Impermeable Spheres Research Articles

Slow Translation of a Composite Sphere in an Eccentric Spherical Cavity

This semi-analytical study is presented examining the quasi-steady creeping flow caused by a soft (composite) spherical particle, which is a hard (impermeable) sphere core covered by a porous (permeable) layer, translating in an incompressible Newtonian fluid within a non-concentric spherical cavity along the line joining their centers. To solve the Brinkman and Stokes equations for the flow fields inside and outside the porous layer, respectively, general solutions are constructed in two spherical coordinate systems attached to the particle and cavity individually. The boundary conditions at the cavity wall and particle surface are fulfilled through a collocation method. Numerical results of the normalized drag force exerted by the fluid on the particle are obtained for numerous values of the ratios of core-to-particle radii, particle-to-cavity radii, the distance between the centers to the radius difference of the particle and cavity, and the particle radius to porous layer permeation length. For the translation of a soft sphere within a concentric cavity or near a small-curvature cavity wall, our drag results agree with solutions available in the literature. The cavity effect on the drag force of a translating soft sphere is monotonically increasing functions of the ratios of core-to-particle radii and the particle radius to porous layer permeation length. While the drag force generally rises with an increase in the ratio of particle-to-cavity radii, a weak minimum (surprisingly, smaller than that for an unconfined soft sphere) may occur for the case of low ratios of core-to-particle radii and of the particle radius to permeation length. This drag force generally increases with an increase in the eccentricity of the particle position, but in the case of low ratios of core-to-particle radii and particle radius to permeation length, the drag force may decrease slightly with increasing eccentricity.

Study of Conducting Fluid Flow in Composite Regions Past an Impermeable Sphere in the Presence of Magnetic Field

A steady, two dimensional, incompressible, viscous and conducting fluid flow over a fixed rigid sphere has been considered under the effect of magnetic force applied normal to flow direction. The fluid flow occurs in three multiple regions namely fluid, porous and fluid region respectively. The governing equations are reduced into linear PDEs in terms of dimensionless parameters which intern converted into linear ODEs by similarity transformation method. The impact of Hartmann number and porosity on the fluid flow has been analyzed graphically. It is observed that as the Hartmann number increases for fixed porosity, the flow of fluid is well controlled in porous and non-porous regions. Further, as porosity increases for fixed Hartmann number, fluid flow over a porous region is observed. Also, diminishes the fluid velocity in the porous region due to the suppression of the fluid flow as ‘σ’ increases when magnetic field is fixed to finite constant. The same observation is made when the Hartmann number is intensified for the fixed porosity ‘σ’.

Effective hydraulic properties of 3D virtual stony soils identified by inverse modeling

Abstract. Stony soils that have a considerable amount of rock fragments (RFs) are widespread around the world. However, experiments to determine the effective soil hydraulic properties (SHPs) of stony soils, i.e., the water retention curve (WRC) and hydraulic conductivity curve (HCC), are challenging. Installation of measurement devices and sensors in these soils is difficult, and the data are less reliable because of their high local heterogeneity. Therefore, effective properties of stony soils especially under unsaturated hydraulic conditions are still not well understood. An alternative approach to evaluate the SHPs of these systems with internal structural heterogeneity is numerical simulation. We used the Hydrus 2D/3D software to create virtual stony soils in 3D and simulate water flow for different volumetric fractions of RFs, f. Stony soils with different values of f from 11 % to 37 % were created by placing impermeable spheres as RFs in a sandy loam soil. Time series of local pressure heads at various depths, mean water contents, and fluxes across the upper boundary were generated in a virtual evaporation experiment. Additionally, a multistep unit-gradient simulation was applied to determine effective values of hydraulic conductivity near saturation up to pF=2. The generated data were evaluated by inverse modeling, assuming a homogeneous system, and the effective hydraulic properties were identified. The effective properties were compared with predictions from available scaling models of SHPs for different values of f. Our results showed that scaling the WRC of the background soil based on only the value of f gives acceptable results in the case of impermeable RFs. However, the reduction in conductivity could not be simply scaled by the value of f. Predictions were highly improved by applying the Novák, Maxwell, and GEM models to scale the HCC. The Maxwell model matched the numerically identified HCC best.

Settling of highly porous and impermeable particles in linear stratification: implications for marine aggregates

The settling velocity of porous particles in linear stratification is affected by the diffusive exchange between interstitial and ambient water. The extent to which buoyancy and interstitial mass adaptation alters the settling velocity depends on the ratio of the diffusive and viscous time scales. We conducted schlieren experiments and lattice Boltzmann simulations for highly porous (95 %) but impermeable spheres settling in linear stratification. For a parameter range that resembles marine porous particles, ‘marine aggregates’, i.e. low Reynolds numbers ($0.05\leq \textit {Re}\leq 10$), intermediate Froude numbers ($0.1\leq \textit {Fr}\leq 100$) and Schmidt number of salt ($\textit {Sc}=700$), we observe delayed mass adaptation of the interstitial fluid due to lower-density fluid being dragged by a particle that forms a density boundary layer around the particle. The boundary layer buffers the diffusive exchange of stratifying agent with the ambient fluid, leading to an enhanced density contrast of the interstitial pore fluid. Stratification-related drag enhancement by means of additional buoyancy of dragging lighter fluid and buoyancy-induced vorticity resembles earlier findings for solid spheres. However, the exchange between density boundary layer and pore fluid substantially increases stratification drag for small$\textit {Fr}$. To estimate the effect of stratification on marine aggregates settling in the ocean, we derived scaling laws and show that small particles ($\leq$0.5 mm) experience enhanced drag which increases retention times by 10 % while larger porous particle (>0.5 mm) settling is dominated by delayed mass adaptation that diminishes settling velocity by 10 % up to almost 100 %. The derived relationships facilitate the integration of stratification-dependent settling velocities into biogeochemical models.

Effect of interfacial transport on the diffusivity of highly filled polymers

The diffusivity of substances, such as moisture, through polymer composites is often described by an effective macroscopic quantity, even though microscopically the diffusivity might be far from uniform. In this work, we study the theoretical example of a permeable matrix containing equal-sized impermeable spheres. We assume that, due to interface effects, the diffusivity of the matrix in the vicinity of the spheres is higher than its bulk matrix diffusivity. Using numerical simulations of the composite's diffusivity, we show that upon the formation of large clusters of the highly permeable interfaces, i.e. near percolation of the spheres, the diffusivity of the composite rises sharply. For even higher values of the volume fraction of the spheres, up to the close-packing limit, the diffusivity decreases due to the increased tortuosity. This effect is well described by an analytical solution for the composite's diffusivity.

Cell Models for Non-Newtonian Fluid Past a Semipermeable Sphere

This paper is focused on investigating the boundary effects of the steady translational motion of a semipermeable sphere located at the center of a spherical envelope filled with an incompressible micropolar fluid. Stokes equations of micropolar fluid are employed inside the spherical envelope and Darcy’s law governs in semipermeable region. On the surface of semipermeable sphere, the boundary conditions used are continuity of normal velocity, vanishing of tangential velocity of micropolar fluid, and continuity of pressure. On the surface of the spherical envelope, the Happel’s, Kuwabara’s, Kvashnin’s, and Cunningham’s boundary conditions, are used along with no spin boundary condition. The expression for the hydrodynamic normalized drag force acting on the semipermeable sphere is obtained. The limiting cases of drag expression exerted on the semipermeable sphere and impermeable solid sphere in cell models filled with Newtonian fluid are obtained. Also, in absence of envelope, the drag expression for the micropolar fluid past a semipermeable sphere is obtained.

Assessment of the interplay between scaffold geometry, induced shear stresses, and cell proliferation within a packed bed perfusion bioreactor.

By favoring cell proliferation and differentiation, perfusion bioreactors proved efficient at optimizing cell culture. The aim of this study was to quantify cell proliferation within a perfusion bioreactor and correlate it to the wall shear stress (WSS) distribution by combining 3-D imaging and computational fluid dynamics simulations.NIH-3T3 fibroblasts were cultured onto a scaffold model made of impermeable polyacetal spheres or Polydimethylsiloxane cubes. After 1, 2, and 3 weeks of culture, constructs were analyzed by micro-computed tomography (μCT) and quantification of cell proliferation was assessed. After 3 weeks, the volume of cells was found four times higher in the stacking of spheres than in the stacking of cube.3D-μCT reconstruction of bioreactors was used as input for the numerical simulations. Using a lattice-Boltzmann method, we simulated the fluid flow within the bioreactors. We retrieved the WSS distribution (PDF) on the scaffolds surface at the beginning of cultivation and correlated this distribution to the local presence of cells after 3 weeks of cultivation. We found that the WSS distributions strongly differ between spheres and cubes even if the porosity and the specific wetted area of the stackings were very similar. The PDF is narrower and the mean WSS is lower for cubes (11 mPa) than for spheres (20 mPa). For the stacking of spheres, the relative occupancy of the surface sites by cells is maximal when WSS is greater than 20 mPa. For cubes, the relative occupancy is maximal when the WSS is lower than 10 mPa. The discrepancies between spheres and cubes are attributed to the more numerous sites in stacking of spheres that may induce 3-D (multi-layered) proliferation.

MOTION OF A POROUS SPHERICAL SHELL IN A SPHERICAL CONTAINER

In this paper slow motion of a porous spherical shell with radially varying permeability in a spherical container at the instant it passes through the center of the spherical container is discussed. The container is filled by a viscous incompressible fluid. Flow outside the porous spherical shell is governed by the Stokes equation and inside the spherical shell by the Brinkman equation. The boundary conditions used at the surface of the porous spherical shell are the continuity of velocity and stress and the no-slip condition is used on qn impermeable sphere. Exact solution of the problem is obtained and relevant quantities such as stream lines, velocity, pressure, wall correction factor, and drag on surface of the spherical shell are obtained. The influence of the permeability parameter on the flow has been discussed and exhibited graphically. We found that decrease in permeability of the porous shell caused increase in drag and wall correction factor. In the limit case obtained results reduce to the well known classical results.

Viscous Fluid Flows Induced by Translational-Oscillatory Motion of a Submerged Porous Sphere

Viscous fluid flows induced by a translational-oscillatory motion of a submerged porous sphere are determined. The fluidmotion inside and outside the sphere is considered in amoving noninertial reference frame fitted to the sphere. Exact analytical solutions of the Navier–Stokes equation outside the sphere and the nonstationary Brinkman equation inside the sphere are found. Examples of streamline patterns for some values of governing parameters are presented. It is shown that the results obtained earlier for particular cases of viscous flow around an impermeable solid sphere can be obtained from the solution presented.

Dipolic Flows Relevant to Aquifer Storage and Recovery: Strack’s Sink Solution Revisited

Steady, 2-,3-D Darcian flows generated by a dipole (a pair of horizontal or vertical injection–abstraction wells closely placed one above another), with circulation of fresh water inside an interface confined lens or “bubble” underneath an impermeable caprock, surrounded by a static saline groundwater, are analytically studied. For 2-D dipole, the complex potential domain is a plane with a horizontal cut. This domain is conformally mapped onto a reference half-plane where the Keldysh–Sedov formula is used to obtain the complex physical coordinates. Explicit closed-form expressions for the vase-shaped interface, flow net, isohypses, magnitudes of the Darcian velocity and Riesenkampf’s resultant force are obtained, depending on the dipole moment, its position with respect to the caprock, and the ratio of densities of the two fluids. It is shown that for sufficiently small injection-pumping rates the fresh water “vase” separates from the caprock and becomes a circle, inside which streamlines are Newtons’ loops of monodiametral degenerate hyperbolae (cubics). Two numerical codes, MT3DMS and SEAWAT, are also used for delineation of isoconcentric lines, which qualitatively corroborate the analytical solutions in delineation of the “bubble” in the part where the sharp interface model predicts stable free boundaries and evidencing “dimples” on the boundary of the “bubble” where the saline water overlies the fresh one. For 3-D dipole not bounded by the caprock, the analytical fresh water “bubble” is a sphere and solution follows, mutatis mutandis, from the textbook formulae for flow of an ideal fluid past an impermeable sphere. The Stokes streamlines inside the sphere are sixtics; isotachs are plotted in an axial section. Stability of the soil matrix near the wells is also discussed.

Experimental validation of a bias in apparent exchange rate measurement

Abstract Double diffusion encoding (DDE) MRI is a technique in which two diffusion weighted periods are applied between excitation and acquisition. This allows to acquire structural information (as pore size and shape) and molecular exchange between compartments, in terms of apparent exchange rate (AXR). However, it has not been well investigated yet whether the diffusion restriction underlying DDE size estimation may compromise AXR results. Numerical simulations for impermeable spheres have shown that in case of small pores (diameter < 10 μm) the restriction effect may be neglected. Yet, for larger pores, the restriction effect can be relevant and needs to be considered in the analysis of experiments. Here, we present experiments performed on a 3 T clinical MRI system using a water-in-oil emulsion in which no exchange between compartments is expected. The results show that a nonzero apparent exchange rate may be estimated in a sample with impermeable pores. This shows that, as predicted in simulations, the restriction effect affects the AXR results. The use of perpendicular diffusion gradients may be a solution for this problem in isotropic samples.

The larmor frequency shift in magnetically heterogeneous media depends on their mesoscopic structure.

Recent studies have addressed the determination of the NMR precession frequency in biological tissues containing magnetic susceptibility differences between cell types. The purpose of this study is to investigate the dependence of the precession frequency on medium microstructure using a simple physical model. This dependence is governed by diffusion of NMR-visible molecules in magnetically heterogeneous microenvironments. In the limit of fast diffusion, the precession frequency is determined by the average susceptibility-induced magnetic field, whereas in the limit of slow diffusion it is determined by the average local phase factor of precessing spins. The main method used is Monte Carlo simulation of isotropic suspensions of impermeable magnetized spheres. In addition, NMR spectroscopy was performed in aqueous suspensions of polystyrene microbeads. The precession frequency depends on the structural organization of magnetized objects in the medium. Monte Carlo simulations demonstrated a nonmonotonic transition between the regimes of fast and slow diffusion. NMR experiments confirmed the transition, but were unable to confirm its precise form. Results for a given pattern of structural organization obey a scaling law. The NMR precession frequency exhibits a complex dependence on medium structure. Our results suggest that the commonly assumed limit of fast water diffusion holds for biological tissues with small cells. Magn Reson Med 79:1101-1110, 2018. © 2017 International Society for Magnetic Resonance in Medicine.

Functional regression-based fluid permeability prediction in monodisperse sphere packings from isotropic two-point correlation functions

We study fluid permeability in random sphere packings consisting of impermeable monodisperse hard spheres. Several different pseudo-potential models are used to obtain varying degrees of microstructural heterogeneity. Systematically varying solid volume fraction and degree of heterogeneity, virtual screening of more than 10,000 material structures is performed, simulating fluid flow using a lattice Boltzmann framework and computing the permeability. We develop a well-performing functional regression model for permeability prediction based on using isotropic two-point correlation functions as microstructural descriptors. The performance is good over a large range of solid volume fractions and degrees of heterogeneity, and to our knowledge this is the first attempt at using two-point correlation functions as functional predictors in a nonparametric statistics/machine learning context for permeability prediction.

Heat transfer characteristics of slip flow over solid spheres

In this study, heat transfer characteristics of slip flow over an isolated impermeable solid sphere are investigated numerically. An isothermal solid sphere is considered at intermediate Reynolds numbers (0 ≤ Re ≤ 50) for Prandtl numbers in the range of 0.7–7.0. The Navier–Stokes and energy equations are solved by a control volume technique in conjunction with the velocity slip and temperature jump boundary conditions. It was found that the size of the thermal wake region according to the Knudsen number depends on the Prandtl number. At lower Prandtl numbers (0.7 ≤ Pr ≤ 2.0), the thermal wake region shrinks as the Knudsen number increases, while at higher Prandtl numbers, it grows as the Knudsen number increases. The maximum temperature jump occurs at the front stagnation point where the local Nusselt is itself maximum, owing to the maximum temperature gradient at this point. The results show that due to the opposing effects of the velocity slip and temperature jump, the average Nusselt number variation with the Knudsen number depends nonlinearly on both the Prandtl and Reynolds numbers. Furthermore, for the limiting case of Re → 0, an analytical solution for the problem is presented which has also served as a validation case.

Brinkman – Forchheimer – Darcy Flow Past an Impermeable Sphere Embedded in a Porous Medium

Abstract The flow past an impermeable sphere embedded in a fluid saturated porous medium was studied numerically considering valid the Brinkman-Forchheimer-Darcy (or Brinkman-Hazen-Dupuit-Darcy) model. The flow is viscous, laminar, axisymmetric, steady and incompressible. The porous medium is isotropic, rigid and homogeneous. The stream function - vorticity equations were solved numerically in spherical coordinates system by a multigrid method. The influence of the Darcy number and Forchheimer term on the velocities field was investigated for two boundary conditions on the surface of the sphere: slip and no - slip.

A Numerical Study of the Flow Past an Impermeable Sphere Embedded in a Porous Medium

The flow past an impermeable sphere embedded in a fluid-saturated porous medium was investigated numerically. The flow is considered viscous, laminar, axisymmetric, steady and incompressible. The generalized model for fluid flow through an isotropic, rigid and homogeneous porous medium was used. The stream function–vorticity equations were solved numerically in spherical coordinates system by a defect correction multigrid method. The computations were focused on the influence of the Darcy number (Da) and Forchheimer constant on the flow field for two boundary conditions on the surface of the sphere: slip and no-slip. The conditions when the macroscopic inertial terms of the generalized model can be neglected are presented. For high-porosity porous media, the macroscopic inertial terms of the generalized model can be neglected if Da \(Re< 1\) (Re is the sphere Reynolds number).

Analytical solutions of compacting flow past a sphere

AbstractA series of analytical solutions are presented for viscous compacting flow past a rigid impermeable sphere. The sphere is surrounded by a two-phase medium consisting of a viscously deformable solid matrix skeleton through which a low-viscosity liquid melt can percolate. The flow of the two-phase medium is described by McKenzie’s compaction equations, which combine Darcy flow of the liquid melt with Stokes flow of the solid matrix. The analytical solutions are found using an extension of the Papkovich–Neuber technique for Stokes flow. Solutions are presented for the three components of linear flow past a sphere: translation, rotation and straining flow. Faxén laws for the force, torque and stresslet on a rigid sphere in an arbitrary compacting flow are derived. The analytical solutions provide instantaneous solutions to the compaction equations in a uniform medium, but can also be used to numerically calculate an approximate evolution of the porosity over time whilst the porosity variations remain small. These solutions will be useful for interpreting the results of deformation experiments on partially molten rocks.

Hydrodynamic Properties of Aerobic Granules Cultivated on Phenol as Carbon Source

The cultivation and hydrodynamic properties such as morphology, fractal dimension, porosity, size distribution, and settling velocity of stable aerobic granules, developed in a column type sequencing batch reactor SBR was investigated in this study. A column type SBR was operated with organic loading rate of 1.8kg phenol/m3day with phenol as a sole carbon source. The granules were fractal and porous aggregates and had a fractal dimension and porosity of 2.47 and 0.7-0.9 respectively. The settling velocities of aerobic granules were in the range of 2.38x10-02m/s-7.1x10-02m/s. This was in good agreement with the settling velocities predicted by Stoke's law for porous but impermeable spheres. This may be due to the EPS (Extracellular Polymeric Substances) produced by bacteria form a gel matrix that clogs the pores within the granules which resulted in reduced permeation and settling velocities.

Electric-field-controlled unpinning of scroll waves

Three-dimensional excitation vortices exist in systems such as chemical reactions and the human heart. Their one-dimensional rotation backbone can pin to unexcitable heterogeneities, which greatly affect the structure, dynamics, and lifetime of the vortex. In experiments with the Belousov-Zhabotinsky reaction, we demonstrate vortex unpinning from a pair of inert and impermeable spheres using externally applied electric fields. Unpinning occurs abruptly but is preceded by a slow reorientation and deformation of the initially circular vortex loop. Our experimental findings are reproduced by numerical simulations of an excitable reaction-diffusion-advection model.

Electrophoresis of a Charged Soft Particle in a Charged Cavity with Arbitrary Double-Layer Thickness

An analysis for the quasi-steady electrophoretic motion of a soft particle composed of a charged spherical rigid core and an adsorbed porous layer positioned at the center of a charged spherical cavity filled with an arbitrary electrolyte solution is presented. Within the porous layer, frictional segments with fixed charges are assumed to distribute uniformly. Through the use of the linearized Poisson-Boltzmann equation and the Laplace equation, the equilibrium double-layer potential distribution and its perturbation caused by the applied electric field are separately determined. The modified Stokes and Brinkman equations governing the fluid flow fields outside and inside the porous layer, respectively, are solved subsequently. An explicit formula for the electrokinetic migration velocity of the soft particle in terms of the fixed charge densities on the rigid core surface, in the porous layer, and on the cavity wall is obtained from a balance between its electrostatic and hydrodynamic forces. This formula is valid for arbitrary values of κa, λa, r0/a, and a/b, where κ is the Debye screening parameter, λ is the reciprocal of the length characterizing the extent of flow penetration inside the porous layer, a is the radius of the soft particle, r0 is the radius of the rigid core of the particle, and b is the radius of the cavity. In the limiting cases of r0 = a and r0 = 0, the migration velocity for the charged soft sphere reduces to that for a charged impermeable sphere and that for a charged porous sphere, respectively, in the charged cavity. The effect of the surface charge at the cavity wall on the particle migration can be significant, and the particle may reverse the direction of its migration.