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Transient dynamics and structure of optimal excitations in thermocapillary spreading: Precursor film model
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10.1063/1.2345372
/content/aip/journal/pof2/18/9/10.1063/1.2345372
http://aip.metastore.ingenta.com/content/aip/journal/pof2/18/9/10.1063/1.2345372

Figures

Image of FIG. 1.
FIG. 1.

Optical micrograph reproduced from Fig. 2 of Ref. 23 showing the evolution of a fingering instability at the advancing front of a silicon oil film on a silicon wafer. The applied thermal stress was . The instability wavelength, which remains constant in time, was approximately . The time corresponding to each image is (a) , (b) , (c) , and (d) .

Image of FIG. 2.
FIG. 2.

Schematic diagram of a liquid film climbing a vertical substrate under the action of a constant shear stress, . Thicker climbing films are also subject to gravitational drainage represented by the opposing acceleration, .

Image of FIG. 3.
FIG. 3.

Numerical solution of the dimensionless, steady state profile, . Films driven strictly by Marangoni stresses with precursor film thicknesses and develop pronounced capillary ridges. The film subject to gravitational drainage with and has a monotonically decreasing front.

Image of FIG. 4.
FIG. 4.

Dispersion curves, , from eigenvalue analysis corresponding to base state profiles shown in Fig. 3. Thinner film exhibits unstable flow for . The inset contains results for the thicker film (without a capillary ridge), which exhibits stable flow for all disturbance wave numbers .

Image of FIG. 5.
FIG. 5.

Maximum possible amplification of disturbances within a time interval . (a) Thin film driven strictly by thermocapillary stresses for and . (b) Thicker film subject to gravitational drainage where and .

Image of FIG. 6.
FIG. 6.

(a) Normalized optimal initial excitation, , and (b) the evolved state after time , , for a disturbance of wave number, , applied to the base state with shown in Fig. 3. The eigenfunction found from linear stability theory is plotted alongside the curves in (b).

Image of FIG. 7.
FIG. 7.

(a) Normalized optimal initial excitation, , and (b) the evolved state after time , , for a disturbance of wave number, , applied to the base state with shown in Fig. 3. The eigenfunction found from linear stability theory is plotted alongside the curves in (b).

Image of FIG. 8.
FIG. 8.

(a) Normalized optimal initial excitation, , and (b) the evolved state after time , , for a disturbance of wave number, , applied to the base state with and shown in Fig. 3. The eigenfunction found from linear stability theory is plotted alongside the curves in (b).

Image of FIG. 9.
FIG. 9.

Comparison of the optimal excitations, , from the slip and precursor film models for the unstable wavenumber at times and . The parameters are and for the precursor film model. The calculations for the slip model (Ref. 10) use a slip coefficient of 0.1 and a contact slope of 0.1.

Image of FIG. 10.
FIG. 10.

Amplification of vs for a thin film and . The transient amplification ceases as soon as the contact line passes over the disturbance. Similar curves result from using a short length of precursor film to compute , which is equivalent to limiting the maximum extent of the initial disturbance.

Tables

Generic image for table
Table I.

Parameter values used to estimate the characteristic shear time, , and the observation time, , for fingering onset or stable flow. The last column represents the number of shear times that have elapsed by finger onset (in unstable flows) or during the observation period (for stable flows). Remaining variables are described in the text.

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/content/aip/journal/pof2/18/9/10.1063/1.2345372
2006-09-05
2014-04-20
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Transient dynamics and structure of optimal excitations in thermocapillary spreading: Precursor film model
http://aip.metastore.ingenta.com/content/aip/journal/pof2/18/9/10.1063/1.2345372
10.1063/1.2345372
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