase in M ₁, M ₃, and decrease in χ, Λ, σ and Bi is to decrease the size of convection cells.     

A theoretical analysis of forced convection in a porous-saturated circular tube: Brinkman–Forchheimer model

Fully developed 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–Forchheimer model. The matched asymptotic expansion method is applied at small Darcy numbers. For large Darcy numbers, the solution for the Brinkman–Forchheimer momentum equation is found in terms of an asymptotic expansion. Once the velocity distribution is determined, the energy equation is solved using the same asymptotic technique. The results for the two limiting cases of clear fluid and Darcy flow conditions show good agreement with those available in the literature.     

Natural Convection in a Cavity Filled with Porous Layers on the Top and Bottom Walls

Natural convection in a partially filled porous square cavity is numerically investigated using SIMPLEC method. The Brinkman-Forchheimer extended model was used to govern the flow in the porous medium region. At the porous-fluid interface, the flow boundary condition imposed is a shear stress jump, which includes both the viscous and inertial effects, together with a continuity of normal stress. The thermal boundary condition is continuity of temperature and heat flux. The results are presented with flow configurations and isotherms, local and average Nusselt number along the cold wall for different Darcy numbers from 10⁻¹ to 10⁻⁶, porosity values from 0.2 to 0.8, Rayleigh numbers from 10³ to 10⁷, and the ratio of porous layer thickness to cavity height from 0 to 0.50. The flow pattern inside the cavity is affected with these parameters and hence the local and global heat transfer. A modified Darcy–Rayleigh number is proposed for the heat convection intensity in porous/fluid filled domains. When its value is less than unit, global heat transfer keeps unchanged. The interfacial stress jump coefficients β ₁ and β ₂ were varied from  −1 to +1, and their effects on the local and average Nusselt numbers, velocity and temperature profiles in the mid-width of the cavity are investigated.     

Natural convection due to heat and mass transfer in a composite system

A numerical study is performed to analyse heat and mass transfer phenomena due to natural convection in a composite cavity containing a fluid layer overlying a porous layer saturated with the same fluid. The flow in the porous region is modelled using Brinkman–Forchheimer-extended Darcy model that includes both the effect of macroscopic shear (Brinkman effect) and flow inertia (Forchheimer effect). The vertical walls of the two-dimensional enclosure are isothermal whilst the horizontal walls are adiabatic. The two regions are coupled by equating the velocity and stress components at the interface. The resulting coupled equations in non-dimensional form are solved by an alternating direction implicit method by transforming them into parabolic form by the addition of false transient terms. The numerical results show that the amount of fluid penetration into the porous layer depends strongly upon the Darcy, thermal and solutal Rayleigh numbers. Average Nusselt number decreases while average Sherwood number increases with an increase of the Lewis number. The transfer of heat and mass on the heated wall near the interface depends strongly on the Darcy number. Received on 11 May 1998     

Analysis of Momentum Transfer in a Lid-Driven Cavity Containing a Brinkman–Forchheimer Medium

The CE/SE (the space-time conservation element and solution element method) scheme with the second-order accuracy has been proposed. And the pretreatment method has been introduced to convert the parabolic equations to the hyperbolic equations, which are accurately solved by the CE/SE method. The lid-driven rectangular cavity containing a porous Brinkman–Forchheimer medium is studied in this numerical investigation. The Brinkman–Forchheimer equation is used such that both the inertial and viscous effects are incorporated. The governing equations are solved by the improved CE/SE approach. The characteristics of the flow are analyzed with emphasis on the influence of the Darcy number and the cavity depth. It is found that the porous medium effect decreases both the strength and the number of eddies, especially for deep cavities.     

A Summary of New Predictive High Frequency Thermo-Vibrational Models in Porous Media

In this paper, we consider the effect of mechanical vibration on the onset of convection in porous media. The porous medium is saturated either by a pure fluid or by a binary mixture. The importance of a transport model on stability diagrams is presented and discussed. The stability threshold for the Darcy–Brinkman case in the Ra _Tc -R and k _c -R diagrams is presented (where Ra _Tc , k _c and R are the critical Rayleigh number, the critical wave number and the vibration parameters, respectively). It is shown that there is a significant deviation from the Darcy model. In the thermo-solutal case with the Soret effect, the influence of vibration on the reduction of multi-cellular convection is emphasized. A new analytical relation for obtaining the threshold of mono-cellular convection is derived. This relation shows how the separation factor Ψ is related to the controlling parameters of the problem, Ψ = f (R, ε*, Le), when the wave number k → 0. The importance of vibrational parameter definition is highlighted and it is shown how, by using a proper definition for vibrational parameter, we may obtain compact relationship. It is also shown how this result may be used to increase component separation.     

Effects of Chemical Reaction and Double Dispersion on Non-Darcy Free Convection Heat and Mass Transfer

In this article, the effects of chemical reaction and double dispersion on non-Darcy free convection heat and mass transfer from semi-infinite, impermeable vertical wall in a fluid saturated porous medium are investigated. The Forchheimer extension (non-Darcy term) is considered in the flow equations, while the chemical reaction power–law term is considered in the concentration equation. The first order chemical reaction (n = 1) was used as an example of calculations. The Darcy and non-Darcy flow, temperature and concentration fields in this study are observed to be governed by complex interactions among dispersion and natural convection mechanisms. The governing set of partial differential equations were non-dimensionalized and reduced to a set of ordinary differential equations for which Runge–Kutta-based numerical technique were implemented. Numerical results for the detail of the velocity, temperature, and concentration profiles as well as heat transfer rates (Nusselt number) and mass transfer rates (Sherwood number) are presented in graphs.     

A general treatment for non-Darcy film condensation within a porous medium in the presence of gravity and forced flow

Non-Darcy film condensation over a vertical flat plate within a porous medium is considered. The Forchheimer extended Darcy model is adopted to account for the non-Darcy effects on film condensation in the presence of both gravity and externally forced flow. A general similarity transformation is proposed upon introducing a modified Peclet number based on the total velocity of condensate, resulting from both gravitational force and externally forced flow. This general treatment makes it possible to obtain all possible similarity solutions including the asymptotic results in the four different limiting regimes, namely, Darcy forced convection regime, Forchheimer forced convection regime, Darcy body force predominant regime and Forchheimer body force predominant regime. Appropriate dimensionless groups for distinguishing these asymptotic regimes are found to be the micro-scale Grashof and Reynolds numbers based on the square root of the permeability of the porous medium. Correspondingly, the non-Darcy effect on the heat transfer rate are investigated in terms of these micro-scale dimensionless numbers.     

Effects of Viscous Dissipation on Unsteady Free Convection in a Fluid Past a Vertical Plate Immersed in a Porous Medium

The effects of viscous dissipation on unsteady free convection from an isothermal vertical flat plate in a fluid saturated porous medium are examined numerically. The Darcy–Brinkman–Forchheimer model is employed to describe the flow field. A new model of viscous dissipation is used for the Darcy–Brinkman–Forchheimer model of porous media. The simultaneous development of the momentum and thermal boundary layers are obtained by using a finite difference method. Boundary layer and Boussinesq approximation have been incorporated. Numerical calculations are carried out for various parameters entering into the problem. Velocity and temperature profiles as well as local friction factor and local Nusselt number are shown graphically. It is found that as time approaches infinity, the values of friction factor and heat transfer coefficient approach steady state.     

Finite Element Modelling of Flow Through a Porous Medium Between Two Parallel Plates Using The Brinkman Equation

Despite the widespread use of the Darcy equation to model porous flow, it is well known that this equation is inconsistent with commonly prescribed no slip conditions at flow domain walls or interfaces between different sections. Therefore, in cases where the wall effects on the flow regime are expected to be significant, the Darcy equation which is only consistent with perfect slip at solid boundaries, cannot predict velocity and pressure profiles properly and alternative models such as the Brinkman equation need to be considered. This paper is devoted to the study of the flow of a Newtonian fluid in a porous medium between two impermeable parallel walls at different Darcy parameters (Da). The flow regime is considered to be isothermal and steady. Three different flow regimes can be considered using the Brinkman equation: free flow (Da > 1), porous flow (high permeability, 1 > Da > 10⁻⁶) and porous flow (low permeability Da < 10⁻⁶). In the present work the described bench mark problem is used to study the effects of solid walls for a range of low to high Darcy parameters. Both no-slip and slip conditions are considered and the results of these two cases are compared. The range of the applicability of the Brinkman equation and simulated results for different cases are shown.     

Effect of Conduction in Bottom Wall on Darcy–Bénard Convection in a Porous Enclosure

Conjugate natural convection-conduction heat transfer in a square porous enclosure with a finite-wall thickness is studied numerically in this article. The bottom wall is heated and the upper wall is cooled while the verticals walls are kept adiabatic. The Darcy model is used in the mathematical formulation for the porous layer and the COMSOL Multiphysics software is applied to solve the dimensionless governing equations. The governing parameters considered are the Rayleigh number (100 ≤ Ra ≤ 1000), the wall to porous thermal conductivity ratio (0.44 ≤ K _r ≤ 9.90) and the ratio of wall thickness to its height (0.02 ≤ D ≤ 0.4). The results are presented to show the effect of these parameters on the heat transfer and fluid flow characteristics. It is found that the number of contrarotative cells and the strength circulation of each cell can be controlled by the thickness of the bottom wall, the thermal conductivity ratio and the Rayleigh number. It is also observed that increasing either the Rayleigh number or the thermal conductivity ratio or both, and decreasing the thickness of the bounded wall can increase the average Nusselt number for the porous enclosure.