# Fingering Instability of a Gravity-Driven Thin Film Flowing Down a Vertical Tube with Wall Slippage
## 2 Mathematical Modeling
### Two Dimensional Simulations  
The most interesting problem we want to know is whether the traveling contact line is stable. Previous studies has discovered that the steady state shape of 2D flow matters a lot to the surface instability [13; 15]. In other words, fingering instability in \(\theta\) direction has essential relationships with the steady base flow independent of coordinate \(\theta\). Besides, the base flow plays an important role in traveling wave solutions and linear stability analysis. Therefore, the two dimensional flow is firstly investigated here, assuming that the evolution equation is independent of the coordinate \(\theta\). Then the evolution equation is simplifies to:  
\[\frac{\partial h}{\partial t}+\frac{\partial}{\partial z}\left(\frac{h^{3}}{3 }+\beta_{z}h^{2}\right)+\frac{\partial}{\partial z}\left[\left(\frac{h^{3}}{3 }+\beta_{z}h^{2}\right)\frac{\partial}{\partial z}\left(-\frac{h}{R^{2}}+ \
abla^{2}h\right)\right]=0 \tag{34}\]An initial condition profile \(h_{0}\) is applied to start the thin film flow. The initial condition is a smooth curve in the form of hyperbolic tangent function [13]:  
\[h_{0}=\frac{1+b}{2}-\frac{1-b}{2}{\tanh(z-z_{0})} \tag{35}\]  
where \(z_{0}\) is the position of the contact line. From the profile, it is seen that \(h\) is constant far away from the contact line. A constant flux boundary condition is approximated well by setting \(h(0,t)=1\) and \(h(z_{max},t)=b\), where \(b\) is the thickness of the precursor layer. Equation (34) is solved using the finite difference method by a MATLAB code. Firstly, the influence of the slippery length is discussed. The time evolution of the fluid profile with different slippery length is visualized in Figure 2, giving the screenshots for \(t=70\) and \(t=140\). In the simulation, the slippery length is set to \(\beta_{z}=0\), \(\beta_{z}=0.025\), \(\beta_{z}=0.05\), \(\beta_{z}=0.075\) and \(\beta_{z}=0.1\), respectively. Theoretically, a more significant influence will be induced to the responses if we give a larger slippery length \(\beta_{z}\). However, in fact, slip usually only occurs very close to the contact line, and this condition is a means of alleviating the stress singularity. Hence the value of the slippery length is given in a reasonable range \([0,0.1]\) in this study. From Figure 2, we note the flow evolves into a traveling wave state after a period, and the traveling wave solutions are to be used in the linear stability analysis (LSA). Resulting from the capillary effect, the steady traveling wave profile has a large ridge at the fluid front, and a higher capillary ridge implies a stronger contact line instability. It has been stated that the profile will tend to the traveling wave shape regardless of the initial condition as long as the boundary conditions were satisfied [16]. Comparing the two graphs, it is seen that the flow travels faster with a bigger value of \(\beta_{z}\), and the bump becomes lower while the axial slippery effect is larger. Thin films are inclined to break when the bump is high, hence the fluid profile predicts that the thin film flows more stably when the axial slippery length is longer. Figure 3 shows the situation for different substrate curvature. Tubes of different radius are used to test the influence of the substrate curvature. In the calculation, \(R=0.9\), \(R=1\), \(R=2\), \(R=10\) and \(R=\infty\) are employed and \(R=\infty\) also corresponds to a vertical plane. It is seen that the capillary ridge appears at the same position after a specified time, indicating that radius of the tubes has no relationship with the traveling wave speed. Also, it can be found that a larger \(R\) corresponds to a larger capillary bump, contrary to the phenomenon in the thin film flow coating on a cylinder surface [16; 25], because the substrate curvature contributes contrarily in the two cases. For a very large \(R>10\), the influence of the substrate curvature is insignificant.