Schematic diagram of a thin liquid film flowing over a heater. The Marangoni stress at the upstream edge of the heater opposes the gravitational flow, which leads to the formation of a capillary ridge.
Base profiles for for different values of and . The curves with symbols are the corresponding temperature profiles at the interface.
(a) Leading eigenvalues of for , , and . (b) Real part of the leading eigenvalues of for , , and .
Leading eigenfunctions for , , and .
Maximum transient amplification of spanwise perturbations to the film with and (a) and and (b) and .
(a) Maximal nonmodal amplification, , vs and (b) vs for the linearly stable film with , , and .
Optimal perturbations for the unstable base state with , , and . (a) Optimal initial perturbation and (b) corresponding evolved state after time for a perturbation with wave number . (c) and (d) for .
(a) Optimal initial perturbation and (b) corresponding evolved state after time for a perturbation with wave number applied to the linearly stable base state with , , and .
Plot of the pseudospectra given by for the linearly stable film with , , , and (a) (least stable eigenvalue) and (b) (repeated eigenvalue). The abscissa is , and the ordinate is . Contours are plotted for [the contour is not visible in (a)]. The symbols are the leading eigenvalues of .
(a) Maximum instantaneous growth rate vs for , , and several . (b) Corresponding eigenvectors for , , and . The curve with symbols is .
(a) Nonlinear amplification ratio of perturbations to the linearly unstable film with , , and . (b) Contour plot of of the nonlinear film evolution after rivulet formation. Contours are shown for , 0.75, 1.25, 1.5, 2.0, 2.5, and 3.0. (c) Nonlinear amplification ratio for the linearly stable film with , , and . (d) Magnitude of perturbation that destabilizes the linearly stable film with , , and .
(a) Oscillations above the heater after a perturbation is applied to the steady film profile at . The dashed curves indicate the maximum and minimum values of the film thickness after the oscillations evolve into a time-periodic profile. (b) Profiles at intervals of after the film reaches the time-periodic state. The dark curve is the steady base state. (c) Phase space plots at . (d) Dimensionless heat transfer coefficient after the time-periodic state is reached relative to that for the steady profile at intervals of .
Asymptotic exponential growth rate of transverse perturbations to the time-periodic film.
Definition and physical interpretation of dimensionless groups.
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