Stability analysis of a thin liquid film on an axially oscillating cylindrical surface in the high-frequency limit

We consider an axisymmetric liquid film on a horizontal cylindrical surface subjected to axial harmonic oscillation in the high-frequency limit. We derive and analyze the nonlinear evolution equation describing the nonlinear dynamics of this physical system in terms of the averaged film thickness. The method used for the derivation of the evolution equation is based on long-wave theory and the separation of the relevant fields into fast and slow components. We carry out the linear stability analysis for a film of a constant thickness which shows that axial forcing of the cylinder may result in either stabilization or destabilization of the axisymmetric flow with respect to the unforced one, depending on the choice of the parameter set. The analysis is extended to the weakly nonlinear stage and it reveals that the system bifurcates subcritically from the equilibrium.

Figure 1

Variation of the normalized dimensionless growth rate ${\lambda }_n=\frac{\lambda }{{\lambda }_{\text{max}}}$, with the normalized wave number $k_n=\frac{k}{k_c}$, as given by Eq. (35) and Eqs. (17, 18, 19) of Goren [9] in the static case, $B=0$. The solid curve represents the long-wave result obtained from Eq. (36), whereas the dotted, dashed, and dot-dashed curves, respectively, correspond to the general results of Goren [9] for $a=0.1,0.2$, and $a=0.3$. As a result of normalization for the wave number and the growth rate, the curves for the two different liquids considered here collapse on a single respective curve depending only on $a$.

Figure 2

Left panel: variation of $S(n_0,\Omega )$ with $\Omega$ for various values of $a=0.1,0.2,0.3,0.4$, and $0.5$, as represented by curves 1–5, respectively. Right panel: Variation of $S(n_0,\Omega )$ with $a$ for various values of $\Omega =1,4,8,9,9.5$, and 10, as represented by curves 1–6, respectively.

Figure 3

Variation of the three separate components contributing to the film evolution by vibration. Curves 1, 2, and 3 correspond, respectively, to the $f_1$ and $f_2$ components in the $J\left(n\right)$ term, and the $D\left(n\right)$ term in Eq. (29a). The combination of these terms represents the vibrational component of the function $L_1\left(n\right)$ in Eq. (31a). The solid and dashed curves correspond to $\Omega =8$ and $\Omega =9.6$, respectively.

Figure 4

Linear stability analysis for a $33\%$ water-glycerin solution film with a thickness of $h_0=0.2$ mm. The dashed curves correspond to the static case $b=0$. Left panel: variation of the cutoff wave number $k_c$ with the parameter $a$ for a fixed forcing amplitude $b=1$ mm ($B=5$). In this case ${\rm Y}=1.6378$. Solid curves 1–8 show, respectively, the cases of $\omega =7100,7000,6950,6900,6800,6000,2000$, and 1000 Hz ($\Omega =9.7931,9.6552,9.5862,9.5172,9.3793,8.2785,2.7586$, and $1.3793$). Right panel: the corresponding growth rate $\lambda$ as a function of the wave number $k$ for $a=0.3$ and the values of $b$ and $\omega$ as presented for the left panel.

Figure 5

Linear stability analysis for a film of $33\%$ water-glycerin solution with a thickness of $h_0=0.2$ mm. The dashed curves correspond to the static case, $b=0$. Left panel: variation of the cutoff wave number $k_c$ with the parameter $a$ for two values of the fixed forcing frequency. Dot-dashed curves 1 and 2 correspond to $\omega =6000$ Hz with $b=0.1$ mm and $0.2$ mm, respectively. Solid curves 1–6 correspond, respectively, to $\omega =7000$ Hz, with the forcing amplitudes $b=1,0.9,0.7,0.5,0.4$, and $0.1$ mm ($B=5,4.5,3.5,2.5,2$, and $0.5,\Omega =9.6552$). Right panel: the corresponding growth rate $\lambda \left(k\right)$ for $a=0.2$. Curves 1–5 display, respectively, the cases of $\omega =7000$ Hz, $b=0.9,0.7,0.5,0.4$, and $0.1$ mm ($B=4.5,3.5,2.5,2$, and $0.5,\Omega =9.6552)$.

Figure 6

Linear stability analysis for a film of $80\%$ water-glycerin solution with a thickness of $h_0=0.2$ mm. In this case, ${\rm Y}=10.9106$. The dashed curves correspond to the case of a static system $b=0$. Solid curves 1–4 show, respectively, the cases of $\omega =16380,16385,16390$, and 16395 Hz ($\Omega =9.4957,9.4986$, $9.5014$, and $9.5043$). Left panel: variation of the cutoff wave number $k_c$ with the parameter $a$ for a fixed forcing amplitude $b=1$ mm $(B=5)$. Right panel: variation of the growth rate $\lambda \left(k\right)$ with the disturbance wave number for a fixed $a=0.3$.

Figure 7

Linear stability analysis for a film of a $80\%$ water-glycerin solution with a thickness of $h_0=0.2$ mm. The dashed curves correspond to the case of a static system $b=0$. Left panel: variation of the cutoff wave number $k_c$ with the parameter $a$ for a fixed forcing frequency $\omega =16390$ Hz $(\Omega =9.5014)$. Solid curves 1–6 show, respectively, the effect of varying forcing amplitude for $b=0.1,0.6,0.9,0.95,0.98$, and 1 mm $(B=0.5,3,4.5,4.75,4.9,5)$. Right panel: variation of the growth rate $\lambda \left(k\right)$ with the disturbance wave number for a fixed $a=0.3$. Curves 1–4 show, respectively, $b=0.1,0.6,0.95$, and 1 mm ($B=0.5,3,4.75$, and 5).