Steady-state Concentration Distribution Research Articles

A membrane’s blueprint: In silico investigation of fluid flow and molecular transport as a function of membrane design parameters in organ-on-a-chip

Following the rapid growth of Organ-on-a-Chip (OoaC) technology, porous membranes have become essential components for in vitro tissue barrier models. Nonetheless, literature highlights lacking knowledge on their integration and effect on microfluidic devices. Therefore, we conducted finite element modelling (FEM) to characterize the influence of membrane, channel geometry, flow and diffusion parameters, in modelling flow rate, shear stress, transient transport and steady state molecular concentration. This analysis was performed for four different conditions based on single channel (SCP) and parallel perfusion (PP). It was found that membrane and geometry parameters are crucial in determining flow and shear for SCP. However, for PP, flow and shear are predominantly governed by the inlet flow rate. Although the transient behaviour is well-controlled within SCP and PP, only PP allows modelling the steady state concentration distribution. It is highlighted that: (1) the pore radius has great influence on flow and shear; (2) a shallow cell channel and a long membrane are capable of establishing different levels of shear on opposing surfaces of the same channel; (3) the membrane thickness, membrane length, height of the cell and flow channels, and inlet flow rate provide good control over transient transport; (4) the membrane length and inlet flow rate enable changing the concentration from a uniform distribution to a complete heterogeneous state across the device. Experimental assays were performed to support the FEM and evidence its significance for OoaC applications. Ultimately, extensive, and systematic guidelines are provided on designing future OoaC devices with integrated porous membranes.

Accumulation and Stretching of DNA Molecules in Temperature-Induced Concentration Gradients.

Temperature fields provide a noninvasive approach for manipulating individual macromolecules in solution. Utilizing thermophoresis and other secondary effects resulting from the inhomogeneous distribution of crowding agents, one may gain valuable insights into the interactions of molecular mixtures. In this report, we examine the steady-state concentration distribution and dynamics of DNA molecules in a poly(ethylene glycol) (PEG)/water solution when exposed to localized temperature gradients generated by optical heating of a thin chrome layer at a liquid-solid boundary. This allowed us to experimentally investigate the interplay between DNA thermophoresis and PEG-induced entropic depletion effects. Our quantitative analysis demonstrates that the depletion effects dominate over DNA thermophoresis, causing the DNA polymers to migrate toward the heat source. Additionally, we explore the transient stretching of individual DNA molecules in thermally induced PEG gradients and estimate the contributing forces.

Determination of constant diffusion coefficient for error function solution of diffusion equation using film-roll method

The error function solution of the diffusion equation with the constant surface concentration was derived by the heat kernel to determine the constant diffusion coefficient for diffusion satisfying the infinite dye-bath condition. For the sublimation diffusion of disperse dye in paste into PET film using the film-roll method, the constant surface concentration was determined from the concentration distribution, and the diffusion coefficients of each layer were obtained by the constant surface concentration from the error function solution. When comparing the diffusion coefficients between the layers and comparing the mean diffusion coefficients for different times at a specific temperature, the constant surface concentrations determined from the quadratic regression curves for concentration–distance plots were more appropriate than those determined from the steady-state concentration distributions, which was also confirmed by the plots of concentrations obtained from the error function solution. At a specific temperature, the average of the mean diffusion coefficients obtained by the constant surface concentration of the quadratic regression curve at three specific times matched well with the constant total-amount diffusion coefficient obtained by the slope of the linear regression line for the plot of total amount against square root of time, which was confirmed by their Arrhenius plots.

Experimental and numerical study on the microstructure and chloride ion transport behavior of concrete-to-concrete interface

Concrete-to-concrete interfaces present both in new and existing construction, are commonly regarded as the weak link. Durability research of interfaces is a vital prerequisite for structural rehabilitation and strengthening applications under marine and coastal environments. Hence, to clarify the chloride ion transport behavior between the new-old concrete composite system and its constituents, microstructural characteristics of the interfaces with various preparation methods were determined via field emission scanning electron microscopy. Then, the time-dependent diffusion chloride ion penetration together with its steady-state concentration distribution evaluated by using AgNO3 chromogenic, electron microprobe, and energy spectrum methods, as well as the numerical simulation based on the real interface roughness from the 3D laser scanning optical system. Results show that microcracks and a large amount of CaCO3 precipitations were formed in the concrete-to-concrete interface, where its porosity was higher compared to that of the matrix and interface transition zone around coarse aggregate; Therefore, the interface regions have higher tortuosity of water transport channel and strengthen the solution absorption rate, resulting in the internal chloride ions not only diffuse deeper perpendicular to the exposed surface but also ingress into the concrete on both sides as the concentration gradient simultaneously, leading to a “diffusion peak” formation. Furthermore, the effects of groove characteristics (such as orientations, clearance, width, and depth) on chloride transport were modeling evaluated in this study, in which the interfacial relationship could be optimized setting the reasonable groove and achieving the goal of durability equivalent to the cast-in-place structure.

Diffusion coefficient calculated by time-lag using the film-roll method

A method for determining the diffusion coefficient by time-lag using the film-roll method for the sublimation diffusion of disperse dye was proposed. A polyethylene terephthalate film-roll coated with dye paste was treated at 170–190°C for various times. A solution consisting of the sum of a steady-state solution and a transient solution was obtained by the homogeneous boundary value problem from a trigonometrical series. The boundary conditions of the steady-state first layer and the steady-state first layer amount of dye were determined from the steady-state concentration distribution. For various diffusion times, the steady-state first layer-passed total amounts of dye that passed through the first layer in the steady-state condition were obtained by subtracting the steady-state first layer amounts from the total amounts. The time-lag was calculated from the linear regression line for the plot of the steady-state first layer-passed total number(X) of positive values against time. The diffusion coefficient was calculated by the boundary conditions of the steady-state first layer and the time-lag. For diffusion at 170°C, 180°C, and 190°C, the correlations of the steady-state first layer-passed total amounts with respect to time were very linear and the reliability of the diffusion coefficients obtained by the time-lag was proved by the good linearity of the Arrhenius plot. The activation energy obtained was 36.8 kcal[Formula: see text]mol−1.

On the reversibility of membrane fouling by deposits produced during crossflow ultrafiltration of casein micelle suspensions

The formation of a deposit layer on membrane surfaces is an important problem, particularly in milk filtration. Properties of the deposit and efficiency of deposit removal can be studied using various indirect methods based mainly on average membrane resistance in deposit formation–deposit removal cycles, but their conclusions on deposit reversibility depend on conditions surrounding deposit analysis.Here we present a method for direct characterization of the reversibility of deposit compression via analysis of local solid concentration distribution during deposit formation at constant pressure and cross-flow velocity and deposit removal after pressure relaxation. Two models are proposed for deposit characterization: a model of crossflow filtration with deposit formation to characterize deposit compressibility-permeability from the data on concentration distribution at steady-state of deposit formation, and a model for evaluating deposit swelling kinetics from the data on steady-state compressibility-permeability.The method was applied here for analysis of the deposit formed on the surface of a polymeric ultrafiltration membrane during crossflow filtration of casein micelle suspensions at 25°C. Local solid concentration distribution in the deposit and concentration polarization layer was probed in situ using small-angle X-ray scattering (SAXS). Experiments were performed in a special-purpose SAXS–filtration cell. It was demonstrated that casein micelle deposits obtained at 1.1 bar transmembrane pressure swell after pressure relaxation. However, swelling rate was significantly lower than that obtained by modeling using deposit compressibility–permeability obtained from the analysis of steady-state concentration distribution. We thus find evidence that casein micelle deposits produced during crossflow membrane filtration undergo compression that is partially irreversible.

On random walk models for simulation of particle-laden turbulent flows

In this investigation, the accuracy of the discrete and continuous random walk (DRW, CRW) stochastic models for simulation of fluid (material) point particle, as well as inertial and Brownian particles, was studied. The corresponding dispersion, concentration, and deposition of suspended micro- and nano-particles in turbulent flows were analyzed. First, the DRW model used in the ANSYS-Fluent commercial CFD code for generating instantaneous flow fluctuations in inhomogeneous turbulent flows was evaluated. For this purpose, turbulent flows in a channel were simulated using a Reynolds-averaged Navier–Stokes (RANS) approach in conjunction with the Reynolds Stress Transport turbulence model (RSTM). Then spherical particles with diameters in the range of 30 µm to 10 nm were introduced uniformly in the channel. Under the assumption of one-way coupling, ensembles of particle trajectories for different sizes were generated by solving the particle equation of motion, including the drag and Brownian forces. The DRW stochastic turbulence model of the software was used to include the effects of instantaneous velocity fluctuations on particle motion, and the steady state concentration distribution and deposition velocity of particles of various sizes were evaluated. In addition, the improved CRW model based on the normalized Langevin equation was used in an in-house Matlab code. Comparisons of the predicted results of the DRW model of ANSYS-Fluent with the available experimental data and the DNS simulation results and empirical predictions showed that this model is not able to accurately predict the flow fluctuations seen by the particles in that it leads to unreasonable concentration profiles and time-varying deposition velocities. However, the predictions of the improved CRW model were in good agreement with the experimental data and the DNS results. Possible reasons causing the discrepancies between the DRW predictions and the experimental data were discussed. The improved CRW model was also implemented through user-defined functions into the ANSYS-Fluent code, which resulted in accurate concentration distribution and deposition velocity for different size particles.

Convection–diffusion molecular transport in a microfluidic bilayer device with a porous membrane

The field of human cell research is rapidly changing due to the introduction of microphysiological systems, which commonly feature two stacked microchannels separated by a porous membrane for in vitro barrier modeling. An essential component to adequately representing a subset of human organ or tissue functions in these microfluidic systems is the concentration distribution of the biospecies involved. In particular, when different cell types are cultured, a delicate balance between media mixing and cellular signaling is required for long-term maintenance of the cellular co-culture. In this work, we experimentally measured the effects of various control parameters on the transient and steady average molecular concentration at the bilayer device outlet. Using these experimental results for validation, we then numerically investigated the concentration distributions due to the convection–diffusion mass transport in both microchannels. The effects of media flow rate, separation membrane porosity, molecular size, microchannel dimensions and flow direction have been systematically characterized. The transient response is found to be negligible for cell co-cultures lasting several days, while the steady-state concentration distribution is dominated by the media flow rate and separation membrane porosity. Numerically computed concentration profiles reveal self-similarity characteristics featuring a diffusive boundary layer, which can be manipulated for successful maintenance of cell co-culture with limited media mixing and enhanced cell signaling.

Cumulative relative reactivity: A concept for modeling aquifer-scale reactive transport

We simulate aquifer-scale reactive transport using an approach based on travel times and relative reactivity. The latter quantifies the intensity of the chemical reaction relative to a reference reaction rate with identical concentrations and can be interpreted as the strength of electron-donor (or electron-acceptor) release by the matrix, scaled by a reference release. In general, the relative reactivity is a spatially variable property reflecting the geology of the formation. In the proposed approach, we track the path of individual water parcels through the aquifer and evaluate the age of the water parcels and the relative reactivity integrated along their trajectories. By switching from spatial discretization to cumulative relative reactivity, advective-reactive transport can be simulated by solving a single system of ordinary differential equations for each combination of concentrations in the inflow. We test the validity of the approach in a two-dimensional test case of steady-state groundwater flow and reactive transport involving aerobic respiration and denitrification. Here we compare steady-state concentration distributions of the spatially explicit virtual truth, accounting for dispersive mixing, with the approximation based on cumulative relative reactivity and show that the errors introduced by neglecting dispersive mixing are minor if the target quantities are the mass fluxes crossing a control plane or being collected by a well. We further demonstrate the efficiency of the approach in a synthetic three-dimensional case study. The proposed approach is computationally so efficient, that ensemble runs to assess statistical distributions of concentration time series of reactive solutes become feasible, which is not practical with a spatially explicit model. This article is protected by copyright. All rights reserved.

Observation of Bi Coarsening and Dissolution Behaviors in Melting Al–Bi Immiscible Alloy

Using synchrotron radiography, a melting process of the Al-10 wt% Bi immiscible alloy was fully observed, and Bi solute coarsening and dissolution behaviors as well as their effect on the microstructural evolution were also investigated. We found that two different steps including coarsening and dissolution processes can be used to describe the structure evolution in melting Al-10 wt% Bi immiscible alloy. And diffusion coalescence controlled the coalescence of Bi droplets at the early stage of the melting Al-10 wt% Bi immiscible alloy, while the coagulation during movement did at later stage. Also, the larger the droplet, the larger the distance of the steady-state concentration distribution around the droplet.

Damage-repair model for mutagenic effects of plant induced by ionizing radiation

The plant mutation effects induced by ionizing radiation involve a rather complex process which is composed of physical, chemical, biochemical and biological stages. Nowadays, although ionizing radiation has been widely used in plant mutation breeding, the theoretical explanations for the mechanism of the ionizing radiation caused plant mutation effects are insufficient. Especially, a saddle shape relationship between the plant survival rate and radiation dose is found in the mutagenis effect of ionizing radiation on plants. The underlying mechanism of the saddle shape relationship remains unclear and challenges to all extant models.To explain this relationship, a damage-repair model for the plant mutation effects induced by ionizing radiation is proposed in the present work. Our model is based on the rate theory of ionizing radiation in which the cell damage and repair are taken into account simultaneously together with the micro-and macro-biological mutation effects of plant caused by ionizing radiation. The states of the radiated plant individuals are grouped into three categories: normal, damaged and lethal categories in our model. The evolution dynamics of the relative concentrations of the three categories are determined by a set of coupled equations which are mathematically the same as the Crow-Kimura equations in species evolution theories. With the numerical solution of our model in its steady state, the relative steady state concentration distributions of different categories of the radiated plants with increasing radiation dose are obtained. It is shown that without the plant repair effect, the relationship between the plant survival rate and radiation dose appears to be a conventional shoulder type one. With the plant repair effect, our model gives a saddle shape survival-dose relationship which has been observed commonly in the experiments on the radiated plants by ionizing radiation. To further test the model, the experimental data on the inbred lines of maizes radiated by heavy ion 7Li are used to determine the parameters of the model. It is shown that the theoretical results are basically consistent with the experimental ones. In addition, the mutation characteristic of the survival plants also appears to be a saddle effect-dose relationship, for which the theoretical model could also give a reasonable explanation. Our damage-repair model explains the saddle shape relationship between the plant survival rate and radiation dose, which indeed illuminates its power. And it provides a theoretical basis and reference for studying the biological effect mechanism of plants induced by ionizing radiation and conducting ionizing radiation plant breeding.

Spatial periodicity in bed form‐scale solute and thermal transport models of the hyporheic zone

AbstractSpatially periodic solute boundaries force symmetry across a model domain by ensuring that concentrations and concentration gradients are identical at the same location on opposite boundaries. They have been used in multiple publications on a hyporheic zone model of a single ripple or dune style bed form, including variable density flow and reactive transport variants. We evaluate simulations of multibed form models without imposing spatially periodic transport to demonstrate that nonphysical solute distributions arise from the periodic solute transport assumption. That is, the flow field within the single bed form model leads to a transport scenario that violates the forced symmetry of periodic solute boundary conditions, culminating in a physically unrealistic solute distribution. Our results show that lack of symmetry between boundaries is a function of the vertical concentration gradient and two‐dimensionless parameters characterizing the hyporheic and underflow flow regimes, and the solute exchange between them. We assess the error associated with the spatially periodic assumption based on an analysis of solute fluxes across the lateral bed form model boundaries. While the focus is on steady state concentration distributions, the implications for transient solute transport models are also discussed. We conclude that periodic solute transport boundary conditions should be applied only to bed form models that have minimal vertical dispersive and diffusive solute transfer. This includes gaining systems and tracers such as temperature, for which a temporally periodic flux reversal occurs across the top boundary.

Effect of pollutant decay on steady-state concentration distributions in variable depth flow

Pollutants discharged together in a variable-depth flow can separate laterally if the decay mechanisms differ e.g. evaporation is more effective in shallow water, but floc breakup is more effective in deeper faster flow. Ray tracing of concentration information away from a point source (similar to mathematical optics) is used to quantify the effects of decay. The concentration reduction rate along a ray increases with decay rate divided by flow speed and with angle between the flow and information directions. Pollutant concentrations tend to fractionate towards regions of smaller decay rate divided by flow speed.

Simple models for dynamic melting in an upwelling heterogeneous mantle column: Analytical solutions

Several lines of evidence suggest that the melt generation and segregation regions of the mantle are heterogeneous, consisting of chemically and lithologically distinct domains of variable size and dimension. Partial melting of such heterogeneous mantle source regions gives rise to a diverse range of basaltic magmas. In order to better assess the role of source heterogeneity during mantle melting, we have undertaken a theoretical study of trace element distribution and fractionation during concurrent melting and melt migration in an upwelling, chemically heterogeneous, two-porosity double lithology melting column. Analytical solutions for the abundance of a trace element in the matrix and channel were obtained under the assumptions that the porosity, melt and solid velocities, and solid–melt partition coefficients are constant and uniform. For simplicity, we neglected diffusion and dispersion in the melt. Chemical source heterogeneities of arbitrary size and shape were integrated into the simple melting models by allowing trace element abundance in the source region to vary as a function of time and space. Concurrent melting and melt migration in an upwelling heterogeneous mantle may be approximated as a quasi-steady state problem in which time-dependent concentration patterns produced by melting of heterogeneous source regions are superimposed on a reference steady-state concentration distribution established by melting of the ambient or background mantle. Chromatographic fractionation is especially important for the matrix melt and solid when chemical heterogeneities are involved during melting and melt migration in the mantle, giving rise to significant phase-shift between two incompatible trace elements in the matrix melt and scattered correlations among incompatible trace elements in residual peridotites. Mixing is the chief mass transfer process in the dunite channel where the chromatographic effect is negligible for most of the incompatible trace elements. The lack of chromatographic fractionation among incompatible trace elements and isotopic ratios in MORB suggests either most MORB are channel melts or mixing in magma conduit and chamber is very efficient such that the phase-shift is averaged out during magma transport and storage processes. Advection brings melt produced by smaller-degree of melting in the deeper part of the melting column to the overlying melting region, increasing the incompatible trace element abundance in the matrix and the channel. This advection-induced self-enrichment is especially important when heterogeneous sources are involved and may account for some of the enriched incompatible trace element patterns observed in residual peridotite that were previously interpreted to be a result of mantle metasomatism. Systematic studies of high-resolution spatially correlated mantle samples may help to constrain the melting history and the size and nature of chemical heterogeneities in the mantle.

Two-dimensional concentration distribution for mixing-controlled bioreactive transport in steady state

Under steady-state conditions, the degradation of contaminant plumes introduced continuously into an aquifer is controlled by transverse dispersion when the other reacting compound is provided from ambient groundwater. Given that the reaction is instantaneous and longitudinal dispersion can be neglected, the length of the plume is inversely proportional to the transverse dispersion coefficient. In typical scenarios of natural attenuation, however, the considered reaction is biotic and kinetic. The standard model of bioreactive transport relies on double-Monod kinetics and pseudo first-order biomass decay. Under these conditions, a fraction of the injected mass flux remains beyond the length of the plume determined for the instantaneous reaction. We present an analytical framework to derive the steady-state concentration distributions of the dissolved compounds and the biomass from the concentration distribution of a conservative compound, assuming double-Monod kinetics and two different models describing biomass decay. The first biomass-decay model assumes a constant first-order decay coefficient, while the second assumes that the decay coefficient depends upon the electron-acceptor concentration. We apply the method to the case of a line-injection in two-dimensional uniform flow. In general, the bioreactive concentration distributions are similar to the distributions computed for an instantaneous reaction. The similarity is greater when the biomass decay coefficient is assumed to depend on the electron-acceptor concentration rather than being constant.

Micro-Thermal Field-Flow Fractionation of Colloidal Particles: Effect of Temperature on Retention and Relaxation Processes

An important decrease in the heat flux across the micro-thermal field-flow fractionation (micro-TFFF) channel compared with the standard size channel allowed to control independently the temperatures of the cold and hot walls and to study the behaviour of colloidal particles within an extended temperature range. The only limitation was imposed by the freezing and boiling points of the carrier liquid. The retention and the relaxation processes were found to be influenced by temperature. A decrease in the viscosity of the carrier liquid with increasing temperature results in an increase in the diffusion coefficient of the retained species. Consequently, the relaxation processes associated with the establishment of the steady-state concentration distribution are accelerated, the time to reach the steady state decreases with increasing temperature, and the retention ratio of moderately retained particles decreases to reach a minimum value. The fractograms of colloidal samples of narrow particle size distribution (PSD) obtained at lowest temperatures can exhibit two peaks; the first corresponding to the unrelaxed particles eluted even at higher than the average velocity of the carrier liquid and the second one corresponding to the particles retained less compared with the zone eluted at a steady state from the very beginning. The further increase in temperature above that at which the retention ratio reaches the mentioned minimum produces inversion of the retention ratio, i.e. its increase. The peak corresponding to the unrelaxed part of the particles progressively disappears with increasing temperature. If the flow of the carrier liquid is stopped immediately after the injection of the sample for a time to reach a steady-state concentration distribution of the particles across the channel and restarted after that period, the retention ratio does not exhibit a pronounced minimum. On the other hand, the width of the zone generally decreases with increasing temperature but less if the stop-flow procedure is applied. These findings confirm that the retention ratio is influenced by temperature in agreement with the theoretical prediction. A practical conclusion is that higher resolution and reduced time of the separation can be achieved at higher temperatures of the accumulation wall. The micro-TFFF thus becomes a highly competitive method of separation and determination of the PSD of colloidal particles.

Measuring and modeling the dissolution of nonideally shaped dense nonaqueous phase liquid pools in saturated porous media

A three‐dimensional physical aquifer model was used to study the dissolution of a dense nonaqueous phase liquid (DNAPL) pool. The model aquifer comprised a packing of homogeneous, medium‐sized sand and conveyed steady, unidirectional flow. Tetrachloroethene (PCE) pools were introduced within model aquifers atop glass‐ and clay‐lined aquifer bottoms. Transient breakthrough at an interstitial velocity of 7.2 cm/h, and three‐dimensional steady state concentration distributions at velocities ranging from 0.4 to 7.2 cm/h were monitored over periods of 59 and 71 days for the glass‐ and clay‐bottom experiments, respectively. Pool‐averaged mass transfer coefficients were obtained from the observations via a single‐parameter fit using an analytical model formulated with a second type boundary condition to describe pool dissolution [Chrysikopoulos, 1995]. Other model parameters (interstitial velocity, longitudinal and transverse dispersion coefficients, and pool geometry) were estimated independently. Simulated and observed dissolution behavior agreed well, except for locations relatively close to the pool or the glass‐bottom plate. Estimated mass transfer coefficients ranged from 0.15 to 0.22 cm/h, increasing weakly with velocity toward a limiting value. Pool mass depletions of 31 and 43% for the glass‐ and clay‐bottom experiments failed to produce observable changes in the plumes and suggested that changes in pool interfacial area over the period of the experiment were negligible. Dimensionless mass transfer behavior was quantified using a modified Sherwood number (Sh*). Observed Sh* values were found to be about 2–3 times greater than values predicted by an existing theoretical mass transfer correlation, and 3–4 times greater than those estimated previously for an ideally configured trichloroethene (TCE) pool (circular and smooth). It appeared that the analytical model's failure to account for pore‐scale pool‐water interfacial characteristics and larger scale pool shape irregularities biased the Sh* estimates toward greater values.

Steady-state concentration distribution of artificial receptor and target analyte in plasticized PVC membrane between solutions differing in target analyte concentration.

A barbiturate receptor has proven effective in improving selectivity in solid-phase microextraction of barbiturates when doped into plasticized poly(vinyl chloride) (PVC). It would be beneficial to have selective extractions for any given organic species; however, the receptors do not exist. They will be found by screening of libraries of potential receptors; thus, a screening method is needed. It is important to screen the receptors in the medium in which they will work: plasticized PVC. We hypothesize that we can make receptors move in solution in response to the presence of a solute to which they bind. This work examines whether we can establish a sufficient free energy gradient for a good receptor to move to a predetermined place in space. A difference in the barbiturate solute (substrate or guest) concentration in solutions bathing the two sides of a plasticized PVC membrane containing the barbiturate receptor (or host) creates a spatial concentration gradient of the substrate in the membrane. This causes the receptor's chemical potential to vary across the membrane. Upon binding to the analyte, the receptor undergoes a local activity drop, which decreases its free energy. This process produces a flux of receptor to accumulate at place where there is a high substrate concentration. A concentration gradient of substrate can be maintained across the membrane at steady state. In membranes for which the formation of the complex is favored, the receptor responds to the gradient of substrate. In membranes for which binding is not favored, a gradient of substrate is completely ignored by the receptor. Thus, the receptor does respond to the gradient but only if the concentration gradient of guest corresponds to a chemical potential gradient.

Transport of a decay chain in homogenous porous media: analytical solutions

With the aid of integral transforms, analytical solutions for the transport of a decay chain in homogenous porous media are derived. Unidirectional steady-state flow and radial steady-state flow in single and multiple porosity media are considered. At least in Laplace domain, all solutions can be written in closed analytical formulae. Partly, the solutions can also be inverted analytically. If not, analytical calculation of the steady-state concentration distributions, evaluation of temporal moments and numerical inversion are still possible. Formulae for several simple boundary conditions are given and visualized in this paper. The derived novel solutions are widely applicable and are very useful for the validation of numerical transport codes.

Measurement of shear-induced dispersion in a dilute emulsion

The time-dependent drop distribution of a dilute, polydisperse emulsion is measured in a simple shear flow. The suspending fluid is much more viscous than the dispersed phase (1:1000). Drops are found to drift away from either bounding wall and accumulate near the center of the gap, due to the anisotropy of droplet–plane interactions. An expression for this drift velocity has been derived for single drops by Chan and Leal [J. Fluid Mech. 92, 131 (1979)] and was in agreement with isolated drop migration observed in our work. Eventually the inward drift is balanced by a shear-induced gradient diffusivity, and a steady-state concentration distribution is reached. When the drops are sufficiently far from either wall a self-similar, parabolic concentration profile is predicted at all times. Droplet diffusivities were determined for capillary numbers Ca=γ̇āμ/σ between 0.17 and 0.92, where γ̇ is the shear rate, ā is the mean drop radius, μ is the viscosity of the suspending fluid, and σ the interfacial tension. The values obtained are an order of magnitude lower than theoretical predictions of Loewenberg made in the limit of small deformation.