Mass Transport Problem Research Articles

COMPENSATORY REVERSE FLOW OF PROGRESSIVE WAVES WITH FINITE AMPLITUDE

This paper is devoted to problem of mass transport of fluid for the surface progressive waves. Both Stokes and cnoidal waves are considered. New solutions for the transitional current are obtained. It is discovered that the mass transport of fluid in the direction of wave propagation exists only in the top layer. In the underlying layers a compensatory reverse flow is formed. The existence of a compensatory flow was verified experimentally. It is revealed that theoretical results duly conform to experimental data.

Surface modification by self-assembled monolayer and carbon nanotubes

The rule that governs surface reactions provides very important information for fundamental studies in chemistry and biochemistry. The potential to define the chemical and physical properties of solid surfaces is decisive for chemical sensing, electronics and many other applications. The application of the technique in interfacial surfaces such as one-molecule-thick films has led to a feasible and significant tool for modern scientific studies. A monomolecular film of a surfactant formed spontaneously on a substrate upon exposure to a surfactant solution constitutes a self-assembled monolayer (SAM). SAM technology provides a powerful means to generate monomolecular films of biological molecules on a variety of solid substrates. SAMs give quick and accurate response to stimuli without having any mass transport problem due to the intimate contact with the supporting surface. The environment of an immobilized recognition center and the density over the surface will also be controlled by SAMs. Carbon nanotubes (CNTs) have become intensely capable materials for biomolecular attachment due to their properties such as large length-to-diameter aspect ratios, which provide high surface-to-volume ratios. The attachment of CNTs to the interfacial surface and to biomolecules plays an important role in formation of functional assemblies on the solid surface.

Optimal Transport Over a Linear Dynamical System

We consider the problem of steering an initial probability density for the state vector of a linear system to a final one, in finite time, using minimum energy control. In the case where the dynamics correspond to an integrator ( $\dot{x}(t) = u(t)$ ) this amounts to a Monge-Kantorovich Optimal Mass Transport (OMT) problem. In general, we show that the problem can again be reduced to solving an OMT problem and that it has a unique solution. In parallel, we study the optimal steering of the state-density of a linear stochastic system with white noise disturbance; this is known to correspond to a Schrodinger bridge . As the white noise intensity tends to zero, the flow of densities converges to that of the deterministic dynamics and can serve as a way to compute the solution of its deterministic counterpart. The solution can be expressed in closed-form for Gaussian initial and final state densities in both cases.

Thickness Optimisation of Sealed Seams in Respect of Insulating Properties

The main goal is to determine the optimal thickness of material layers in seams in respect of their insulating properties. The impact of structure on heat insulation and moisture resistance is also described. The insulation was analysed for different types of sealed seams which are typically used to connect the different parts of clothing. The coupled heat and mass transport problem is defined by the transport equations and a set of boundary and initial conditions. Based on the variational approach, we introduce the typical objective functionals and the optimality conditions. The problem was solved using the finite element method at the analysis stage and sensitivity approach at the synthesis stage.

Probing Electrode Heterogeneity using Fourier-Transformed Alternating Current Voltammetry: Protocol Development

Fourier-transformed large amplitude alternating current voltammetry (FTACV) provides a sensitive analytical tool for the discrimination of electrode reactions that are complicated by surface heterogeneity. In this paper, it is shown how the FTACV response at a dual-electrode system comprised of different electrode materials having different heterogeneous charge transfer (k01 and k02) can be resolved into its individual electrode kinetics components without prior knowledge of the electrode size ratio (θ1:θ2). This is possible when one process is reversible and the other is quasi-reversible; achievable by careful selection of the FTACV frequency. The applicability of the FTACV method over a wide range of electrode kinetic values and size ratios is considered for conditions under which numerical simulations based on a 1D diffusion model are adequate to describe the mass transport problem.

High-Yield Paste-Based Synthesis of Thiolate-Protected Silver Nanoparticles

Nanoparticles rival pharmaceuticals in synthetic inefficiency, in particular as measured by process mass intensity, with by far the largest contribution to waste being from solvents. We have therefore developed a greener method of synthesizing silver nanoparticles using a paste format instead of a metal salt solution. The paste-based synthesis requires 87% less solvent yet still produces exclusively Na4Ag44(p-MBA)30 nanoparticles, without size sorting, with 89% yield. By using a stoichiometric silver-thiolate polymer as a precursor to intimately mix the metal atoms and ligands, and by using a small amount of liquid to form a paste to promote mass transport, the heterogeneity and kinetics problems that are associated with entirely solid-state syntheses were avoided. Because the nanoparticle product was also a paste, solvent use for postprocessing was minimized. Use of the silver-thiolate polymer can also reduce health risks associated with hazardous free thiols in conventional solution-phase syntheses. Usi...

Transport Phenomena in Eccentric Cylindrical Coordinates

Studies in transport phenomena have been limited to a select few coordinate systems. Specifically, Cartesian, cylindrical, spherical, Dijksman toroidal, and bipolar cylindrical coordinates have been the primary focus of transport work. The lack of diverse coordinate systems, for which the equations of change have been worked out, limits the diversity of transport phenomena problem solutions. Here, we introduce eccentric cylindrical coordinates and develop the corresponding equations of change (continuity, motion, and energy). This new coordinate system is unique, distinct from bipolar cylindrical coordinates, and does not contain cylindrical coordinates as a special case. We find eccentric cylindrical coordinates to be more intuitive for solving transport problems than bipolar cylindrical coordinates. Specific applications are given, in the form of novel exact solutions, for problems important to chemical engineers, in momentum, heat and mass transfer. We complete our analysis of eccentric cylindrical coordinates by using the new equations to solve one momentum, one energy, and one mass transport problem exactly. © 2017 American Institute of Chemical Engineers AIChE J, 63: 3563–3581, 2017

Probing Electrode Heterogeneity Using Fourier-Transformed Alternating Current Voltammetry: Application to a Dual-Electrode Configuration.

Quantitative studies of electron transfer processes at electrode/electrolyte interfaces, originally developed for homogeneous liquid mercury or metallic electrodes, are difficult to adapt to the spatially heterogeneous nanostructured electrode materials that are now commonly used in modern electrochemistry. In this study, the impact of surface heterogeneity on Fourier-transformed alternating current voltammetry (FTACV) has been investigated theoretically under the simplest possible conditions where no overlap of diffusion layers occurs and where numerical simulations based on a 1D diffusion model are sufficient to describe the mass transport problem. Experimental data that meet these requirements can be obtained with the aqueous [Ru(NH3)6]3+/2+ redox process at a dual-electrode system comprised of electrically coupled but well-separated glassy carbon (GC) and boron-doped diamond (BDD) electrodes. Simulated and experimental FTACV data obtained with this electrode configuration, and where distinctly different heterogeneous charge transfer rate constants (k0 values) apply at the individual GC and BDD electrode surfaces, are in excellent agreement. Principally, because of the far greater dependence of the AC current magnitude on k0, it is straightforward with the FTACV method to resolve electrochemical heterogeneities that are ∼1-2 orders of magnitude apart, as applies in the [Ru(NH3)6]3+/2+ dual-electrode configuration experiments, without prior knowledge of the individual kinetic parameters (k01 and k02) or the electrode size ratio (θ1:θ2). In direct current voltammetry, a difference in k0 of >3 orders of magnitude is required to make this distinction.

Generalized Sinkhorn Iterations for Regularizing Inverse Problems Using Optimal Mass Transport

The optimal mass transport problem gives a geometric framework for optimal allocation, and has recently gained significant interest in application areas such as signal processing, image processing, and computer vision. Even though it can be formulated as a linear programming problem, it is in many cases intractable for large problems due to the vast number of variables. A recent development to address this builds on an approximation with an entropic barrier term and solves the resulting optimization problem using Sinkhorn iterations. In this work we extend this methodology to a class of inverse problems. In particular we show that Sinkhorn-type iterations can be used to compute the proximal operator of the transport problem for large problems. A splitting framework is then used to solve inverse problems where the optimal mass transport cost is used for incorporating a priori information. We illustrate the method on problems in computerized tomography. In particular we consider a limited-angle computerized tomography problem, where a priori information is used to compensate for missing measurements.

Heat/mass transport in shear flow over a reactive surface with inert defects

We study the problem of mass transport to surfaces with heterogeneous reaction rates in the presence of shear flow over these surfaces. The reactions are first order and the heterogeneity is due to the existence of inert regions on the surfaces. Such problems are ubiquitous in the field of heterogeneous catalysis, electrochemistry and even biological mass transport. In these problems, the microscale reaction rate is characterized by a Damköhler number $\unicode[STIX]{x1D705}$, while the Péclet number $P$ is the dimensionless ratio of the bulk shear rate to the inverse diffusion time scale. The area fraction of the reactive region is denoted by $\unicode[STIX]{x1D719}$. The objective is to calculate the yield of reaction, which is directly related to the mass flux to the reactive region, denoted by the dimensionless Sherwood number $S$. Previously, we used boundary element simulations and examined the case of first-order reactive disks embedded in an inert surface (Shah & Shaqfeh J. Fluid Mech., vol. 782, 2015, pp. 260–299). Various correlations for the Sherwood number as a function of $(\unicode[STIX]{x1D705},P,\unicode[STIX]{x1D719})$ were obtained. In particular, we demonstrated that the ‘method of additive resistances’ provides a good approximation for the Sherwood number for a wide range of values of $(\unicode[STIX]{x1D705},P)$ for $0<\unicode[STIX]{x1D719}<0.78$. When $\unicode[STIX]{x1D719}\approx 0.78$, the reactive disks are in a close packed configuration where the inert regions are essentially disconnected from each other. In this work, we develop an understanding of the physics when $\unicode[STIX]{x1D719}>0.78$, by examining the inverse problem of inert disks on a reactive surface. We show that the method of resistances approach to obtain the Sherwood number fails in the limit as $\unicode[STIX]{x1D719}\rightarrow 1$, i.e. in the dilute limit of periodic inert disks, due to the existence of a surface concentration boundary layer around each disk that scales with ($1/\unicode[STIX]{x1D705}$). This boundary layer controls the screening length between inert disks and allows us to introduce a new theory, thus providing new correlations for the Sherwood number that are highly accurate in the limit of $\unicode[STIX]{x1D719}\rightarrow 1$.

Pollution Transfer as Optimal Mass Transport Problem

In this paper, we use mass transportation theory to study pollution transfer in porous media. We show the existence of a $L^2-$regular vector field defined by a $W^{1, 1}-$ optimal transport map. A sufficient condition for solvability of our model, is given by a (non homogeneous) transport equation with a source defined by a measure. The mathematical framework used, allows us to show in some specifical cases, existence of solution for a nonlinear PDE deriving from the modelling. And we end by numerical simulations.

Problems of optimal transportation on the circle and their mechanical applications

We consider a mechanical problem concerning a 2D axisymmetric body moving forward on the plane and making slow turns of fixed magnitude about its axis of symmetry. The body moves through a medium of non-interacting particles at rest, and collisions of particles with the body's boundary are perfectly elastic (billiard-like). The body has a blunt nose: a line segment orthogonal to the symmetry axis. It is required to make small cavities with special shape on the nose so as to minimize its aerodynamic resistance. This problem of optimizing the shape of the cavities amounts to a special case of the optimal mass transportation problem on the circle with the transportation cost being the squared Euclidean distance. We find the explicit solution for this problem when the amplitude of rotation is smaller than a fixed critical value, and give a numerical solution otherwise. As a by-product, we get explicit description of the solution for a class of optimal transportation problems on the circle.

EXTREMAL DEPENDENCE FOR BILATERAL CREDIT VALUATION ADJUSTMENTS

Recognizing counterparty default risk as integral part of the valuation process of financial derivatives has changed the classical view on option pricing. Calculating the bilateral credit valuation adjustment (BCVA) including wrong way risk (WWR) requires a sound model for the dependence structure between three quantities: the default times of the two contractual parties and the derivative/portfolio value at the first of the two default times. There exist various proposals, but no market consensus, on how this dependence structure should be modeled. Moreover, available mathematical tools depend strongly on the marginal models for the default times and the model for the underlying of the derivative. In practice, independence between all (or some) quantities is still a popular (over-)simplification, which completely misses the root of WWR. In any case, specifying the dependence structure imposes one to model risk and even within some parametric model one typically obtains a considerable interval of BCVA values when the parameters are taken to the extremes. In this work, we present a model-free approach to identify the dependence structure that implies the extremes of BCVA. This is achieved by solving a mass-transportation problem using tools from optimization.

Optimal mass transport-based adaptive mesh method for phase-field models of two-phase fluid flows

In this paper, we present an efficient adaptive mesh method for solving phase-field model of a mixture of two-phase incompressible fluid flows. The adaptive mesh is generated by a coordinate transformation that is determined as the solution of the classical problem of the optimal mass transportation, also known as Monge–Kantorovich problem (MKP). The numerical solution of the MKP is computed by taking the gradient of the steady state solution of a parabolic Monge–Ampère equation (PMAE). Several numerical experiments are conducted to demonstrate the performance of the PMAE method for solving phase-field models. The numerical results illustrate that the adaptive mesh computed with PMAE method captures well the moving interface of the phase-field function.

Duality for the $L^{\infty }$ optimal transport problem

We derive the dual of the relaxed Monge-Kantorovich optimal mass transport problem in $L^{\infty }$ in which one seeks to minimize $\mu$-$\mathrm {ess sup}_{(x,y)\in \mathbb {R}^N \times \mathbb {R}^N} c(x,y)$ over Borel probability measures $\mu$ with given marginals $P_0, P_1.$ Several formulations of the dual problem are obtained using various techniques including quasiconvex duality. We also consider weighted optimal transport in $L^{\infty }$ and we identify the form of the dual in the Lagrangian cost setting for both integral and essential supremum costs along trajectories. Finally, we prove a duality formula that relates a maximization problem which arises naturally in the $L^{\infty }$ calculus of variations with a family of optimal partial transport problems.

Validation of a Central Approximation in Theories of Regular and Random Electrochemical Electrode Arrays. Extraction of Statistical Information from Amperometric Response of Random Electrode Arrays

Regular electrode arrays generally possess a symmetry which (under certain conditions) allows to reduce their mathematical description to consideration of a single electrode of this array within the laterally bounded domain (known as ‘unit cell’ or ‘diffusion domain’) [1, 2]. In this case by multiplying the simulated current of the single cell by the number of electrodes in the array one would obtain an electrochemical response of the whole array. The electrodes in arrays are generally ordered in a hexagonal or square patterns, which result in hexagonal or square cross-sections of the unit cells, respectively. A straightforward approach to the mass-transport problem in such cells would require complex and time-consuming 3D simulations. Therefore almost always an approximation is made in simulations by replacing the original unit cell by the one with circular cross-section of the same surface area. At one hand, this greatly facilitates the simulations due to the possibility to formulate the mass-transport problem in two dimensions (2D). On the other hand, although this approach was proven to be effective in practice, the error introduced by this approximation, to the best of our knowledge, was never been studied or even estimated. The error associated with the approximation is based on the fact that it is impossible to pave the plane with non-intersecting circles contrary to the perfect plane paving with hexagons or squares as well as due to the different possible diffusion pathways within the single cells with different cross-sections. In order to assess quantitatively this approximation we performed 3D Brownian motion simulations for the single cells of square, hexagon and circular cross-sections equipped with disk electrode. Moreover the situations with inlaid, recessed and protruding electrodes were also considered for each cross-section shape. Notwithstanding the completely different diffusion patterns observed in these systems, the obtained results showed that the approximation gives rise to an experimentally indistinguishable error (less than 5%) for all considered systems [2]. The only exception (i.e. when the error is larger than 5%) was the case when the electrode radius was extremely close to the characteristic size of the cross-section, i.e. the situation when the neighbouring electrodes in the array almost touch each other. Such systems are of low experimental interest and, in addition, exhibit the same behavior as a planar electrode with the surface area of the whole array (except of the very initial times of the experiment). With the same idea in mind, to check the validity of the circular diffusion domain approximation, the random arrays were considered. The arrays with randomly distributed electrodes of the same size were partitioned by means of Voronoi tessellation, which creates a unit cell with polygonal cross-section around each electrode. These polygonal unit cells were replaced then by the unit cells with circular cross-section having the same surface area. Taking into account distribution of the unit cell sizes one can compute the response of the whole array. By comparison the outcome of this procedure with the full scale 3D Brownian motion simulations of the whole array we showed that employment of the circular unit cells in simulations of the responses of the random electrode arrays is validated [3]. These results allow us to develop an easy to evaluate analytical approximation for the electrochemical current of the random array requiring only a distribution of sizes of the unit cells as an input. The analytical approximation in turn makes possible retrieval of the unit cell size distribution from the amperometric response of the random array.

Optimal mass transportation for costs given by Finsler distances viap-Laplacian approximations

AbstractIn this paper we approximate a Kantorovich potential and a transport density for the mass transport problem of two measures (with the transport cost given by a Finsler distance), by taking limits, aspgoes to infinity, to a family of variational problems ofp-Laplacian type. We characterize the Euler–Lagrange equation associated to the variational Kantorovich problem. We also obtain different characterizations of the Kantorovich potentials and a Benamou–Brenier formula for the transport problem.

Analytical solutions for species transport in a T‐sensor at low peclet numbers

A 3D analytical solution is presented for the problem of mass transport in a T‐sensor by taking the axial diffusion effects into account. The solution methodology is based on an eigenfunction expansion of the solute concentration and enjoys the variational calculus for the solution of the associated eigenvalue problem. The method is capable of handling a mixed electroosmotic and pressure‐driven velocity profile and is executed assuming a rectangular channel cross‐section although it can be easily extended to more complex geometries. Two simplified models, one based on a uniform velocity profile, valid for the channel half height to Debye length ratios of above 100, and the other based on a depthwise averaging of the species concentration to be used for cases in which the channel width to height ratio is above 5 are also presented. As a part of the latter, expressions are derived for the Taylor dispersion coefficient of the mixed flow in a slit microconduit. The most interesting finding of this study is that, when the diffusion mechanism significantly contributes to the axial movement of the species, the well‐known heterogeneous mass transport evolves into a nearly uniform pattern in the depthwise direction and the mixing length noticeably increases. © 2016 American Institute of Chemical Engineers AIChE J, 62: 4119–4130, 2016

An explicit martingale version of the one-dimensional Brenier theorem

By investigating model-independent bounds for exotic options in financial mathematics, a martingale version of the Monge–Kantorovich mass transport problem was introduced in (Beiglbock et al. in Finance Stoch. 17:477–501, 2013; Galichon et al. in Ann. Appl. Probab. 24:312–336, 2014). Further, by suitable adaptation of the notion of cyclical monotonicity, Beiglbock and Juillet (Ann. Probab. 44:42–106, 2016) obtained an extension of the one-dimensional Brenier theorem to the present martingale version. In this paper, we complement the previous work by extending the so-called Spence–Mirrlees condition to the case of martingale optimal transport. Under some technical conditions on the starting and the target measures, we provide an explicit characterization of the corresponding optimal martingale transference plans both for the lower and upper bounds. These explicit extremal probability measures coincide with the unique left- and right-monotone martingale transference plans introduced in (Beiglbock and Juillet in Ann. Probab. 44:42–106, 2016). Our approach relies on the (weak) duality result stated in (Beiglbock et al. in Finance Stoch. 17:477–501, 2013), and provides as a by-product an explicit expression for the corresponding optimal semi-static hedging strategies. We finally provide an extension to the multiple marginals case.

Lattice Boltzmann method for the fractional advection-diffusion equation

Mass transport, such as movement of phosphorus in soils and solutes in rivers, is a natural phenomenon and its study plays an important role in science and engineering. It is found that there are numerous practical diffusion phenomena that do not obey the classical advection-diffusion equation (ADE). Such diffusion is called abnormal or superdiffusion, and it is well described using a fractional advection-diffusion equation (FADE). The FADE finds a wide range of applications in various areas with great potential for studying complex mass transport in real hydrological systems. However, solution to the FADE is difficult, and the existing numerical methods are complicated and inefficient. In this study, a fresh lattice Boltzmann method is developed for solving the fractional advection-diffusion equation (LabFADE). The FADE is transformed into an equation similar to an advection-diffusion equation and solved using the lattice Boltzmann method. The LabFADE has all the advantages of the conventional lattice Boltzmann method and avoids a complex solution procedure, unlike other existing numerical methods. The method has been validated through simulations of several benchmark tests: a point-source diffusion, a boundary-value problem of steady diffusion, and an initial-boundary-value problem of unsteady diffusion with the coexistence of source and sink terms. In addition, by including the effects of the skewness β, the fractional order α, and the single relaxation time τ, the accuracy and convergence of the method have been assessed. The numerical predictions are compared with the analytical solutions, and they indicate that the method is second-order accurate. The method presented will allow the FADE to be more widely applied to complex mass transport problems in science and engineering.