# Fingering Instability of a Gravity-Driven Thin Film Flowing Down a Vertical Tube with Wall Slippage
## 2 Mathematical Modeling  
### Governing Equations and Boundary Conditions  
Figure 1 shows a gravity-driven liquid film flowing down a vertical tube, where the interior surface of the tube is assumed to be a directional slippery liquid-infused surface. A cylindrical coordinate \((r,\theta,z)\) is applied here. The radius of the tube is \(R\) and the different slippery lengths in azimuthal and axial directions are denoted by \(\beta_{\theta}\) and \(\beta_{z}\), respectively. Under the driven force of gravity \(g\), the liquid film flows and breaks into fingers. The thickness of the film is defined by \(h(\theta,z,t)\), where \(t\) is time. To study thin film flows with contact lines, a given slip boundary condition or a precursor model are both effective [13]. Considering that the substrate is slippery, we use the precursor model to drive the governing equation, and the thickness of the precursor film is defined by \(b\). The liquid is viscous and incompressible, and the constant density and viscosity of the fluid are \(\rho\) and \(\mu\), respectively.  
For the liquid, the continuity equation and Navier-Stokes equations are satisfied:  
\[\frac{1}{r}\frac{\partial(ru)}{\partial r}+\frac{1}{r}\frac{\partial v}{ \partial\theta}+\frac{\partial w}{\partial z}=0 \tag{1}\]  
\[\rho\left(\frac{Du}{Dt}-\frac{v^{2}}{r}\right)=-\frac{\partial p}{\partial r} +\mu\left(\
abla^{2}u-\frac{u}{r^{2}}-\frac{2}{r^{2}}\frac{\partial v}{ \partial\theta}\right) \tag{2}\]  
\[\rho\left(\frac{Dv}{Dt}+\frac{uv}{r}\right) = -\frac{1}{r}\frac{\partial p}{\partial\theta}+\mu\left(\
abla^{2 }v-\frac{v}{r^{2}}+\frac{2}{r^{2}}\frac{\partial u}{\partial\theta}\right) \tag{3}\]  
\[\rho\frac{Dw}{Dt} = -\frac{\partial p}{\partial z}+\mu\
abla^{2}w+\rho g \tag{4}\]  
where \(D/Dt\) represents the material derivative and \(\
abla^{2}\) is the Laplace operator in cylindrical coordinate, given as follows:  
\[\frac{D}{Dt}=\frac{\partial}{\partial t}+u\frac{\partial}{\partial r}+\frac{ v}{r}\frac{\partial}{\partial\theta}+w\frac{\partial}{\partial z} \tag{5}\]\[\
abla^{2}=\frac{\partial^{2}}{\partial r^{2}}+\frac{1}{r}\frac{\partial}{ \partial r}+\frac{1}{r^{2}}\frac{\partial^{2}}{\partial\theta^{2}}+\frac{ \partial^{2}}{\partial z^{2}} \tag{6}\]  
where \(\mathbf{u}=(u,v,w)\) are velocities related to coordinates \((r,\theta,z)\) and \(p\) is the pressure. In the following text, the pressure of the ambient gas is denoted by \(p_{g}\).  
For the directional slippery wall, slip boundary conditions are applied in \(\theta\) direction and \(z\) direction while the non-slip boundary condition is applied in the \(r\) direction:  
\[u=0,v=\beta_{1}v_{r},w=\beta_{2}w_{r} \tag{7}\]  
Here, \(\beta_{1}\) and \(\beta_{2}\) are the slippery length in \(\theta\) direction and \(z\) direction, respectively, and \(v_{r}=\frac{\partial v}{\partial r}\) and \(w_{r}=\frac{\partial w}{\partial r}\) are partial derivative of velocity with respect to the coordinate \(r\). Unless stated otherwise, the subscripts in this paper denote the partial derivative.  
At the free surface \(r=R-h\), the balances of normal and tangential stresses should be satisfied:  
\[(\mathbf{P}-\mathbf{P}_{\infty})\cdot\mathbf{n}\ =\ -\
abla_{s}\sigma+\sigma(\
abla\cdot \mathbf{n})\mathbf{n} \tag{8}\]  
where \(\mathbf{P}=\mu[\
abla\mathbf{n}+\
abla\mathbf{n}^{T}]\) and \(\mathbf{P}_{\infty}=(p-p_{g})\mathbf{I}\) are the stress tensor and pressure tensor, respectively. \(\
abla_{s}\) is the surface gradient and \(\mathbf{n}\) is the unit vector normal to surface. \(\sigma\) is the surface tension and \(-\
abla_{s}\cdot\mathbf{n}\) means the averaged curvature of the free surface.  
The kinematic condition is given at the free interface:  
\[\frac{\partial h}{\partial t}+\frac{1}{r}\frac{\partial}{\partial\theta}\int_ {R-h}^{R}vdr+\frac{\partial}{\partial z}\int_{R-h}^{R}vdr=0 \tag{9}\]