Electrohydrodynamic instability in a thin fluid layer with an electrical conductivity gradient

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∈E₀²Δσ/μK _effσ₀, an electrical Rayleigh number; Δσ/σ₀, the relative conductivity increment: and α, the transverse wave number of the disturbance. Here d is the thickness, ∈ is the dielectric constant, and μ is the viscosity of the layer, E₀ 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 1.416×10⁴ at a critical transverse wave number of 1.90 when Δσ/σ₀ is 8. As Δσ/σ₀ 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 Δσ/σ₀ ≲8, the critical Rayleigh number increases as Δσ/σ₀, decreases and the layer is only sensitive to long wavelength disturbances (α<0.1) for Δσ/σ₀ below 4. Similar trends were obtained for liquid layers with other boundary conditions; e.g., rigid boundaries and constant potential boundaries.