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Annual Review of Fluid Mechanics

Volume 51, 2019

Film Flows in the Presence of Electric Fields

Abstract

The presence of electric fields in immiscible multifluid flows induces Maxwell stresses at sharp interfaces that can produce electrohydrodynamic phenomena of practical importance. Electric fields can be stabilizing or destabilizing depending on their strength and orientation. In microfluidics, fields can be used to drive systems out of equilibrium to produce hierarchical patterning, mixing, and phase separation. We describe nonlinear theories of electrohydrodynamic instabilities in immiscible multilayer flows in several geometries, including flows over or inside planar or topographically structured substrates and channels and flows in cylinders and cylindrical annuli. Matched asymptotic techniques are developed for two- and three-dimensional flows, and reduced-dimension nonlinear models are derived and studied. When all regions are slender, electrostatic extensions to lubrication or shallow-wave theories are derived. In the presence of nonslender layers, nonlocal terms emerge naturally to modify the evolution equations. Analysis and computations provide a plethora of dynamics, including nonlinear traveling waves, spatiotemporal chaos, and singularity formation. Direct numerical simulations are used to evaluate the models and go beyond their range of validity to quantify phenomena such as electric field–induced directed patterning, suppression of Rayleigh–Taylor instabilities, and electrostatically induced pumping in microchannels. Comparisons of theory and simulations with available experiments are included throughout.

Keyword(s): electrohydrodynamic instabilities, interfacial phenomena, long-wave theories, thin films
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2019-01-05
2025-04-25
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Film Flows in the Presence of Electric Fields
Annual Review of Fluid Mechanics 51, 155 (2019); https://doi.org/10.1146/annurev-fluid-122316-044531
/content/journals/10.1146/annurev-fluid-122316-044531
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Supplementary Data

  • Supplemental Video 1: Experiments on the effect of electric fields on pressure-driven two-fluid channel flow. Top fluid - glycerine; bottom fluid - corn oil. Channel dimensions: width 1.5mm, depth 0.25mm. The electrode has length 38mm, i.e. about 25 channel widths long. Before field is turned on, flow is stable. Instability creates a mono disperse array of separated drops. The separation is viewed (i) under the electrode, (ii) beyond the electrode where the field is absent and Bretherton drops form. An increase in the voltage causes a decrease in the resulting droplet volumes.

    Supplemental Video 2: Direct numerical simulation of lithographically induced self assembly (LISA) for dielectric fluids. Dimensionless system based on a channel with height 350nm. Top to bottom fluid properties: density ratio 1/5, viscosity ratio 1/2, permittivity ratio 1/5. Top electrode dimensionless voltage V̅ = 7.5, bottom electrode voltage zero. (Definition of V̅ - see Cimpeanu et al. (2014), Phys. Fluids 26:022105.) The instability selects the natural most unstable mode and pillars are formed guided by this scale.

    Supplemental Video 3: Direct numerical simulation of lithographically induced self construction (LISC) for dielectric fluids. Dimensionless system based on a maximum channel with height 350nm. Top electrode structured topographically and held at potential V̅ = 7.5. Bottom electrode is grounded. Top to bottom fluid properties: density ratio 1/5, viscosity ratio 1/2, permittivity ratio 1/5. Non-uniformity in the electric field generates interfacial deformations that seek the topographical protrusions.

    Supplemental Video 4: Direct numerical simulation showing stabilization of Rayleigh-Taylor instability using a horizontal electric field - see Cimpeanu et al. (2014), Phys. Fluids 26:022105. Top fluid is 5 times more dense than the bottom one, and gravity is acting. Panels from left to right have different strengths of the applied horizontal electric field V̅ = 0, 1, 2, 3, 4. As the field increases the instability is reduced and for the rightmost panel having V̅ = 4 we obtain complete stabilization.

    Supplemental Video 5: Simulation of the 2D electrically modified Kuramoto-Sivashinsky equation. Subcritical β = 0.01, γ = 2, domain 13 x 13. Time-periodic solution comprised of a long-lived 1D state (∂x ≠ 0, ∂y = 0), that looses stability to spanwise disturbances before returning to the 1D state.

    Supplemental Video 6: Simulation of the 2D electrically modified Kuramoto-Sivashinsky equation. Subcritical β = 0.01, γ = 2, domain 18.85 x 18.85. Quasi-periodic in time homoclinic behavior connecting long-lived rivulet like structures along the flow direction (roughly ∂x = 0, ∂y ≠ 0) with 2D cellular structures.

    Supplemental Video 7: Simulation of the 2D electrically modified Kuramoto-Sivashinsky equation. Subcritical β = 0.5, γ = 2, domain 7.2 x 7.2. Time periodic 2D solution visiting the underlying 1D attractor (yet remaining 2D) over part of the cycle.

    Supplemental Video 8: Simulation of the 2D electrically modified Kuramoto-Sivashinsky equation. Subcritical β = 0.5, γ = 2, domain 19 x 19. Doubly periodic cellular structures that weakly oscillate in time quasiperiodically. In a stationary frame this is a weakly modulated traveling wave.

    Supplemental Video 9: Simulation of the 2D electrically modified Kuramoto-Sivashinsky equation. Supercritical β = 2, domain 30 x 30. L = 30 supports chaotic solutions for the 1D KS. Start with no field γ = 0 for time interval I1 : 0 ≤ t < 20, switch it on to γ = 1 for I2 : 20 ≤ t < 40, then increase it to γ = 2 for I3 : 40 ≤ t ≤ 60. In I1 quasi-1D chaotic solutions; in I2 spanwise modulations but the underlying 1D structures strongly present but of larger amplitudes; in I3 the field increases the amplitude further and generates 3D cellular structures that move and merge in the plane - the 1D scaffold is lost.

    Supplemental Video 10: Direct numerical simulation of electrostatically induced pumping using traveling wave voltage condition on the lower wall (see equation 39). Constant voltage is applied initially to attract the interface to the walls, and at t ≈ 0.25 the computation switches to the traveling voltage. Traveling voltage characteristics: Background value C = 3; hump amplitude A = 1; travel speed Ur = 1.

    Supplemental Video 11: Direct numerical simulation of electrostatically induced pumping using traveling wave voltage condition on the lower wall (see equation 39). Constant voltage is applied initially to attract the interface to the walls, and at t ≈ 0.25 the computation switches to the traveling voltage. Traveling voltage characteristics: Background value C = 3; hump amplitude A = 1.5; travel speed Ur = 1.

    Supplemental Video 12: Direct numerical simulation of electrostatically induced pumping using traveling wave voltage condition on the lower wall (see equation 39). Constant voltage is applied initially to attract the interface to the walls, and at t ≈ 0.25 the computation switches to the traveling voltage. Traveling voltage characteristics: Background value C = 3; hump amplitude A = 1; travel speed Ur = 2.

    Supplemental Video 13: Simulation of a two-fluid electrified pressure-driven flow in a channel with undulating walls. The lower wall is given by z = -Zc + db cos(4x) and held at voltage V̅. The upper wall is flat and grounded. Wall parameters: Zc= 0.9, db = 0.1 (relatively small amplitude), applied voltage is Vb = 3; pillar-like traveling wave structures emerge that are subharmonic with respect to the boundary, span the channel and slide over the walls.

    Supplemental Video 14: Simulation of a two-fluid electrified pressure-driven flow in a channel with undulating walls. The lower wall is given by z = -Zc + db cos(4x) and held at voltage V̅. The upper wall is flat and grounded. Wall parameters: Zc= 0.9, db = 0.1, voltage Vb = 5.4 (to the left of a pitchfork bifurcation found from the Floquet analysis). The flow is initially nonuniform and quasi-static before instability sets in to produce channel-spanning traveling wave structures.

    Supplemental Video 15: Simulation of a two-fluid electrified pressure-driven flow in a channel with undulating walls. The lower wall is given by z = -Zc + db cos(4x) and held at voltage V̅. The upper wall is flat and grounded. Wall parameters: Zc= 0.9, db = 0.1, voltage Vb =5.6 (just to the right of a pitchfork bifurcation found from the Floquet analysis). The flow is initially nonuniform and quasi-static before instability sets in; now it produces a traveling wave that approaches the wall but then peels away from it in a time-periodic fashion.

    Supplemental Video 16: Simulation of a two-fluid electrified pressure-driven flow in a channel with undulating walls. The lower wall is given by z = -Zc + db cos(4x) and held at voltage V̅. The upper wall is flat and grounded. Wall parameters: Zc= 0.9, db= 0.1, relatively large voltage Vb = 8.5. The nonlinear evolution begins with a time-periodic standing wave along the channel with its troughs trapped in the topography. This state looses stability to pillar-like traveling wave structures as found previously.

  • Article Type: Review Article

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