1887
banner image
No data available.
Please log in to see this content.
You have no subscription access to this content.
No metrics data to plot.
The attempt to load metrics for this article has failed.
The attempt to plot a graph for these metrics has failed.
Dynamics of a horizontal thin liquid film in the presence of reactive surfactants
Rent:
Rent this article for
USD
10.1063/1.2775938
/content/aip/journal/pof2/19/11/10.1063/1.2775938
http://aip.metastore.ingenta.com/content/aip/journal/pof2/19/11/10.1063/1.2775938
View: Figures

Figures

Image of FIG. 1.
FIG. 1.

Two-dimensional thin film of thickness on a horizontal solid substrate. denotes the flat film thickness. The free surface of the film is covered by a mixture of two chemical species undergoing a chemical reaction with FitzHugh-Nagumo kinetics. The effective concentrations of the two species are and with the latter acting as a surfactant. Depending on the values of the physical parameters, the flat film may be unstable exhibiting different types of waves on its surface.

Image of FIG. 2.
FIG. 2.

The nullclines of the FitzHugh-Nagumo equations (4). The intersection points define the homogeneous steady states of the chemical system [Eq. (29)]. (a) Single intersection with and corresponding to the case considered in Figs. 7 and 8. (b) Three intersections with and corresponding to the case considered in Fig. 12.

Image of FIG. 3.
FIG. 3.

Case Ia for , , , , , , , and . (a) Dispersion curves for (solid lines), (dashed lines), and (dotted lines). As the modified Marangoni number increases, the range of unstable modes becomes smaller. A bullet on a curve indicates that for this curve the growth rate is complex. Only two growth rates for each are shown; the third one is the complex conjugate of the complex growth rate. (b) Corresponding stability map.

Image of FIG. 4.
FIG. 4.

Case Ib, defined by , , , , , , , and . (a) Dispersion curves for (solid lines), (dashed lines), and (dotted lines). As in Fig. 3, the Marangoni stresses shorten the range of unstable modes. A circle on a curve indicates that for this curve the growth rate is real. (b) Corresponding stability map.

Image of FIG. 5.
FIG. 5.

Case Ib (cf. Fig. 4). (a) Free-surface profile for different values of the modified Marangoni number: , 1.5, and 3. (b) Corresponding streamlines for . The flow induced by the Marangoni stresses gives rise to a stationary pattern for the film surface.

Image of FIG. 6.
FIG. 6.

Case Ib (cf. Fig. 4). Profile of (a) the activator and (b) the inhibitor for parameters as in Fig. 5.

Image of FIG. 7.
FIG. 7.

Case IIa for , , , , , , , and . (a) Dispersion curves for two different values of the modified Marangoni number: (solid lines) and (dashed lines). For all modes are stable (with the mode being neutrally stable) whereas for a range of unstable modes exists. A circle or bullet on a curve indicates real or complex growth rates, respectively. (b) Corresponding stability map.

Image of FIG. 8.
FIG. 8.

Case IIb for , , , , , , , and . (a) Dispersion curves for two different values of the modified Marangoni number: (solid lines) and (dashed lines). For the steady state is linearly stable whereas for it is unstable. A circle or bullet on a curve indicates real or complex growth rates, respectively. (b) Corresponding stability map.

Image of FIG. 9.
FIG. 9.

Case Ib (cf. Fig. 4). Time-dependent computations for modified Marangoni numbers (a) and (b) . In both cases the initial condition is a sinusoidal perturbation of amplitude 0.1 superimposed to the uniform state. The domain size is ( times the fastest growing wavelength obtained in the linear stability analysis; see Fig. 4), the number of spatial grid points is , and the time interval between two consecutive profiles is . For the chemical subsystem evolves into a Turing pattern, whereas for a breather-like pattern is found.

Image of FIG. 10.
FIG. 10.

Case IIa (cf. Fig. 7). Evolution of (a) the free surface and (b) the activator for . The uniform state is disturbed by random noise of maximal amplitude 0.1. The domain size is ( times the fastest growing wavelength obtained in the linear stability analysis; see Fig. 7), the number of spatial grid points is and the time interval between two consecutive curves is .

Image of FIG. 11.
FIG. 11.

Case IIb (cf. Fig. 8). Evolution of (a) the free surface and (b) the activator for . The initial condition is as in Fig. 10. The domain size is [only the interval is displayed], the number of spatial grid points is , and the time interval between two consecutive curves is .

Image of FIG. 12.
FIG. 12.

Case IIIa: The large/small front concentrations are the more stable/less stable states, respectively. , , , , , , , and . (a) Free surface for several values of the modified Marangoni number between and 10. (b) Corresponding profiles of (solid lines) and (dashed lines).

Image of FIG. 13.
FIG. 13.

Case IIIb: The large/small front concentrations are the less stable/more stable states, respectively. , , , , , , , and . (a) Free surface for several values of the modified Marangoni number between and 10. (b) Corresponding profiles of (solid lines) and (dashed lines).

Image of FIG. 14.
FIG. 14.

Case IV: Solitary pulse for the chemical subsystem with , , , , , , , and . (a) Free surface for several values of the modified Marangoni number between and 10. (b) Corresponding profiles of (solid lines) and (dashed lines).

Image of FIG. 15.
FIG. 15.

Case IIIa (cf. Fig. 12). (a) Bifurcation diagram for the speed as a function of the modified Marangoni number . (b) Amplitude of the corresponding free-surface wave as a function of .

Image of FIG. 16.
FIG. 16.

Case IIIb (cf. Fig. 13). (a) Bifurcation diagram for the absolute value of the speed as a function of the modified Marangoni number . (b) Amplitude of the corresponding free-surface wave vs .

Image of FIG. 17.
FIG. 17.

Case IV (cf. Fig. 14). (a) Bifurcation diagram for the speed as a function of the modified Marangoni number . (b) Amplitude of the corresponding free-surface wave vs .

Image of FIG. 18.
FIG. 18.

Bifurcation diagram in the front case for the speed as a function of with . The values of the remaining dimensionless groups for (a) and (b) are given in Figs. 12 and 13, respectively. In contrast to Figs. 15(a) and 16(a), a symmetry exists between (a) and (b).

Image of FIG. 19.
FIG. 19.

Eigenvalues associated with the fixed point corresponding to the base state of the system in case IV (see Fig. 14). The corresponding wave speeds that enter the computation are given in Fig. 17. For a given value of there are eight eigenvalues. For example, for we obtain four real eigenvalues (denoted by dotted lines), three positive ones and one equal to zero, and two complex pairs (denoted by solid lines) with negative real parts.

Image of FIG. 20.
FIG. 20.

Case IIIa (see Fig. 12): Time evolution for . The domain size is , the number of spatial grid points is 1000, and the time interval between two consecutive curves is . (a) Free surface: two positive-hump solitary waves with opposite speeds travel along the surface. (b) The activator: two fronts with opposite speeds propagate in the system.

Image of FIG. 21.
FIG. 21.

Case IIIb (see Fig. 13): Time evolution for . The domain size is , the number of spatial grid points is 1000, and the time interval between two consecutive curves is . (a) Free surface: two negative-hump solitary waves with opposite speeds travel along the surface. (b) The activator: two fronts with opposite speeds propagate in the system.

Image of FIG. 22.
FIG. 22.

Case IV (see Fig. 14): Time evolution for . The domain size is , the number of spatial grid points is 1000, and the time interval between two consecutive curves is . (a) Free surface: two solitary waves with opposite speeds travel along the surface. Note the presence of a slowly decaying residual dip due to the initial condition. (b) The activator: the resting pulse formed at the beginning splits into two counterpropagating solitary pulses.

Loading

Article metrics loading...

/content/aip/journal/pof2/19/11/10.1063/1.2775938
2007-11-09
2014-04-20
Loading

Full text loading...

This is a required field
Please enter a valid email address
752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Dynamics of a horizontal thin liquid film in the presence of reactive surfactants
http://aip.metastore.ingenta.com/content/aip/journal/pof2/19/11/10.1063/1.2775938
10.1063/1.2775938
SEARCH_EXPAND_ITEM