Contact line instability and pattern selection in thermally driven liquid films

Liquids spreading over a solid substrate under the action of various forcesare known to exhibit a long wavelength contact line instability. We use anexample of thermally driven spreading on a horizontal surface to study how thestability of the flow can be altered, or patterns selected, using feedbackcontrol. We show that thermal perturbations of certain spatial structureimposed behind the contact line and proportional to the deviation of thecontact line from its mean position can completely suppress the instability.Due to the presence of mean flow and a spatially nonuniform nature of spreadingliquid films the dynamics of disturbances is governed by a nonnormal evolutionoperator, opening up a possibility of transient amplification and nonlinearinstabilities. We show that in the case of thermal driving the nonnormality canbe significant, especially for small wavenumber disturbances, and trace theorigin of transient amplification to a close alignment of a large group ofeigenfunctions of the evolution operator. However, for values of noise likelyto occur in experiments we find that the transient amplification is notsufficiently strong to either change the predictions of the linear stabilityanalysis or invalidate the proposed control approach.Comment: 13 pages, 14 figure