Velocity Slip Boundary Condition Research Articles

Numerical Simulation of Rarefied Gas Flow Through a Porous Medium.

A rarefied gas flow in a porous medium has been numerically investigated for the slip-flow regime, using a two-dimensional periodic model of porous structure. A macrmoscopically uniform flow is assumed to pass through a collection of square rods placed regularly in an infinite space, where a temperature gradient is imposed perpendicularly to the flow direction. Continuity, momentum and energy equations are solved numerically, using the slip velocity and temperature boundary conditions, to describe the microscopic velocity and temperature fields at a pore scale. The numerical results thus obtained are integrated over a unit structure to determine the effective permeability and the molecular diffusion due to tortuosity. The effects of Knudsen number on the permeability and thermal conductivity are elucidated

Application of Differing Forcing Function Models on Simulated Flow Past an Oscillating Cylinder in a Uniform Low Reynolds Number Flow

A Computational Fluid Dynamics (CFD) model is presented for the uniform viscous two dimensional flow past an oscillating cylinder at low Reynolds number. Numerical simulations are made to study the effect of differing forced induced oscillation mechanisms with a large range of cylinder forcing frequencies. In the first case sinusoidal velocity slip boundary conditions are adopted for the cylinder surface to simulate cylinder oscillation. The implication suggests that no modification or additional term need to be added to the Navier-Stokes equations. In the second case this time extra body force terms which are assumed to account for velocity effects due to cylinder movement are included in the Navier-Stokes equations with the imposition of same boundary conditions. Drag and lift coefficients are extracted from present numerical results and other detailed computations of these coefficients are made at a Reynolds number of 80 and an amplitude-to diameter ratio 0.14. The results are found to be in agreement with each other at low force driving frequencies below and near lock-in. However, differences are found at higher frequencies above lock-in. Agreement are also found with experimental results at some frequency ranges.

Gaseous slip flow in long microchannels

An analytic and experimental investigation into gaseous flow with slight rarefaction through long microchannels is undertaken. A two-dimensional (2-D) analysis of the Navier-Stokes equations with a first-order slip-velocity boundary condition demonstrates that both compressibility and rarefied effects are present in long microchannels. By undertaking a perturbation expansion in /spl epsiv/, the height-to-length ratio of the channel, and using the ideal gas equation of state, it is shown that the zeroth-order analytic solution for the streamwise mass flow corresponds well with the experimental results. Also, the effect of slip upon the pressure distribution is derived, and it is obtained that this slip velocity leads directly to a wall-normal migration of mass. The fabrication of wafer-bonded microchannels that possess well-controlled surface structure is described, and a means for accurately measuring the mass how through the channels is presented. Experimental results obtained with this mass-flow measurement technique for streamwise helium mass flow through microchannels 52.25-/spl mu/m wide, 1.33-/spl mu/m deep, and 7500-/spl mu/m long for a pressure range of 1.6-4.2 atmospheres (outlet pressures at atmospheric) are presented and shown to compare favorably with the analysis.

Nonmonotonic relationship between the Reynolds number and the lift coefficient of a plate in a hypersonic rarefied gas flow

The aerodynamic coefficients of a plate in a hypersonic stream at low Reynolds numbers are studied over a wide range of similarity parameters. The dependence of the lift coefficientCY on the tangential force coefficient, the finite Mach number at the outer edge of the boundary layer and the velocity-slip and temperature-jump boundary conditions is taken into consideration. The nonmonotonic character of the relationship betweenCY and the Reynolds number, revealed previously in experiments, is explained within the framework of the viscous hypersonic interaction model.

Slip and no-slip velocity boundary conditions at interface of porous, plain media

The hydrodynamic boundary condition at the interface between a porous and a plain medium is examined by direct simulation of the two-dimensional flow field near the interface of a porous medium made of cylinders. The existing slip boundary condition, which contains a slip coefficient α, and the noslip boundary condition, which contains an effective viscosity μ', are examined. The dependence of α on the direction of the flow (with respect to the interfacial plane), the porosity, the Reynolds number (based on the unit cell length and the Darcean velocity), the selection of the interfacial location, and the arrangement of the cylinders (structure) is examined in detail. The numerical results show that α is not only a function of the structure but also depends on the flow direction, the Reynolds number, the extent of the plain medium, and the nonuniformities in the arrangement of the surface particles. It is also shown that for an accurate prediction of the local velocity near the interface (inside the porous medium), μ' must vary within the porous medium. This shows that the Brinkman extension based on a uniform μ', and the associated screening distance, do not satisfactorily model the flow field in the porous medium.

The statistical distribution of the mean squared weighted deviation — Comment: Isochrons, errorchrons, and the use of MSWD-values

What role does the progressive geometric evolution of subduction-related mélange shear zones play in the development of strain transients? We use a “virtual shear box” experiment, based on outcrop-scale observations from an ancient exhumed subduction interface – the Chrystalls Beach Complex (CBC), New Zealand – to constrain numerical models of slip processes within a meters-thick shear zone. The CBC is dominated by large, competent clasts surrounded by interconnected weak matrix. Under constant slip velocity boundary conditions, models of the CBC produce stress cycling behavior, accompanied by mixed brittle-viscous deformation. This occurs as a consequence of the reorganization of competent clasts, and the progressive development and breakdown of stress bridges as clasts mutually obstruct one another. Under constant shear stress boundary conditions, the models show periods of relative inactivity punctuated by aseismic episodic slip at rapid rates (meters per year). Such a process may contribute to the development of strain transients such as slow slip.

On the performance of narrow porous journal bearing lubricated with couple stress fluid

In the present analysis an attempt has been made to study the performance characteristics of a narrow porous journal bearing lubricated with couple stress fluid. A modified form of Reynolds equation is derived for the lubrication of porous journal bearings with couple stress fluid as lubricant. The analysis takes into account the velocity slip at the surface of a porous medium by using Beavers-Joseph criterion. The governing equation for flow in the porous media and the modified Reynolds equation derived from the Stokes [1] constistutive equations for the couple stress fluid satisfying the velocity slip boundary condition, are solved analytically for the film pressure distribution. Eigen type of expansions for the field variations are obtained. The dimensionless load capacity, attitude angle and coefficient of friction are presented for different operating parameters. The effect of couple stress and velocity slip on the dynamic characteristics of narrow porous journal bearings are discussed. It is observed that the bearings with couple stress fluid as lubricant provide significant load carrying capacity and ensure considerable reduction in the coefficient of friction compared with viscous lubricants.

Sliding effect of gas-lubricated porous rectangular thrust bearings

A theoretical model has been developed to investigate the dynamic characteristics of porous rectangular thrust bearings lubricated with gas in which pressurized gas is pumped through a porous pad to lubricate the sliding elements. In the model a uniform temperature distribution in the gas lubricating film is assumed and a modified form of the Beaver-Joseph slip velocity boundary condition is applied at the interface between the porous pad and the lubricating film. The pressure in the porous pad is obtained in a closed form when the pad thickness is very small compared with its other two dimensions. A modified Reynold's equation was solved numerically using a finite difference method. The load capacity, mass flow rate of gas and stiffness are obtained in dimensionless forms and calculated numerically for different operating parameters. A higher load capacity is observed for a thinner film with and without relative motion. The effect of sliding is to reduce the load capacity and the stiffness, whereas it increases the mass flow rate of gas. It is also noted that when the pad shape ratio is very small, the effect introduced by sliding becomes less significant.

Supersonic low-density flow over airfoils

Two-dimensional, Reynolds-averaged, full Navier-Stokes equations for a perfect gas are solved for supersonic, low-density flow at Reynolds numbers ranging from 4 x 10-3.5 x 10. Slip-velocity boundary conditions based on various Knudsen numbers are introduced. This formulation is referred to as the extended NavierStokes method. Numerical results are obtained for three symmetric airfoils (NACA 0009, NACA 0012, and a 9%-thick, circular-arc airfoil) traveling at supersonic speeds at an angle of attack of 1.25 deg. The effect of the slip velocity becomes noticeable when the Knudsen number is greater than 0.01 and becomes important for Kn > 0.1. In a low-density atmosphere, the airfoil lift coefficients are low and drag coefficients are high because of strong viscous effects. The effect of the slip velocity tends to reduce both the lift and drag coefficients, but to a much smaller degree than the effect due to viscosity.

Analysis of Ultra-Thin Gas Film Lubrication Based on Linearized Boltzmann Equation: First Report—Derivation of a Generalized Lubrication Equation Including Thermal Creep Flow

A generalized Reynolds-type lubrication equation valid for arbitrary Knudsen numbers, defined as the ratio of the molecular mean free path to the film thickness, is derived from a linearized Boltzmann equation by semi-numerically calculating the flow rates of fundamental flows in the lubrication film: Poiseuille flow, Couette flow, and thermal creep flow. Numerical analysis of the equation for high Knudsen numbers reveals three principal results. First, Burgdorfer’s modified Reynolds equation featuring the first-order velocity slip boundary condition overestimates load carrying capacities, while the approximation equation including both the first- and second-order velocity slip boundary condition underestimates them. Second, since the flow rate of the Couette flow, which is independent of Knudsen numbers, becomes dominant as the bearing number increases, all the lubrication equation results tend toward the same asymptotic value for an infinite bearing number. Third, a new kind of load carrying capacity caused by thermal creep flow occurs if temperature gradients at the boundaries exist in the flow direction.

Analysis of lubricated bearing performance with very low clearances (Calculation by the finite element method)

The floating gap of a floating head slider for large capacity magnetic disk files is becoming smaller and smaller, the present values being from 0.3 to 0.5 μm. To accurately predict the floating gap, numerical analysis by the finite element method has been carried out using the modified Reynolds equation with 2nd order velocity slip boundary conditions. The higher order effects of molecular rarefaction are taken into consideration. The calculated results are in exellent agreement with the experimental ones at low sliding speeds and low ambient pressures.

Hybrid Porous Gas Journal Bearings: Steady State Solution Incorporating the Effect of Velocity Slip

A theoretical analysis is presented to predict the steady state performance characteristics of externally pressurized rotating gas journal bearings incorporating the effect of velocity slip at the porous interface. The governing equation for flow in the porous media and the modified Reynolds equation derived from the Navier-Stokes equations satisfying the velocity slip boundary condition, are solved simultaneously for film pressure distribution. Due to the nonlinearity of modified Reynolds equation the solution is obtained by perturbation method using finite difference technique. The dimensionless load capacity, attitude angle and mass rate of flow are computed numerically for different operating parameters. The effect of slip on the static characteristic is discussed. Comparison of the results with similar available solution for the no-slip case shows good agreement.

An Experimental Investigation of Molecular Rarefaction Effects in Gas Lubricated Bearings at Ultra-Low Clearances

The experimental investigation discussed here gives experimental confirmation of the slip-flow theory for modeling hydrodynamic gas bearings with clearances below 0.25 microns. An interferometric technique employing two CW lasers is used to measure the small clearances with an accuracy of 0.025 microns. The effects of molecular rarefaction are studied by operating the slider bearing in different gas media of different mean free paths. Bearings operating at extremely high local Knudsen numbers are studied without approaching excessively high bearing numbers. Experimentally measured trailing edge clearances and pitch angles are compared with theoretical predictions using the modified Reynolds equation with velocity slip boundary conditions. Excellent agreement between experiment and theory is found for clearances as high as 1.60 microns to as low as 0.075 microns with corresponding ambient Knudsen numbers of 0.04 and 2.51, respectively.

Analysis of aerostatic porous journal bearings using the slip velocity boundary conditions

Abstract A theoretical analysis of the static characteristics of aerostatic porous journal bearings is presented using the slip velocity boundary condition at the interface of the lubricating film and porous surface of the bearing. With thin wall bearings, the pressure in the porous bushing is obtained in closed form and the modified Reynolds equation is solved numerically using the finite difference method. The load capacity and the mass rate of flow are expressed non-dimensionally and are calculated numerically for different operating conditions. Variation in W and G with the feeding parameter and other parameters is shown graphically. The effect of velocity slip on W and G for different operating conditions was studied. The theoretical results are compared with a similar available solution using a three-dimensional flow model in the bushing. There is near-perfect agreement.

Molecular Mean Free Path Effects in Gas Lubricated Slider Bearings : 2nd Report, Experimental Studies

The latest magnetic disk files require lightly loaded air-lubricated slider bearings operating with film thicknesses less than 0.5μm to achieve high recording density. This paper reports results on flying characteristics obtained from experiments with such air bearings, which are influenced considerably by slip-flow effects. Experiments are carried out in the range of 30 to 70% load reductions due to slip-flow. White light interferometric techniques for measuring the spacing between a rotating glass disk and the slider are improved. For load reductions less than 60%, experimental data agree well with the numerical solution of a modified Reynolds equation which takes into account the velocity-slip boundary conditions. It is verified that the modified Reynolds equation can be used to predict the static behavior of slider bearings in designing magnetic flying heads. An effective calculation technique for the Reynolds equation by the finite element method is presented.

Effect of slip flow in aerostatic porous rectangular thrust bearings

A theoretical analysis of the static characteristics of aerostatic porous rectangular thrust bearings with uniform film thickness is presented. A modified form of the Beavers-Joseph slip velocity boundary condition is applied at the interface of the porous pad and the lubricating film. When the thickness of the pad is small compared with the other two dimensions, the pressure distribution in the porous media is obtained in closed form and the modified Reynolds equation can be solved numerically by the finite difference method. The load capacity, mass flow rate of the gas and static stiffness are expressed in dimensionless form and calculated numerically for different operating parameters. Velocity slip reduces the load capacity and the mass flow rate of the gas and depends on the feed parameter for its effect on stiffness.

Flow of viscous stratified fluid of variable viscosity past a porous bed

To study the effects of stratification and slip velocity on the flow of fluid of variable viscosity over a permeable bed, we divide the flow into two zones called zone 1 and zone 2. Zone 1 pertains to the flow called the free flow governed by the Navier-Stokes equations in the region between the impermeable upper plate and the porous bed.. Zone 2 pertains to the flow in the bed governed by the modified Darcy Law. Using the slip velocity boundary condition, velocity distributions in zones 1 and 2 are obtained and are matched at the interface. The boundary layer thickness just beneath the permeable interface and the friction factor are also obtained.

Rarefaction Effects of Gas-Lubricated Bearings in a Magnetic Recording Disk File

The load versus spacing characteristics of a self-acting, gas-lubricated slider bearing similar to that used in a magnetic recording disk file have been investigated experimentally under sub-atmospheric ambient conditions. Interferometric techniques are used to measure the steady spacing between a rotating glass disk and the slider over a wide range of ambient pressures and disk speeds. For local Knudsen number less than 0.1, excellent agreement is found to exist between experimental data and numerical solutions of the Reynolds lubrication equation taking into account the velocity-slip boundary conditions. Effects of rarefaction on the bearing performance for a range of pertinent bearing parameters (i.e., bearing number and inlet-to-outlet ratio) are presented.

The Slip-Dominated Merged Layer Regime for Newtonian Hypersonic Flow Past a Flat Plate

The rarefied uniform hypersonic flow past the leading edge of a sharp flat plate at zero angle of attack is analyzed on the basis of a continuum model consisting of the Navier–Stokes equations and the velocity-slip and temperature-jump plate boundary conditions. The model fluid is a perfect gas having constant specific heats, a constant Prandtl number of order unity, and first and second viscosity coefficients varying as a power of the absolute temperature. For this flow, it is taken that the Newtonian parameter, $\varepsilon = ( {\gamma - 1} )/( {\gamma + 1} )$, goes to zero, and that the freestream Mach number, $M = ( \rho _\infty u_\infty ^2 /\gamma p_\infty )^{1/2} $, the stagnation temperature parameter, $\theta _S = \{ 1 + \varepsilon )/( 1 - \varepsilon )\}\varepsilon M^2 $, and the free-stream Reynolds number (based on the characteristic axial length from the leading edge), $R_L = \rho _\infty u_\infty L/\mu _\infty $, go to infinity.For the viscosity-temperature exponent, $\omega $, satisfying $1...

On a New Approach to Slip Flow Using Rayleigh’s Problem

A new approach, involving a perturbation of the Navier-Stokes equations, is used to analyze the phenomenon of slip flow over a flat plate. An expression for the coefficient of drag is derived and compared to the drag coefficient obtained by the traditional approach of solving the Navier-Stokes equations with a slip-velocity boundary condition.