Large Peclet Numbers Research Articles

MLPG method for convection-dominated flow problems

In this paper, the Meshless Local Petrov-Galerkin (MLPG) method is applied to compute convection-dominated flow problems. The results of the MLPG method are compared with the results of the finite volume method. The results show that the first-order upwind (FUD) scheme exhibits the false diffusion at a larger-Peclet number; the QUICK scheme and the MLPG method can obtain much closed solutions; but they have small overshoots produced at larger-Peclet number. The results also indicate that the MLPG method is a highly effective and accurate numerical method to deal with convection-dominated flow problems and can eliminate the effect of the false diffusion.

Hydraulic Characteristics of a Pilot-Scale Vertical Subsurface Flow Constructed Wetland

The hydraulic characteristics of constructed wetland are vital to its performance for pollutant removal. Hydraulic characteristics of a pilot scale vertical flow constructed wetland were studied by tracer experiments with the method of moments (MoM) analyzing. From the residence-time distribution (RTD) of tracer during the experiment, the mean detention time, average water velocity and dispersion coefficients were obtained for the constructed wetland. The results showed that the mean residence time was approximately 15% shorter than nominal residence time indicating 85% of total available pore volume was active in pollutant removal processes. The two peaks observed on the RTD curves suggested short-circuiting flows or dead zones exist in the constructed wetland. The relatively large Peclet number of 11.8 suggested that dispersive processes dominated in the tracer movement. Comparison of tracer profiles at different sampling locations indicated that large amount of tracer transported through the upper portion of the constructed wetland.

Fast Evaporation of Spreading Droplets of Colloidal Suspensions

When a coffee droplet dries on a countertop, a dark ring of coffee solute is left behind, a phenomenon often referred to as the coffee-ring effect. A closely related yet less-well-explored phenomenon is the formation of a layer of particles, or skin, at the surface of the droplet during drying. In this work, we explore the behavior of a mathematical model that can qualitatively describe both phenomena. We consider a thin axisymmetric droplet of a colloidal suspension on a horizontal substrate undergoing spreading and evaporation. In contrast to prior work, precursor films (rather than pinned contact lines) are present at the droplet edge, and evaporation is assumed to be limited by how quickly molecules can transfer out of the liquid phase (rather than by how quickly they can diffuse through the gas phase). The lubrication approximation is applied to simplify the mass and momentum conservation equations, and the colloidal particles are allowed to influence the droplet rheology through their effect on the viscosity. By describing the transport of the colloidal particles with the full convection-diffusion equation, we are able to capture depthwise gradients in particle concentration and thus describe skin formation, a feature neglected in prior models of droplet evaporation. The highly coupled model equations are solved for a range of problem parameters using a finite-difference scheme based on a moving overset grid. The presence of evaporation and a large particle Peclet number leads to the accumulation of particles at the liquid-air interface. Whereas capillarity creates a flow that drives particles to the droplet edge to produce a coffee ring, Marangoni flows can compete with this and promote skin formation. Increases in viscosity due to particle concentration slow down droplet dynamics and can lead to a reduction in the spreading rate.

Mass/heat transfer from a neutrally buoyant sphere in simple shear flow at finite Reynolds and Peclet numbers

AbstractTheoretical analyses of mass/heat transfer from a neutrally buoyant particle in simple shear flow indicate that mass/heat must diffuse across a region of closed streamlines of finite thickness at zero Reynolds number, whereas spiraling streamlines allow the formation of a thin mass transfer boundary layer at small but non‐zero Reynolds numbers (Subramanian and Koch, Phys Rev Lett. 2006;96:134503; Subramanian and Koch, Phys Fluids. 2006;18: 073302). This article presents the first numerical results for mass/heat transfer at finite Reynolds and Peclet numbers. The simulations indicate that fluid particles in the flow‐gradient plane spiral away from the particle for Reynolds numbers smaller than about 2.5 while they spiral toward the particle for higher Reynolds numbers. Solutions of the Navier‐Stokes equations coupled with a boundary layer analysis of mass transfer yield predictions for the rate of mass transfer at asymptotically large Peclet numbers and Reynolds numbers up to 10. Simulations of mass transfer for zero Reynolds number and finite Peclet numbers confirm Acrivos' (Acrivos, J Fluid Mech. 1971;46:233–240) prediction that the Nusselt number approaches a finite value with increasing Peclet number. Simulations at finite Reynolds numbers and Peclet numbers up to 10,000 confirm the theoretical predictions for the concentration gradient at the particle surface at angular positions away from the flow‐gradient plane. However, the wake near the flow‐gradient plane remains too large at this Peclet number to yield a quantitative agreement of the overall rate of mass transfer with the theory for asymptotically large Peclet number. © 2010 American Institute of Chemical Engineers AIChE J, 2011

An Experimental Study for Mixed Convection through a Circular Tube Filled with Porous Media and Fixed Horizontally and Inclined

The porous media have a great influence on the heat transfer and storaging it characteristics. In this study an experimental work has been conducted to measure the mixed convection though the filled circular tubes. First, the test apparatus has been manufactured; filled circular tubes with a porous media and fixed horizontally first then tilted with different angles. Secondly, the circular tube is heated with a constant heat flux and, thermocouples have been stickled in proper positions. Then many readings, for temperatures, have been registered for each thermocouple with different speeds of the outlet air (from the circular tube) and the inclination angle of the tube has been changed many times to be 0o, 30o,45o & 60o. These experiments have been conducted for the range (108.54£Ra£907.73) of Rayleigh number and for the range (29.31£Pe£516.94) of Peclet number. Three tests for the heat flux have been conducted, for each five Peclet number has been used. Finally, the experimental result shows that the surface temperature of pipe has a proportional relation with the tube length for all values of the heat flux and Peclet number. In addition, the heat transfer is achieved by free convection for small Peclet number, by forced convection for large Peclet number and by mixed convection for medium value of Peclet number.

How well do mean breakthrough curves predict mixing‐controlled reactive transport?

Macroscopic transport models calibrated by flux‐averaged breakthrough curves of conservative compounds do not necessarily characterize mixing well because such breakthrough curves do not provide information on fluctuations of concentration within the solute flux, which may influence mean reaction rates. We numerically examine the validity of macroscopic transport models, which are capable of describing all details of flux‐averaged breakthrough curves, for predicting a mixing‐controlled bimolecular precipitation reaction in heterogeneous media. We consider a homogeneous, isotropic medium with an elliptical, low‐permeability inclusion and random heterogeneous fields. For the single‐inclusion case, slow advection through the inclusion results in a multimodal breakthrough curve with enhanced tailing. We vary the hydraulic conductivity contrast and Peclet number to investigate the performance of a “perfect” macroscopic transport model for predicting the total precipitated mass within the domain and the peak concentration difference between the conservative and reactive cases at the outflow boundary. The results indicate that such a model may perform well in media with either very small or very high permeability contrast or at low Peclet number. In the high‐contrast case, most flow takes place in preferential flow paths, resulting in a small variance of the flux‐weighted concentration, even though the offset in the breakthrough between the slow and fast travel paths is substantial. Maximum relative errors in terms of total precipitated mass and the peak concentration difference between the conservative and reactive cases occur at intermediate permeability contrasts and large Peclet numbers. Numerical simulations on random heterogeneous fields confirm the finding of the single‐inclusion case. Thus, in cases with intermediate hydraulic conductivity contrast, making macroscopic models fit flux‐averaged concentration breakthrough curves better may not improve the prediction of mixing‐controlled reactive transport, and it becomes necessary to quantify and account for the variability of conservative concentrations in the flux in order to formulate an appropriate macroscopic transport model that predicts mixing‐controlled reactive transport.

Theoretical and experimental study of nonlinear electrophoresis

Linear and nonlinear electrophoresis of conducting and nonconducting spherical disperse particles is studied. A comparative analysis is performed for the regularities of nonlinear electrophoresis in the stationary and pulsed modes at small and large Peclet numbers.

A positivity-preserving ALE finite element scheme for convection–diffusion equations in moving domains

A new high-resolution scheme is developed for convection–diffusion problems in domains with moving boundaries. A finite element approximation of the governing equation is designed within the framework of a conservative Arbitrary Lagrangian Eulerian (ALE) formulation. An implicit flux-corrected transport (FCT) algorithm is implemented to suppress spurious undershoots and overshoots appearing in convection-dominated problems. A detailed numerical study is performed for P 1 finite element discretizations on fixed and moving meshes. Simulation results for a Taylor dispersion problem (moderate Peclet numbers) and for a convection-dominated problem (large Peclet numbers) are presented to give a flavor of practical applications.

Nonlocal Four-Point Boundary Value Problem for the Singularly Perturbed Semilinear Differential Equations

This paper deals with the existence and asymptotic behavior of the solutions to the singularly perturbed second-order nonlinear differential equations. For example, feedback control problems, such as the steady states of the thermostats, where the controllers add or remove heat, depending upon the temperature detected by the sensors in other places, can be interpreted with a second-order ordinary differential equation subject to a nonlocal four-point boundary condition. Singular perturbation problems arise in the heat transfer problems with large Peclet numbers. We show that the solutions of mathematical model, in general, start with fast transient which is the so-called boundary layer phenomenon, and after decay of this transient they remain close to the solution of reduced problem with an arising new fast transient at the end of considered interval. Our analysis relies on the method of lower and upper solutions.

Laminar mixed convection on a horizontal plate of finite length in a channel of finite width

The steady two-dimensional laminar mixed-convection flow past a horizontal plate of finite length is analysed for large Peclet numbers, small Prandtl numbers and weak buoyancy effects. The plate is placed in a channel of finite width, with the plane walls of the channel being parallel to the plate. The temperature of the plate is assumed to be constant. The hydrostatic pressure difference across the wake behind the plate is compensated by a perturbation of the inviscid channel flow. This outer flow perturbation affects the temperature distribution in the thermal boundary layer at the plate and the heat transfer rate, respectively. Solutions in closed form are given. The forces acting on the plate due to the potential flow perturbation are also determined.

Nonlocal Four-Point Boundary Value Problem for the Singularly Perturbed Semilinear Differential Equations

This paper deals with the existence and asymptotic behavior of the solutions to the singularly perturbed second-order nonlinear differential equations. For example, feedback control problems, such as the steady states of the thermostats, where the controllers add or remove heat, depending upon the temperature detected by the sensors in other places, can be interpreted with a second-order ordinary differential equation subject to a nonlocal four-point boundary condition. Singular perturbation problems arise in the heat transfer problems with large Peclet numbers. We show that the solutions of mathematical model, in general, start with fast transient which is the so-called boundary layer phenomenon, and after decay of this transient they remain close to the solution of reduced problem with an arising new fast transient at the end of considered interval. Our analysis relies on the method of lower and upper solutions.

Mass transfer from a contaminated fluid sphere

The unsteady mass transfer from a contaminated fluid sphere moving in an unbounded fluid is examined numerically for unsteady-state transfer. The effect of the interface contamination and the flow regime on the concentration profiles, inside and outside a fluid sphere, is investigated for different ranges of Reynolds number (0 < Re < 200) and Peclet number (0 < Pe < 105), viscosity ratio between the dispersed phase and the continuous phase (0 < κ < 10), and the stagnant-cap angle (0° < θcap < 180°). It was found that the stagnant-cap angle significantly influences the mass transfer from the sphere to a surrounding medium. For all Peclet and Reynolds numbers and κ, the contamination reduces the mass transfer flux. The average Sherwood number increases with an increase of stagnant-cap angle and reaches a maximum equal to the average one for a clean fluid sphere at low viscosity ratio and large Peclet numbers. A predictive equation for the Sherwood number is derived from these numerical results. © 2010 American Institute of Chemical Engineers AIChE J, 2011

Incorporation of Effects of Diffusion into Advection-Mediated Dispersion in Porous Media

The distribution of solute arrival times, W(t;x), at position x in disordered porous media does not generally follow Gaussian statistics. A previous publication determined W(t;x) in the absence of diffusion from a synthesis of critical path, percolation scaling, and cluster statistics of percolation. In that publication, W(t;x) as obtained from theory, was compared with simulations in the particular case of advective solute transport through a two-dimensional model porous medium at the percolation threshold for various lengths x. The simulations also did not include the effects of diffusion. Our prediction was apparently verified. In the current work we present numerical results related to moments of W(x;t), the spatial solute distribution at arbitrary time, and extend the theory to consider effects of molecular diffusion in an asymptotic sense for large Peclet numbers, Pe. However, results for the scaling of the dispersion coefficient in the range 1<Pe<100 agree with those of other authors, while results for the dispersivity as a function of spatial scale also appear to explain experiment.

The asymptotic expansions of the solution for the boundary value problem to a convective diffusion equation with volume chemical reaction near a spherical drop

The paper deals with a boundary value problem for a quasilinear partial elliptical equation. The equation describes a stationary process of convective diffusion near a spherical drop and takes into account the value of a chemical reaction for large Peclet numbers ( Pe). For the problem considered, an asymptotic expansions (AE) of solution on the small parameter ( ε = Pe −1/2) is obtained near a spherical drop.

Mass transfer around oblate spheroidal drops at low Reynolds numbers

Steady and unsteady mass transfer in the continuous phase around slightly deformed oblate spheroidal drops at low (but not zero) Reynolds numbers was investigated theoretically. Asymptotic analytical solutions for short and long times, at large Peclet numbers, were obtained by the useful equations derived by Lochiel and Calderbank and by Favelukis and Mudunuri for axisymmetric drops of revolution, with the only requirements being the shape of the drop and the tangential velocity at the surface of the drop. As expected, the result, although complicated, represents a small correction to the classical problem of mass transfer around a spherical drop under creeping flow conditions, since the physical problem presented here requires both the Reynolds and the Weber number to be much smaller than one.

A hybrid numerical method for interfacial fluid flow with soluble surfactant

We address a significant difficulty in the numerical computation of fluid interfaces with soluble surfactant that occurs in the physically representative limit of large bulk Peclet number Pe. At the high values of Pe in typical fluid-surfactant systems, there is a transition layer near the interface in which the surfactant concentration varies rapidly, and large gradients at the interface must be resolved accurately to evaluate the exchange of surfactant between the interface and bulk flow. We use the slenderness of the layer to develop a fast and accurate ‘hybrid’ numerical method that incorporates a separate, singular perturbation analysis of the dynamics in the transition layer into a full numerical solution of the interfacial free boundary problem. The accuracy and efficiency of the method is assessed by comparison with a more ‘traditional’ numerical approach that uses finite differences on a curvilinear coordinate system exterior to the bubble, without the separate transition layer reduction. The traditional method implemented here features a novel fast calculation of fluid velocity off the interface.

Particle trapping by an external body force in the limit of large Peclet number: applications to magnetic targeting in the blood flow

Motivated by the technology of magnetically targeted drug and gene delivery, in which a magnetic field is used to direct magnetic carrier particles from the circulation to a target site, we develop a continuum model for the motion of particles (magnetic carriers) subject to an external body force (magnetic field) in a flow of a concentrated suspension of a species of neutrally buoyant particles (blood). An advection–diffusion equation describes the evolution of the carrier particles as they advect in the flow under the action of an external body force, and diffuse as a result of random interactions with the suspension of neutrally buoyant particles (shear-induced diffusion). The model is analysed for the case in which there is steady Poiseuille flow in a cylindrical vessel, the diffusive effects are weak and there is weak carrier uptake along the walls of the vessel. The method of matched asymptotic expansions is used to show that carriers are concentrated in a boundary layer along the vessel wall and, further, that there is a carrier flux along this layer which results in a sub-layer, along one side of the vessel, in which carriers are even more highly concentrated. Three distinguished limits are identified: they correspond to cases for which (i) the force is sufficiently weak that most particles move through the vessel without entering the boundary layers along the walls of the vessel and (ii) and (iii) to a force which is sufficiently strong that a significant fraction of the particles enter the boundary layers and, depending upon the carrier absorption from the vessel walls, there is insignificant/significant axial carrier flux in these layers.

Models &amp; measures of mixing &amp; effective diffusion

Mixing a passive scalar field by stirring can be measured in a variety of ways including tracer particle dispersion, via the flux-gradient relationship, or by suppression of scalar concentration variations in the presence of inhomogeneous sources and sinks. The mixing efficiency or efficacy of a particular flow is often expressed in terms of enhanced diffusivity and quantified as an effective diffusion coefficient. In this work we compare and contrast several notions of effective diffusivity. We thoroughly examine the fundamental case of a steady sinusoidal shear flow mixing a scalar sustained by a steady sinusoidal source-sink distribution to explore apparent quantitative inconsistencies among the measures. Ultimately the conflicts are attributed to the noncommutative asymptotic limits of large Peclet number and large length-scale separation. We then propose another approach, a generalization of Batchelor's 1949 theory of diffusion in homogeneous turbulence, that helps unify the particle dispersion and concentration variance suppression measures.

Effect of Temperature on High Shear-Induced Gelation of Charge-Stabilized Colloids without Adding Electrolytes

We demonstrated previously (Wu, H.; Zaccone, A.; Tsoutsoura, A.; Lattuada, M.; Morbidelli, M. Langmuir 2009, 25, 4715) that, for a colloid stabilized by charges from both polymer chain-end groups and adsorbed sulfonate surfactants, when the surfactant surface density reaches a certain critical value, the shear-induced gelation becomes unachievable at room temperature, even at an extremely large Peclet number, Pe = 4.6 x 10(4). This is due to the presence of the short-range, repulsive hydration force generated by the adsorbed surfactant. In this work, we investigate how such hydration force affects the shear-induced gelation at higher temperatures, in the range between 303 and 338 K. It is found that a colloidal system, which does not gel at room temperature in a microchannel at a fixed Pe = 3.7 x 10(4), does gel when temperature increases to a certain value. The critical initial particle volume fraction for the gelation to occur decreases as temperature increases. These results indicate that the effect of the hydration force on the gelation decreases as temperature increases. Moreover, we have observed that at the criticality only part of the primary particles is converted to the gel network and the effective particle volume fraction forming the gel network does not change significantly with temperature. The effective particle volume fraction is also independent of the surfactant surface coverage. Since the effective particle volume fraction corresponds to space filling requirement of a standing gel network, which is mainly related to the clusters structure, this result indicates that at a given shear rate the cluster structure does not change significantly with the surfactant surface coverage. On the other hand, since the cluster morphology is a strong function of the shear rate, we have observed that when the Peclet number is lowered from Pe = 3.7 x 10(4) to 1.7 x 10(4), the effective particle volume fraction reduces from 0.19 to 0.12 at 313 K.

Enhancement of transport from drops by steady and modulated electric fields

We consider the problem of transport of heat or mass from circulating droplets that are both settling and subject to an axial electric field. The electric field can be either steady or oscillatory in time and drives an electrohydrodynamic flow, called the Taylor circulation, which augments the Hadamard circulation caused by steady translation. The problem is governed by four dimensionless groups: the Peclet number Pe, the dimensionless amplitudes of both the steady and unsteady electric field, and the dimensionless frequency ω of the modulation. The convective diffusion equation is solved numerically by an efficient finite-difference scheme that allows a wide range of parameters—in particular, very large Peclet numbers—to be covered. The results are characterized by the asymptotic rate of extraction of heat or mass from the droplet, which is found to be exponential in time. The enhancement factor, defined as the ratio of this rate to that of a stagnant drop, is studied as a function of parameters. For steady drops, we find that transport remains diffusion controlled, but the enhancement factor is significantly higher with the Taylor flow than without. For modulated electric field the enhancement factor is not a simple function of parameters and exhibits spectral “resonant peaks” at particular values of ω for which the enhancement factor is extremely large. Movies of the simulations are used to study the underlying time-periodic spatial structures of the concentration field (so-called strange eigenmodes) and the complex time dependence that is responsible for these resonances.