[Paper Review] Electroconvective instability of self-similar equilibria

E. A. Demekhin, S. V. Polyanskikh+1 more|arXiv (Cornell University)|Jan 25, 2010

Membrane-based Ion Separation Techniques33 references3 citations

TL;DR

This paper introduces a novel class of one-dimensional, unsteady, self-similar equilibrium solutions for electroconvective flows between ion-exchange membranes, derived from asymptotic analysis and validated numerically. It demonstrates that these transient equilibria exhibit electroconvective instability under specific conditions, with marginal stability curves showing fair agreement between asymptotic and numerical approaches, particularly at intermediate to large times and small Debye lengths.

ABSTRACT

Stability of electro-hydrodynamic processes between ion-exchange membranes is investigated. Solutions of the equilibrium problem are commonly described in the one-dimensional (1D) steady-state approximation. In the present work, a novel class of 1D unsteady self-similar equilibrium solutions is developed, whose existence is supported by recent experiments. 1D unsteady two-parametric family of self-similar equilibrium solutions and their stability are studied both asymptotically in small dimensionless Debye length, and numerically. The self-similar solutions and marginal stability curves obtained in both approaches are in a fair agreement with each other at intermediately large times for which the dimensionless distance between membranes is large.

Motivation & Objective

• To develop a new class of one-dimensional, unsteady, self-similar equilibrium solutions for electrohydrodynamic systems between ion-exchange membranes.
• To analyze the linear stability of these self-similar equilibria and identify conditions leading to electroconvective instability.
• To compare asymptotic solutions in the limit of small dimensionless Debye length with direct numerical simulations of the full system.
• To construct marginal stability curves and determine the onset of instability in terms of key dimensionless parameters such as $1/ ilde{ u}$, $ ilde{ u}$, and $ ilde{ u}$.
• To reconcile discrepancies near critical points by identifying limitations of the asymptotic approach in the transition region between under-limiting and limiting current regimes.

Proposed method

• Derives a simplified system of equations from the full electrohydrodynamic model, incorporating Poisson and Stokes equations for ion transport and fluid flow.
• Applies matched asymptotic expansions in the limit of small dimensionless Debye length ($ ilde{ u} o 0$) to obtain outer solutions, neglecting inner boundary layer contributions.
• Uses patching conditions at $y = 0$ and $y = y_m$ to approximate the outer solution without solving for thin boundary layers.
• Solves the resulting nonlinear system asymptotically to obtain analytic expressions for ion concentrations $c^{/pm}$, electric field $F$, and potential drop $ ilde{ u}$.
• Performs numerical simulations of the full problem to validate the asymptotic results, particularly in the limiting current regime.
• Constructs marginal stability curves in the $(1/ ilde{ u}, ilde{ u})$-plane by analyzing the growth rate of perturbations, with asymptotic approximations for short- and long-wave limits.

Experimental results

Research questions

• RQ1 Can self-similar unsteady solutions be derived for the 1D electroconvective system between ion-exchange membranes in the limit of small Debye length?
• RQ2 How do the stability characteristics of these self-similar equilibria compare between asymptotic analysis and direct numerical simulation?
• RQ3 What are the critical parameters ($1/ ilde{ u}_0$, $ ilde{ u}_0$, $ ilde{ u}_0$) that separate stable and unstable regions in the parameter space?
• RQ4 Why does the asymptotic approach show discrepancies near the critical point of the marginal stability curve?
• RQ5 What are the scaling laws for the upper and lower branches of the marginal stability curves in the short- and long-wave limits?

Key findings

• A two-parametric family of 1D unsteady self-similar equilibrium solutions is derived, valid for intermediate to large times and small Debye length.
• The asymptotic solution for the electric field and ion concentrations shows fair agreement with numerical simulations, especially in the limiting current regime.
• Marginal stability curves from asymptotic and numerical methods align well except near the critical point, where the asymptotic method breaks down due to proximity to the under-limiting regime.
• The upper branch of the marginal stability curve scales as $ ilde{ u} o Q( ilde{ u}) (1/ ilde{ u})^{2/3}$, with $Q( ilde{ u})$ tabulated numerically for various $ ilde{ u}$ values.
• The lower branch scales as $ ilde{ u} o R( ilde{ u})$, with $R( ilde{ u})$ also tabulated, and the time gap $ ilde{ ilde{t}}$ during which perturbations grow is estimated as $ ilde{ ilde{t}} o rac{k^4 Q^6 - R^2}{4 ilde{ u}^2 ilde{D}}$.
• The critical parameters $1/ ilde{ u}_0$, $ ilde{ u}_0$, and $ ilde{ u}_0$ are computed for different $ ilde{ u}$, showing that instability arises when the perturbation wave number $k < k_0$.