Stability of surfactant-laden liquid film flow over a cylindrical rod

The stability of surfactant-laden liquid film flow over a cylindrical rod is examined in creeping flow limit using standard temporal linear stability analysis. The clean film flow configuration (i.e., in absence of surfactant) is well-known to become unstable owing to Rayleigh-Plateau instability of cylindrical liquid interfaces. Previous studies demonstrated that for a static liquid film (i.e., zero basic flow) coating a rod, the presence of interfacial surfactant decrease the growth of Rayleigh-Plateau instability, but is unable to suppress it completely. Further, the presence of interfacial surfactant is known to introduce an additional mode, referred to as surfactant mode in the present work. To the best of our knowledge, the stability of surfactant mode has not been analyzed in the context of cylindrical film flows. Thus, we reexamined the stability of surfactant-laden cylindrical liquid film flow to analyze the stability behavior of the above said two modes when the basic flow is turned on. The present study reveals that the incorporation of basic flow in stability analysis leads to the complete suppression of Rayleigh-Plateau instability due to the presence of interfacial surfactants as compared to the partial suppression obtained for a stationary liquid film. Three nondimensional parameters appear for this problem: Bond number (denoted as Bo) which characterizes the strength of basic flow, Marangoni number (denoted as Ma) which signifies the presence of surfactant, and ratio of rod radius to film thickness denoted as $S$. In creeping flow limit, the characteristic equation is quadratic with one root belonging to Rayleigh-Plateau mode and the other to surfactant mode. We first carried out an asymptotic analysis to independently capture the eigenvalues corresponding to both the modes in limit of long-wave disturbances. The long-wave results show that the Rayleigh-Plateau instability is completely suppressed on increasing the Marangoni number above a critical value while the surfactant mode always remains stable in low wave-number limit. The continuation of long-wave results to arbitrary wavelength disturbances show that the suppression of Rayleigh-Plateau instability mode still holds, however, the surfactant mode becomes unstable at sufficiently high values of Marangoni number. Further, this surfactant mode instability shifts toward low wave numbers with critical Marangoni number for instability scaling with wave number in a particular fashion. We used this scaling and carried out an asymptotic analysis to capture this instability in low wave-number limit. Depending on $S$ and Bo, we observed the existence of a stable gap in terms of Ma where both the eigen-modes remain stable. Our results indicate that for a given Bond number, the width of stable gap in terms of Ma decreases with decrease in $S$ and the stable gap vanishes when $S$ is sufficiently small. The effect of increasing Bond number (or equivalently, the strength of basic flow) is found to be stabilizing for the film flow configuration.

Figure 1

Surfactant-laden Newtonian liquid film flowing outside a rigid vertical cylinder.

Figure 2

Growth-rate vs. wave-number data for a stationary film ($\mathrm{Bo}=0$). The continuous lines (solid, dashed, dash-dotted, etc.) correspond to present results while symbols (open circle, open square, star, etc.) represent data points obtained using Carroll and Lucassen's growth-rate Eq. (37).

Figure 3

Growth-rate vs. wave-number data for $\mathrm{Bo}=0.25$, and $S=1$.

Figure 4

Growth-rate vs. wave-number data for $\mathrm{Bo}=0.04$.

Figure 5

Neutral stability diagram in Ma vs. $k_1$ plane. “$S$” denotes surfactant mode.

Figure 6

Neutral stability diagram in Ma vs. $k_1$ plane.

Figure 7

Comparison of surfactant mode neutral curve branch in low-wave-number limit with asymptotic result. The dashed lines with symbols (open circle, star, open square, and open triangle) represent the Marangoni number evaluated using Eq. (38).

Figure 8

Neutral stability curves at $S=1$ and with varying Bond number.