The electroconvective instability (ECI) in an electrodialysis channel under a strong electric field is studied here. The phenomenon of ECI with extreme depletion (ECI-HD) is reported; that is, the overlapping vortices cause the extreme depletion zone to propagate in the horizontal direction. Using scaling theory and direct numerical simulation, we indicate a series of features under the ECI-HD. The decrease in ion transport rate with voltage in ECI-HD is different from the enhancement in the ECI with moderate depletion (ECI-MD), which results in a unique peak in the voltage-current curve. More importantly, we reveal that the ECI is regulated by a scaling factor consisting of the electric field, hydrodynamic coupling coefficient, and Péclet number. For the ECI-HD, the scaling factor has an opposite effect on the vortex size and overlimiting current as that on the ECI-MD. The extreme depletion zone of the ECI-HD also has an uncommon diffusion self-similar dynamics. These unique scaling laws allow one to establish the quantitative bridge between the ion concentration, electric field, and vortex size by the overlimiting current.

The transient thermal fracture problem of a crack (perpendicular to the gradient direction) in a graded orthotropic strip is investigated. Most of the materials properties are assumed to vary as an exponential function of thickness direction. The transient two-dimensional temperature problem is analyzed by the methods of Laplace and Fourier transformations. A system of singular integral equations are obtained and solved numerically. Numerical results are figured out to show the variation of the temperature on the crack faces and extended line and stress intensity factors for different material parameters with dimensionless time.

The contact problem for thermoelectric materials with functionally graded properties is considered. The material properties, such as the electric conductivity, the thermal conductivity, the shear modulus, and the thermal expansion coefficient, vary in an exponential function. Using the Fourier transform technique, the electro-thermoelastic problems are transformed into three sets of singular integral equations which are solved numerically in terms of the unknown normal electric current density, the normal energy flux, and the contact pressure. Meanwhile, the complex homogeneous solutions of the displacement fields caused by the gradient parameters are simplified with the help of Euler’s formula. After addressing the non-linearity excited by thermoelectric effects, the particular solutions of the displacement fields can be assessed. The effects of various combinations of material gradient parameters and thermoelectric loads on the contact behaviors of thermoelectric materials are presented. The results give a deep insight into the contact damage mechanism of functionally graded thermoelectric materials (FGTEMs).

Ion and water are transported by electroconvection near permselective membranes, resulting in complex phenomena associated with the flow–fines interaction. Sheltering the flow chaos by the shear flow is a common strategy in plasma fluids and has recently been successfully applied to control ionic fluids. The paper herein reveals the critical selection of shear velocity regarding the fluid from a chaotic to a steady state through numerical and theoretical analyses. For the shear sheltering, the dimensionless Debye length ${\lambda _D}$ with varying channel height is introduced to achieve a comprehensive discussion of the factors and laws affecting the shear vortex state. Based on an analysis of the vortex driving mechanism, the scaling of the slip velocity ${u_s}\sim {(\lambda _D^{ - 1}\Delta {\phi ^4})^{1/3}}$ is recommended as the critical selection factor for the steady and chaotic state under a fixed shear flow velocity, which involves the dimensionless Debye length ${\lambda _D}$ and voltage difference $\Delta \phi $ . Furthermore, for ionic fluid control by shear flow, a critical shear velocity ${U_{HPC}}$ is proposed to distinguish the electroconvective flow from a chaotic state to a steady state. When the shear flow velocity ${U_{HP}} > {U_{HPC}}$ , the shear flow shelters chaos, and the scaling law is also recommended for the regulation of the critical shear flow velocity ${U_{HPC}}$ jointly by ${\lambda _D}$ and $\Delta \phi $ . The analysis is confirmed by direct numerical simulation and existing experimental data (J. Fluid Mech, vol. 813, 2017, pp. 799–823). This work provides a more comprehensive physical insight for shear sheltering and affects the design of electromembrane microfluidics.