^{1}, P. M. J. Trevelyan

^{1,a)}, U. Thiele

^{2,b)}and S. Kalliadasis

^{3}

### Abstract

We investigate the interplay between a stable horizontal thin liquid film on a solid substrate and an excitable or bistable reactive mixture on its free surface. Their coupling is twofold. On the one hand, flow in the film transports the reacting surfactants convectively. On the other hand, gradients in the surfactant concentration exert Marangoni stresses on the free surface of the film. A reduced model is derived based on the long-wave approximation. We analyze the linear stability of the coupled system as well as the nonlinear behavior, including the propagation of solitary waves, fronts, and pulses. We show, for instance, that the coupling of thin filmhydrodynamics and surfactant chemistry can either stabilize instabilities occurring in the pure chemical system, or in a regime where the pure hydrodynamic and chemical subsystems are both stable, the coupling can induce instabilities.

We acknowledge financial support from the Engineering and Physical Sciences Research Council of the UK (EPSRC) through Grant Nos. GR/S7912 and GR/S01023. S.K. acknowledges financial support from the EPSRC through an Advanced Fellowship, Grant No. GR/S49520. U.T. acknowledges support by the European Union and Deutsche Forschungsgemeinschaft through Grant Nos. MRTN-CT-2004-005728 and SFB 486 B13, respectively.

I. INTRODUCTION

II. PROBLEM DEFINITION AND GOVERNING EQUATIONS

A. Hydrodynamics

B. The reaction-diffusion system

C. The surface transport equations

III. SCALINGS AND DIMENSIONLESS EQUATIONS

IV. LONG-WAVE EXPANSION

V. ANALYSIS OF THE UNIFORM STATE

VI. FRONT- AND PULSE-DRIVEN HYDRODYNAMIC INSTABILITIES

VII. CONCLUSION

### Key Topics

- Hydrodynamics
- 32.0
- Surfactants
- 29.0
- Free surface
- 28.0
- Hydrodynamic waves
- 25.0
- Liquid thin films
- 23.0

## Figures

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

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

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.

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.

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.

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.

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.

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.

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 .

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 .

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 .

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 .

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).

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).

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).

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).

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).

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).

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 .

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 .

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 .

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 .

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 .

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 .

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).

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).

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.

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.

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.

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.

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.

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.

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.

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.

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