Stochastic process model for solute transport and the associated transport equation

A mathematical model of Lagrangian motions of a particle in turbulent flows is developed on the basis of a stochastic differential equation. The model expresses uncertainties involved in turbulence by standard Brownian motion. Because the model does not guarantee smoothness of the path of the particle, local velocity is newly defined so as to be suitable for observation of a velocity time series at a fixed point. Then, it is shown that the newly defined local velocity is governed by a Gaussian distribution. In addition, an estimation method of the turbulent diffusion coefficient involved in the model is proposed by using the local velocity. The estimation method does not require tracer experiments. In order to assess the validity of the proposed local velocity, velocity measurements with three-dimensional acoustic Doppler velocimeters were conducted in agricultural drainage canals. Also, the turbulent diffusion coefficient was estimated by the derived time series of the observed local velocity. Finally, a transport equation of conservative solute is derived by using the linearity of the Kolmogorov forward equation without using gradient-type lows.     

A mathematical model for solute coupled water transport in the production of aqueous humor

A simple mathematical model for the transport of solute and water in the production of aqueous humor by ciliary epithelium in the eye has been developed. The model introduces the intercellular channel, caped with a leaky (porous) tight junction between the layers of non-pigmented ciliary epithelium, as a bisectional channel which consists of two sections: one representing the tight junction which constitutes the blood-aqueous barrier and the other intercellular space with the active solute transport pumps on its lateral surfaces near the junction. The intercellular space and porous tight junction are modeled as electroneutral, uniform, semi-permeable channels of unequal cross-sectional area. Both the cylindrical pore- and rectangular-slit models for the transport through the channels are simultaneously introduced. The approximate analytical solutions to the governing non-linear coupled equations are obtained in normalized forms by employing Segal’s “Isotonic Convection Approximation”. The computational results for the scaled variables are presented through the graphs. The effects of important parameters on the flow/transport produced by (1) the hydrostatic pressure difference alone, (2) the concentration difference alone, and (3) the active transport alone, are examined and discussed. The results of the model may contribute to the present understanding of the mechanisms governing transport processes involved in the aqueous production.     

An analytical model for carrier-facilitated solute transport in weakly heterogeneous porous media

Carrier-facilitated solute transport in heterogeneous aquifers is studied within a Lagrangian framework. Dissolved solutes and carriers are advected by steady random groundwater flow, which is modeled by Darcy's law with uncertain hydraulic conductivity that is treated as a stationary random space function. We derive general expressions for the spatial moments of the dissolved concentration and the concentration associated with the carrier phase. In order to reduce the computational effort, we use previously derived solutions for the flow field. This enables us to obtain closed-form solutions for the spatial moments of the two concentration fields. The mass and center of gravity of the two propagating plumes depend only on the mean velocity field and chemical/degradation processes. The higher (second and third) moments are affected by the coupling between reactions (sorption/desorption and degradation) among the three phases (i.e., dissolved, carrier and sorbed concentrations) and the aquifer’s heterogeneity. We investigate the potentially enhancing effect of carriers by comparing spatial moments of the two propagating plumes. The forward/backward mass transfer rates between the liquid and carrier phases, and the degradation coefficients are identified as critical parameters. The carrier's role is most prominent when detachment from carrier sites is slow, provided that degradation on the carriers is smaller than that in the liquid phase.     

A complete steady state model of solute and water transport in the kidney

The purpose of this paper is to incorporate a detailed model, along with an optimized set of parameters for the proximal tubule, into J. L. Stephenson's current central core model of the nephron. In this model a set of equations for the proximal tubule are combined with Stephenson's equations for the remaining four tubules and interstitium, to form a complete nonlinear system of 34 ordinary differential and algebraic equations governing fluid and solute flow in the kidney. These equations are then discretized by the Crank-Nicholson scheme to form an algebraic system of nonlinear equations for the unknown concentrations, flows, hydrostatic pressure, and potentials. The resulting system is solved via factored secant update with a finite-difference approximation to the Jacobian. Finally, numerical simulations performed on the model showed that the modeled behavior approximates, in a general way, the physiological mechanisms of solvent and solute flow in the kidney.     

A path probability approach to the solute transport problem

An alternative solute transport algorithm, termed the path probability approach, is presented. The method may be more effective than typical finite methods in some numerical solution problems. The method differs from predominant numerical techniques in that the conventional conservation differential is replaced by a series of equations defining solute migration in terms of a finite number of representative particle position histories and corresponding probabilities. The governing physical assumptions applied in model development are conditionally consistent with those of conventional Lagrangian and Eulerian conceptualizations, but mathematical expression and development of the problem differ from traditional approaches. The method represents dispersion orthogonal to the primary axis of advective transport explicitly, according to arbitrary user-defined probability functions; longitudinal dispersion effects result from the existence of velocity shear effects along the primary axis of movement. It is possible to incorporate Fickian dispersive processes, and thereby reproduce results obtainable with traditional stream tube models; but non-Fickian alternatives can also be explored. Simulation results of the path probability model are compared to analogous results from a representative finite difference model for a hypothetical test flow channel. The comparison demonstrated the ability of the new model to effectively generate simulation results with a computational effeciency considerably higher than conventional techniques under the conditions tested.     

A numerical method for mass conservative coupling between fluid flow and solute transport

We present a new coupled discretization approach for species transport in an incompressible fluid. The Navier-Stokes equations for the flow are discretized by the divergence-free Scott-Vogelius element on barycentrically refined meshes guaranteeing LBB stability. The convection-diffusion equation for species transport is discretized by the Voronoi finite volume method. In accordance to the continuous setting, due to the exact integration of the normal component of the flow through the Voronoi surfaces, the species concentration fulfills discrete global and local maximum principles. Besides of the numerical scheme itself, we present important aspects of its implementation. Further, for the case of homogeneous Dirichlet boundary conditions, we give a convergence proof for the coupled scheme. We report results of the application of the scheme to the interpretation of limiting current measurements in an electrochemical flow cell with cylindrical shape.     

A coupled phase transformation and solute diffusion model for bainitic transformation

In materials science one distinguishes between upper and lower bainite, where both microstructures develop due to different diffusion processes. In this work we describe these different mechanisms with a new diffusion model coupled to a multiphase-field equation. Numerical examples demonstrate the expected behavior. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A reduced finite element formulation based on POD method for two-dimensional solute transport problems

A proper orthogonal decomposition (POD) method is applied to a usual finite element (FE) formulation for two-dimensional solute transport problems with real practical applied background such that it is reduced into a reduced FE formulation with lower dimensions and high enough accuracy. The error estimates between the reduced POD FE solutions and the usual FE solutions are provided. It is shown by numerical examples that the results of numerical computation are consistent with theoretical conclusions. Moreover, it is also shown that this validates the feasibility and efficiency of POD FE method.     

Discontinuous Galerkin relaxation algorithm for solute transport in porous media

We consider a degenerate parabolic convection dominated equation which models the transport of contaminant in porous media. The numerical scheme is fulfilled by combining the DG – Discontinuous Galerkin Method with and an efficient relaxation algorithm that recently developed. Numerical results show the efficiency of our scheme. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical simulation based on POD for two-dimensional solute transport problems

A proper orthogonal decomposition (POD) method is applied to a usual finite element scheme for two-dimensional solute transport problems such that it is reduced into a reduced finite element formulation with lower dimensions and high enough accuracy. Numerical examples show that the results of numerical computations are consistent with accurate solutions. Moreover, this validates the feasibility and efficiency of POD method.     

Modeling of solute transport into sub-aqueous sediments

A model for investigating the solute transport into a sub-aqueous sediment bed, under an imposed standing water surface wave, is developed. Under the assumption of Darcy flow in the bed, a model based on a two-dimensional, unsteady advection–diffusion equation is derived; the relative roles of the advective and diffusive transport are characterized by a Peclet number, Pe. Two solutions for the equation are developed. The first is a basic control volume method using the power-law scheme. The second is a smear-free, modified upwind solution for the special case of Pe → ∞. Results, at a given time step, are reported in terms of a laterally averaged solute verse depth profile. The main result of the paper is to demonstrate that the one-dimensional solute concentration verse depth profile is essentially independent of any numerical dissipation present in the solute field predictions. This demonstration is achieved by (i) using an extensive grid refinement study, and (ii) by comparing Pe → ∞ predictions obtained with the basic and smear-free solutions.     

A network model for nursing staff scheduling

The staffing of hospital nurses has become critical in recent years. As a means of containing skyrocketing costs, many hospitals are reducing nursing staff to a bare minimum. At the same time, the hospitals must maintain some minimal staffing level as insufficient staff could lead to a life threatening situation possibly with detrimental social, economic, and legal consequences. This article will present a model which can be used to determine the optimal scheduling of nursing staff under varying conditions. A typical problem will illustrate its use.     

A neural network model for scheduling problems

Artificial neural networks (ANNs) have been successfully applied to solve a variety of problems. This paper proposes a new neural network approach to solve the single machine mean tardiness scheduling problem and the minimum makespan job shop scheduling problem. The proposed network combines the characteristics of neural networks and algorithmic approaches. The performance of the network is compared with the existing scheduling algorithms under various experimental conditions. A comprehensive bibliography is also provided in the paper.     

The role of epithelial ion transport in taste transduction: A network thermodynamic model

The sensation of taste is mediated by electrical events in taste bud cells, which are embedded in the tongue epithelium. Prior to about 1981, many taste researchers regarded the tongue epithelium as virtually impermeable to most substances including small electrolytes. Taste transduction was assumed to be due to equilibrium adsorption of tastants to receptors on the taste cell membrane. More recent experiments show that, in the case of salt taste, the electrical events associated with transduction are due to ion flow across membranes rather than ion binding to a receptor. We wished to determine if standard assumptions about topology and membrane ion transport processes in high resistance (“tight”) epithelia apply to the tongue for a range of salt concentrations on the tongue surface. A model of the tongue epithelium was developed based on an earlier network model of ion transport in a kidney tight epithelium. The model has 4 membranes with series and parallel pathways and 3 transported ions. The model successfully simulates some aspects of both steady-state and transient transepithelial electrical measurements observed in vitro for a wide range (50–2000 mM) of NaCl concentrations. This is the concentration range of interest for gustation. The earlier kidney model had not been applied to hyperosmotic NaCl concentrations. The simulations indicate that common epithelial transport mechanisms can account for some phenomena important in taste transduction. They also indicate that the Na⁺ channels in taste cells have different properties than those typical in other tight epithelia.     

On reactive solute transport through a curved pipe

The transport of a reactive solute by diffusion and convection in a thin (or long) curved pipe is considered. Using asymptotic analysis with respect to the pipe’s thickness, the effective model for solute concentration is formally derived. A simple approximation is computed, showing explicitly the effects of the pipe’s geometry in nature and magnitude.     

一种Tandem网络模型

本文考虑的是一个MAP输入的tandem队列，运用一个新颖的方法把它模拟成一个带爆炸块依赖阶数的拟生灭过程，这样使得我们可以很好地分析它的联合时间分布和平稳状态的停留时间分布布。     

Newton method for reactive solute transport with equilibrium sorption in porous media

We present a mass conservative numerical scheme for reactive solute transport in porous media. The transport is modeled by a convection-diffusion-reaction equation, including equilibrium sorption. The scheme is based on the mixed finite element method (MFEM), more precisely the lowest-order Raviart-Thomas elements and one-step Euler implicit. The underlying fluid flow is described by the Richards equation, a possibly degenerate parabolic equation, which is also discretized by MFEM. This work is a continuation of Radu et al. (2008) and Radu et al. (2009) [1] and [2] where the algorithmic aspects of the scheme and the analysis of the discretization method are presented, respectively. Here we consider the Newton method for solving the fully discrete nonlinear systems arising on each time step after discretization. The convergence of the scheme is analyzed. In the case when the solute undergoes equilibrium sorption (of Freundlich type), the problem becomes degenerate and a regularization step is necessary. We derive sufficient conditions for the quadratic convergence of the Newton scheme.     

On the evaluation of moments for solute transport by random velocity fields

In this paper, we consider the random linear transport equation. We show that standard averaging approaches to obtain an equation for the evolution of the statistical mean of the solution may also be valid for all the statistical moments of the solution. With this result we can obtain more statistical information about the random solution, as illustrated in two particular examples.