Electrocapillary, thermocapillary, and buoyancy convection driven flows in the Melcher-Taylor experimental setup

Electrocapillary driven two-phase flows in a confined configuration of a classical experiment of Melcher and Taylor are studied. The computed streamlines of the flow of the heavier dielectric liquid (corn oil) qualitatively represent the corresponding experimental image. With the increase of electrocapillary forcing, the flow pattern changes, so that the main circulation localizes near a boundary with a larger electric potential. When a dielectric liquid is replaced by a poorly conducting one, the system becomes nonisothermal owing to the Joule heating. Then the flow is driven also by buoyancy and thermocapillary convection, whose effect becomes noticeably stronger than the electrocapillary one. With the increase of electric conductivity, the electrocapillary effect is further weakened compared to the two others, while the electrocapillary and thermocapillary forces remain comparable at the central part of the interface. The results show that consideration of the two-phase model is mandatory for obtaining correct flow patterns in the lower, heavier fluid. The Lippmann equation, connecting electrically induced surface tension with nonuniform surface electric potential, is numerically verified for both isothermal and nonisothermal formulations and is found to hold in both of them.

Figure 1

Sketch of the problem. All the boundaries are assumed to be no slip and isothermal.

Figure 2

(a) Flow image of the experiment [1] at $V_0=20\mathrm{kV}$. Streamlines of the computed flow (b) with a constant surface Coulomb tension (13); ${\psi }_{\min }=-0.000523{\mathrm{m}}^2/\mathrm{s},{\psi }_{\max }=0.000211{\mathrm{m}}^2/\mathrm{s}$ and (c) with a complete leaky dielectric model (1)–(11a); ${\psi }_{\min }=-0.000\frac{516{\mathrm{m}}^2}{\mathrm{s}},{\psi }_{\max }=0.000213{\mathrm{m}}^2/\mathrm{s}$. Along the interface the flow is directed from the right to the left boundary.

Figure 3

Equipotential lines (a) and distribution of the surface charge density along the surface (b).

Figure 4

Change in streamlines with the increase of applied voltage. Along the interface the flow is directed from the right to the left boundary.

Figure 5

Illustration of validity of the Lippmann equation: comparison of potential dependence of the surface tension and distribution of the surface charge.

Figure 6

Isotherms resulting from Joule heating in the considered geometry, not affected by flow.

Figure 7

Streamlines and isotherms of flows driven by the electrocapillary force only.

Figure 8

Streamlines and isotherms of flows driven by the electrocapillary and thermocapillary forces with neglected buoyancy force.

Figure 9

Streamlines and isotherms of flows driven by all three forces.

Figure 10

Dimensional values of electrocapillary and thermocapillary forces in the different cases considered. Electrocapillary, thermocapillary, and buoyancy forces: comparison of potential dependence of the surface tension and distribution of the surface charge.

Figure 11

Verification of the Lippmann equation for nonisothermal flows in electrolytes driven by electrocapillary, thermocapillary, and buoyancy forces: comparison of potential dependence of the surface tension and distribution of the surface charge.