# Fingering Instability of a Gravity-Driven Thin Film Flowing Down a Vertical Tube with Wall Slippage
## 3 Linear Stability Analysis
(**a**) \(R=1,b=0.05\) corresponding to different \(\beta_{z}\), (**b**) \(R=1,\beta_{z}=0\) corresponding to different \(b\), (**c**) \(\beta_{z}=0,b=0.05\) corresponding to different \(R\). where \(\Theta_{z}(\bar{H})=\frac{\bar{H}^{3}}{3}+\beta_{z}\bar{H}^{2}\) and \(\Theta_{\theta}(\bar{H})=\frac{\bar{H}^{3}}{3}+\beta_{\theta}\bar{H}^{2}\).The perturbation \(H^{\prime}(t,\xi,\theta)\) can be expressed as a superposition of a series of Fourier transforms [17; 26], expressed as \(H^{\prime}(t,\xi,y)=\hat{H}e^{iq\theta+\sigma(q)t}\), where \(q\) is the wavenumber and \(\sigma(q)\) relates to the growth rate. For every given wavenumber \(q\), substituting \(H^{\prime}(t,\xi,y)=\hat{H}e^{iq\theta+\sigma(q)t}\) into Equation (38), Equation (38) transforms into a standard eigenvalue equation:  
\[\sigma(q)\hat{H}\ =\ -\ L\hat{H} \tag{39}\]  
Here only the largest real \(\sigma\) is to be used representing the largest eigenvalue of \(\mathbf{L}\) and \(\mathbf{L}\) is a linear operator with the matrix form:  
\[\mathbf{L} =-c\mathbf{D}_{1}+[(2\bar{H}\mathbf{D}_{1}\bar{H}+2\beta_{z}\mathbf{D}_{1} \bar{H})(1+\kappa\mathbf{D}_{1}\bar{H}+\mathbf{D}_{3}\bar{H})]+ \tag{40}\] \[\qquad\left[(\bar{H}^{2}\mathbf{D}_{1}\bar{H}+2\beta_{z}\bar{H}\mathbf{D} _{1}\bar{H})(1+\kappa\mathbf{D}_{1}+\mathbf{D}_{3}-q^{2}\mathbf{D}_{1})\right]+\] \[\qquad\left(\bar{H}^{2}+2\beta_{z}\bar{H}\right)\left[(\kappa\bm {D}_{1}\bar{H}+\mathbf{D}_{3}\bar{H})\mathbf{D}_{1}+(\kappa\mathbf{D}_{2}\bar{H}+\mathbf{D}_{ 4}\bar{H})\right]+\] \[\Theta_{x}(\bar{H})(\kappa\mathbf{D}_{2}+\mathbf{D}_{4}-q^{2}\mathbf{D}_{2})+ \Theta_{\theta}(\bar{H})(-\kappa q^{2}\mathbf{I}-q^{2}\mathbf{D}_{2}+q^{4}\mathbf{I})\]  
where \(\kappa=-\frac{1}{R^{2}}\); \(\mathbf{I}\) represents the identity matrix, and \(\mathbf{D}_{i}(i=1,2,3,4)\) are finite difference matrix of \(i-\)th order. Using the difference matrix, we have \(\bar{H}_{\xi}=\mathbf{D}_{1}\bar{H}\), \(\bar{H}_{\xi\xi}=\mathbf{D}_{2}\bar{H}\), \(\bar{H}_{\xi\xi\xi}=\mathbf{D}_{3}\bar{H}\) and \(\bar{H}_{\xi\xi\xi\xi}=\mathbf{D}_{4}\bar{H}\). Equation (40) is solved numerically to obtain the values of \(\sigma(q)\) for a given wavenumber \(q\), following the work by [PERSON] [13]. Given a wavenumber, the eigenvalues of the matrix \(\mathbf{L}\) are obtained by a MATLAB code, and the growth rate is related to the largest real value. The growth rate \(\sigma(q)\) associated with different cylinder radius is illustrated in Figure 6. Each curve represents the largest eigenvalue of \(-\mathbf{L}\) versus the wavenumber \(q\), and we can determine the range of unstable wavenumbers as \((0,q_{c})\) with \(\sigma(q_{c})=0\). For small wavenumber \(q<q_{c}\), the flow is always unstable. The dispersion curves illustrate that a smaller cylinder radius results in a larger growth rate and a bigger peak wavenumber (the value corresponding to the maximum growth rate). A smaller cut-off wavenumber \(q_{c}\) is also found when the cylinder radius is smaller. For \(R=\infty\), the cylinder can be considered as a vertical plane, demonstrating that the thin film flow down a vertical tube is much more stable when given a certain perturbation. From the plots, it is seen that the substrate curvature greatly promotes the fingering instability, especially for the case of \(R<1\). The dispersion curves for different thickness of precursor layer \(b\) are visualized in Figure 7. We note that a small thicknesses of the precursor layer is related to a much larger maximum growth rate, showing good consistency with the time responses. Much larger cutoff wavenumbers and peak wavenumbers are found for smaller \(b\), demonstrating that both the strength and the wavenumber range of the surface instability are enhanced by a thinner precursor layer. Figure 8 shows the growth rate for a tube with the isotropic slippery lengths, i.e., \(\beta_{\theta}=\beta_{z}\). As shown in Figure 8, both the growth rate and the peak wavenumber change small with the increase of slippery length, showing that the slippery effect impedes the surface instability.