The onset of electrohydrodynamic instability in isoelectric focusing

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 spatio-temporal 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(sub e) is equivalent to d(epsilon)E(sup 2, sub 0) Delta sigma/mu K(sub eff) sigma(sub 0), an electrical Rayleigh number; delta sigma/sigma(sub 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 layer, E(sub 0) is the applied field strength at the lower conductivity boundary, and K(sub eff) is an effective diffusivity associated with the Brownian motion of the charge-carrying solutes. With viscous-stress-free boundaries, at which the electrical conductivity and the normal component of the electric field are prescribed, the critical Ra(sub e) is 1.504 x 10(exp 4) at a critical transverse wave number of 1.97 when Delta sigma/sigma(sub 0) is 10. As Delta sigma/sigma(sub 0) increases, the critical Ra(sub e) increases and shifts to 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. Similar trends were obtained for liquid layers with rigid boundaries.