Thermocapillary and electrohydrodynamic effects on the stability of dynamic contact lines

Motivated by the need to understand how external fields influence the stability of dynamic contact lines, the linear stability of gravity-driven spreading of a thin liquid film in the presence of electric and temperature fields is studied. The film is confined from below by a flat substrate and from above by an air gap and another flat substrate. An electrostatic potential difference or temperature difference can be applied between the two substrates and the liquid is taken to be a perfect dielectric whose surface tension decreases linearly with temperature. Traveling-wave solutions are found for the film profile, and both electric and temperature fields influence the height of the capillary ridge of liquid that forms near the advancing contact line. The linear stability analysis shows that electric fields destabilize the film front to transverse perturbations and that temperature fields can either stabilize or destabilize the front, depending on the direction of the temperature gradient. An energy analysis reveals that the electric field in the capillary ridge is most responsible for the enhancement of the perturbation growth. For the case of temperature fields, the perturbed temperature gradients are the dominant mechanism through which the perturbation in film height is affected.

Figure 1

Schematic of problem geometry. The $x$ and $z$ axes are shown, while the $y$ axis points into the page. The electrostatic potential and temperature in each layer are denoted by ${\psi }_i$ and ${\theta }_i$, respectively.

Figure 2

(a) Effect of electric field on the traveling wave. The values of other parameters are $\varepsilon =2.5$ and $\text{Ma}=0$. (b) Electrostatic pressure overlaid with the traveling-wave profile for $\text{Co}=0.2$.

Figure 3

Traveling-wave profiles for various values of Ma. The values of the other parameters are $\kappa =5$ and $\text{Co}=0$.

Figure 4

(a) Dispersion relations showing electrohydrodynamic effects on front stability at various values of Co. The values of the other parameters are $\varepsilon =2.5$ and $\text{Ma}=0$. (b) Dispersion relations showing thermocapillary effects on front stability at various values of Ma. The values of the other parameters are $\kappa =5$ and $\text{Co}=0$.

Figure 5

Contributions to the growth rate from each of the first seven terms of operator $L$: (a) terms with no electric or temperature field, (b) terms in the presence of only an electric field ($\text{Co}=0.2$), and (c) terms in the presence of only a temperature field ($\text{Ma}=0.1$). The values of the other parameters are $\varepsilon =2.5$ and $\kappa =5$.

Figure 6

(a) Difference in growth rate between each standard term with $\text{Co}=0.2$ and $\text{Co}=0$ with $\varepsilon =2.5$. (b) Difference in growth rate between each standard term with $\text{Ma}=0.1$ and $\text{Ma}=0$ with $\kappa =5$.

Figure 7

Energy analysis results for (a) $\text{Co}=0.2$ and $\varepsilon =2.5$ ($\text{Ma}=0$) and (b) $\text{Ma}=0.1$ and $\kappa =5$ ($\text{Co}=0$).

Figure 8

Eigenfunctions of the operator $L(q=0.5)$ overlaid with base states for (a) $\text{Co}=0$ and $\text{Ma}=0$, (b) $\text{Co}=0.2$ and $\varepsilon =2.5$ ($\text{Ma}=0$), and (c) $\text{Ma}=0.1$ and $\kappa =5$ ($\text{Co}=0$).

Figure 9

Dispersion relations showing thermocapillary stabilization and electrohydrodynamic destabilization for various values of Ma. The parameter values are $\varepsilon =2.5$, $\kappa =5$, and (a) $\text{Co}=0.05$ and (b) $\text{Co}=0.1$.