On the response of neutrally stable flows to oscillatory forcing with application to liquid sheets

Abstract

Industrial coating processes create thin liquid films with tight thickness tolerances, and thus models that predict the response to inevitable disturbances are essential. The mathematical modeling complexities are reduced through linearization as even small thickness variations in films can render a product unsalable. The signaling problem, considered in this paper, is perhaps the simplest model that incorporates the effects of repetitive (oscillatory) disturbances that are initiated, for example, by room vibrations and pump drives. In prior work, Gordillo and P\'erez (Phys. Fluids 14, 2002) examined the structure of the signaling response for linear operators that admit exponentially growing or damped solutions, i.e., the medium is classified as unstable or stable via classical stability analysis. The signaling problem admits two portions of the solution, the transient behavior due to the start-up of the disturbance and the long-time behavior that is continually forced; the superposition reveals how the forced solution evolves through the passage of a transient. In this paper, we examine signaling for the linear operator examined by King et al. (King et al. 2016, Phys. Rev. Fluids 1(7)) that governs varicose waves in a thin liquid sheet and that can admit solutions having algebraic growth although the underlying medium is classified as being neutrally stable. Long-time asymptotic methods are used to extract critical velocities that partition the response into distinct regions having markedly different location-dependent responses, and the amplitudes of oscillatory responses in these regions are determined. Together, these characterize the magnitude and breadth of the solution response. Results indicate that the signaling response in neutrally stable flows (that admit algebraic growth) is significantly different from that in exponentially unstable systems.