Falling film on a flexible wall in the presence of insoluble surfactant

Abstract

This work investigates the effect of an insoluble surfactant on the gravity-driven flow of a liquid film down a vertical flexible wall. The paper builds upon previous work [Matar et al., Phys Rev E 76(5):056301, 2007; Sisoev et al., Chem Eng Sci 65(2):950–961, 2010] to include the Marangoni effect attributable to the gradient of surfactant concentration on a free surface. Here we employ an integral method to derive a set of asymptotic evolution equations valid for a moderate flow rate, based on a long-wave approximation. A normal-mode approach is used to examine the linear stability of the system. Similar to the work presented by Matar et al., the results show that a flexible wall with weak damping acts to stabilize flow, while wall tension plays an unstable role. The insoluble surfactant, which acts to stabilize film flow, can reduce the effects of wall flexibility (wall damping and tension) on flow linear stability. The nonlinear evolution equations for the system are solved numerically for both a given initial perturbation wave packet and a periodic perturbation at the inlet boundary. The equations are mainly concerned with the evolution of the flow stability and wave interaction processes, during which solitary-like waveforms are observed. When wall damping is weak, it tends to deplete the ripples preceding the solitary-like humps. However, as wall damping increases in strength, the ripples intensify; a similar phenomenon is observed with an increase in wall tension. The surfactant, which reduces the amplitude and traveling speed of the solitary-like waveforms, acts to distinctly weaken the dispersion of the interfacial wave.

Appendix

Equation (32) is substituted into Eqs. (27), (28), (31) and (30) and linearized in the limit $({\psi _\zeta },{\psi _h},{\psi _Q},{\psi _\varGamma }) \rightarrow 0$. The linear stability equations for the system can be expressed as follows:

$$\begin{aligned} \omega {\psi _h} = - \mathrm{i}k{\psi _Q}, \end{aligned}$$

(36)

$$\begin{aligned} \omega \varTheta _{\varepsilon } {\psi _\zeta } = - \frac{{\left( {1 + T_w } \right) {k^2}}}{{5\delta }}{\psi _\zeta } - \frac{{{k^2}}}{{5\delta }}{\psi _h}, \end{aligned}$$

(37)

$$\begin{aligned} \omega {\psi _Q}&= \left( {\frac{3}{{5\delta }} - \frac{{\mathrm{i}{k^3}}}{{5\delta }} + \frac{{6\mathrm{i}k}}{5}} \right) {\psi _h} - \frac{{\mathrm{i}{k^3}}}{{5\delta }}{\psi _\zeta } - \left( {\frac{1}{{5\delta }} + \frac{{12\mathrm{i}k}}{5}} \right) {\psi _Q} - \frac{{{M_\varepsilon }k}}{{20}}\left( {2k + \frac{{5\mathrm{i}}}{\delta }} \right) {\psi _\varGamma }, \end{aligned}$$

(38)

$$\begin{aligned} \omega {\psi _\varGamma } = \frac{{3\mathrm{i}k}}{2}{\psi _h} - \frac{{3\mathrm{i}k}}{2}{\psi _Q} - \left( {\frac{{3\mathrm{i}k}}{2} + \frac{{5{M_\varepsilon }{k^2}}}{8}} \right) {\psi _\varGamma }. \end{aligned}$$

(39)

A generalized eigenvalue system is obtained, which is composed of Eqs. (36)–(39). It can be solved using the software package MATLAB based on the QZ algorithm. The growth rate of perturbation is denoted by ${\omega _r} = \mathrm{Re} \left( \omega \right)$. The system would be linearly unstable with a positive value of ${\omega _r}$, and would be stable with a negative ${\omega _r}$.