Brinkman-extended Darcy Model Research Articles

Direct and Inverse Solutions for Elastohydrodynamic Lubrication of Finite Porous Journal Bearings

In this paper a numerical solution for the elastohydrodynamic lubrication of a finite porous journal bearing is presented. The Brinkman–extended Darcy model was applied within the porous layer to incorporate the viscous shear stresses into the analysis. A simple elastic model is used to describe the elastic deformation of the liner. An inverse model is introduced to estimate the permeability parameter and/or the eccentricity ratio from experimentally obtained pressure data. As experimental measurements were not conducted, these data were simulated numerically. It consists of a direct solution for the pressure, for given parameters (permeability parameter and/or the eccentricity ratio), to which a normally distributed random error is added. The least-squares optimization technique is used to solve the proposed inverse problem. A unique solution was obtained for one-parameter estimation. The two-parameter estimation case, however, leads to multiple solutions lying on a constant load line.

Thermal convection in a sparcely packed porous layer saturated with suspended particles

This analytical study has considered the incipient convection of a sparcely packed porous medium saturated with suspended particles. A linear stability analysis is performed to determine the effects of the permeability of the medium and suspended components on the conditions for incipient instability under a range of thermal boundary conditions. The flow in the medium is modeled by using the Brinkman-extended Darcy model. The analysis reveals that the stability of the system is affected significantly by the presence of the porous medium, state of particles and the thermal boundary conditions. Moreover, the results of this study bridge the gap between thermal convection in a porous medium and in a classical fluid with suspended particles.

Transient film condensation on a vertical plate imbedded in porous medium

The present paper introduces a mathematical model that simulates the transient film condensation on a vertical plate imbedded in a porous medium. In this model, the Brinkman-extended Darcy model is adopted and the local macroscopic inertial term is included. Analytical solutions are presented that describe the transient behavior of the condensate film thickness, condensate mass flow rate and heat transfer coefficient. The current results show the effect of the permeability of the porous material on several issues including the velocity profiles, the film thickness and the time required to reach steady state conditions.

The Brinkman model for thermosolutal convection in a vertical annular porous layer

A numerical study is carried out on double-diffusive natural convection within a vertical circular porous annulus. Motions are driven by the externally applied constant temperature and concentration differences imposed across the vertical walls of the enclosure. In the formulation of the problem, use is made of the Brinkman extended Darcy model which allows the no-slip boundary condition on a solid wall, to be satisfied. The flow is assumed to be laminar and two dimensional. The density variation is taken into account by the Boussinesq approximation. The control-volume approach is used for solving the governing equations. Solutions for the flow fields, temperature and concentration distributions and Nusselt, Nu i and Sherwood, Sh i numbers are obtained in terms of the governing parameters of the problem. The effect of both the Darcy number, Da, and the radius ratio, κ, on Nu i and Sh i is found to be significant. Results for a pure viscous fluid and a Darcy (densely packed) porous medium emerge from the present model as limiting cases.

Natural convection in a cylindrical porous cavity with internal heat source: a numerical study with Brinkman-extended Darcy model

Natural convection in a 2-D vertical cylinder containing an isotropic porous media with internal heat generation was studied numerically for assessing the effect of Darcy’s law with and without the Brinkman extension on the streamlines, isotherms and Nusselt numbers. Two cases were analyzed: (1) insulated walls at the top and bottom of the cylinder and cooled external walls; (2) isothermally cooled walls at all external surfaces. The effect of the Brinkman extension was significant for high values of Da (10 −4–10 −1). In addition, the Nusselt number increases asymptotically as Da decreases. Also, four correlations for the average Nusselt number were derived.

Double-diffusive parallel flow induced in a horizontal Brinkman porous layer subjected to constant heat and mass fluxes: analytical and numerical studies

Fluid flow and heat and mass transfer induced by double-diffusive natural convection in a horizontal porous layer subjected to vertical gradients of temperature and concentration are studied analytically and numerically using the Brinkman-extended Darcy model. Both cases of rigid and free horizontal boundaries are examined in the present study. The parameters governing the problem are the Rayleigh number RT, the Lewis number Le, the buoyancy ratio N, the Darcy number Da and the aspect ratio Ar. The analytical solution is based on the parallel flow approximation. The critical Rayleigh number corresponding to the onset of the parallel flow in this system is determined analytically as a function of Le, N and Da. For sufficiently small Da, both free and rigid boundaries yield results which are identical to those predicted by the Darcy model. The present investigation shows that there exists a region in the plane (N, Le) where the convective flow is not possible in the layer regardless of the Rayleigh and Darcy numbers considered.

Derivation of Modified Reynolds Equation—A Porous Media Model

In this study, a porous media model is developed which can be applied to thin film lubrication problems. The microstructure of bearing surfaces is modeled as porous layers attached to the impermeable substrate. The Brinkman-extended Darcy equations and Stokes’ equations are utilized to model the flow in the porous region and fluid film region, respectively. The stress jump boundary condition at the porous media/fluid film interface and effects of viscous shear are included in deriving the modified Reynolds equation. The present model can correct and modify a previous study based on the Darcy model with slip-fiow effects or another based on the Brinkman-extended Darcy model with stress continuity at the porous media/fluid film interface. In the results, the effects of material properties: viscosity ratio (αi2), thickness of porous layer (Δi), permeability (Ki), stress jump parameter (βi), on the velocity distributions, and performance of one-dimensional converging wedge problems are discussed.

Thermoconvective instability in a horizontal porous cavity saturated with cold water

The onset of convection in a horizontal porous cavity with regard to the density maximum of water at 3.98°C is studied using a linear stability analysis. In the formulation of the problem use is made of the Brinkman-extended Darcy model which is relevant to sparsely packed porous media. A parabolic density-temperature relationship is used to model the effect of density inversion. The perturbation equations are solved with the aid of the Galerkin and finite element methods. The onset of motion is found to be dependent of the aspect ratio A of the cavity, the Darcy number Da, the inversion parameter γ and the hydrodynamic boundary conditions applied on the horizontal walls of the porous layer. The results for a viscous fluid ( Da→∞) and the Darcy porous medium ( Da→0) emerge from the present analysis as limiting cases. Numerical results for finite amplitude convection, obtained by solving the full governing equations, indicate that subcritical convection is possible when the upper stable layer extends over more than the half depth. Also, the existence of multiple solutions for a given range of governing parameters is demonstrated.

Derivation of the modified molecular gas lubrication equation - a porous media model

A porous media model has been developed which can be applied to ultra-thin gas film lubrication problems encountered in the analysis of a magnetic head slider flying over a magnetic disk coated with giant molecule liquid. In this model, the textured surface coated with giant molecule lubricant is replaced with a permeable material. The appropriate slip-flow boundary conditions, i.e. the high-order slip-flow model, and the proper porous flow model, i.e. the Brinkman-extended Darcy model, are utilized in the gas film and porous film, respectively. The modified molecular gas lubrication (modified MGL) equation is then derived by consideration of the permeability of the running surface. Finally, the effects of porous film thickness (), permeability (K) and inverse Knudsen number (D) on the performance of the magnetic head slider are discussed.

Natural convection between coaxial vertical cylinders partially filled with a porous material

This paper presents laminar fully developed free convection flow between two coaxial vertical cylinders partially filled with a porous matrix when the cylinders are kept at different temperatures. The Brinkman-extended Darcy model is used to simulate momentum transfer in the porous region. The two regions are coupled by equating the velocity and by considering shear stress jump condition at the interface. The effect of Darcy number on the velocity throughout porous domain and adjustable coefficient in the stress jump condition at the interface is shown graphically. It is observed that velocity is influenced by the shear stress jump condition at the interface.

Free-convection between vertical walls partially filled with porous medium

This paper presents an analytical study of laminar fully developed free-convection flow between two vertical walls partially filled with porous matrix and partially with a clear fluid having interface vertically. The momentum transfer in porous medium is described by the Brinkman-extended Darcy model and the two regions are coupled by equating the velocity and shear stress at the interface. The governing equations having non-linear nature have been solved by using perturbation method. It has been found that effect of Brinkman term is in entire porous domain for large values of Darcy number while its effect is confined nearer to interface and wall for small values of Darcy number.

Unsteady non-darcian fluid flow in parallel-plates channels partially filled with porous materials

Using Green's function method, analytical solutions are obtained for the problem of transient fluid flow in parallel-plate channels partially filled with porous materials. The unsteadiness in the fluid flow is caused either by a sudden change in the imposed pressure gradient or (and) by a sudden change in the velocity of the channel boundaries. The Brinkman-extended Darcy model is used to model the flow inside the porous domain.

Boundary-layer regime in a vertical porous layer with anisotropic permeability and boundary effects

Abstract In this paper, natural convection heat transfer in a vertical porous layer is studied both analytically and numerically. A constant heat flux is applied for heating and cooling the long side walls of the rectangular cavity, while the other two walls are insulated. The porous medium is assumed to be hydrodynamically anisotropic with its principal axes oriented in a direction that is oblique to the gravity vector. In the formulation of the problem, use is made of the Brinkman-extended Darcy model which allows the no-slip boundary condition on a solid wall, to be satisfied. The governing equations are solved analytically, under the boundary-layer approximations, using the modified Oseen technique. The effects of varying the Rayleigh and Darcy numbers, the anisotropic permeability ratio, and the orientation angle of the principle axes on the flow and heat transfer are investigated. A numerical study of the same phenomenon, obtained by solving the complete system of governing equations, is also conducted. Results are reported in the range 500 ≤ R ≤ 1000, 10 −7 ≤ Da ≤ 10 3 , 0.25 ≤ K ∗ ≤ 4, 0° ≤ φ ≤ 90°, and A = 1 and 4. Good agreement is found between the analytical prediction and the numerical solution.

Lubrication of Long Porous Slider Bearings. (Use of the Brinkman-Extended Darcy Model).

We apply the Brinkman-extended Darcy model to analyze the hydrodynamic lubrication of long porous slider bearings. The solutions are obtained and compared with those based on the Darcy model. The results show that the effects of viscous shear given by the Brinkman-extended Darcy model lead to a higher load capacity and a lower friction parameter. It is found that a thicker porous bearing gives a greater effect of viscous shear on load capacity and friction parameter.

Analytical solution of non-Darcian forced convection in an annular duct partially filled with a porous medium

An analytical solution is obtained for a fully developed, forced convection in a gap between two concentric cylinders. The inner is exposed to a constant heat flux and the outer is thermally insulated. A porous layer is attached to the inner cylinder. The effects of the permeability, thermal conductivity and the thickness of the porous material are investigated using a Brinkman-extended Darcy model. It is shown that there exists a critical thickness of the porous layer at which heat transfer is minimum in the case of low thermal conductivity materials, while this does not show for highly conducting materials. The obtained results show that increasing either the permeability or the thermal conductivity improves the heat transfer. Further, for highly permeable and conducting porous media, it may not be necessary to fill the gap completely to attain the maximum heat transfer.

Linear stability analysis of finite porous journal bearings—Use of the Brinkman-extended Darcy model

On the basis of the Brinkman-extended Darcy model (BEDM), this paper examines the effects of viscous shear stresses on the linear stability of finite porous journal bearings. The result shows that the influences of viscous shear stresses on the dynamic performance of porous journal-bearing systems are apparent and not negligible. In certain regions of parameter space the BEDM predicts quite different stability characteristics to those derived using either the slip-flow model (SFM) or Darcy model (DM). In particular, a higher threshold speed (increased stability) is predicted using the BEDM for porous bearings with (i) small length-to-diameter ratios ( λ < 1) or (ii) low permeability parameters and high steady-state eccentricity ratios.

Hydrodynamic lubrication of finite porous journal bearings—Use of the Brinkman-extended Darcy model

This paper applies the Brinkman-extended Darcy model (BEDM) to investigate the hydrodynamic lubrication of finite porous journal bearings. A finite difference technique employing the Preconditioned Conjugate Gradient method (PCGM) of iteration is used to solve the modified Reynolds equation. The results show that the influences of viscous shear stresses on the lubrication performance of porous journal-bearing systems are apparent and not negligible. Comparing with those of the Darcy model (DM), the effects of viscous shear stresses of the BEDM provide an increase in the load capacity, as well as a reduction in the friction parameter; and the quantitative effects of the viscous shear stresses on the static characteristics are more pronounced for the cases of large length-to-diameter ratios.

Static and dynamic characteristics of long porous journal bearings: use of the Brinkman-extended Darcy model

Based on the Brinkman-extended Darcy model, the main objective of this paper is to predict theoretically the effects of viscous shear stresses on the static and dynamic characteristics of long porous journal bearings. The static and dynamic characteristics obtained from the model, as the value of permeability parameter (D approaches zero,give good agreement with those of a solid bearing and hence provide support for the present predictions. The results show that the influences of viscous shear stresses on the static performance and the stability limits of porous journal-bearing systems are apparent and not negligible. In comparison with those of the slip-flow model and the Darcy model, the viscous shear effects of our model significantly provide an increase in the load capacity and static attitude angle as well as a reduction in the friction parameter of long porous journal bearings. Moreover, the long bearing system using our model under small disturbances is more stable than those of the slip-flow and Darcy models; and the quantitative effects on the stability criterion are more pronounced for the low-upsilon and high-epsilons. cases.

Forced convection heat transfer to an elastic fluid of constant viscosity flowing through a channel filled with a Brinkman-Darcy porous medium

An analysis is presented of fully developed flow and heat transfer in a channel confined by two parallel walls subjected to uniform heat flux in a highly porous medium saturated with an elastic fluid of constant viscosity. The Brinkman-extended Darcy model is used for studying the effect of the boundary viscous frictional drag on the heat transfer characteristics. The approximate integral method is employed to obtain a solution.

Lubrication of short porous journal bearings — use of the Brinkman-extended Darcy model

We apply the Brinkman-extended Darcy model within the porous medium to analyze the hydrodynamic lubrication of short porous journal bearings. The results show that the effects of viscous shear will increase the load capacity and decrease the coefficient of friction. Furthermore, the present analysis can correct and modify the previous study based on the Darcy model with slip-flow effects.