Poiseuille Flow of a Third Grade Fluid in a Porous Medium

Effects of porous medium have been investigated on the steady flow of a third grade fluid between two stationary porous plates. The continuity and momentum equations along with modified Darcy??s law are used for the development of mathematical problem. The governing nonlinear problem is solved by a homotopy analysis method. The dimensionless velocity and shear stresses at the plates are analyzed.     

Unsteady Flow of an Oscillating Porous Disk and a Fluid at Infinity

An exact analytic solution of the unsteady Navier–Stokes equations is obtained for the flow caused by the non-coaxial rotations of a porous disk and a fluid at infinity. The porous disk is executing oscillations in its own plane with superimposed injection or suction. An increasing or decreasing velocity amplitude of the oscillating porous disk is also discussed. Further, it is shown that a combination of suction/injection and decreasing/increasing velocity amplitude is possible as well. In addition, the flow due to porous oscillating disk and a fluid at infinity rotating about an axis parallel to the z-axis is attempted as a second problem. Sommario. Si studia il flusso non stazionario prodotto dall'oscillazione di un disco poroso in un fluido e si fornisce una soluzione analitica delle equazioni di Navier–Stokes. Si discute l'effetto di una suzione/iniezione e di una variazione sull'ampiezza della velocità' di oscillazione. Infine si studia il flusso dovuto alle oscillazioni non coassiali di un disco poroso e di un fluido all'infinito.     

Brinkman Flow of a Viscous Fluid Through a Spherical Porous Medium Embedded in Another Porous Medium

An analytical investigation for a two-dimensional steady, viscous, and incompressible flow past a permeable sphere embedded in another porous medium is presented using the Brinkman model, assuming a uniform shear flow far away from the sphere. Semi-analytical solutions of the problem are derived and relevant quantities such as velocities and shearing stresses on the surface of the sphere are obtained. The streamlines inside and outside the sphere and the radial velocity are shown in several graphs for different values of the porous parameters \({\sigma _1 =(\mu /\tilde {\mu }) (a/\sqrt{K_1 })}\) and \({\sigma _2 =(\mu /\tilde {\mu }) (a/\sqrt{K_2 })}\) , where a is the radius of the sphere, μ is the dynamic viscosity of the fluid, \({\tilde {\mu }}\) is an effective or Brinkman viscosity, while K ₁ and K ₂ are the permeabilities of the two porous media. It is shown that the dimensionless shearing stress on the sphere is periodic in nature and its absolute value increases with an increase of both porous parameters σ ₁ and σ ₂.     

Hall effects on oscillating flow due to eccentrically rotating porous disk and a fluid at infinity

Hall effects on the viscous incompressible fluid due to non-coaxial rotations of an oscillating porous disk and a fluid at infinity are studied. The velocity field, shear stresses and temperature distribution are obtained in closed form. It is found that with increase in frequency parameter, the primary velocity increases near the disk and becomes almost stationary away from the disk. The secondary velocity also increases with increase in frequency parameter. It is seen that with increase in Hall parameter, the primary velocity increases near the disk and decreases away from the disk. The reversed effect is observed for the secondary velocity. The shear stresses at the disk are also obtained. It is found that the shear stresses due to the primary and the secondary velocities decrease with increase in Hall parameter. The heat transfer characteristic is also studied on taking viscous dissipation into account. It is found that the mean temperature at the disk decreases with increase in Hall parameter.     

Oscillatory Motion of a Viscous Fluid in Contact with a Flat Layer of a Porous Medium

Analytical solutions are obtained for two problems of transverse internal waves in a viscous fluid contacting with a flat layer of a fixed porous medium. In the first problem, the waves are considered which are caused by the motion of an infinite flat plate located on the fluid surface and performing harmonic oscillations in its plane. In the second problem, the waves are caused by periodic shear stresses applied to the free surface of the fluid. To describe the fluid motion in the porous medium, the unsteady Brinkman equation is used, and the motion of the fluid outside the porous medium is described by the Navier–Stokes equation. Examples of numerical calculations of the fluid velocity and filtration velocity profiles are presented. The existence of fluid layers with counter-directed velocities is revealed.     

Static Deformations of a Linear Elastic Porous Body Filled with an Inviscid Fluid

We study infinitesimal deformations of a porous linear elastic body saturated with an inviscid fluid and subjected to conservative surface tractions. The gradient of the mass density of the solid phase is also taken as an independent kinematic variable and the corresponding higher-order stresses are considered. Balance laws and constitutive relations for finite deformations are reduced to those for infinitesimal deformations, and expressions for partial surface tractions acting on the solid and the fluid phases are derived. A boundary-value problem for a long hollow porous solid cylinder filled with an ideal fluid is solved, and the stability of the stressed reference configuration with respect to variations in the values of the coefficient coupling deformations of the two phases is investigated. An example of the problem studied is a cylindrical cavity leached out in salt formations for storing hydrocarbons.     

膨胀性非牛顿流体多孔介质流动的模拟分析

在表征体元尺度采用格子Boltzmann方法分析膨胀性非牛顿流体在多孔介质中的流动,基于二阶矩模型在演化方程中引入表征介质阻力的作用力项,求解描述渗流模型的广义Navier-Stokes方程．采用局部法计算形变速率张量,通过循环迭代得到非牛顿粘度和松弛时间．对多孔介质的Poiseuille流动进行分析,通过比较发现结果与孔隙尺度的解析解十分吻合,并且收敛较快,表明方法合理有效．分析了渗透率和幂律指数对速度和压力降的影响,研究结果表明,膨胀性流体的多孔介质流动不符合达西规律,压力降的增加幅度小于渗透率的减小幅度．当无量纲渗透率Da小于10-5时,流道中的速度呈现均匀分布,并且速度分布随着幂律指数的减小趋于平滑．压力降随着幂律指数的增加而增加,Da越大幂律指数对压力降的影响越明显．     

Oscillatory flow due to eccentrically rotating porous disk and a fluid at infinity

An analytical solution of the unsteady Navier–Stokes equations is obtained for the flow due to non-coaxial rotations of an oscillating porous disk and a fluid at infinity, rotating about an axis parallel to the axes of rotation of the disk through a fixed point. The velocity distributions and the shear stresses at the disk are obtained for three different cases when the frequency parameter is greater than, equal to or less than the rotation parameter. The flow has a boundary layer structure even in the case of blowing at the disk.     

Unsteady motion of a non-linear viscoelastic fluid

In this paper, we study the unsteady flow of a generalized second grade fluid. Specifically, we solve numerically the linear momentum equations for the flow of this viscoelastic shear-thinning (shear-thickening) fluid surrounding a solid cylindrical rod that is suddenly set into longitudinal and torsional motion. The equations are made dimensionless. The results are presented for the shear stresses at the wall, related to the drag force; these are physical quantities of interest, especially in oil-drilling applications.     

Fluid flow simulation in a random elliptical porous medium by using the lattice Boltzmann method

A fluid flow through an isotropic porous medium with randomly arranged elliptical particles is simulated by the lattice Boltzmann method. The dimensionless pressure drop and the dimensionless permeability are evaluated as functions of the Reynolds number. The effect of the aspect ratio of the major to minor semi-axis of the ellipse on the dimensionless permeability is considered for different values of porosity. The pressure drop is thoroughly investigated as a function of fluid viscosity for different values of the aspect ratio and porosity. The influence of various parameters of the problem on the mean tortuosity of the medium is considered.     

Homotopy Solutions for a Generalized Second-Grade Fluid Past a Porous Plate

The flow of a second-grade fluid past a porous plate subject to either suction or blowing at the plate has been studied. A modified model of second-grade fluid that has shear-dependent viscosity and can predict the normal stress difference is used. The differential equations governing the flow are solved using homotopy analysis method (HAM). Expressions for the velocity have been constructed and discussed with the help of graphs. Analysis of the obtained results showed that the flow is appreciably influenced by the material and normal stress coefficient. Several results of interest are deduced as the particular cases of the presented analysis.     

Flow of a Nonlinearly Viscous Fluid over the Surface of a Rotating Flat Disk

The flow of a nonlinearly viscous (power-law) fluid over the surface of a rotating flat disk is investigated. A solution form which makes it possible to reduce the complete system of partial differential equations to a system of ordinary differential equations is found. This system is integrated using the Runge-Kutta method and reduction to a Cauchy problem on the basis of Newton's method. The velocity and pressure fields in a power-law fluid film flowing over the surface of a rotating flat disk are found numerically.     

Compliance effects on the torsional flow of a viscoelastic fluid

The effects of transducer compliance on transient stress measurements in torsional flows of a viscoelastic fluid are investigated theoretically. The analysis is based on the torsional flow of an upper-convected Maxwell fluid between a rotating and ‘stationary’ disk, which is allowed to twist and displace axially as a result of the stresses exerted on the disk by the fluid. An approximate analytical solution to the governing equations is obtained using a standard perturbation method. Results of the analysis are used to examine how the fluid velocity is altered by the motion of the stationary disk and to gain insight on how transient stress measurements are affected by transducer compliance. The analysis shows that compliance effects increase with applied shear rate and that the effects of torsional and axial compliance are coupled in measurements of the shear stress and first normal stress difference.     

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.     

Theoretical Approach to the Onset of Convection in a Porous Layer

We examine the effect of viscous forces on the displacement of one fluid by a second, immiscible fluid along parallel layers of contrasting porosity, absolute permeability and relative permeability. Flow is characterized using five dimensionless numbers and the dimensionless storage efficiency, so results are directly applicable, regardless of scale, to geologic carbon storage. The storage efficiency is numerically equivalent to the recovery efficiency, applicable to hydrocarbon production. We quantify the shock-front velocities at the leading edge of the displacing phase using asymptotic flow solutions obtained in the limits of no crossflow and equilibrium crossflow. The shock-front velocities can be used to identify a fast layer and a slow layer, although in some cases the shock-front velocities are identical even though the layers have contrasting properties. Three crossflow regimes are identified and defined with respect to the fast and slow shock-front mobility ratios, using both theoretical predictions and confirmation from numerical flow simulations. Previous studies have identified only two crossflow regimes. Contrasts in porosity and relative permeability exert a significant influence on contrasts in the shock-front velocities and on storage efficiency, in addition to previously examined contrasts in absolute permeability. Previous studies concluded that the maximum storage efficiency is obtained for unit permeability ratio; this is true only if there are no contrasts in porosity and relative permeability. The impact of crossflow on storage efficiency depends on the mobility ratio evaluated across the fast shock-front and on the time at which the efficiency is measured.     

Fluid pressure response in poroelastic materials subjected to cyclic loading

When cyclic loading is applied to poroelastic materials, a transient stage of interstitial fluid pressure occurs, preceding a steady state. In each stage, the fluid pressure exhibits a characteristic mechanical behavior. In this study, an analytical solution for fluid pressure in two-dimensional poroelastic materials, which is assumed to be isotropic, under cyclic axial and bending loading is presented, based on poroelasticity. The obtained analytical solution contains transient and steady-state responses. Both of these depend on three dimensionless parameters: the dimensionless stress coefficient; the dimensionless frequency; and, the axial-bending loading ratio. We focus particularly on the transient behavior of interstitial fluid pressure with changes in the dimensionless frequency and the axial-bending loading ratio. The transient properties, such as half-value period and contribution factor, depend largely on the dimensionless frequency and have peak values when its value is about 10. This suggests that, under these conditions, the transient response can significantly affect the mechanical behavior of poroelastic materials.     

Experiment on Vortex Formation in a Fluid Flowing out of a Reservoir

Experimental data on vortex formation near the sinkhole in a fluid flowing out of a reservoir are analyzed. An experimental facility which makes it possible to study the influence of various factors on vortex formation near a sinkhole has been designed and built. In the experiments performed, a vortex was observed only if before the sinkhole was opened a rotating disk was immersed in the fluid. In the steady-state flow regime, a brief rotation of the disk resulted in the formation and subsequent damping of a vortex. For a preswirled fluid, the formation of a vortex crater depends on the sinkhole location.     

Flow of a Weakly Conducting Fluid in a Channel Filled with a Porous Medium

We investigate the fully developed flow in a fluid-saturated porous medium channel with an electrically conducting fluid under the action of a parallel Lorentz force. The Lorentz force varies exponentially in the vertical direction due to low fluid electrical conductivity and the special arrangement of the magnetic and electric fields at the lower plate. Exact analytical solutions are derived for fluid velocity and the results are presented in figures. All these flows are new and are presented for the first time in the literature.     

Perturbation solution of rotating solid disks with nonlinear strain-hardening

In order to determine the stresses and displacement in rotating disks with nonlinear strain-hardening, a polynomial stress-plastic strain relation is proposed and dimensionless quantities are adopted. A dimensionless governing equation for rotating disks is derived from the deformation theory of plasticity, Von Mises' yield criterion, proposed stress-plastic strain relation, equilibrium and compatibility equations. Its perturbation solution is obtained and applied to compute a rotating solid disk with nonlinear strain-hardening. The results are compared with those from the finite element calculation and other existing approaches, which confirms the validity of the method presented in this paper.     

Vibrational Dynamics of a Centrifuged Fluid Layer

The dynamics of a low-viscosity fluid layer inside a rotating cylinder under transverse translational vibration relative to the rotation axis is investigated experimentally. A novel vibrational effect, the generation of intense azimuthal fluid flows with velocities comparable with the cavity rotation velocity, is revealed. The structure and intensity of the vibrational flows and the flow transformation with variation of the determining dimensionless parameters (frequency and vibrational acceleration) are studied.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, 2005, pp. 147–156.Original Russian Text Copyright © 2005 by Ivanova, Kozlov, and Polezhaev.