We have modeled the electrohydrodynamic behavior of emulsions somewhat simplistically ( 2) using the Navier -Stokes equation for vis-We present theoretical evidence from computational fluid dycous Newtonian flow, a Fickian diffusion equation incorporating gravitanamics for transient isothermal roll-cell convection induced by tional and dielectrophoretic fluxes for the droplets, the hydrodynamic very strong vertical electric fields in thin horizontal layers of watercontinuity equation for mass conservation, and the Maxwell equation, in-oil emulsions initially at sedimentation equilibrium under grav-Γr( e 0 E / P ) Γ
0, consistent with the absence of free charges. The last ity. This convection is driven by the Kelvin body force, which condition requires the emulsion to be stabilized by nonionic surfactants. depends on the concentration gradient of droplets and on differ-Our two-dimensional model employs the barycentric velocity ( 3 ) v , ences between the isotropic dielectric permittivities of the two which is a function of the horizontal coordinate x , the vertical coordinate phases of a nonionic emulsion. Provided that the applied voltage y and the time t . We describe the incompressible flow in terms of a and hence a nondimensional electric Rayleigh number N is suffistream function c according to ciently large compared to a gravitational Rayleigh number R, rollcell convection sets in, grows, and then decays. If N is sufficiently v Γ
curl(ck), [1] large this roll-cell convection disturbs the local composition of the suspension and increases the spatial homogeneity of the emulsion. where (i, j, k) is the triad of unit vectors along the x, y, and z axes. Thus This electroconvective effect, which can accompany dielectrothe vorticity W Γ
krcurl v Γ
0Γ 2 c. Simple linear constitutive expressions phoresis, is of interest as a novel form of transient pattern formaare assumed for the density, r Γ
r 2 (1 / ac), of the suspension and for tion and as an example of electrically induced stirring. Similar its relative permittivity, e Γ
e 2 (1 / bc). In these expressions the subscript 2 refers to the continuous phase and 1 refers to the droplets. Thus, a Γ
r 1 / convection patterns should be observable in related colloidal susr 2 0 1 measures the ratio of the densities of the two phases, b Γ
e 1 /e 2 0 pensions of fine particles with high dielectric permittivities dis-1, and c Γ
c 1 is the local mass-fraction of the droplets in the emulsion.
persed in carrier fluids with lower dielectric permittivities and
The nondimensional governing equations for a dilute emulsion are the densities.