Electrostatic control of the Navier-Stokes equations for thin films

A robust control scheme is derived and tested for the Navier-Stokes equations for two-dimensional multiphase flow of a thin film underneath an inclined solid surface. Control is exerted via the use of an electrode parallel to the substrate, which induces an electric field in the gas phase, and a resultant Maxwell stress at the liquid-gas interface. The imposed potential at the second electrode is derived using a model predictive control loop, together with optimal control of a high-fidelity reduced-dimensional model. In this implementation the interfacial shape of the fluid is successfully controlled; however, the algorithm is sufficiently general to control any other quantity of interest.

Figure 1

Geometry of the hanging liquid film described by interfacial height $h(x,t)$, with the multifluid system confined between two electrodes placed at $y=0$ and $y=d$, at inclination angle $\alpha$. All quantities are dimensionless.

Figure 2

(a) Iterative loop used for OC [forms MPC controller in (b) for control of DNS]. (b) Loop used for MPC.

Figure 3

Regulation control of the RDM towards (15). (a) Interface height $h$ over time (highlighted: interface at $t=0.4t_{\mathrm{f}}$); (b) $ln\left|\right|H-H_{\mathrm{reg}}{\left|\right|}_2$ (solid line; left axis) and $ln\left|\right|{\varphi }_d{\left|\right|}_2$ (dashed line; right axis); vertical dotted line: $t=0.4t_{\mathrm{f}}$; (c) voltage potential ${\varphi }_d$ over time.

Figure 4

Regulation MPC control of DNS towards (a), (d), (g) a uniform state ($e_{\mathrm{crit}}=1.5\times {10}^{-5/2}$); (b), (e), (h) a smoothed top-hat shape ($e_{\mathrm{crit}}=1.5\times {10}^{-2}$); and (c), (f), (i) a “W” -like shape ($e_{\mathrm{crit}}=1.5\times {10}^{-2}$). (a), (b), (c) DNS interfaces over time; highlighted time $t=0.4t_{\mathrm{f}}$. (d), (e), (f) DNS interface (solid line) and target state (dashed) at $t=0.4t_{\mathrm{f}}$. (g), (h), (i) $ln\left|\right|H-H_{\mathrm{reg}}{\left|\right|}_2$ over time; dashed line: RDM; solid line: DNS; vertical dotted line: $t=0.4t_f$. Filled symbols indicate control recomputations.