Hydrodynamic permeability of a membrane composed of porous spherical particles in the presence of uniform magnetic field

This work concerns the flow of an incompressible viscous fluid past a porous sphere in presence of transverse applied uniform magnetic field, using particle-in-cell method. The Brinkman equations are used in porous region and the Stokes equations for non-porous region. At the fluid-porous interface, the stress jump boundary condition for tangential stresses along with continuity of normal stress and velocity components are used. Four known boundary conditions on the hypothetical surface are considered and compared: Happel’s, Kuwabara’s, Kvashnin’s and Cunningham’s (Mehta-Morse’s condition). The hydrodynamic drag force experienced by a porous spherical particle in a cell and hydrodynamic permeability of membrane built up by porous spherical particles are evaluated. The patterns of streamlines are also obtained and discussed. The effect of stress jump coefficient, Hartmann number, dimensionless specific permeability of the porous particle and particle volume fraction on the hydrodynamic permeability and streamlines are discussed. Some previous results for hydrodynamic drag force and dimensionless hydrodynamic permeability have been verified.     

Drag and Torque on Clusters of N Arbitrary Spheres at Low Reynolds Number

Hydrodynamics of particle clusters suspended in viscous fluids is a subject of considerable theoretical and practical importance. Using a multipole expansion of the flow velocity in a series of spherical harmonics, Lamb's fundamental solution of the Stokes flow outside a single sphere is generalized in this work to the case of N nonoverlapping spheres of arbitrary size with slip boundary conditions. The expansion coefficients are found by transforming the boundary conditions to the Lamb form and by transforming the spherical coordinates and solid spherical harmonics centered at different spheres. The problem is reduced to the solution of the linear system of equations for the expansion coefficients, which is carried out numerically. Based on the developed theory, the relation between the hydrodynamic and gyration radius of fractal-like aggregates with different structure is established. In another application, an asymptotic slip-regime dependence of the aggregate hydrodynamic radius on the Knudsen number and the number of particles is found by performing calculations of drag forces acting on the gas-borne fractal-like and straight chain aggregates. A good agreement is shown in comparing predictions of the described theory with available experimental and theoretical results on motion of various small sphere clusters in viscous fluid. Copyright 2000 Academic Press.     

On hydrodynamic permeability of a membrane built up by porous deformed spheroidal particles

This paper concerns the slow viscous flow of an incompressible fluid past a swarm of identically oriented porous deformed spheroidal particles, using particle-in-cell method. The Brinkman’s equation in the porous region and the Stokes equation for clear fluid region in their stream function formulations are used. Explicit expressions are investigated for both the inside and outside flow fields to the first order in a small parameter characterizing the deformation. The flow through the porous oblate spheroid is considered as the particular case of the porous deformed spheroid. The hydrodynamic drag force experienced by a porous oblate spheroid and permeability of a membrane built up by porous oblate spheroids having parallel axis are evaluated. The dependence of the hydrodynamic drag force and the hydrodynamic permeability on particle volume fraction, deformation parameter and viscosity of porous fluid are also discussed. Four known boundary conditions on the hypothetical surface are considered and compared: Happel’s, Kuwabara’s, Kvashnin’s and Cunningham’s (Mehta-Morse’s condition). Some previous results for hydrodynamic drag force and hydrodynamic permeability have been verified. The model suggested can be used for evaluation of changing hydrodynamic permeability of a membrane under applying unidirectional loading in pressure-driven processes (reverse osmosis, nano-, ultra- and microfiltration).     

Faxen's Laws of a Composite Sphere under Creeping Flow Conditions

Under creeping flow conditions, Faxen's laws are derived for a composite sphere comprising a solid core covered by a permeable layer of arbitrary thickness. The derivations are carried out by applying reciprocal theorem in combination with fluid velocity and pressure distributions in certain simple flow as a comparison field. In this regard, the fluid velocity disturbances caused by a composite sphere subject to a simple shear flow and a rotational flow are solved individually. In the limiting case where the solid core vanishes, the resulting Faxen expressions for the drag force, torque, and stresslet compare very well with the existing Faxen's law for a porous sphere. It is found that when the porous layer is thick enough and its permeability is sufficiently low, the hydrodynamic behavior of a composite sphere can be approximated by that of a porous particle with equal permeability. This can be explained by the fact that the fluid cannot penetrate deeply into a porous layer of low permeability to flow through the pores near the core surface, and thereby the fluid can hardly feel the resistance from the core surface. Copyright 2000 Academic Press.     

Direct measurement of the viscous force between two spherical particles trapped in a thin wetting film

Here we present the first direct measurement of the viscous drag force between two spherical particles of millimeter size trapped in a thin wetting film. Each particle is constrained by the liquid/air interface and the solid substrate. The viscous force is counterbalanced by another known force, the attractive capillary immersion force between identical particles protruding from the film surface. The results of the measurements provide evidence for an increased hydrodynamic force due to a non-Stokesian resistance to the particle motion. Our findings can be applied to the self-assembly of colloidal particles in a two-dimensional array for coating and to the friction between small species and a solid. Received: 19 March 1999 Accepted in revised form: 11 May 1999     

Influence of particle-particle interactions and particles rotational motion in traveling wave dielectrophoresis

Traveling wave dielectrophoresis provides an interesting method for the controlled movement of microsized particles in suspended mixtures, and as such is a promising tool in microfluidic technology. In this case, the electrostatic force acting on the particles has two components: one due to the spatially varying magnitude of the electric field and the other due to the spatially varying phase. The actual movement of the particle is determined by the combined effect of these two forces and corresponding torques, the viscous drag exerted by the fluid on the particle, and the electrostatic and hydrodynamic particle-particle interactions. This paper presents the first numerical simulations of the motion of particles subjected to all previous forces and torques. Our technique is based on a finite-element scheme in which the particles are moved using a direct simulation scheme respecting the fundamental equations of motion for both the fluid and the solid particles. The fluid-particle motion is resolved by the method of distributed Lagrange multipliers and the electrostatic forces are computed using the point-dipole approximation. Our simulations show that the particle behavior strongly depends on the mismatch of the dielectric properties between the particles and the fluid, and that the particle-particle interaction force as well as particles rotation speeds play crucial roles in the various regimes.     

The drag of the row of parallel chains of spherical particles in the Stokes flow

The drag of thin-layered porous deposit consisting of dendrites of identical spherical particles with respect to the flow of viscous incompressible liquid is calculated. The deposit is approximated by a model system, a row of parallel chains of particles oriented perpendicular to a flow direction. The expression is derived for the dimensionless drag force acting on the unit chain length as a function of the ratio of a particle radius to a half-distance between chain axes, a/h. It is shown that, at a/h < 0.5, the hydrodynamic equivalent of the chains is the smooth cylinder whose radius is 1.16 times smaller than the particle radius that agrees with the experiment. It is also shown that, at a/h = 1, the drag force of a particle contacting with four adjacent particles in the layer with square packing is equal to F = 44F _St, where F _St is the Stokes drag force of a spherical particle. The pressure drop in this single layer is by 3.5% higher than in the layer of spherical particles with cubic packing. At a/h = 2/√3, drag force F of the particle contacting with six particles in a single layer with hexagonal packing is equal to 340F _St.     

Drag force of a particle moving axisymmetrically in open or closed cavities

Hydrodynamic resistance to particle transport arising from the solid mass in porous media is of fundamental importance. We investigate an axisymmetric creeping flow caused by a spherical particle migrating in a spherical cavity or connected cavities of equal size by a boundary element method. Each cavity has either one or two circular apertures, through which a sufficiently small particle can pass. Drag force on the particle is calculated to determine the correction factor to the Stokes law. It is found that when passing through an aperture, the particle experiences a local maximum drag force larger than that located in the cavity center. This force is also greater than that for the particle near the closed end at the same smallest surface-to-surface distance. For connected cavities open to the exterior fluid, the drag force is smaller than that in the corresponding closed system.     

Dynamic Mobility of Two Spherical Particles with Thick Double Layers

The dynamic electrophoretic mobility of a pair of nearby spherical particles is analyzed in the case when the thickness of the electrical double layer around each particle is comparable to the particle radius. By means of an integral reciprocal relation, a formal expression is obtained for the force and torque on N spheres subject to an oscillating electric field which may be spatially varying. Upon linearizing in the surface potential, this expression is shown to depend upon a set of purely hydrodynamic problems involving N neutral spheres, the calculation of the electric field around N neutral spheres, and the equilibrium charge distribution around N charged spheres. In the case of a single particle, the known analytic formula for the dynamic mobility is recovered. For a pair of identical particles, the dynamic mobility is calculated numerically, using known solutions to the required subproblems. An analytical expression for the mobility of a pair of widely separated spheres is also obtained by a method of reflections, and this is in excellent agreement with the numerical results outside the range of double layer overlap. Copyright 2000 Academic Press.     

Porous agglomerates in the general linear flow field

We calculate the flow within and around a porous spherical agglomerate suspended in the general linear flow field, and also the flow induced by its rotation. We use the Stokes equations exterior to the particle and the Brinkman equations inside it. The effect of particle permeability on the flow is expressed via the Brinkman parameter beta = r(0)/square root of k, where r0 is particle radius and k is its permeability. With translational creeping motion of porous spheres in a quiet fluid investigated by Debye and Bueche [P. Debye, A.M. Bueche, J. Chem. Phys. 16 (6) (1943) 573-579], this study provides information necessary for investigating dynamics of porous particles moving in creeping shear flows under the action of external forces and torques. The agglomerate flow field solutions are used to calculate the effective viscosity of a dilute suspension of porous solid aggregates, which generalizes the well-known Einstein's equation for solid suspensions. The agglomerate effective viscosity diameter is proposed which allows using the Einstein's formula evaluation of the agglomerates suspension viscosity.     

Diffusiophoresis in suspensions of highly charged soft particles

Diffusiophoresis phenomenon of aoft particles suspended in binary electrolyte solutions is explored theoretically in this study based on the spherical cell model, focusing on the chemiphoresis component in absence of diffusion potential. Both the electrostatic and hydrodynamic aspects of the boundary confinement, or steric effect, due to the presence of neighboring particles are examined extensively under various electrokinetic conditions. Significant local extrema are found in mobility profiles expressed as functions of the Debye length in general, synchronized with the strength of the motion-inducing double layer polarization. Moreover, a seemingly peculiar phenomenon is observed that the soft particles may move faster in more concentrated suspensions. The competition between the simultaneous enhancement of the motion-inducing electric driving force and the motion-retarding hydrodynamic drag force from the boundary confinement effect of the neighboring particles is found to be responsible for it. The above findings are also demonstrated experimentally in a very recent study on the diffusiophoretic motion of soft particles through porous collagen hydrogels. The results presented here are useful in various practical applications of soft particles like drug delivery.     

Diffusion, sedimentation, and rheology of concentrated suspensions of core-shell particles

Short-time dynamic properties of concentrated suspensions of colloidal core-shell particles are studied using a precise force multipole method which accounts for many-particle hydrodynamic interactions. A core-shell particle is composed of a rigid, spherical dry core of radius a surrounded by a uniformly permeable shell of outer radius b and hydrodynamic penetration depth κ(-1). The solvent flow inside the permeable shell is described by the Brinkman-Debye-Bueche equation, and outside the particles by the Stokes equation. The particles are assumed to interact non-hydrodynamically by a hard-sphere no-overlap potential of radius b. Numerical results are presented for the high-frequency shear viscosity, η(∞), sedimentation coefficient, K, and the short-time translational and rotational self-diffusion coefficients, D(t) and D(r). The simulation results cover the full three-parametric fluid-phase space of the composite particle model, with the volume fraction extending up to 0.45, and the whole range of values for κb, and a/b. Many-particle hydrodynamic interaction effects on the transport properties are explored, and the hydrodynamic influence of the core in concentrated systems is discussed. Our simulation results show that for thin or hardly permeable shells, the core-shell systems can be approximated neither by no-shell nor by no-core models. However, one of our findings is that for κ(b - a) ? 5, the core is practically not sensed any more by the weakly penetrating fluid. This result is explained using an asymptotic analysis of the scattering coefficients entering into the multipole method of solving the Stokes equations. We show that in most cases, the influence of the core grows only weakly with increasing concentration.     

Effects of pre-oxidation temperature on coal secondary spontaneous combustion

The hydrodynamic force (drag) on spherical and irregularly shaped particles significantly increases when the particles move close to solid and permeable boundaries. The overall effect of the increased hydrodynamic drag is to hinder the particle movement in the vicinity of boundaries and this includes the Brownian movement and electrophoresis. The Monte Carlo simulation method is used to model the Brownian movement, the resulting diffusion, and the electrophoresis of spherical particles in narrow, cylindrical pores, filled with Newtonian fluids. It is observed that the effect of the pore walls is a significant reduction of the space-averaged electrophoretic velocity of the particles, which implies reduced particle flux through the pores. The hindered electrophoresis is primarily a geometric phenomenon, caused by the increased drag and depends on the size of the particles and the pore-to-particle diameter ratio. The temperature of the fluid slightly affects the hindered electrophoresis through its effect on the viscosity, which is a determinant of the Brownian force, the diffusivity and the electrophoretic velocity. The hindered electrophoresis is almost independent of the other fluid and particle properties, such as density. Based on the simulation results a non-linear correlation for the flux of particles is derived, valid in the ranges 5?<?R/α?<?120, 5 nm?<?α?<?100 nm and 273 K?<?T?<?355 K.

     

Theoretical investigation of diffusion along columns packed with fully and superficially porous particles

Chromatographic columns packed with shell particles are now nearly twice more efficient than columns packed with conventional, fully porous particles. Shell particles are made of a solid core surrounded by a porous shell of constant thickness. Diffusion through the bed of packed columns is complex due to their heterogeneity. It involves diffusion through the external and the internal fluid, and surface diffusion. Six diffusion models are compared that combine these diffusion mechanisms. They involve the external porosity of the bed (?(e)), the ratio of the core to the particle diameters (ρ), and the ratio of the shell diffusivity to the bulk diffusion coefficient (Ω). Four different theoretical approaches were considered. They are based on (1) the additivity of the mass flux densities modulated by the obstruction factors caused by non-porous spherical inclusions; (2) the effective medium theory of Landauer; (3) the effective medium theory of Garnett for spherical inclusions; and (4) the probabilistic theory of Torquato (for binary composite materials only). The two Landauer models fail because they cannot account for the obstruction factor imposed by the presence of non-porous spherical inclusions. The ternary Garnett model (3) provides an excellent approximation of the actual diffusion mechanism but the most physically relevant model seems to be the one derived from a combination of the Garnett model for a binary core-shell particle and of the Torquato model for random dispersion of contacting spheres in a matrix. Accurate measurements of axial dispersion coefficients are needed to validate or reject the semi-empirical parallel diffusion models and to select the most appropriate one. The results of such measurements made with the peak parking method for various compounds are reported in the companion paper.     

Effects of dielectric gradients-mediated ions partitioning on the electrophoresis of composite soft particles: An analytical theory

In this work, we report original analytical expressions defining the electrophoretic mobility of composite soft particles comprising an inner core and a surrounding polymer shell with differentiated permeabilities to ions from aqueous background electrolyte and to fluid flow developed under applied DC field conditions. The existence of dielectric permittivity gradients operational at the core/shell and shell/solution interfaces is accounted for within the Debye–Hückel approximation and flat plate configuration valid in the thin double layer regime. The proposed electrophoretic mobility expressions, applicable to weakly to moderately charged particles with size well exceeding the Debye layer thickness, involve the relevant parameters describing the particle core/shell structure and the electrohydrodynamic features of the core and shell particle components. It is shown that the analytical expressions reported so far in literature for the mobility of hard (impermeable) or porous particles correspond to asymptotic limits of the more generic results detailed here. The impacts of dielectric-mediated effects of ions partitioning between bulk solution and particle body on the electrophoretic response are further discussed. The obtained expressions pave the way for a refined quantitative, analytical interpretation of electrophoretic mobility data collected on soft (nano)particles (e.g., functionalized dendrimers and multilayered polyelectrolytic particles) or biological cells (e.g., viruses) for which the classical hard core-soft shell representation is not appropriate.     

3D simulations of hydrodynamic drag forces on two porous spheres moving along their centerline

This paper numerically evaluates the hydrodynamic drag force exerted on two highly porous spheres moving steadily along their centerline through a quiescent Newtonian fluid over a Reynolds number ranging from 0.1 to 40. At creeping-flow limit, the drag forces exerted on both spheres were approximately identical. At higher Reynolds numbers the drag force on the leading sphere (sphere #1) was higher than the following sphere (sphere #2), revealing the shading effects produced by sphere #1 on sphere #2. At dimensionless diameter beta<2 (beta=d(f)/2k(0.5), d(f) and k are sphere diameter and interior permeability, respectively), the spheres can be regarded as "no-spheres" limit. At increasing beta for both spheres, the drag force on sphere #2 was increased because of the more difficult advective flow through its interior, and at the same time the drag was reduced owing to the stronger wake flow produced by the denser sphere #1. The competition between these two effects leads to complicated dependence of drag force on sphere #2 on beta value. These effects were minimal when beta became low.     

Cooperative motion of spheres arranged in periodic grids between two parallel walls

In this article, we analyze the collective motion of a two-dimensional periodic array of spheres in a slit-pore confined by two parallel planar walls. We determine the friction coefficient of the spheres when all particles move with the same velocity along a particular direction and cooperate with each other in their motion. In order to solve this many-body problem, we use Stokesian dynamics algorithm and resolve multiparticle hydrodynamic interactions in wall-bounded geometry. Apart from particle-particle interactions, we also recognize that the aforementioned collective motion of all particles creates a cumulative effect on the fluid medium. This effect is manifested as either a net induced flow for a periodic pressure field or an additional pressure gradient for quiescent fluid. In our analysis, we focus on both periodic pressure and no-flow conditions. For both cases, the hydrodynamic friction on the translating particles is calculated using our multiparticle Stokesian dynamics simulation. The simulation for the no-flow condition is relatively straightforward-we only need to compute the multiparticle hydrodynamic interactions in quiescent fluid. However, for the periodic pressure condition, the net induced flow dragged by the particles has to be evaluated also. We express this net induced flow in terms of an additional pressure-driven velocity field. We present the hydrodynamic friction as a function of the dimensions of the two-dimensional periodic lattice. For closely packed arrays, the results show a considerable reduction in friction coefficients that usually increase with interparticle distance. Hence, our work renders the theoretical justification for other recent findings that indicate the importance of interparticle mutual cooperation.     

Deposition of spherical particles onto cylindrical solid surfaces. I. Numerical simulations

The permeability of fractal porous aggregates with realistic three-dimensional structure is investigated theoretically using model aggregates composed of identical spherical primary particles. Synthetic aggregates are generated by several techniques, including a lattice-based method, simulation of aggregation by differential settling and turbulent shear, and the specification of simple cubic structures, resulting in aggregates characterized by the number of primary particles, solid fraction, characteristic radius, and fractal dimension. Stokesian dynamics is used to determine the total hydrodynamic force on and the distribution of velocity within an aggregate exposed to a uniform flow. The aggregate permeability is calculated by comparing these values with the total force and velocity distribution calculated from the Brinkman equation applied locally and to the entire aggregate using permeability expressions from the literature. The relationship between the aggregate permeability and solid fraction is found to be best predicted by permeability expressions based on cylindrical rather than spherical geometrical elements, the latter tending to underestimate the aggregate permeability significantly. The permeability expressions of Jackson and James or Davies provide good estimates of the force on and flow through porous aggregates of known structure. These relationships are used to identify a number of general characteristics of fractal aggregates.