Marangoni instability in a thin film heated from below: Effect of nonmonotonic dependence of surface tension on temperature

We investigate Marangoni instability in a thin liquid film resting on a substrate of low thermal conductivity and separated from the surrounding gas phase by a deformable free surface. Considering a nonmonotonic variation of surface tension with temperature, here we analytically derive the neutral stability curve for the monotonic and oscillatory modes of instability (for both the long-wave and short-wave perturbations) under the framework of linear stability analysis. For the long-wave instability, we derive a set of amplitude equations using the scaling $k\sim {\left(\mathrm{Bi}\right)}^{1/2}$, where $k$ is the wave number and Bi is the Biot number. Through this investigation, we demonstrate that for such a fluid layer upon heating from below, both monotonic and oscillatory instability can appear for a certain range of the dimensionless parameters, viz., Biot number $\left(\mathrm{Bi}\right)$, Galileo number $\left(\mathrm{Ga}\right)$, and inverse capillary number $\left(\mathrm{\Sigma }\right)$. Moreover, we unveil, through this study, the influential role of the above-mentioned parameters on the stability of the system and identify the critical values of these parameters above which instability initiates in the liquid layer.

Figure 1

Schematic of the physical system under consideration with the imposed boundary conditions. The deformable interface is located at $z=h\left(x,y,t\right)$. A constant heat flux provided at the solid substrate yields the temperature gradient –A at the $z=0$ plane.

Figure 2

Neutral stability curves for the long-wave monotonic (solid lines) and oscillatory (dashed lines) modes at $\mathrm{Pr}=1$, $\mathrm{Ga}=10$, and $\mathrm{\Sigma }=1000$. Panel (a) corresponds to $\mathrm{Bi}=0.05$; (b) corresponds to $\mathrm{Bi}=0.5$. The inset in (a) shows the zoomed-in view of the stability curve for the monotonic mode of instability.

Figure 3

Neutral stability curves for the long-wave monotonic (solid and dash-dotted lines) and oscillatory modes (dotted and dashed lines) for different values of Ga at $\mathrm{Pr}=1$, $\mathrm{\Sigma }=1000$, and $\mathrm{Bi}=0.1$. Lines marked by 1 and 2 correspond to $\mathrm{Ga}=1$ and $\mathrm{Ga}=10$, respectively. The inset shows the zoomed-in view of the stability curve at higher wave number.

Figure 4

Neutral stability curves for the long-wave monotonic (solid and dash-dotted lines) and oscillatory modes (dotted and dashed lines) for different $\mathrm{\Sigma }$ at $\mathrm{Pr}=1$, $\mathrm{Ga}=10$, and $\mathrm{Bi}=0.1$. Lines marked by 1 and 2 correspond to $\mathrm{\Sigma }=1000$ and $\mathrm{\Sigma }=10000$, respectively.

Figure 5

Neutral stability curves for the monotonic and oscillatory modes of instability at $\mathrm{Pr}=1$, $\mathrm{Ga}=10$, and $\mathrm{\Sigma }=1000$. Panel (a) crresponds to $\mathrm{Bi}=0.05$; (b) corresponds to $\mathrm{Bi}=0.5$. (a), (b) LM, LO, SM, and SO represents the stability curves for the long-wave monotonic, long-wave oscillatory, short-wave monotonic, and short-wave oscillatory modes, respectively.

Figure 6

Variation of the (a) critical Marangoni number, and (b) the critical wave number with $\mathrm{Bi}$ at $\mathrm{Pr}=1$, $\mathrm{Ga}=10$, and $\mathrm{\Sigma }=1000$. (a), (b) LM, LO, SM, and SO represent the long-wave monotonic, long-wave oscillatory, short-wave monotonic, and short-wave oscillatory modes, respectively. The inset in (a) shows the zoomed-in view at small Biot number.

Figure 7

Variation in the domain of oscillatory instability for (a) $\mathrm{\Sigma }$ vs $\mathrm{Bi}$ at $Pr=1$, $Ga=10$, and (b) $\mathrm{Ga}$ vs $\mathrm{Bi}$ at $Pr=1$, $\mathrm{\Sigma }=1000$. Domains 1 and 2 in (a), (b) correspond to the long-wave oscillatory (LO) and short-wave oscillatory (SO) modes, respectively.

Figure 8

Neutral stability curves for the short-wave monotonic (solid and dash-dotted lines) and oscillatory mode (dotted and dashed lines) for different values of Ga at $\mathrm{Bi}=0.1$, $\mathrm{Pr}=1$, and $\mathrm{\Sigma }=1000$. Lines marked by 1 and 2 correspond to $\mathrm{Ga}=1$ and $\mathrm{Ga}=10$, respectively. The inset shows the zoomed-in view of the neutral stability curve at higher wave number.

Figure 9

(a) Variation of the critical Marangoni number and (b) the critical wave number with Galileo number at $\mathrm{Bi}=0.05$, $\mathrm{P}\mathrm{r}=1$, and $\mathrm{\Sigma }=1000$. (a), (b), LM, LO, SM, and SO represent the long-wave monotonic, long-wave oscillatory, short-wave monotonic, and short-wave oscillatory modes, respectively.

Figure 10

Neutral stability curves for the short-wave monotonic (solid and dash-dotted lines) and oscillatory modes (dotted and dashed lines) for different values of $\mathrm{\Sigma }$ at $\mathrm{Bi}=0.1$, $\mathrm{Pr}=1$, and $\mathrm{Ga}=10$. Lines marked by 1 and 2 correspond to $\mathrm{\Sigma }=1000$ and $\mathrm{\Sigma }=10000$, respectively.

Figure 11

Variation of (a) the critical Marangoni number and (b) the critical wave number with $\mathrm{\Sigma }$ at $\mathrm{Bi}=0.1$, $\mathrm{Pr}=1$, and $\mathrm{Ga}=10$. (a), (b); LM, LO, SM, and SO represent the long-wave monotonic, long-wave oscillatory, short-wave monotonic, and short-wave oscillatory modes, respectively.

Figure 12

Neutral stability curves for an aqueous 1-butanol fluid layer of thickness 0.05 mm, $\mathrm{Bi}=0.025$, $\mathrm{Ga}=6$, and $\mathrm{\Sigma }=5000$. The solid line represents the monotonic mode, and dashed one corresponds to the oscillatory mode of instability.