Sliding of fine particles on the slip surface of rising gas bubbles: Resistance of liquid shear flows

In this paper a model was developed to describe the shear flow resistance force and torque acting on a fine particle as it slides on the slip surface of a rising gas bubble. The shear flow close to the bubble surface was predicted using a Taylor series and the numerical data obtained from the Navier–Stokes equations as a function of the polar coordinates at the bubble surface, the bubble Reynolds number, and the gas hold-up. The particle size was considered to be sufficiently small relative to the bubble size that the bubble surface could be locally approximated to a planar interface. The Stokes equation for the disturbance shear flows was solved for the velocity components and pressure using series of bispherical coordinates and the boundary conditions at the no-slip particle surface and the slip bubble surface. The solutions for the disturbance flows were then used to calculate the flow resistance force and torque on the particle as a function of the separation distance between the bubble and particle surfaces. The resistance functions were determined by dividing the actual force and torque by the corresponding (Stokes) force and torque in the bulk phase. Finally, numerical and simplified analytical rational approximate solutions for force correction factors for sliding particles as a function of the (whole range of the) separation distance are presented, which are in good agreement with the exact numerical result and can be readily applied to more general modelling of the bubble–particle interactions.     

The effect of the slip condition on Stokes and Couette flows due to an oscillating wall: exact solutions

Stokes and Couette flows produced by an oscillatory motion of a wall are analyzed under conditions where the no-slip assumption between the wall and the fluid is no longer valid. The motion of the wall is assumed to have a generic sinusoidal behavior. The exact solutions include both steady periodic and transient velocity profiles. It is found that slip conditions between the wall and the fluid produces lower amplitudes of oscillations in the flow near the oscillating wall than when no-slip assumption is utilized. Further, the relative velocity between the fluid layer at the wall and the speed of the wall is found to overshoot at a specific oscillating slip parameter or vibrational Reynolds number at certain times. In addition, it is found that wall slip reduces the transient velocity for Stokes flow while minimum transient effects for Couette flow is achieved only for large and small values of the wall slip coefficient and the gap thickness, respectively. The time needed to reach to steady periodic Stokes flow due to sine oscillations is greater than that for cosine oscillations with both wall slip and no-slip conditions.     

Darcy–Brinkman Flow Through a Corrugated Channel

A perturbation analysis is carried out to the second order to give effective equations for Darcy–Brinkman flow through a porous channel with slightly corrugated walls. The flow is either parallel or normal to the corrugations, and the corrugations of the two walls are either in phase or half-period out of phase. The present study is based on the assumptions that the corrugations are periodic sinusoidal waves of small amplitude, and the channel is filled with a sparse porous medium so that the flow can be described by the Darcy–Brinkman model, which approaches the Darcian or Stokes flow limits for small or large permeability of the medium. The Reynolds number is also assumed to be so low that the nonlinear inertia can be ignored. The effects of the corrugations on the flow are examined, quantitatively and qualitatively, as functions of the flow direction, the phase difference, and the wavelength of the corrugations, as well as the permeability of the channel. It is found that the corrugations will have greater effects when it is nearer the Stokes’ flow limit than the Darcian flow limit, and when the wavelength is shorter. For the same wavelength and phase difference, cross flow is more affected than longitudinal flow by the corrugations. Opposite effects can result from 180° out-of-phase corrugations, depending on the flow direction, the wavelength, as well as the permeability.     

AN ARTIFICIAL BOUNDARY CONDITION FOR TWO-DIMENSIONAL INCOMPRESSIBLE VISCOUS FLOWS USING THE METHOD OF LINES

We design an artificial boundary condition for the steady incompressible Navier–Stokes equations in streamfunction–vorticity formulation in a flat channel with slip boundary conditions on the wall. The new boundary condition is derived from the Oseen equations and the method of lines. A numerical experiment for the non-linear Navier–Stokes equations is presented. The artificial boundary condition is compared with Dirichlet and Neumann boundary conditions for the flow past a rectangular cylinder in a flat channel. The numerical results show that our boundary condition is more accurate.     

Slip–Brinkman Flow Through Corrugated Microannulus with Stationary Random Roughness

This paper presents a boundary perturbation method of the Brinkman-extended Darcy model to investigate the flow in corrugated microannuli cylindrical tubes with slip surfaces. The stationary random model is used to mimic the surface roughness of the cylindrical walls. The tube is filled with a porous medium. We shall consider the two cases where corrugations are either perpendicular or parallel to the flow, and particular attention is given to the effect of the phase shift. The effects of the corrugations on the flow rate and pressure gradient are investigated as functions of wavelength, the permeability of the medium, the radius ratio and the slip parameter. Particular surface roughnesses are examined as special cases of stationary random surface. It is found that the effect of the partial slip is significant on the corrugation functions. The limiting cases of Stokes and Darcy’s flows and no-slip case are discussed.     

Time‐dependent flow across a step: the slip with friction boundary condition

The paper studies numerically the slip with friction boundary condition in the time‐dependent incompressible Navier–Stokes equations. Numerical tests on two‐ and three‐dimensional channel flows across a step using this boundary condition on the bottom wall are performed. The influence of the friction parameter on the flow field is studied and the results are explained according to the physics of the flow. Due to the stretching and tilting of vortices, the three‐dimensional results differ in many respects from the two‐dimensional ones. Copyright © 2005 John Wiley & Sons, Ltd.     

A Mathematical Model for Determining Oil Migration Characteristics in Low-Permeability Porous Media Based on Fractal Theory

Shale reservoirs are characterized by very low permeability in the scale of nano-Darcy. This is due to the nanometer scale of pores and throats in shale reservoirs, which causes a difference in flow behavior from conventional reservoirs. Slip flow is considered to be one of the main flow regimes affecting the flow behavior in shale gas reservoirs and has been widely studied in the literature. However, the important mechanism of gas desorption or adsorption that happens in shale reservoirs has not been investigated thoroughly in the literature. This paper aims to study slip flow together with gas desorption in shale gas reservoirs using pore network modeling. To do so, the compressible Stokes equation with proper boundary conditions was applied to model gas flow in a pore network that properly represents the pore size distribution of typical shale reservoirs. A pore network model was created using the digitized image of a thin section of a Berea sandstone and scaled down to represent the pore size range of shale reservoirs. Based on the size of pores in the network and the pore pressure applied, the Knudsen number which controls the flow regimes was within the slip flow regime range. Compressible Stokes equation with proper boundary conditions at the pore’s walls was applied to model the gas flow. The desorption mechanism was also included through a boundary condition by deriving a velocity term using Langmuir-type isotherm. It was observed that when the slip flow was activated together with desorption in the model, their contributions were not summative. That, is the slippage effect limited the desorption mechanism through a reduction of pressure drop. Eagle Ford and Barnett shale samples were investigated in this study when the measured adsorption isotherm data from the literature were used. Barnett sample showed larger contribution of gas desorption toward gas recovery as compared to Eagle Ford sample. This paper has produced a pore network model to further understand the gas desorption and the slip flow effects in recovery of shale gas reservoirs.

     

The effect of slip distribution on flow past a circular cylinder

A slip boundary has been shown to have a significant impact on flow past bluff bodies. In this work and using a circular cylinder as a model system, the effects of various slip configurations on the passing flow are investigated. A theoretical analysis using matched-asymptotic expansion is first performed in the small-Reynolds number regime following Stokes and Oseen. A slip boundary condition is shown to lead to only higher-order effects (~1/ln(Re)) on the resulting drag coefficient. For higher Reynolds numbers (100–500), the effects of five types of symmetric slip boundary conditions, namely, no slip, fore-side slip, aft-side slip, flank slip, and all slip on the flow field and pertinent parameters are investigated with numerical simulations. Detailed results on the flow structure and force distribution are presented. Flank slip is found to have the best effect for drag reduction with comparable coverage of slip area. For asymmetric slip distributions, torque and lift are found to generally occur.     

Boundary conditions for stokes flows near a porous membrane

A theoretical development is carried out to model the boundary conditions for Stokes flows near a porous membrane, which, in general, allows non-zero slip as well as normal flow at the surface. Two types of models are treated: an infinitesimally thin plate with a periodic array of circular apertures and a series of parallel slits. For Stokes flows, the mean normal flux and slip velocity are proportional to the pressure difference across the membrane and the average shear stress at the membrane, respectively. The appropriate proportionality constants which depend on the membrane geometry are calculated as functions of the porosity. An interesting feature of the results is that the slip at the membrane has, in general, a direction different from that of the applied shear for these models.     

Investigation of the stability of a plane-channel suspension flow with account for finite particle volume fraction

Within the framework of the two-fluid approach, a variant of a heterogeneous-medium model which takes into account a finite volume fraction of the inclusions and a small but finite phase velocity slip is proposed. The interphase momentum exchange is described by the Stokes force with the Brinkman correction for the finite particle volume fraction. The suspension viscosity depends on the particle volume fraction in accordance with the Einstein formula. Within the framework of the model constructed, a formulation of the problem of linear stability of plane-parallel two-phase flows is proposed. As an example, the stability of a channel suspension flow is considered. The system of equations for small disturbances with the boundary conditions is reduced to an eigenvalue problem for a fourth-order ordinary differential equation. Using the orthogonalization method, the dependence of the critical Reynolds number on the governing nondimensional parameters of the problem is studied numerically. It is shown that taking a finite volume fraction of the inclusions into account significantly affects the laminar-turbulent transition limit.     

Oscillatory channel flow in a rotating system

Oscillatory channel flow in a rotating system is considered. The Navier–Stokes equations reduce to the Ekman equations that are solved exactly. The results show the interaction between oscillation frequency and rotation rate. Resonance occurs when the oscillation frequency is twice the rotation rate.     

Resistance of rough plates in a rotating fluid

Generalizing Navier’s partial slip condition, the flow due to a rough or striated plate moving in a rotating fluid is studied. It is found that the motion of the plate, the fluid surface velocity, and the shear stress are in general not in the same direction. The solution is extended to the case of finite depth, or Couette slip flow in a rotating system. In this case an optimum depth for minimum drag is found. The solutions are also closed form exact solutions of the Navier–Stokes equations. The results are fundamental to flows with Coriolis effects.     

Hydromagnetic flow through uniform channel bounded by porous media

The combined effects of the magnetic field, permeable walls, Darcy velocity, and slip parameter on the steady flow of a fluid in a channel of uniform width are studied. The fluid flowing in the channel is assumed to be homogeneous, incompressible,and Newtonian. Analytical solutions are constructed for the governing equations using Beavers-Joseph slip boundary conditions. Effects of the magnetic field, permeability,Darcy velocity, and slip parameter on the axial velocity, slip velocity, and shear stress are discussed in detail. It is shown that the Hartmann number, Darcy velocity, porous parameter, and slip parameter play a vital role in altering the flow and in turn the shear stress.     

Flow of Particulate-Fluid Suspension in a Channel with Porous Walls

The problem of the dispersed particulate-fluid two-phase flow in a channel with permeable walls under the effect of the Beavers and Joseph slip boundary condition is concerned in this paper. The analytical solution has been derived for the longitude pressure difference, stream functions, and the velocity distribution with the perturbation method based on a small width to length ratio of the channel. The graphical results for pressure, velocity, and stream function are presented and the effects of geometrical coefficients, the slip parameter and the volume fraction density on the pressure variation, the streamline structure and the velocity distribution are evaluated numerically and discussed. It is shown that the sinusoidal channel, accompanied by a higher friction factor, has higher pressure drop than that of the parallel-plate channel under fully developed flow conditions due to the wall-induced curvature effect. The increment of the channel’s width to the length ratio will remarkably increase the flow rate because of the enlargement of the flow area in the channel. At low Reynolds number ranging from 0 to 65, the fluids move forward smoothly following the shape of the channel. Moreover, the slip boundary condition will notably increase the fluid velocity and the decrease of the slip parameter leads to the increment of the velocity magnitude across the channel. The fluid-phase axial velocity decreases with the increment of the volume fraction density.     

Asymptotic theory of the Boltzmann system,for a steady flow of a slightly rarefied gas with a finite Mach number: General theory

A steady rarefied gas flow with Mach number of the order of unity around a body or bodies is considered. The general behaviour of the gas for small Knudsen numbers is studied by asymptotic analysis of the boundary-value problem of the Boltzmann equation for a general domain. The effect of gas rarefaction (or Knudsen number) is expressed as a power series of the square root of the Knudsen number of the system. A series of fluid-dynamic type equations and their associated boundary conditions that determine the component functions of the expansion of the density, flow velocity, and temperature of the gas is obtained by the analysis. The equations up to the order of the square root of the Knudsen number do not contain non-Navier–Stokes stress and heat flow, which differs from the claim by Darrozes (in Rarefied Gas Dynamics, Academic Press, New York, 1969). The contributions up to this order, except in the Knudsen layer, are included in the system of the Navier–Stokes equations and the slip boundary conditions consisting of tangential velocity slip due to the shear of flow and temperature jump due to the temperature gradient normal to the boundary.     

A lattice Boltzmann model for diffusion of binary gas mixtures that includes diffusion slip

This paper describes the development of a lattice Boltzmann (LB) model for a binary gas mixture, and applications to channel flow driven by a density gradient with diffusion slip occurring at the wall. LB methods for single component gases typically use a non‐physical equation of state in which the relationship between pressure and density varies according to the scaling used. This is fundamentally unsuitable for extension to multi‐component systems containing gases of differing molecular masses. Substantial variations in the species densities and pressures may exist even at low Mach numbers; hence, the usual linearized equation of state for small fluctuations is unsuitable. Also, existing methods for implementing boundary conditions do not extend easily to novel boundary conditions, such as diffusion slip. The new model developed for multi‐component gases avoids the pitfalls of some other LB models. A single computational grid is shared by all the species, and the diffusivity is independent of the viscosity. The Navier–Stokes equation for the mixture and the Stefan–Maxwell diffusion equation are both recovered by the model. Diffusion slip, the non‐zero velocity of a gas mixture at a wall parallel to a concentration gradient, is successfully modelled and validated against a simple one‐dimensional model for channel flow. To increase the accuracy of the scheme, a second‐order numerical implementation is needed. This may be achieved using a variable transformation method that does not increase the computational time. Simulations were carried out on hydrogen and water diffusion through a narrow channel for varying total pressure and concentration gradients. Copyright © 2011 John Wiley & Sons, Ltd.     

Slip and induced magnetic field effects on peristaltic transport of Johnson-Segalman fluid

The peristaltic flow of a Johnson-Segalman fluid in a planar channel is investigated in an induced magnetic field with the slip condition.The symmetric nature of the flow in a channel is utilized.The velocity slip condition in terms of shear stresses is considered.The mathematical formulation is presented,and the equations are solved under long wavelength and low Reynolds number approximations.The perturbation solutions are established for the pressure,the axial velocity,the micro-rotation component,the stream function,the magnetic-force function,the axial induced magnetic field,and the current distribution across the channel.The solution expressions for small Weissenberg numbers are derived.The flow quantities of interest are sketched and analyzed.     

Effective slip for flow through a channel bounded by lubricant-impregnated grooved surfaces

This study aims to investigate effective slip arising from pressure-driven flow through a slit channel bounded by lubricant-impregnated grooved surfaces. The problem for flow over longitudinal grooves is solved analytically using the methods of domain decomposition and eigenfunction expansion, while that for flow over transverse grooves is solved numerically using the front tracking method. It is found that the effective slip length and the lubricant flow rate can depend strongly on the geometry of the microstructure, the direction of flow, and the lubricant viscosity. In particular, the effective slip can be effectively enhanced by increasing the thickness of a lubricating film atop the ribs. Under the same conditions, a flow that is parallel to the lubricant-impregnated grooves will have a larger effective slip, but also a larger lubricant flow rate, when compared with the case of flow normal to the grooves. It is also shown that, in the case of transverse grooves, because of the downward displacement of the interface between the working/lubricating fluids, the effective slip length and lubricant flow rate may vary non-monotonically with the groove depth.