The onset of electrohydrodynamic motion associated with the imposition of an electric field across a thin layer of liquid has been investigated for the case in which the electrical conductivity varies linearly over the depth of the layer. The variation of the conductivity is due to concentration gradients in the charge-carrying solutes and its spatiotemporal evolution is represented by a convective-diffusion equation. When the viscous relaxation time is long compared to the time for charge relaxation, the analysis reveals that the neutral stability curves for the layer can be characterized by three dimensionless parameters: Ra-e=d epsilon E-0(2) Delta sigma/mu K(eff)sigma(0), an electrical Rayleigh number; Delta sigma/sigma(0), the relative conductivity increment; and alpha, the transverse wave number of the disturbance. Here d is the thickness, epsilon is the dielectric constant, and mu is the viscosity of the layer, E-0 is the applied field strength at the lower conductivity boundary, and K-eff is an effective diffusivity associated with the Brownian motion of the charge-carrying solutes. With stress-free boundaries, at which the electrical conductivity and current are prescribed, the critical Ra-e is 1.416x10(4) at a critical transverse wave number of 1.90 when Delta sigma/sigma(0) is 8. As Delta sigma/sigma(0) increases, the critical Ra-e increases and shifts to slightly shorter wavelength disturbances; the critical imposed field strength, however, passes through a minimum because the lower-conductivity boundary exerts a considerable stabilizing influence in the presence of steep conductivity gradients. For Delta sigma/sigma(0) less than or similar to 8, the critical Rayleigh number increases as Delta sigma/sigma(0) decreases and the layer is only sensitive to long wavelength disturbances (alpha<0.1) for Delta sigma/sigma(0) below 4. Similar trends were obtained for liquid layers with other boundary conditions; e.g., rigid boundaries and constant potential boundaries. (C) 1998 American Institute of Physics.
