# Fingering Instability of a Gravity-Driven Thin Film Flowing Down a Vertical Tube with Wall Slippage
## 3 Linear Stability Analysis
From the curves, we can find that with a larger slippery length, a smaller unstable range is determined as the \(q_{c}\) is smaller. For \(\beta_{z}=\beta_{\theta}=0\), the range is \([0,0.32]\) and for \(\beta_{z}=\beta_{\theta}=0.1\), the range is \([0,0.48]\). Figure 8: Effect of slippery length on the growth rate of the perturbation (\(\beta_{z}=\beta_{\theta}\)). Figure 6: Effect of radius of the vertical tube on the growth rate of the perturbation. \[\sigma(q)\hat{H}\ =\ -\ L\hat{H} \tag{39}\]0 Effect of thickness of the precursor layer on the growth rate of the perturbation. As introduced above, the slippery lengths are often different in the azimuthal and axial directions, as the properties of the substrate material are anisotropic, such as fibers and composite materials. Figures 9 and 10 display the dispersion relations for different slippery lengths while \(\beta_{z}\
eq\beta_{\theta}\) is taken into consideration. In Figure 9, the slippery effect in only one direction is considered while in Figure 10, the slippery effect in two directions is considered. From the two figures, the change of the growth rate displays similarly with a difference in the values. It is notable that the two slippery lengths \(\beta_{\theta}\) and \(\beta_{z}\) play different roles in the flow stability. From Figures 9a and 10a, a larger axial slippery length corresponds to a smaller growth rate of perturbation. Therefore, the surface instability is suppressed by the axial slippery length. In addition, the change in the growth rate is much more interesting when \(\beta_{\theta}\) increases. It can be seen that the growth rate changes slightly in the range \([0,q_{t}]\), where \(q_{t}\) is a turning point. If \(q<q_{t}\), the growth rate increases with the increase of azimuthal slippery length, while \(q>q_{t}\), the growth rate decreases with the increase of azimuthal slippery length. In other words, if the wavenumber of the perturbation is less than \(q_{t}\), the azimuthal slippery effect slightly enhances the surface instability, while the wavenumber of the perturbation exceeds \(q_{t}\), the azimuthal slippery effect suppresses the surface instability. Besides, for the two cases, both the peak wavenumber and the range of \(\sigma>0\) decreases with the increase of slippery length. wavenumber \(q\). Using the wavenumber, we could calculate the most unstable mode of wavelength \(\lambda=2\pi/q\) in the azimuthal direction. Here, a similar initial condition used in 2D simulation is chosen, i.e., a smooth surface consisting of two flat regions. The two regions are connected by a tanh function at the initial front position \(z_{0}\). Far behind the contact line, the thickness is set to \(h(\theta,0,t)=1\) and the thickness is set to \(h(\theta,z_{max},t)=b\) far front of the contact line. A single mode perturbation is applied at the contact line [17], impelling the contact line to convex slightly into the streamwise direction, given as:  
\[\sigma(q)\hat{H}\ =\ -\ L\hat{H} \tag{39}\]1  
where \(A_{0}\) is the amplitude of the small perturbation. Figure 11 shows the single-mode perturbed thin film flows for different \(\beta_{z}\). The time profiles for \(t=200\) are shown in Figure 11a and the time profiles for \(t=300\) are shown in Figure 11b, and the thickness of the falling films is expressed using the color bar. To compare the influence of the two slippery lengths, a wavenumber of \(q=0.2\) is chosen, and related wavenumber length is \(10\pi\), meaning that the radius of the tube is \(R=5\). For \(\beta_{z}=0\), \(\beta_{z}=0.05\) and \(\beta_{z}=0.1\), the growth rates \(\sigma(q)\) are 0.01058, 0.00678 and 0.00499, receptively. It has been concluded that the axial slippery length causes a larger traveling wave speed and suppresses the flow instability. From the 3D simulations, the flow travels much faster with a larger \(\beta_{z}\) as expected, and the fingers for \(\beta_{z}=0\) are longer than that for \(\beta_{z}=0.05\) and \(\beta_{z}=0.1\), showing good consistency with the prediction of the linear stability analysis.