Small Peclet Numbers Research Articles

Boundary effects on the creeping-flow and thermophoretic motions of an aerosol particle in a spherical cavity

An analytical study is presented for the quasisteady translation, rotation, and thermophoresis of an aerosol sphere located at the center of a spherical cavity. The Knudsen number is assumed to be small so that the fluid flow is described by a continuum model with a temperature jump, a thermal creep, and a frictional slip at the solid surfaces. The particle and the cavity surroundings may differ in thermal conductivity and in surface properties. In the limit of small Reynolds and Peclet numbers, the appropriate energy and momentum equations are solved for each system and the particle velocities for an applied force, torque, and temperature gradient are derived individually in closed forms. In general, the boundary effects of the cavity wall on the isothermal creeping-flow and thermophoretic motions of an aerosol particle can be quite significant in appropriate situations. In practical aerosol systems, the boundary effect on thermophoresis is much weaker than that on the motion driven by a gravitation field.

Analytical Solutions for Solute Transport in Finite Soil Columns with Arbitrary Initial Distributions

AbstractAnalytical expressions were derived for the solute concentration in soils with a nonuniform initial distribution and a finite outlet condition assuming a zero gradient at the soil outlet. The advection‐dispersion equation (ADE) was solved by successively using a Laplace transform with respect to time, variation of parameters, and two different inversion procedures. Solutions for two resident concentrations and a flux‐averaged concentration were derived for arbitrary and pulse‐type initial distributions. The solutions are applicable for a wide range of conditions. Particular attention was paid to cases involving small Peclet numbers and large dispersive fluxes associated with steep fronts in the initial distribution. Flux‐averaged concentrations may become negative or exceed the initial concentration, while solutions for the resident concentration, derived for first‐ and third‐type inlet conditions, can predict a substantial amount of upstream solute transport due to dispersion. This suggests that the conventional ADE is not always suitable to describe initial value problems where travel times are typically short and large concentration gradients may occur.

Comparative studies on closure approximations in flows of liquid crystal polymers: I. elongational flows

We derive nine approximate models for flows of liquid crystal polymers by applying different closure approximations to the kinetic theory developed by Bhave et al. We then benchmark the approximate models by comparing their uniaxial steady state solutions to the corresponding ones of the kinetic theory in elongational flows. For prolate steady states, we find that all the models give qualitatively acceptable approximations. In particular, when the flow is uniaxially stretching, the models in which the flow-orientation interaction term is approximated by the second Hinch-Leal closure approximation approximate the kinetic theory well in the range of small Peclet numbers; likewise, the models in which the molecule-molecule interaction term is approximated by the Doi closure approximation approximate the kinetic theory well in the range of large Peclet numbers. In the oblate phase, however, all models except for the DD (Doi closure approximation) and DHL2 (Hybrid Doi and HL2 closure approximation) model may produce either fictitious or nonphysical oblate steady states with out-of-range order parameter values. Therefore, we caution the use of the models in this phase. In contrast, the DD and DHL2 model are applicable to all phases for any Peclet numbers and polymer concentration values. In addition, they approximate the kinetic theory well in the oblate phase when the flow is biaxially stretching. In the end, we find that the best overall approximation to the kinetic theory is given by the DHL2 model.

Heat dispersion in stationary mixed convection flow about horizontal surfaces in porous media

An analysis has been performed to study the influence of velocity dependent dispersion on transverse heat transfer in mixed convection flow above a horizontal wall of prescribed temperature in a saturated porous medium. The Boussinesq approximation and boundary layer analysis were used to numerically obtain gravity affected temperature and velocity distributions within the frames of Darcy's law and a total thermal diffusivity tensor comprising both of constant coefficient heat conduction and velocity proportional mechanical heat dispersion. Dependending on Pe∞, the molecular Peclet number basing on the effective thermal diffusivity and the velocity of the oncoming flow, density coupling has distinct influences on heat transfer rates between the wall surface and the porous medium flow region. For small Peclet numbers, when heat conduction is the prevailing mechanism, wall heat fluxes are the higher the larger the density difference between the oncoming and the near wall fluid is. The opposite is true for larger Peclet numbers, when mechanical heat dispersion is the main cause of heat spreading. For Pe∞ tending to infinity these wall heat fluxes approach finite maximum values in the total heat diffusivity model, they grow beyond any limit if only constant coefficient heat conduction is considered. Thus, the inclusion of mechanical heat dispersion effects yields physically more realistic predictions.

Deep-biofilm kinetics of substrate utilization in anaerobic filters

Acetate fermentation in anaerobic filters was performed to study the deep-biofilm kinetics of substrate utilization. Small Peclet number ranging from 0.01 to 1.5 found in the tracer test indicates that the flow regime in anaerobic filters is close to complete-mix. An axial dispersion model coupled with deep-biofilm kinetics is simulated along the biofilter height. A good agreement between the calculated and experimental results indicates that the proposed deep-biofilm model can be used to estimate the removal efficiency of the readily degradable substrate in anaerobic filters if the ratio of voidage to specific surface area is greater than several millimeters. Also, a higher influent substrate concentration gives a higher substrate removal efficiency if the surface loading is kept the same.

Transient heat transfer to a spheroidal liquid drop suspended in an electric field

Heat transfer to a spheroidal drop of a dielectric fluid suspended in another dielectric fluid in the presence of an electric field is investigated in this paper. The effect of drop deformation on the heat transport is analyzed. A range of prescribed drop deformations from ba=0.99 to ba=0.4 is considered. The electric potential distribution is numerically obtained for each deformed drop configuration. The resulting flow field is determined by the solution of the Navier-Stokes equations in the continuous and the dispersed phase. The transient heat transfer is studied in the limit of bulk of the resistance to the heat transport being in the droplet. An alternating-direction-implicit (ADI) method is used to obtain the transient temperature field for drop Peclet number from 5 to 1500. In the limiting case of negligible drop deformation, we recover the flow field and the electrically induced interfacial stresses for a liquid sphere calculated by analytical methods. Heat transfer results for a spherical drop (ba=0.99) show excellent agreement with results available in the published literature. Study of drop deformation reveals interesting features in the flow and heat transport. The location of the maximum surface velocity moves toward the equatorial plane for a deformed drop compared to that for a liquid sphere. It is found that the increase in drop deformation results in higher steady state Nusselt numbers for very large as well as very small equivalent Peclet numbers. However, for intermediate equivalent drop Peclet numbers, the equivalent steady state Nusselt numbers for a deformed drop may be lower than for a sphere.

Rheology of highly aligned nematic liquid crystals

Highly aligned nematic phases arise in systems of charged rodlike molecules or in highly concentrated solutions of uncharged rods. A multivariate Gaussian distribution is used to represent the orientation distribution of the rods and obtain predictions for the structure and rheology of the nematic solution. In weak flows, the director tumbles in one of a set of paths similar to the Jeffery orbits of a rigid rod with an effective aspect ratio equal to the reciprocal of the angular spread of the orientation distribution. The director migrates slowly across the orbits. An analytical expression for the rate of migration in the limit of small rotary Peclet number is obtained. The director may spiral toward the vorticity axis or the flow-gradient plane of the simple shear flow depending on its initial orientation. As the angular spread of the rods decreases, the domain of attraction of the vorticity axis becomes small.

Unsteady heat and mass transfer from a spheroid

AbstractThe unsteady heat‐transfer processes from oblate or prolate spheroids, at the limit of very small Peclet numbers is examined. A perturbation technique for the temperature and the geometry of the particle is used to obtain the rates of heat and mass transfer, first in the Laplace and then in the time domain. A solution to the problem is obtained, including the ϵ2 contribution (ϵ is the eccentricity). The solution reveals the existence of several history terms, which are analogous to the history terms of the creeping flow equation of motion. One of these terms is solely due to the eccentricity of the spheroid. This is an indication that the shape of the particle is a factor of the existence and from of history terms. In addition, an exact expression for the steady‐state heat transfer from a spheroid is obtained using a convenient transformation of the heat‐transfer integral.

Laplacian and diffusional growth: A unified theoretical description for symmetrical and parity-broken patterns

Nowadays, Laplacian growth patterns such as the Saffman-Taylor instability are well understood, mainly because of exact results derived by conformal mapping techniques. This is not the case for crystal growth processes except for the free needle-crystal. Normally, it is thought that the crystal growth process can be understood by an analogy with the equivalent steady Laplacian process. The results that follow from this analogy, however, are in contradiction with numerical and experimental results even at very low velocities. We present here a way to mimic the diffusion instabilities in a simple way which allow to describe the formation and the dynamics of new patterns. We show that the problem of crystal growth in a channel is closely related to the Saffman-Taylor problem in a sector. By using this analogy we obtain the family of zero-surface-tension solutions for small Peclet numbers including an asymmetric degree of freedom. We study analytically the selection problem and confirm that parity-broken solutions originate from the symmetrical solutions through a bifurcation in agreement with recent numerical investigations. We obtain scaling laws for the parity-broken solutions for a wide range of parameters. We extend these results to the infinite geometry and show that parity-broken double fingers can exist even in the range of small undercooling with fully isotropic surface tension. The scaling law for the selected velocity of double-fingering structures is very different from the scaling law of a free dendrite with anisotropy.

A Laplace transform solution for tracer tests in a radially convergent flow field with upstream dispersion

When tracers are introduced into an injection borehole, noticeable concentration gradients at the injection well may cause a backward spreading of the initial plume during radially convergent tracer tests. Based on this concept a non-rigorous mathematical model is developed to estimate the effect of backward spreading. The injection well with an instantaneous slug input is treated as a mathematical source and the initial plume at the injection well is allowed to move upstream. The model assumes that advection and longitudinal dispersion are the transport mechanisms in a radially converging flow field. The Laplace transform solution for solute concentration in an infinite porous medium is obtained using the method of Green's function. Breakthrough curves are computed by numerically inverting the Laplace transform solution. As compared with the solution of Moench, it is concluded that the presence of backward movement of initial plume yields a decrease of peak concentration and spreading out of breakthrough curve tails. This effect is significant for small Peclet numbers and can be neglected for large Peclet numbers. A field tracer test is carried out to demonstrate the applicability of the model. The experimental data are fitted with the type curves of this study, and those of Moench's study, to determine dispersivity and aquifer porosity. The results show that dispersivity values estimated by matching the two type curves are different. Because it neglects backward dispersion, Moench's solution overestimates the dispersivity and effective aquifer porosity.

Boundary effects on the sedimentation and hindered diffusion of charged particles

AbstractA general formalism is developed to examine the sedimentation of a charged particle in a bounded system at small Peclet numbers and small particle‐surface potentials. The excess viscous force is evaluated using a generalized form of the Lorentz reciprocal theorem, eliminating the need to calculate the detailed fluid flow around the charged particle. Specific calculations are provided for the sedimentation of a charged sphere in a concentric (uncharged) spherical cavity. Boundary interactions increase the magnitude of the excess force at small Debye lengths, with the opposite effect seen at very large Debye lengths. A general solution is presented for the sedimentation velocity for an arbitrary particle in both unbounded and bounded systems in the limit of very thin double layers. The excess force under these conditions is proportional to the square of the Debye length, with the proportionality constant being a function of the detailed system geometry. These results provide important insights into the transport of charged particles in porous membranes and chromatographic materials.

Reactive Solute Transport in a Single Fracture

The transport of reactive and nonreactive solutes under the action of fully developed laminar flow in a parallel plate fracture has been analyzed. Three types of surface reactions are examined: (1) irreversible first‐order kinetic, (2) instantaneous reversible, and (3) reversible first‐order kinetic. Approximate analytical models for the dispersion of nonreactive and reactive solutes with an irreversible first‐order reaction are developed, and numerical “exact” solutions for the three surface reaction types are obtained. The conditions under which the approximate solutions hold are determined by comparison with the numerical solutions. The solute migration patterns and the extent of retardation due to surface reactions are then analyzed and compared with the nonreactive case. Dimensionless reaction coefficients incorporated into the solutions are estimated from data available in the literature. It is found that the approximate solutions are consistent with the numerical solutions; that the criterion for applicability of a local equilibrium assumption is based on the values of dimensionless time and a dimensionless number representing the ratio of the rate at which the surface reaction reaches equilibrium to the rate of the molecular diffusion in the transverse direction; and that the effect of surface reaction can be described by a retardation factor Rf = 1 + Kd* for the case in which the reaction can be assumed at equilibrium, with small Peclet number,small dimensionless distribution coefficient Kd*, and large dimensionless time. Conditions are also found under which reactive solute transport can be well approximated by nonreactive solute transport and reversible reactions can be treated as irreversible. These results and the dimensionless analysis method employed herein may serve as a basis for development of more complex models of reactive solute transport in saturated fractured media.

Unsteady Heat Transfer From a Sphere at Small Peclet Numbers

In a recent paper by Michaelides and Feng (1994) it was discovered that there are history terms in the unsteady heat conduction equation from a small sphere. The history terms are analogous to the terms found in the equation of motion of the sphere, which sometimes are called “Basset terms.” An analysis is presented here for the unsteady heat transfer from a sphere, when its surface temperature undergoes a step change. The singular perturbation technique is used to obtain short and longtime solutions, first in the vicinity of the sphere and then far from the sphere. The solution obtained is for finite but low Peclet numbers. It appears that in the case of the step temperature, change, the history terms are reduced to an analytical expression of the error function.

The Coagulation Rate of Small Particles in a Sedimenting Dispersion

The rate of coagulation is considered for small particles when both convection and Brownian motions simultaneously exist. By solving the equation for the pair-distribution function for the case of large and small Peclet numbers, which provides the measure of the ratio of the gravity-induced convection to Brownian diffusion between two rigid particles, it is shown that there are nonlinear interactions between Brownian diffusion and the gravity-induced convection, so the additivity assumption is not valid in these regimes. Based on the above work and using an interpolation method, we obtain the complete pictures of the dimensionless coagulation rate Nu (Nusselt number) as a function of the dimensionless parameter Pe (Peclet number). The dependence of Nu curves on the size ratio λ and reduced density ratio γ and the numerical errors made by the additivity assumption are also discussed. It is only a theoretical result and needs to be verified by experiments.

Multiple-timescale analysis of Taylor dispersion in converging and diverging flows

A multiple-timescale analysis is employed to analyse Taylor-dispersion-like convective-diffusive processes in converging and diverging flows. A long-time asymptotic equation governing the cross-sectionally averaged solute probability density is derived. The form of this equation is shown to be dependent upon the number of spatial dimensions characterizing the duct or ‘cone’. The two-dimensional case (non-parallel plates) is shown to be fundamentally different from that for three dimensions (circular cone) in that, in two dimensions, a Taylor dispersion description of the process is possible only for small Peclet numbers or angles of divergence. In contrast, in three dimensions, a Taylor dispersion description is always possible provided sufficient time has passed since the initial introduction of solute into the system. The convective Taylor dispersion coefficients versus the half-vertex angle θ0, the respective dispersivity results for the two cases hardly differ from one another, increasing monotonically from 1.0 for θ0 = 0 to approximately 2.6 for a fully flared duct, θ0 = θ/2. Lastly, the techniques developed above for the case of rectilinear channel and duct boundaries are extended to the case of curvilinear boundaries, and an illustrative calculation performed for the case of axisymmetric flow in a flared Venturi tube.

Diffusional deposition of aerosol particles in model filter at small Peclet numbers

Diffusional deposition of aerosol particles in model filter at small Peclet numbers

The effect of weak Shear-induced motion on brownian coagulation of aerosol particles

The coagulation rate of a dilute polydisperse aerosol dispersion of particles is considered for small Peclet number, which provides a measure of the ratio of the relative shear-induced motion to Brownian motion between two rigid spherical aerosol particles. The asymptotic form of the relative velocity of two unequal particles immersed in a simple shear flow when they are far apart is obtained. Using a singular perturbation method, a two term expansion for the dimensionless coagulation rate (Nusselt number) as function of the Peclet number is developed. In the limit of the radius of one of the two spheres becoming small, the result agrees with the dimensionless mass transfer rate to an aerosol particle at small Peclet number.

Migration of Aerosol Spheres under the Combined Action of Thermophoretic and Gravitational Effects

The migration of an aerosol spherical particle subject to the combined action of gravity and a vertical temperature gradient is considered. In the limit of small Reynolds and Peclet numbers, the solution for the stream function of the fluid flow and the migration velocity of particle is obtained by superposition of the individual solutions for sedimentation and ther-mophoresis. The flow structures for the fluid phase, being illustrated via streamlines in meridian section, reveal complex topology whether in the laboratory frame of reference or a reference frame traveling with the particle. The results demonstrate that fresh intuition must be developed regarding this class of problems.

Application of flux‐corrected transport to the Las Cruces Trench site

A flux‐corrected transport (FCT) based numerical algorithm is developed to model the solute transport equation in heterogeneous variably saturated soils. The fully Eulerian algorithm is designed to model very steep gradients without the characteristic numerical diffusion/dispersion found in standard finite difference schemes. This algorithm can handle both small and infinite grid Peclet numbers but is limited to grid Courant numbers less than or equal to 0.5. Since useful results can be obtained for infinite Peclet numbers, the algorithm can be used to trace water movement when coupled with a numerical model for Richards' equation. We used the FCT based algorithm in conjunction with Richards' equation to model a hypothetical experiment at the Las Cruces Trench site. In this numerical experiment, water with tracer is applied across the top of a 24‐m‐wide domain. The two‐dimensional distribution for the hydraulic properties estimated during site characterization is used to define the soil model. The predicted flow and tracer transport is quite complex. The movement of the wetting front appears to be heavily influenced by the old water, whereas the new water tends to bypass much of the old water indicating preferential flow.

Numerical integration of the multidimensional convective diffusion equation

The mixed boundary-value problem is investigated for the stationary equation of convective diffusion in a rectangular region for small Peclet numbers. A conservative difference scheme is constructed and its stability is analyzed. The question of solving a system of difference equations is considered. A numerical example is presented.