Transient dynamics and structure of optimal excitations in thermocapillary spreading: Precursor film model

Abstract

Linearized modal stability theory has shown that the thermocapillary spreading of a liquid film on a homogeneous, completely wetting surface can produce a rivulet instability at the advancing front due to formation of a capillary ridge. Mechanisms that drain fluid from the ridge can stabilize the flow against rivulet formation. Numerical predictions from this analysis for the film speed, shape, and most unstable wavelength agree remarkably well with experimental measurements even though the linearized disturbance operator is non-normal, which allows transient growth of perturbations. Our previous studies using a more generalized nonmodal stability analysis for contact lines models describing partially wetting liquids ͑i.e., either boundary slip or van der Waals interactions͒ have shown that the transient amplification is not sufficient to affect the predictions of eigenvalue analysis. In this work we complete examination of the various contact line models by studying the influence of an infinite and flat precursor film, which is the most commonly employed contact line model for completely wetting films. The maximum amplification of arbitrary disturbances and the optimal initial excitations that elicit the maximum growth over a specified time, which quantify the sensitivity of the film to perturbations of different structure, are presented. While the modal results for the three different contact line models are essentially indistinguishable, the transient dynamics and maximum possible amplification differ, which suggests different transient dynamics for completely and partially wetting films. These differences are explained by the structure of the computed optimal excitations, which provides further basis for understanding the agreement between experiment and predictions of conventional modal analysis.

Adaptive rational spectral methods for the linear stability analysis of nonlinear fourth-order problems

“…Note, however, that reasonably accurate results can be obtained with a few hundred grid points for saturations as low as u + = 10 −4 , which is two orders of magnitude smaller than the typical precursor size used in the thin films literature for analogous stability analyses [26,44]. The effect of conditioning on the accuracy of the scheme is shown in Figure 8.…”

“…In addition, the presence of fourth-order derivatives renders the numerical treatment of the model quite challenging, due to the explosive growth of the condition number of the Chebyshev differentiation matrices, and the associated finite precision effects. We discuss the effectiveness and conditioning of the proposed adaptive discretization, and show that it allows the computation of accurate traveling waves and eigenvalues for small values of the initial water saturation/film precursor, outperforming existing approaches for the linear stability analysis of similar equations [18,26,44].…”

“…The model predicts a saturation overshoot at the tips of the fingers, a feature that is believed to be essential for the instability of the infiltration front [70,42,30,71,31]. The mathemat-ical structure of the model is analogous to that describing the flow of thin films and driven contact lines [50,77,49,18,86,52,27,26,75,44,53], and to phase-field models of epitaxial growth of surfaces, binary transitions and solidification [21,22,19,40].…”

“…Chebyshev-Padé approximation is used to approximate the locations of the singularities of the solution in the complex plane. We show that the proposed discretization allows us to compute accurate traveling waves and eigenvalues for very small values of the initial water saturation/film precursor, several orders of magnitude smaller than the values considered previously in analogous stability analyses of thin film flows [18,26,44], using just a few hundred grid points.…”

Nonmodal and nonlinear dynamics of a volatile liquid film flowing over a locally heated surface

“…[53][54][55][56][57] The oscillations then include the ridge, are largest above the heater, and decay as they are convected downstream of the heater as waves. Shown in Fig.…”