Stokes flow past a porous sphere using Brinkman's model

A general non-axisymmetric Stokes flow past a porous sphere in a viscous, incompressible fluid is considered. The flow inside the sphere is governed by Brinkman's equations. A representation for velocity and pressure for the Brinkman's equations is suggested and a method of finding the flow quantities is given. Faxén's laws for drag and torque for the flow past a porous sphere are also given.     

Stokes flow inside a porous spherical shell: Stress jump boundary condition

An arbitrary Stokes flow of a viscous, incompressible fluid inside a sphere with internal singularities, enclosed by a porous spherical shell, using Brinkmans equation for the flow in the porous region is discussed. At the interface of the clear fluid and porous region stress jump boundary condition for tangential stresses is used. The drag and torque are found by deriving the corresponding Faxens laws. It is found that drag and torque not only change with the varying permeability, but also change for different values of stress jump coefficient. Critical permeability is found for which drag and torque change their behavior. As a limiting case the corresponding Faxens laws for the rigid spherical shell with internal singularities has been obtained.Received: December 17, 2002; revised: February 3, 2004     

Some exact solutions for fractional generalized Burgers’ fluid in a porous space

This work is concerned with deriving the equation for describing the magnetohydrodynamic (MHD) flow of a fractional generalized Burgers’ fluid in a porous space. Modified Darcy's law has been taken into account. Closed form solutions for velocity are obtained in three problems. The solutions for Navier–Stokes, second grade, Maxwell, Oldroyd-B and Burgers’ fluids appear as the limiting cases of the obtained solutions. A parametric study of some physical parameters involved in the problems is performed to illustrate the influence of these parameters on the velocity profiles.     

A solution for the problem of stokes flow past a porous sphere

The problem of a general non-axisymmetric Stokes flow of a viscous fluid past a porous sphere is considered. The expressions for the velocity and pressure, both inside and outside the sphere are given, when the flow outside satisfies the Stokes equations and the flow inside the sphere is governed by Darcy's law. The expressions for drag and torque are given. It is found that the drag is greater or smaller than the drag in the rigid case, depending on whether the undisturbed velocity is a pure biharmonic or a harmonic respectively. The torque is same as in the rigid case.     

Slip at the surface of a sphere translating perpendicular to a plane wall in micropolar fluid

The Stokes axisymmetrical flow caused by a sphere translating in a micropolar fluid perpendicular to a plane wall at an arbitrary position from the wall is presented using a combined analytical-numerical method. A linear slip, Basset type, boundary condition on the surface of the sphere has been used. To solve the Stokes equations for the fluid velocity field and the microrotation vector, a general solution is constructed from fundamental solutions in both cylindrical, and spherical coordinate systems. Boundary conditions are satisfied first at the plane wall by the Fourier transforms and then on the sphere surface by the collocation method. The drag acting on the sphere is evaluated with good convergence. Numerical results for the hydrodynamic drag force and wall effect with respect to the micropolarity, slip parameters and the separation distance parameter between the sphere and the wall are presented both in tabular and graphical forms. Comparisons are made between the classical fluid and micropolar fluid.      

Flow in a porous medium induced by torsional oscillation of a disk near its surface

Torsional oscillation of an infinite disk in a viscous liquid bounded by a porous medium fully saturated with the liquid has been discussed. It is assumed that the flow between the disk and the porous medium is governed by Navier-Stokes equation and that in the porous medium by Brinkman equation. Flows in the two regions are matched at the interface by assuming that the velocity and stress components are continuous at it. It is found that the depth of penetration of the flow in the porous medium is proportional to the square root of the permeability of the medium. The oscillation of the disk induces a steady radial-axial flow in both the regions in such a way that there is a steady axial flow of the fluid from the porous medium to the free flow region i.e. the fluid is expelled out from the porous medium. The steady flow in the porous medium increases with the increase of the permeability of the medium and with the decrease of the distance between the oscillating disk and porous surface.     

The lift on a small sphere in a linear shear flow near the interface of two immiscible fluids

Analytical expressions have been derived which predict, to lowest order, the inertial lift and the lateral migration velocity of a rigid sphere translating and rotating in a linear shear flow field near the flat interface of two immiscible fluids. This asymptotic analysis is primarily based on the assumption that the two Reynolds numbers defined by the gap width between the interface and the sphere, the shear rate and the translational slip velocity with which the spherical particle moves parallel to the interface are small. Furthermore, the radius of the sphere is assumed to be small compared to the gap width. To leading order in this creeping flow regime, the linear Stokes equations are obtained and a symmetry argument can be used to show that the Stokes solution does not predict any lift force. The transverse force experienced by the sphere and its migration velocity are due to the small but finite inertial terms in the Navier-Stokes equations, which can be studied by perturbation techniques. By applying a Green's function approach and matched asymptotic methods, which also incorporate the effects of the outer Oseen-like flow regime, the three components comprising the lift velocity have been calculated in closed form: the one induced by the shear rate only, the purely slip induced one and the one due to the interaction of the slip velocity with the shear flow field. The thus obtained expressions for the case of two immiscible fluids with arbitrary density and viscosity ratios extend the results that already exist in the literature for other flow configurations, such as an unbounded shear flow field [1] or a wall-bounded one, where the wall lies either within the leading order Stokes region [2] or in the outer Oseen region [3]. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

混合微通道中不稳定的广义Couette流

在一个平行板通道中,部分充满了均匀的多孔介质,部分为纯流体的流动区,对其微通道中完全发展的不稳定层流进行了数值分析,流动由其中一块板的运动和压力梯度所引起.多孔介质区域的流动,采用扩展的Brinkman模型,即Darcy模型,纯净流动区域的流动,采用Stokes方程.还对稳定的完全发展流进行了理论分析,给出了分界面速度、边界板处的速度和表面摩擦的闭式解.通过数值计算发现,稳定完全发展流的闭式解,和不稳定流动的数值解,在所有时间点上得到很好地吻合.     

Dynamic permeability of an assemblage of soft spherical particles

Under oscillatory Stokes flow, dynamic permeability of assemblage of soft spherical particles is derived. For the bed of soft particles, the fluid‐particle system is represented as an assemblage of uniform permeable spheres fixed in space. Each sphere, with a surrounding envelope of fluid, is uncoupled from the system and considered separately. This model is popularly known as cell model. Oscillatory Stokes equations are employed inside the fluid envelope, and oscillatory Brinkman equations are used inside the porous region. Four known boundary conditions namely: Happel's, Kuwabara's, Kvashnin's, and Cunningham's are considered on the outer boundary and results are compared. The behavior of dynamic permeability is analyzed with various parameters such as Darcy number (Da), frequency parameter (?), porosity (φ), and viscosity ratio (δ). Copyright © 2013 John Wiley & Sons, Ltd.     

Slow flow past a heterogeneous porous sphere

Stokes’ flow past a heterogeneous porous sphere has been studied, adopting the boundary conditions modified by Jones (1973) for curved surfaces at the interface of the free fluid region and porous material. The porous sphere is made up ofn + 1 concentric spheres of different permeability. The results for drag experienced by the sphere has been discussed and the following cases of interest are deduced:

1. WhenK ₁=K ₂=...=K _n+1=K.
2. WhenK _i is very small for eachi.

     

The slow stationary flow of a viscous liquid about porous spheroids

The slow stationary flow of a viscous liquid about porous spheroids is examined. The solution describing the Stokes stream function is found to be a one-parameter family of surfaces and for special values of the parameter we recover the flow past impermeable spheroids. The ranges of the parameter are determined and the case of the porous sphere as well as that of a porous disc are deduced as limiting case.     

A Robin-Robin Domain Decomposition Method for a Stokes-Darcy Structure Interaction with a Locally Modified Mesh

A new numerical method based on locally modified Cartesian meshes is proposed for solving a coupled system of a fluid flow and a porous media flow. The fluid flow is modeled by the Stokes equations while the porous media flow is modeled by Darcy's law. The method is based on a Robin-Robin domain decomposition method with a Cartesian mesh with local modifications near the interface. Some computational examples are presented and discussed.     

Retracted: Unsteady flow and heat transfer on a rotating disk in a porous medium

This article has been retracted. See retraction notice DOI: 10.1002/mma.850 . An unsteady flow and heat transfer in a porous medium of a viscous incompressible fluid over a rotating disk in an otherwise ambient fluid are studied. The unsteadiness in the flow field is caused by the angular velocity of the disk which varies with time. The new self‐similar solution of the Navier–Stokes and energy equations is obtained numerically. The solution obtained here is not only the solution of the Navier–Stokes equations, but also of the boundary layer equations. Also, for a simple scaling factor, it represents the solution of the flow and heat transfer in the forward stagnation‐point region of a rotating sphere or over a rotating cone. The asymptotic behaviour of the solution for a large porosity or for a large independent variable is also examined. The surface shear stresses in the radial and tangential directions and the surface heat transfer increase as the acceleration parameter increases. Also, the surface shear stress in the radial direction and the surface heat transfer decrease with increasing porosity, but the surface shear stress in the tangential direction increases. Copyright © 2006 John Wiley & Sons, Ltd.     

Slow motion of a micropolar fluid through a porous sphere bounded by a solid sphere

The paper examines the slow motion of a micropolar fluid produced by the relative motion of a solid sphere to an inside porous sphere. The result extends the Cunningham’s problem to micropolar fluid when the inner sphere is porous with prescribed radial suction/injection velocity at the surface of the sphere. The result can also be taken as an extension of the work of Ramkissoon and Majumdar when the fluid is bounded at a radiusr=b (b>a) but the solid sphere is replaced by a porous sphere. The force experienced by the inner sphere has been calculated and particular cases of interest have been deduced.     

Arbitrary Motions of a Viscous Incompressible Liquid in a Finite Region

By using an entirely constructive analysis it is shown that there exist solutions of the viscous flow equations in which two of the Cartesian components of velocity are arbitrary and the third component can be constructed uniquely by the application of integrability conditions. The solutions are in general defined in a finite region of the fluid and can be matched onto a well-defined deterministic solution of the Navier–Stokes equations such that all of the relevant physical quantities are continuous at the interface.     

A study of an arbitrary Stokes flow past a fluid coated sphere in a fluid of a different viscosity

A general method to discuss the problem of an arbitrary Stokes flow (both axisymmetric and non-axisymmetric flows) of a viscous, incompressible fluid past a sphere with a thin coating of a fluid of a different viscosity is considered. We derive the expressions for the drag and torque experienced by the fluid coated sphere and also discuss the conditions for the reduction of the drag on the fluid coated sphere. In fact, we show that the drag reduces compared to the drag on a rigid sphere of the same radius when the unperturbed velocity is either harmonic or purely biharmonic, i.e., of the form ${r^2\vec{\textbf{v}}}$ , where ${\vec{\textbf{v}}}$ is a harmonic function. Previously Johnson (J Fluid Mech 110:217–238, 1981), who considered a uniform flow showed that the drag on the fluid coated sphere reduces compared to the drag on the uncoated sphere when the ratio of the surrounding fluid viscosity to the fluid-film viscosity is greater than 4. We show that this result is true when the undisturbed velocity is harmonic or purely biharmonic, uniform flow being a special case of the former. However, we illustrate by an example that the drag may increase in a general Stokes flow even if this ratio is greater than 4. Moreover, when the unperturbed velocity is harmonic or purely biharmonic, and the ratio of the surrounding fluid viscosity to the fluid-film viscosity is greater than 4 for a fixed value of the viscosity of the ambient fluid, we determine the thickness of the coating for which the drag is minimum.     

Fluid flow in an inhomogeneous granular medium

The seepage of a compressible fluid in an inhomogeneous undeformable granular medium is investigated. It is assumed that the fluid flow in a porous space is described by the Navier–Stokes equations. It is shown that, in the case of an inhomogeneous velocity field, a tensor of additional effective stresses occurs in connection with the transfer of fluid particles in a transverse direction when flow occurs around the granules of the medium in a longitudinal direction. Using the fundamental propositions of Reynolds’ averaging theory and Prandtl's mixing path, the structure of the effective viscosity coefficient is determined and hypotheses are formulated which enable it to be assumed to be independent of the flow velocity. It is established by comparison with experimental data that the effective viscosity coefficient can exceed the viscosity coefficient of the flowing fluid by an order of magnitude. The equations of average motion are obtained, which in the case of an incompressible fluid have the form of the Navier–Stokes equations with body forces proportional to the velocity. It is established that, in addition to the well-known dimensionless flow numbers, there is a new number which characterizes the ratio of the Darcy porous drag forces to the effective viscosity forces. The proposed equations are extended to the case of the flow of an aerated fluid. The components of the angular momentum vector are used as the required functions instead of the components of the velocity vector. This enables a solving system of equations to be obtained, which, apart from the notation, is identical with the similar equations for the case of an incompressible fluid. The solution of a new problem of the fluid flow in a plane channel with permeable walls is presented using three models: Darcy's law for an incompressible and aerated fluid, and also of an aerated fluid taking the effective viscosity into account. It is established that, for the same pressure drop, the maximum flow rate corresponds to Darcy's law. Compressibility leads to its reduction, but by simultaneously taking into account the compressibility and the effective viscosity one obtains minimum values of the flow rate. The effective viscosity and aeration of the fluid has a considerable effect on the flow parameters.     

New Exact Solutions of the Navier–Stokes Equations for Axisymmetric Automodel Flows of Fluids

We investigate self-similar solutions of the Navier–Stokes equations for the axisymmetric flow of a viscous incompressible fluid. The original equations are transformed by the Slezkin method. On the basis of analysis of physical properties of the flow and the Slezkin general equation, we show that, in parallel with the known solutions of this equation, there exist several other solutions with physical meaning. We consider the simplest case of irrotational flows for which current lines may be circles, ellipses, parabolas, and hyperbolas. Unlike the Landau and Squire solutions, these flows are interpreted as nonjet flows of fluid flowing into and out of a homogeneous porous axially symmetric body.     

SPH modelling of fluid at the grain level in a porous medium

Computational modelling of the flow of fluids in porous media has traditionally been at a macroscopic level where the medium’s permeability and porosity are an input (from experiments for example). In many cases this is difficult, especially if the porous medium changes its solid structure as a function of time. This situation occurs in reactive systems such as “heap-leaching”, where biological and/or chemical solutions are introduced into the heap to dissolve or react with valuable materials. In this case, modelling fluid flow at the grain level is paramount and we show how this can be done with the SPH technique. We present three-dimensional SPH simulations of fluid flow in an idealised porous medium and show that the technique yields flows which are physically realistic. The permeability of the medium is then predicted.     

On slow motion of a sphere in a viscous liquid

Stokes’ and Seth’s solutions for the slow motion of a sphere in a viscous, incompressible liquid have been discussed from the viewpoint of the structure of the velocity field and its relation to the drag of the sphere. The problem is analysed from a different angle in this paper. It is believed that it throws more light on thephysics of the problem.