# Fingering Instability of a Gravity-Driven Thin Film Flowing Down a Vertical Tube with Wall Slippage
## 1 Introduction  
Thin film flowing and droplet spreading are commonly encountered in chemical engineering, nuclear industry and materials sciences. The coating examples show that the free interface of a liquid film easily breaks into fingers resulting from the capillary effect [1; 2; 3]. The understanding of the dynamics and flow stability of liquid films is essential to control heat and mass transport processes.  
To model the thin film flows, lubrication approximation, a thin film model and weighted residual model have been introduced in previous studies [4; 5; 6]. In the modeling of the liquid film, assumption of a small averaged thickness is crucial [7; 8]. These modeling methods were also applied in dynamical contact line problems [9; 10; 11; 12]. [PERSON] [13] and [PERSON] [14; 15] provided a systematic analysis to deal with the contact line instability problem, and the method was easy to implement [16; 17]. Based on the governing equations, the relationship between different parameters and the surface instability was analyzed by linear stability analysis (LSA) [18; 19].  
In recent years, the spreading and falling of the liquid films on different substrates have been received much attention. It was found the curvature and property of the substrate played important roles in the film flows [20; 21; 22; 23; 24]. The liquid film flowing down a cylinder/fiber was more unstable than that flowing on an inclined plane [25]. As a common situation in interstitial flow, thin liquid film flowing on porous medium exhibits interesting dynamical phenomena. [PERSON] and [PERSON] [26; 27] discovered that the film flowing on a porous vertical cylinder was more unstable and the increasingpermeability of the porous medium significantly enhanced the destabilizing effect. The slippery effect often has a connection with wettability and porosity of the materials [28; 29; 30]. Recently, inspired by the good antiwetting property of neepenthes pitcher plants, specialists designed various slippery liquid-infused porous substrates similar to the microstructures and functions of pitcher plants [31; 32; 33]. Moreover, liquid-infused surface with directional slippery lengths was fabricated making use of immobilized lubricant menisci [34]. It is necessary to understand the thin film flow on these biomimetic surfaces, as these structures are usually designed for water collection and transportation.  
In the analysis, the influence of the porous medium was usually reduced as a slip condition [35; 36]. The effect of a slip on the primary instability was investigated by [PERSON] et al. [37; 38]. It was found the slip played a destabilizing role close to the instability onset, while the flow was stabilizing at larger values of the Reynolds number. [PERSON] [39] investigated the coupling action of the thermocapillary effect and slippery effect, concluding that the instability was enhanced as the slippery length increased. [PERSON] [40] studied the dynamics of falling films down a cylinder in the presence of wall slippage, and also found that wall slippage enhanced the capillary instability. All the aforementioned investigations were concentrated on a substrate with slippery length in only one direction. However, the substrates used in biomedical applications usually exhibit anisotropic behavior. Therefore, this paper mainly focuses on the influence of the anisotropic slippery length on the surface instability of thin films flowing down a vertical tube.  
The rest of this paper is organized as follows: In Section 2, the mathematical model is conducted based on the thin film model and two dimensional (2D) evolution is displayed. In Section 3, linear stability analysis is carried out based on traveling wave solutions. To verify the linear stability analysis, three dimensional (3D) flow simulations with random perturbations are displayed in Section 4, and some conclusions are briefly summarized in Section 5.