Flow of a viscous fluid at small Reynolds number past a porous sphere with a solid core

Summary Uniform flow of an incompressible viscous fluid at small Reynolds number past a porous sphere of radius a with a solid concentric spherical core of radius b has been discussed. The region of the porous shell is called zone I which is fully saturated with the viscous fluid, and the flow in this zone is governed by the Brinkman equation. The space outside the shell where clear fluid flows is divided into two zones (II and III). In these zones the flow is discussed following Proudman and Pearson's method of expanding Stokes' stream function in powers of Reynolds number and then matching Stokes' solution with Oseen's solution. The stream function of zone II is matched with that of zone I at the surface of the shell by the condition suggested by Ochoa – Tapia and Whitaker. It is found that the drag on the spherical shell increases with the increase of the λ (=b/a) and also with the increase of the Darcy number. The graph of dimensionless drag against λ for various values of Reynolds number shows that the drag increases with the increase of the Reynolds number for all values of λ.     

Stokes flow past a swarm of porous approximately spheroidal particles with Kuwabara boundary condition

The solution of the problem of symmetrical creeping flow of an incompressible viscous fluid past a swarm of porous approximately spheroidal particles with Kuwabara boundary condition is investigated. The Brinkman equation for the flow inside the porous region and the Stokes equation for the outside region in their stream function formulations are used. As boundary conditions, continuity of velocity and surface stresses across the porous surface and Kuwabara boundary condition on the cell surface are employed. Explicit expressions are investigated for both inside and outside flow fields to the first order in a small parameter characterizing the deformation. As a particular case, the flow past a swarm of porous oblate spheroidal particles is considered and the drag force experienced by each porous oblate spheroid in a cell is evaluated. The dependence of the drag coefficient on permeability for a porous oblate spheroid in an unbounded medium and for a solid oblate spheroid in a cell on the solid volume fraction is discussed numerically an and graphically for various values of the deformation parameter. The earlier known results are then also deduced from the present analysis.     

Estimation of the dynamic permeability of an assembly of permeable spherical porous particles using the cell model

This paper is concerned with the theoretical estimation of the dynamic permeability of an assembly of permeable porous particles subject to an oscillatory Stokes flow. A cell model is used to approximate the hydrodynamic interactions of particles. The flow field inside the representative porous sphere is governed by Darcy’s law and that within the fluid region by the unsteady Stokes equations. Faxén’s laws for drag and torque exerted on the representative particle are derived, and the results are compared in special cases with the existing literature. The dynamic permeability of the bed of particles is obtained and the variation in dynamic permeability with various parameters such as porosity, Darcy number and, frequency is analyzed.     

Stokes resistance of a porous spherical particle in a spherical cavity

The boundary effect on the asymmetrical motion of a porous spherical particle in an eccentricspherical cavity is investigated in the quasi-steady limit under creeping flow conditions. The porous particletranslates and rotates in the viscous fluid, located within the spherical cavity, normal to the line connectingtheir centers. The fluid inside the porous particle is governed by the Brinkman equation. A tangential stressjump condition at the interface between the fluid and the porous particle is applied. A semi-analytical approachbased on a collocation technique is used. Due to the linearity of the present problem, the flow variables for theclear fluid region are constructed by superposing basic solutions of two problems: the first one is the regularsolution inside the cavity region in the absence of the porous particle where a first system of coordinates has itsorigin at the center of the cavity, while the second problem is the regular solution in the infinite region outsidethe spherical porous particle where a second coordinate system with its origin at the center of the porousparticle is used. Numerical results displaying the resistance coefficients acting on the particle are obtainedwith good convergence for various values of the physical parameters of the problem. The results are tabulatedand represented graphically. The findings demonstrate that the collocation results of the resistance coefficientsare in good agreement with the corresponding results for the impermeable solid particle.     

Experimental and numerical analysis on the hydrodynamic behaviors of permeable microbial granules

The microbial granules are found to be porous and permeable, which leads to a different drag force coefficient from the rigid sphere granules. Discrete element method (DEM) was employed to establish geometric models of porous microbial granules for the first time in this study. And computational fluid dynamics (CFD) was applied to simulate the effects of porosity and Reynolds number on the fluid flow, shear stress, pressure and drag force based on the established geometric models. The results showed that both the Reynolds number and the porosity of microbial granules significantly affect the fluid velocity distribution inside the granules. The porosity shows less effect on maximum shear stress than Reynolds number. It s well known that drag force consists form drag and skin drag. The ratio of form drag to drag force increased, while the skin drag force ratio decreased with the increasing Reynolds number. The porosity will enhance the skin drag and weaken the form drag at the same Reynolds number. A drag force coefficient equation was established based on the simulated results in order for engineering application. The correctness of the equation was confirmed by comparing with experimental results. The results from this study may provide valuable information for operation and designing of a granule-based bioreactor in wastewater treatment.     

A coupled discrete element‐finite difference approach for modeling mechanical response of fluid‐saturated porous materials

A model of fluid‐saturated poroelastic medium was developed based on a combination of the discrete element method and grid method. The developed model adequately accounts for the deformation, fracture, and multiscale internal structure of a porous solid skeleton. The multiscale porous structure is taken into account implicitly by assigning the porosity and permeability values for the enclosing skeleton, which determine the rate of filtration of a fluid. Macroscopic pores and voids are taken into account explicitly by specifying the computational domain geometry. The relationship between the stress–strain state of the solid skeleton and pore fluid pressure is described in the approximations of simply deformable discrete element and Biot's model of poroelasticity. The developed model was applied to study the mechanical response of fluid‐saturated samples of brittle material. Based on simulation results, we constructed a generalized logistic dependence of uniaxial compressive strength on loading rate, mechanical properties of fluid and enclosing skeleton, and on sample dimensions. The logistic form of the generalized dependence of strength of fluid‐saturated elastic–brittle porous materials is due to the competition of two interrelated processes, such as pore fluid pressure increase under solid skeleton compression and fluid outflow from the enclosing skeleton to the environment. Copyright © 2015 John Wiley & Sons, Ltd.     

Transient Couette flow in a rotating non-Darcian porous medium parallel plate configuration: network simulation method solutions

The transient, viscous, incompressible, hydrodynamic Couette flow in a rotating porous medium channel is studied in this paper. The channel comprises a pair of infinitely long parallel plates which rotate with uniform angular velocity about an axis normal to the plates. The porous medium is simulated using a Darcy–Forchheimer drag force model which includes both bulk matrix porous drag (dominant at low Reynolds numbers) and second order inertial impedance (dominant at higher Reynolds numbers). The two-dimensional Navier–Stokes equations are reduced to a (z*, t*) coordinate system incorporating Coriolis terms, and appropriate initial and boundary conditions are prescribed. Separate porous drag body force terms are incorporated in both the primary and secondary flow momentum equations. Using a set of transformations, the model is rendered dimensionless and shown to be dictated by the Ekman number, Forchheimer number, Darcy number and Reynolds number in a (z, t) coordinate system. Numerical solutions are obtained for the transformed model using the Network Simulation Method. The influence of the hydrodynamic parameters are computed graphically and also the interaction of parameters on the velocity fields is discussed at length. Excellent agreement is found with earlier non-porous flow studies. The analysis has important applications in geophysics and also chemical engineering systems.     

An improved immersed moving boundary for hydrodynamic force calculation in lattice Boltzmann method

Particles suspension is considerably prevalent in petroleum industry and chemical engineering. The efficient and accurate simulation of such a process is always a challenge for both the traditional computational fluid dynamics and lattice Boltzmann method. Immersed moving boundary (IMB) method is promising to resolve this issue by introducing a particle-fluid interaction term in the standard lattice Boltzmann equation, which allows for the smooth hydrodynamic force calculation even for a large grid size relative to the solid particle. Although the IMB method was proved good for stationary particles, the deviation of hydrodynamic force on moving particles exists. In this work, we reveal the physical origin of this problem first and figure out that the internal fluid effect on the hydrodynamic force calculation is not counted in the previous IMB. An improved immersed moving boundary method is therefore proposed by considering the internal fluid correction, which is easy to implement with the little extra computation cost. A 2D single elliptical particle and a 3D sphere sedimentation in Newtonian fluid is simulated directly for the validation of the corrected model by excellent agreements with the standard data.     

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

Summary 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.     

On self-similarly expanding Eshelby inclusions: Spherical inclusion with dilatational eigenstrain

For a subsonically self-similarly expanding spherical inclusion with dilatational transformation strain in a linear elastic solid, the governing system of partial differential equations is shown to be elliptic under scaling of uniform stretching of the variables, and the resulting elliptic equation is solved by satisfying the Hadamard jump conditions on the moving boundary. The solution has the Eshelby constant stress property for the interior domain, and can thus be used for the expanding inhomogeneity with transformation strain according to Eshelby (1957). The driving force on the moving boundary is also obtained.     

Equivalent inclusion method-based simulation of particle sedimentation toward functionally graded material manufacturing

A novel numerical approach based on Eshelby’s equivalent inclusion method is presented to simulate the Stokes’ flow of many spherical particles moving in a viscous fluid at a small Reynolds number. Many particles fall toward the bottom of the fluid at different velocities and thus form a graded microstructure in terms of the material phase and the particle size. For each particle, an eigenstrain rate, which is given in a polynomial form, is introduced to represent the mismatch between the particle and the rest of the fluid. Rongved’s fundamental solution of a point force in a semi-infinite domain with a fixed boundary (Rongved in J. Appl. Mech. 22, 545–546, 1955) is used to calculate the velocity field caused by the body force and eigenstrain rate. Based on Eshelby’s stress equivalent condition, the eigenstrain rate of each particle can be solved and the sedimentation process of a many-particle system can be simulated as a quasi-equilibrium process. If only one or two particles are considered, the results agree with the finite element results very well. Using a suspension of aluminum and high-density polyethylene (HDPE) powders mixed in ethanol, the microstructural evolution is illustrated along with the sedimentation process, which produces a graded mix collected at the fixed boundary for functionally graded material manufacturing.     

Wave field simulation for heterogeneous transversely isotropic porous media with the JKD dynamic permeability

Poroelastic wave field in a 2D heterogeneous transversely isotropic porous medium is calculated. The Johnson-Koplik-Dashen (JKD) dynamic permeability is assumed with two scalar JKD permeability operators for vertical and horizontal direction, respectively. The time domain expression of drag force in the JKD model is expressed in terms of the shifted fractional derivative of the relative fluid velocity. A method for calculating the shifted fractional derivative without storing and integrating of the entire velocity histories is developed. By using the new method for calculating the shifted fractional derivative, the governing equations for the 2D transversely isotropic porous medium are reduced to a system of first-order differential equations for velocities, stresses, pore pressure and the quadrature variables associated with the drag forces. The spatial derivatives involved in the first-order differential equation system are calculated by the Fourier pseudospectral method, while the time derivatives of the system are discretized by a predictor-corrector method. For the demonstration of our method, some numerical results are given in the paper.     

Modified Reynolds equation for coupled stress fluids – a porous media model

Summary. In this paper, the rheological effects of coupled stress fluids on thin film lubrication modeling are developed. Thin porous layers attached to the impermeable substrate are utilized to model the microstructure of bearing surfaces. In the fluid film region, the constitutive equations for coupled stress fluids proposed by Stokes [1] as well as the continuity and momentum equations are applied to model the flow. In the porous region, the Brinkman-extended Darcy equations are applied to model the flow. Under the usual assumption of hydrodynamic lubrication applicable to thin films, the effects of viscous shear and the stress jump boundary condition at the porous media/fluid film interface are included in deriving the modified Reynolds equation. The effects of material properties such as coupled stress parameter viscosity ratio (_i²), thickness of porous layer (_i), permeability (K_i), and stress jump parameter (_i), on the velocity distributions and load capacities of one-dimensional converging wedge problems are discussed.     

Oscillatory and Steady Dynamics of a Cylindrical Body Near the Border of Vibrating Cavity Filled with Liquid

The results of experimental study of vibrational dynamics of cylindrical solid in a rectangular cavity filled with viscous incompressible fluid are generalized. The cavity performs high-frequency translational oscillations in a horizontal plane. Experiments are carried out with bodies of different relative density: more or less than liquid’s density. The cylinder oscillates in the cavity under the influence of oscillating inertia force. An averaged force repels the body from the boundary and holds a heavy body over the bottom of the cavity and the light one at some distance from the ceiling. The vibrational lift force depends on the amplitude and frequency of vibrations as well as on the properties of liquid. It is shown that the value of the averaged lift force decreases with increasing dimensionless amplitude. Special attention is paid to the oscillatory behavior of a solid. The rotational oscillations of the body, observed in experiments simultaneously with the translational ones, and fluid motion, excited by an oscillating body, are investigated. The different modes of interaction of the body with the container’s boundary were found. The oscillatory dynamics of bodies with different relative density is studied by high-speed video-registration.     

Stokes flow with slip caused by the axisymmetric motion of a sphere bisected by a free surface bounding a semi-infinite micropolar fluid

The first part of this paper investigates the motion of a solid spherical particle in an incompressible axisymmetric micropolar Stokes flow. A linear slip, Basset-type, boundary condition has been used. Expressions for the drag force and terminal velocity has been obtained in terms of the parameter characterizing the slip friction. In the second part, we consider the flow of an incompressible axisymmetrical steady semi-infinite micropolar fluid arising from the motion of a sphere bisected by a free surface bounding a semi-infinite micropolar fluid. Two cases are considered for the motion of the sphere: perpendicular translation to the free surface and rotation about a diameter which is also perpendicular to the free surface. The speed of the translational motion and the angular speed for the rotational motion of the sphere are assumed to be small so that the nonlinear terms in the equations of motion can be neglected under the usual Stokesian approximation. Also a linear slip, Basset-type, has been used. The analytical expressions for velocity and microrotation components are determined in terms of modified Bessel functions of second kind and Legendre polynomials. The drag for the translation case and the couple for the rotational motion on the submerged half sphere are calculated and expressed in terms of nondimensional coefficients whose variation is studied numerically. The variations of the drag and couple coefficients with respect to the micropolarity parameter and slip parameter are tabulated and displayed graphically.     

Mass-transfer kinetics in a shell packing material for chromatography

A shell particle consists of a solid, nonporous core that is surrounded with a shell of a porous solid having essentially the same physicochemical properties as those of the conventional porous particles used as packing media in chromatography. The diameter of the solid core and the thickness of its shell or the external diameter of the particle characterizes the chromatographic properties of the packing material. The potential advantage of this particle structure would be the shorter average path length experienced by solute molecules during their diffusion across the particles of packing material when they are retained. Compounds having slow pore diffusion would exhibit higher efficiencies on columns packed with shell than with conventional, fully porous particles. Using columns packed with Halo, a new type of porous silica shell particles, we assess the gain achieved with this principle for peptides of moderate molecular weights and for small proteins.     

Dynamics of rotating paramagnetic particles simulated by lattice Boltzmann and particle dynamics methods

Novel biochemical sensors consisting of rotating chains of microscale paramagnetic particles have been proposed that would enable convenient, sensitive analyte detection. Predicting the dynamics of these particles is required to optimise their design. The results of lattice Boltzmann (LB) and particle dynamics (PD) simulations are reported, where the LB approach provides a verified solution of the complete Navier-Stokes equations, including the hydrodynamic interactions among the particles. On the other hand, the simpler PD approach neglects hydrodynamic interactions, and does not compute the fluid motion. It is shown that macroscopic properties, like the number of aggregated particles, depend only on the drag force and not on the total hydrodynamic force, making PD simulations yield reasonably accurate predictions. Relatively good agreement between the LB and PD simulations, and qualitative agreement with experimental data, are found for the number of aggregated particles as a function of the Mason number. The drag force on a rotating cylinder is significantly different from that on particle chains calculated from both simulations, demonstrating the different dynamics between the two cases. For microscopic quantities like the detailed force distributions on each particle, the complete Navier-Stokes solution, here represented by the LB simulation, is required.     

Nonisothermal cross-flow around a cylinder with a square cross section and an impermeable core covered with a porous layer

Nonisothermal cross flow of a viscous incompressible fluid around a porous cylinder with a square cross section is considered. Main attention is paid when an impermeable core of the cylinder is surrounded with a porous layer. The full system of Navier–Stokes and energy equations is integrated numerically by the finite-volume method. The hydrodynamic interaction between the flow and the matrix of the porous layer is described by Darcy’s law. At moderate Reynolds numbers, the influence of the permeability of the porous layer on the nature of the flow and the heat exchange between the cylinder and the flow is studied. It is shown that, with increasing permeability, heat transfer from the cylinder increases mainly on its front side. From the analysis of the data obtained, an approximate formula for the mean Nusselt number as a function of the Reynolds and Darcy numbers is derived. The results of the calculation of hydrodynamic and thermal characteristics of the cross-flow around an impermeable and a fully permeable cylinder are also presented.     

Governing equations for unsaturated flow through woven fiber mats. Part 1. Isothermal flows

Correct modeling of resin flow in liquid composite molding (LCM) processes is important for accurate simulation of the mold-filling process. Recent experiments indicate that the physics of resin flow in woven (also stitched or braided) fiber mats is very different from the flow in random fiber mats. The dual length-scale porous media created by the former leads to the formation of a sink term in the equation of continuity; such an equation in combination with the Darcy's law successfully replicate the drooping inlet pressure history, and the region of partial saturation behind the flow-front, for the woven mats. In this paper, the mathematically rigorous volume averaging method is adapted to derive the averaged form of mass and momentum balance equations for unsaturated flow in LCM. The two phases used in the volume averaging method are the dense bundle of fibers called tows, and the surrounding gap present in the woven fiber mats. Averaging the mass balance equation yields a macroscopic equation of continuity which is similar to the conventional continuity equation for a single-phase flow except for a negative sink term on the right-hand side of the equation. This sink term is due to the delayed impregnation of fiber tows and is equal to the rate of liquid absorbed per unit volume. Similar averaging of the momentum balance equation is accomplished for the dual-scale porous medium. During the averaging process, the dynamic interaction of the gap flow with the tow walls is lumped together as the drag force. A representation theorem and dimensional analysis are used to replace this drag force with a linear function of an average of the relative velocity of the gap fluid with respect to the tow matrix for both the isotropic and anisotropic media. Averaging of the shear stress term of the Navier–Stokes equation gives rise to a new quantity named the interfacial kinetic effects tensor which includes the effects of liquid absorption by the tows, and the presence of slip velocity on their surface. Though the gradient of the tensor contributes a finite force in the final momentum balance equation, a scaling analysis leads to its rejection in the fibrous dual-scale porous medium if the permeability of flow through the gaps is small. For such a porous medium, the momentum equation reduces to the Darcy's law for single-phase flow.