# Fingering Instability of a Gravity-Driven Thin Film Flowing Down a Vertical Tube with Wall Slippage
## 2 Mathematical Modeling
### Scalings and Non-Dimensionalization  
In order to nondimensionalize the system, we introduce the following scales:  
\[(r^{\prime},z^{\prime},h^{\prime})=(\frac{r}{R},\frac{z}{R},\frac{h}{R}),(u^{ \prime},v^{\prime},w^{\prime})=(\frac{u}{W},\frac{v}{W},\frac{w}{W}),p^{ \prime}=\frac{p-p_{0}}{\rho gR},t^{\prime}=\frac{t}{T} \tag{10}\]  
The characteristic velocity scale \(W\) and time scale \(T\) are given as:  
\[W=\frac{\rho gR^{2}}{\mu},T=\frac{R}{W} \tag{11}\]  
Figure 1: Schematic of a thin liquid film flowing down a vertical tube with a slippery wall. The wall is directional slippery with a slip length \(\beta_{\theta}\) in azimuthal direction and a slip length \(\beta_{z}\) in the axial direction. The primes are dropped for simplicity and the nondimensionlized continuity equation and the N-S equations become:  
\[\frac{1}{r}\frac{\partial(ru)}{\partial r}+\frac{1}{r}\frac{\partial v}{\partial \theta}+\frac{\partial w}{\partial z}=0 \tag{12}\]  
\[Re\left(\frac{Du}{Dt}-\frac{v^{2}}{r}\right)=-\frac{\partial p}{\partial r}+ \left(\
abla^{2}u-\frac{u}{r^{2}}-\frac{2}{r^{2}}\frac{\partial v}{\partial \theta}\right) \tag{13}\]  
\[Re\left(\frac{Dv}{Dt}+\frac{uv}{r}\right) = -\frac{1}{r}\frac{\partial p}{\partial\theta}+\left(\
abla^{2}v- \frac{v}{r^{2}}+\frac{2}{r^{2}}\frac{\partial u}{\partial\theta}\right) \tag{14}\]  
\[Re\frac{Dw}{Dt} = -\frac{\partial p}{\partial z}+\
abla^{2}w+1 \tag{15}\]  
where \(Re=\frac{gWh_{0}^{2}}{\mu}\) denotes the Reynold number. Dimensionless boundary conditions at the fluid-solid interface are expressed as:  
\[u=0,v=\beta_{\theta}v_{r},w=\beta_{z}w_{r} \tag{16}\]  
where \(\beta_{\theta}=\beta_{1}/R\) and \(\beta_{z}=\beta_{2}/R\) are the dimensionless slippery lengths. The dimensionless kinematic condition (9) at the liquid-air interface in integral form remains the same:  
\[\frac{\partial h}{\partial t}+\frac{1}{r}\frac{\partial}{\partial\theta}\int _{R}^{R-h}vdr+\frac{\partial}{\partial z}\int_{R}^{R-h}wdr=0 \tag{17}\]  
With the scalings, dimensionless boundary conditions for balances of normal and tangential stresses become:  
\[p+(\mathbf{P}\cdot\mathbf{n})\cdot\mathbf{n} = -\frac{1}{C}K(h) \tag{18}\] \[W=\frac{\rho gR^{2}}{\mu},T=\frac{R}{W} \tag{11}\]0  
where \(K(h)\) is the mean curvature, composed by the principal curvature of the interface and the curvature of the substrate. \(C=\mu W/\sigma_{0}\) is the capillary number and \(\tau_{i}(i=1,2)\) denotes the unit vector tangential to surface. Considering that the mean thickness of the liquid film is much smaller than the size in the other two directions, a new coordinate \(r=R-\epsilon x\) is applied in the radial direction. \(\epsilon=h_{0}/R<<1\) is a small parameter, where \(h_{0}\) is the characteristic thickness of the thin film. The variables can be expanded as a perturbation series of \(\epsilon\)[40]:  
\[W=\frac{\rho gR^{2}}{\mu},T=\frac{R}{W} \tag{11}\]1 \[W=\frac{\rho gR^{2}}{\mu},T=\frac{R}{W} \tag{11}\]2  
Normal stress is balanced by the pressure and surface tension, giving:  
\[W=\frac{\rho gR^{2}}{\mu},T=\frac{R}{W} \tag{11}\]3  
where the first and the second terms in the right hand are obtained using the Taylor expansion. The difference between a tube and a cylinder is that the second term in Equation (28) has opposite signs. From Equations (23)-(28), the functions \((v,w)\) of the zeroth order are obtained:  
\[W=\frac{\rho gR^{2}}{\mu},T=\frac{R}{W} \tag{11}\]4  
\[W=\frac{\rho gR^{2}}{\mu},T=\frac{R}{W} \tag{11}\]5  
Substituting \(u_{0}\) and \(v_{0}\) into the kinematic boundary condition of integral form:  
\[W=\frac{\rho gR^{2}}{\mu},T=\frac{R}{W} \tag{11}\]6  
The evolution equation for the thin film flow with dynamical contact line are finally obtained:  
\[W=\frac{\rho gR^{2}}{\mu},T=\frac{R}{W} \tag{11}\]7  
where \(\
abla=[\partial_{\theta},\partial_{z}]\) and \(\Theta(h)\) is a coefficient matrix:  
\[\Theta(h)=\left[ \begin{array}{cc}\frac{h^{3}}{3 Ca}+\frac{\beta_{z}h^{2}}{Ca} &0\\ 0&\frac{h^{3}}{3 Ca}+\frac{\beta_{z}h^{2}}{Ca}\end{array}