Theory of rotating electrohydrodynamic flows in a liquid film

The mathematical model of rotating electrohydrodynamic flows in a thin suspended liquid film is proposed and studied. The flows are driven by the given difference of potentials in one direction and constant external electric field ${\mathit{E}}_{\text{out}}$ in another direction in the plane of a film. To derive the model, we employ the spatial averaging over the normal coordinate to a film that leads to the average Reynolds stress that is proportional to ${\left|{\mathit{E}}_{\text{out}}\right|}^3$. This stress generates tangential velocity in the vicinity of the edges of a film that, in turn, causes the rotational motion of a liquid. The proposed model is used to explain the experimental observations of the liquid film motor.

Figure 1

A sketch of thin film.

Figure 2

The physical mechanism of rotation. The capacitor field creates the opposite electric charges in the fluid near the interface boundaries $y=0$, $Y$. Then the action of electric field related to electrodes (applied in $x$ direction) generates the rotating flow.

Figure 3

The function $G\left(x;X,Y\right)$ for $X=1$ and the different values of $Y$: (1) $Y=1$, (2) $Y=0.5$, (3) $Y=0.2$, and (4) $Y=2$.

Figure 4

The sketch of a rotating flow in the film.

Figure 5

The isolines of the potential $\overline{\phi }\left(x,y\right)$ (left) and $\Vert \overline{\psi }\left(\cdot ,t\right)\Vert$.

Figure 6

The streamlines of $\overline{\psi }\left(x,y,t\right)$ for $t=10$ $\left(\approx 0.77\text{s}\right)$ and $t=30$ $\left(\approx 2.31\text{s}\right)$.

Figure 7

The isolines of $\overline{u}\left(x,y,t\right)$ (left) and $\overline{v}\left(x,y,t\right)$ at $t=30$.

Figure 8

The isolines of $v_r\left(x,y,t\right)$ (left) and $v_{\theta }\left(x,y,t\right)$ at $t=30$.

Figure 9

The isolines of $\omega \left(x,y,t\right)$ and $\omega \left(r,{\theta }_m,t\right)$ for ${\theta }_m=m\pi /4$, $m=0,1,2,3,4$, and $t=30$.

Figure 10

The isolines of $\Omega \left(x,y,t\right)$ and $\Omega \left(r,{\theta }_m,t\right)$ for ${\theta }_m=m\pi /4$, $m=0,1,2,3,4$, and $t=30$.

Figure 11

The streamlines of $\overline{\psi }\left(x,y,t\right)$ for $X=1$, $Y=0.5$ at $t=7$ $\left(\approx 0.539\text{s}\right)$ and $t=20$ $\left(\approx 1.54\text{s}\right)$.

Figure 12

The streamlines for $\overline{\psi }\left(x,y,t\right)$ at $t=10$ $\left(\approx 0.77\text{s}\right)$ and $t=30$ $\left(\approx 2.31\text{s}\right)$.

Figure 13

The isolines of the potential $\overline{\phi }\left(x,y\right)$ (left) and the streamlines for $\overline{\psi }\left(x,y,t\right)$ at $t=200\approx 15.4\text{s}$ (right).

Figure 14

The streamlines for $\overline{\psi }\left(x,y,t\right)$ at $t=10$ $\left(\approx 0.77\text{s}\right)$ and $t=30$ $\left(\approx 2.31\text{s}\right)$.

Figure 15

The isolines of $\Omega \left(x,y,t\right)$ and $\Omega \left(r,{\theta }_m,t\right)$ for ${\theta }_m=m\pi /4$, $m=0,1,2,3,4$, and $t=30$.

Figure 16

The isolines of the concentration $c\left(x,y,t\right)$ at $t=300$, 500, 700, and 1500.

Figure 17

Concentration $c\left(x_n,y_n,t\right)$, $n=1,2$.