Search Results (54)

Aqueous humor (AH) is a watery fluid continuously circulating through the posterior and anterior chambers of the human eye and is essential to maintain a healthy intraocular pressure in the eyeball and keep the eye clean from waste products of metabolism and external agents. This paper presents a stationary compartment model of AH dynamics consisting of three integrated modules (M): M1 for AH production, M2 for AH passive flow and M3 for AH drainage. M1 is a zero-dimensional (0D) reduction of the velocity-extended Poisson-Nernst-Planck model and simulates solute transfer and fluid movement across the cellular structure of the ciliary epithelium (CE). M2 is the electric equivalent representation of Poiseuille flow across the series of two linear hydraulic resistors. M3 is a 0D reduction of the Darcy equations for a porous medium and simulates AH flow across the parallel between a nonlinear and a linear resistor. Compared to existing compartment approaches, the present model integrates at the macroscopic scale the multi-physical description of the human eye at the cellular scale. Numerical simulations suggest that (1) sodium channels in the CE are essential for maintaining proper AH dynamics; and (2) increased episcleral vein pressure reduces AH drainage, potentially explaining the development of secondary open-angle glaucoma. These insights advance the understanding of the mechanisms regulating AH dynamics and offer new perspectives for patient-specific therapies. Full article

This work numerically reveals a novel strategy to modulate two ionic conductance state in a nanochannel via pressure-dependent fluidic motion inside the channel. Steady and transient simulations based on Poisson–Nernst–Planck–Stokes equations demonstrate that the two states with distinct ionic conductance and ion selectivity can be reversibly switched by external pressure, with a characteristic time of ~100 μs. Furthermore, the two conductance states are found to depend on the transversal electric field, which gives rise to two distinct intrachannel fluidic flow patterns, namely laminar flow and vortex flow, respectively. This finding suggests the potential of pressure-controlled ionic conductance switching for applications in nanofluidic ionic circuits, flow-regulated sensing, and integrated micro/nanoscale devices. It also provides insights into nonlinear ionic current–voltage behaviors. Full article

Designing structure-preserving numerical schemes for the generalized Poisson-Nernst-Planck (PNP) system is challenging due to its inherent strong nonlinearity and coupling. In this paper, we propose a class of efficient, unconditional energy-stable schemes based on the Stabilized Scalar Auxiliary Variable (S-SAV) framework combined with the finite element method. We construct both first-order (BE-S-SAV) and second-order (BDF2-S-SAV) fully discrete schemes. A distinguishing feature of our approach is the use of a linear decomposition strategy, which decouples the complex nonlinear system into a sequence of linear, constant-coefficient elliptic equations at each time step. This significantly reduces computational complexity by avoiding expensive nonlinear iterations. We provide rigorous theoretical proofs demonstrating that the proposed schemes are unconditionally energy stable and strictly preserve mass conservation. Numerical experiments satisfy the theoretical analysis, confirming optimal convergence rates and demonstrating robust preservation of mass conservation and modified energy stability in the tested regimes. Full article

We introduce a hybridizable discontinuous Galerkin (HDG) scheme for solving the Poisson–Nernst–Planck (PNP) equations. The log-density formulation as introduced by Metti et al. in their paper “Energetically stable discretizations for charge transport and electrokinetic models. J. Comput. Phys. 2016, 306, 1-18” is utilized to ensure the positivity of the densities of the charged particles. We further prove that our fully discrete scheme is energy stable and mass conserving. Numerical simulations are provided to demonstrate the accuracy of the scheme in one and two spatial dimensions. A derivation of an HDG-DG space–time scheme is given, with implementation and convergence analysis left to future work. Full article

This paper presents the first theoretical investigation of the effect of a variable equilibrium coefficient on the steady-state transport of a binary electrolyte in a desalination channel cross-section of the electrodialyzer. To address this problem, we developed a new mathematical model in the form of a boundary value problem for an extended system of stationary Nernst–Planck–Poisson equations. We obtained a numerical solution to this problem using the finite element method. Analysis of this solution revealed that the channel cross-section has a complex structure: it is divided into seven regions dominated by different processes, and, consequently, the solution to the boundary value problem behaves differently in each of them. Existing models of the diffusion layer or channel cross-section typically assume a constant equilibrium coefficient. In this paper, we demonstrated that in the channel cross-section, the velocity change corresponding to the equilibrium constant is related not only to the field strength but also to the magnitude of the space charge. In the space-charge region, in the boundary layers near the ion-exchange membranes, intense dissociation of water molecules occurs, and the higher the equilibrium coefficient, the more intense this dissociation is. We have shown that an internal boundary layer (recombination region) arises deep within the solution, associated with the recombination reaction of H+ and OH− ions. In this study, we found that with increasing equilibrium coefficient, fluxes increase, while with increasing fluxes, the electric field strength decreases proportionally, and equilibrium is reached. We demonstrate that by calibrating a single fitting parameter in the model, the simulation results can be matched to experimental data with high accuracy. Thus, our proposed model and its numerical solution provide a completely new understanding of the ion transport process in electromembrane systems, taking into account the influence of the dissociation/recombination reaction of water molecules. Full article

Ion channels are protein pores that regulate ionic flow across cell membranes, enabling vital processes such as nerve signaling. They often conduct multiple ionic species simultaneously, leading to complex nonlinear transport phenomena. Because experimental techniques provide only indirect measurements of ion channel currents, mathematical models—particularly Poisson–Nernst–Planck (PNP) equations—are indispensable for analyzing the underlying transport mechanisms. In this work, we examine ionic transport through a one-dimensional steady-state PNP model of a narrow membrane channel containing multiple cation species of different valences. The model incorporates a small fixed charge distribution along the channel and imposes relaxed electroneutrality boundary conditions, allowing for a slight charge imbalance in the baths. Using singular perturbation analysis, we first derive approximate solutions that capture the boundary-layer structure at the channel—reservoir interfaces. We then perform a regular perturbation expansion around the neutral reference state (zero fixed charge with electroneutral boundary conditions) to obtain explicit formulas for the steady-state ion fluxes in terms of the system parameters. Through this analytical approach, we identify several critical applied potential values—denoted $V_k^a$ (for each cation species k), $V^b$, and $V^c$—that delineate distinct transport regimes. These critical potentials govern the sign of the fixed charge’s influence on each ion’s flux: depending on whether the applied voltage lies below or above these thresholds, a small positive permanent charge will either enhance or reduce the flux of each ion species. Our findings thus characterize how a nominal fixed charge can nonlinearly modulate multi-ion currents. This insight deepens the theoretical understanding of nonlinear ion transport in channels and may inform the interpretation of current–voltage relations, rectification effects, and selective ionic conduction in multi-ion channel experiments. Full article

Modeling ion transport through membrane channels is crucial for understanding cellular processes, and Poisson–Nernst–Planck (PNP) equations provide a fundamental continuum framework for such ionic fluxes. We investigate a quasi-one-dimensional steady-state PNP system for two oppositely charged ion species, focusing on how large permanent charges within the channel and realistic boundary conditions impact ion transport. In contrast to classical models that impose ideal electroneutrality at the channel ends (a simplification that eliminates boundary layers near the membrane interfaces), we adopt relaxed neutral boundary conditions that allow small charge imbalances at the boundaries. Using asymptotic analysis treating the large permanent charge as a singular perturbation, we derive explicit first-order expansions for each ionic flux, incorporating boundary layer parameters $(\sigma ,\rho )$ to quantify slight deviations from electroneutrality. This analysis enables a qualitative characterization of individual cation and anion flux behaviors. Notably, we identify two critical transmembrane potentials, $V_{1c}$ and $V_{2c}$, at which the cation and anion fluxes, respectively, vanish, signifying flux-reversal thresholds that delineate distinct monotonic regimes in the flux-voltage response; these critical values depend on the permanent charge magnitude and the boundary layer parameters. We further show that both ionic fluxes exhibit saturation: as the applied voltage becomes extreme, each flux approaches a finite limiting value, with the saturation level modulated by the degree of boundary charge imbalance. Moreover, allowing even small boundary charge deviations reveals non-intuitive discrepancies in flux behavior relative to the ideal electroneutral case. For example, in certain parameter regimes, a large permanent charge that enhances an ionic current under strict electroneutral conditions will instead suppress that current under relaxed-neutral conditions (and vice versa). This new analytical framework exposes subtle yet essential nonlinear dynamics that classical electroneutral assumptions would otherwise obscure. It provides deeper insight into the interplay between large fixed charges and boundary-layer effects, emphasizing the importance of incorporating such realistic boundary conditions to ensure accurate modeling of ion transport through membrane channels. Numerical simulations are performed to provide more intuitive illustrations of our analytical results. Full article

This article proposes a 3D mathematical model of the influence of electrical heterogeneity of the ion exchange membrane surface on the processes of salt ion transfer in membrane systems with axial symmetry; in particular, we investigate an annular membrane disk in the form of a coupled system of Nernst–Planck–Poisson and Navier–Stokes equations in a cylindrical coordinate system. A hybrid numerical–analytical method for solving the boundary value problem is proposed, and a comparison of the results for the annular disk model obtained by the hybrid method and the independent finite element method is carried out. The areas of applicability of each of these methods are determined. The proposed model of an annular disk takes into account electroconvection, which is understood as the movement of an electrolyte solution under the action of an external electric field on an extended region of space charge formed at the solution–membrane boundary under the action of the same electric field. The main regularities and features of the occurrence and development of electroconvection associated with the electrical heterogeneity of the surface of the membrane disk of the annular membrane disk are determined; namely, it is shown that electroconvective vortices arise at the junction of the conductivity and non-conductivity regions at a certain ratio of the potential jump and angular velocity and flow down in the radial direction to the edge of the annular membrane. At a fixed potential jump greater than the limiting one, the formed electroconvective vortices gradually decrease with an increase in the angular velocity of rotation until they disappear. Conversely, at a fixed value of the angular velocity of rotation, electroconvective vortices arise at a certain potential jump, and with its subsequent increase gradually increase in size. Full article

This article is devoted to numerically modeling the effect of the geometric modification of the surfaces of ion-exchange membranes in electromembrane systems (EMSs) on the salt ion transport using a 2D mathematical model of the transport process in the desalination channel based on boundary value problems for the coupled system of Nernst–Planck–Poisson and Navier–Stokes equations. The main patterns of salt ion transport are established taking into account diffusion, electromigration, forced convection, electroconvection, and the geometric modification of the surface of ion-exchange membranes. It is shown that the geometric modification of the surface of ion-exchange membranes significantly changes both the formation and development of electroconvection. A significant combined effect of electroconvection and geometric modification of the surface of ion-exchange membranes in the desalination channel on the salt ion transport is shown, as well as a complex, nonlinear, and non-stationary interaction of all the main effects of concentration polarization in the desalination channel. Full article

This article investigates the boundary value problem for an extended stationary system of Nernst–Planck–Poisson equations, corresponding to a mathematical model of the influence of changes in the equilibrium coefficient on the transport of ions of a binary salt in the diffusion layer. Dimensionless variables were introduced using characteristic parameter values. As a result, a dimensionless boundary value problem was obtained, which is singularly perturbed, containing a small parameter in the derivative of the Poisson equation and, additionally, another regular small parameter. A similarity theory was developed: trivial and non-trivial similarity criteria and their physical meaning were determined, which allowed for the identification of general properties of the solutions. A numerical investigation of the boundary value problem was conducted using the finite element method. With an increase in the initial solution concentration, the value of the small parameter entering singularly decreases, reaching values on the order of 10⁻¹² and below, leading to computational difficulties that prevent a comprehensive analysis of the influence of changes in the equilibrium coefficient on salt ion transport. In this regard, an analytical solution to the problem was constructed, based on dividing the solution domain into several subdomains (regions of electroneutrality, extended space charge region, quasi-equilibrium region, recombination region, intermediate layer), in each of which the problem is solved differently, followed by matching these solutions. Verification of the analytical solution was carried out by comparing it with the numerical solution. The advantage of the obtained analytical solution is the possibility of a comprehensive analysis of the influence of the dissociation/recombination reaction of water molecules on salt ion transport over a wide range of real changes in the concentration and composition of the electrolyte solution and other input parameters. This boundary value problem serves as a benchmark for constructing asymptotic solutions for other singularly perturbed boundary value problems in membrane electrochemistry. Full article

This article proposes various new approximate analytical solutions of the boundary value problem for the non-stationary system of Nernst–Planck–Poisson (NPP) equations in the diffusion layer of an ideally selective ion-exchange membrane at overlimiting current densities. As is known, the diffusion layer in the general case consists of a space charge region and a region of local electroneutrality. The proposed analytical solutions of the boundary value problems for the non-stationary system of Nernst–Planck–Poisson equations are based on the derivation of a new singularly perturbed nonlinear partial differential equation for the potential in the space charge region (SCR). This equation can be reduced to a singularly perturbed inhomogeneous Burgers equation, which, by the Hopf–Cole transformation, is reduced to an inhomogeneous singularly perturbed linear equation of parabolic type. Inside the extended SCR, there is a sufficiently accurate analytical approximation to the solution of the original boundary value problem. The electroneutrality region has a curvilinear boundary with the SCR, and with an unknown boundary condition on it. The article proposes a solution to this problem. The new analytical solution methods developed in the article can be used to study non-stationary boundary value problems of salt ion transfer in membrane systems. The new analytical solution methods developed in the article can be used to study non-stationary boundary value problems of salt ion transport in membrane systems. Full article

A time-dependent electrochemical impedance spectroscopy (EIS) model is presented using the finite element method (FEM) to simulate a 2D interdigitated electrode in an aqueous NaCl electrolyte. Developed in COMSOL Multiphysics, the model incorporates ion transport, electric field distribution, Stern layer effects, and electrode sheet resistance, governed by the Poisson and Nernst–Planck equations. This model can predict the transient current response to an applied excitation voltage, which gives information about the dynamics of the electrochemical system. The simulation results are compared with the experimental data, reproducing key features of the measurements. The transient current response indicates the need for multiple excitation cycles to stabilize the impedance measurement. At low frequencies (<1 kHz), the voltage drop at the Stern layer is significant, while at higher frequencies (>100 kHz), the voltage drop due to sheet resistance dominates. Moreover, the amplitude of the excitation voltage influences the EIS measurement, higher amplitudes (above 0.1 V) lead to non-linear impedance behavior, particularly at low ion concentrations. Discrepancies at low frequencies suggest that Faradaic processes may need to be incorporated for improved accuracy. Overall, this model provides quantitative insights for optimizing EIS sensor design and highlights critical factors for high-frequency and low-concentration conditions, laying the foundation for future biosensing applications with functionalized electrodes. Full article

Pulsed electric field (PEF) modes of electrodialysis (ED) are known for their efficiency in mitigating the fouling of ion-exchange membranes. Many authors have also reported the possibility of increasing the mass transfer/desalination rate and reducing energy costs. In the literature, such possibilities were theoretically studied using 1D modeling, which, however, did not consider the effect of electroconvection. In this paper, the analysis of the ED desalination characteristics of PEF modes is carried out based on a 2D mathematical model including the Nernst–Planck–Poisson and Navier–Stokes equations. Three PEF modes are considered: galvanodynamic (pulses of constant electric current alternate with zero current pauses), potentiodynamic (pulses of constant voltage alternate with zero voltage pauses), and mixed galvanopotentiodynamic (pulses of constant voltage alternate with zero current pauses) modes. It is found that at overlimiting currents, in accordance with previous papers, in the range of relatively low frequencies, the mass transfer rate increases and the energy consumption decreases with increasing frequency. However, in the range of high frequencies, the tendency changes to the opposite. Thus, the best characteristics are obtained at a frequency close to 1 Hz. At higher frequencies, the pulse duration is too short, and electroconvective vortices, enhancing mass transfer, do not have time to develop. Full article

The paper considers the a posteriori error estimates for fully discrete approximations of time-dependent Poisson–Nernst–Planck (PNP) equations, which provide tools that allow for optimizing the choice of each time step when working with adaptive meshes. The equations are discretized by the Backward Euler scheme in time and conforming finite elements in space. Overcoming the coupling of time and the space with a full discrete solution and dealing with nonlinearity by taking G-derivatives of the nonlinear system, the computable, robust, effective, and reliable space–time a posteriori error estimation is obtained. The adaptive algorithm constructed based on the estimates realizes the parallel adaptations of time steps and mesh refinements, which are verified by numerical experiments with the time singular point and adaptive mesh refinement with boundary layer effects. Full article

This article considers a theoretical analysis of the influence of the main coupled effects and spacers on the transfer of salt ions in electromembrane systems (EMS) using a 2D mathematical model of the transfer process in a desalting channel with spacers based on boundary value problems for the coupled system of Nernst–Planck–Poisson and Navier–Stokes equations. The basic patterns of salt ion transport have been established, taking into account diffusion, electromigration, forced convection, electroconvection, dissociation/recombination reactions of water molecules, as well as spacers located inside the desalting channel. It has been shown that spacers and taking into account the dissociation/recombination reaction of water molecules significantly change both the formation and development of electroconvection. This article confirms the fact of the exaltation of the limiting current studied by Harkatz, where it is shown that the current (flux) of salt ions increases when the dissociation reaction begins by a certain value called the exaltation current, which is proportional to the flow of water dissociation products. A significant combined effect of electroconvection and dissociation/recombination reactions as well as the spacer system in the desalting channel on the transport of salt ions are shown. The complex, nonlinear, and non-stationary interaction of all the main effects of concentration polarization and spacers in the desalting channel are also considered in the work. Full article