Weakly nonlinear analysis of long-wave Marangoni convection in a liquid layer covered by insoluble surfactant

We consider the long-wave Marangoni instability in a heated liquid layer covered by insoluble surfactant. The system of nonlinear equations derived in our previous work is regularized in the limit of strong surface tension. Recent research shows that, without the surfactant, a large-scale oscillatory instability mode exists in the interval of wave numbers $k=O\left({\text{Bi}}^{1/2}\right)$ (the Biot number $\text{Bi}\ll 1)$. Here we study the influence of the surfactant on the Marangoni oscillations. The bifurcation analysis for traveling waves and counterpropagating waves is performed. The types of bifurcation and selected pattern depend on the elasticity number and on the Biot number. Specifically, at small elasticity number, both types of waves are supercritical.

Figure 1

A schematic of the physical problem considered in the paper. An insoluble surfactant is adsorbed at the deformable interface, $Z=H(X,Y,\tau )$. At the bottom of the liquid layer $\left(z=0\right)$ the temperature gradient $-a$ is applied.

Figure 2

(a) Marginal stability curves. The solid lines correspond to monotonic mode, the dashed lines correspond to oscillatory mode. Here $G=1$ and $\beta =10$, 40, and 80 for lines 1, 2, and 3, respectively. (b) Growth rate versus Marangoni number at fixed wave number $K=2$. The solid lines correspond to the real part of the growth rate, ${\mathrm{\Lambda }}_r$, the dashed lines correspond to the imaginary part, ${\mathrm{\Lambda }}_i$. Here $G=1$ and $\beta =40$. (c) Growth rate versus Marangoni number at fixed wave number $K=2.5$. The notation and values of parameters are the same as in panel (b).

Figure 3

(a) Marginal stability curves at $G=1N=0.001,L=0.003$, and $\beta =10$, 40, and 80 for lines 1, 2, and 3, respectively. The notations are the same as in Fig. 2. (b) Growth rate versus Marangoni number at fixed wave number $K=2$ at $G=1,S=1,L=0.003,N=0.001$, and $\beta =40$. The notations are the same as in Fig. 2. (c) Growth rate versus Marangoni number at fixed wave number $K=2.5$. The notation and values of parameters are the same as in panel (b).

Figure 4

(a) Variation of critical Marangoni number of the oscillatory mode. (b) Critical wave number $K_c$ with parameter $\beta$. (c) Example of squared frequency jump, the solid line is for $\beta =22$, the dashed line is for $\beta =23$, and the dotted line is for $\beta =21$. Other parameters are $G=1,\tilde{\mathrm{\Sigma }}=1,L=0.003$, and $N=0.001$.

Figure 5

Near the minimum of the critical Marangoni numbers at different $\beta$. The solid line is for $\beta =22.9$, the dashed curve is for $\beta =23$. Other parameters are $G=1,\tilde{\mathrm{\Sigma }}=1,L=0.003$, and $N=0.001$.

Figure 6

(a) Real part of the coefficient ${\kappa }_1$ versus wave number $K$ at $\beta =10$ (solid line) and $\beta =40$ (blue dashed line). (b) The detailed graphs at $K<2$. The critical wave numbers for $\beta =10$ and $\beta =40$ are marked. Other parameters are $G=1,\tilde{\mathrm{\Sigma }}=1,L=0.003$, and $N=0.001$.

Figure 7

(a) Real part of the coefficients ${\kappa }_2$ (solid line) and ${\kappa }_3$ (dashed line) versus the Biot parameter at the critical wave numbers. (b) Detailed fragment of panel (a) at small $\beta$. Other parameters are $G=1,\tilde{\mathrm{\Sigma }}=1,L=0.003$, and $N=0.001$.

Figure 8

(a) Real part of the coefficients ${\kappa }_2$ (solid line) and ${\kappa }_3$ (dashed line) versus the Biot parameter at the critical wave numbers. (b) Detailed fragment of panel (a) at small $\beta$. Other parameters are $G=1,\tilde{\mathrm{\Sigma }}=1,L=0.003$, and $N=0.1$.

Figure 9

(a) Real part of the coefficients ${\kappa }_2$ (solid line) and ${\kappa }_3$ (dashed line) versus the Biot parameter at the critical wave numbers. (b) Detailed fragment of panel (a) at small $\beta$. Other parameters are $G=1,\tilde{\mathrm{\Sigma }}=1,L=0.003$, and $N=0.5$.