ANALYSES AND COMPUTATIONS OF THE NAVIER SLIP MODEL WITH SIGNIFICANT SLIP LENGTH IN STOKES FLOW UNDER THIN-FILM APPROXIMATION

In the moving contact line problem, the slip length $l_s$ plays a crucial role as it defines the distance from the boundary where the linearly extrapolated fluid velocity profile becomes zero. The slip length provides a measure of the boundary slip phenomenon, with a larger slip length indicating a more prominent slip effect. Most Navier slip models for moving contact line problem focus on fluids with small slip length, where the scale of the slip length aligns with the height of the droplet. This criterion is met by many common fluids. However, for fluids with significant slip length, the slip effect on the solid substrate becomes substantial, consequently affecting the dynamic behaviors of the system. 
The active polar gels is such kind of fluid, which play an important role in many biological processes, such as wound healing, connective tissue remodeling, reconstruction of damaged structures, or cancer metastasis formation. Despite the wide range of applications, the dynamic of such fluids have not been thoroughly investigated. 

In this thesis, we contribute to this research area by conducting a systematic study of the Navier slip model with significant slip lengths in Stokes flow under the thin-film approximation, which we refer to as the \emph{slipping thin-film model} or shortly slip regime. We employ two different methods, namely asymptotic analysis and variational analysis, to obtain the same governing equations. 
In the first method, we start from the 3-dimensional incompressible Navier-Stokes equation of two-phase flow and establish boundary conditions based on thermodynamic principles to formulate the original governing system. By applying thin-film approximation and appropriately selecting parameter scales, we successfully match the equations and boundary conditions at different orders, eventually deriving the leading-order governing equations. In the second method, we analyze the total free energy and energy dissipation of the system. Through variational analysis of the leading-order Rayleighian in the bulk and on the interface, we arrive at the same governing equations.

To demonstrate how significant variations in slip length affect fluid dynamics and validate the proposed model, we investigate several applications. Firstly, we examine the spreading of a droplet on a smooth flat substrate with a circular initial shape. By employing model reduction via Onsager's variation principle, we derive a simplified governing equation with a limited number of parameters. Despite being an approximation, this simplified equation accurately captures the evolution characteristics of the fluid. Through this equation, we establish that, in the long-term regime, the contact point $R$ and the evolution time $t$ satisfy a power law relationship, specifically $R \propto t^{1/6}$. Additionally, when the droplet has a rectangular initial shape, we observe the phenomenon of time scale separation and explain it using dimensional analysis.

The second application explores the sliding of a droplet on an inclined substrate, while the third application focuses on the coating problem. In both cases, the slip regime exhibits distinctive phenomena not encountered in classical models. For the sliding problem, we illustrate how the droplet shape changes with the Bond number. In addressing the coating problem, we incrementally increase the dimensionless substrate velocity $U_r$, which is defined as the the ratio of substrate velocity to characteristic velocity. In the context of the slipping model, when $U_r$ is less than a critical value, the droplet eventually evolves into a steady state; however, when $U_r$ exceeds this threshold, a thin film with asymptotically constant thickness is formed during the evolution process. For relatively small $U_r$ values ($O(10^{-2})$), numerical experimentation reveals a power-law relationship between the film thickness $h_f$ and $U_r$, i.e., $h_f \propto U_r^2$. Concurrently, the entire droplet moves backward relative to the substrate during evolution. Contrarily, within the framework of the classical model, for arbitrarily small $U_r$, a thin film with asymptotically constant thickness is formed during evolution, rather than evolving into a steady state. For $U_r$ relatively small ($O(10^{-3})$), there is a well-known power-law relationship between the film thickness $h_f$ and $U_r$, specifically $h_f \propto U_r^{2/3}$. Simultaneously, the entire droplet moves forward relative to the substrate during evolution. These applications provide insights into how changes in slip length affect fluid dynamics and reveal characteristic features of the slip regime. Additionally, we propose a numerical scheme for solving the governing system, and the numerical convergence is verified for all applications.

在移动接触线问题中，滑移长度𝑙𝑠是一个重要的物理量，它被定义为距线性外推流体速度剖面消失的边界的距离。滑移长度提供了边界滑移现象的度量，滑移长度越大表明滑移效应越显著。大多数移动接触线问题的Navier滑移模型关注的是滑移长度较小的流体，这些流体的滑移长度的尺度与液滴高度的尺度相同，许多常见的流体都满足这一要求。然而，对于具有显著滑移长度的流体，随着滑移长度的显著增加，固体基质上的滑移效应将显著影响系统的动态行为。活性极性凝胶就是这样一类流体，它的运动在许多生物学过程中发挥重要作用，如伤口愈合、结缔组织重塑、受损结构重建和癌症转移形成。可以看到，这类流体同样具有广泛的应用背景，但它们的动力学行为尚未得到广泛研究。

作为对这一研究领域的贡献，本论文致力于系统研究薄膜近似下Stokes流中具有显著滑移长度的Navier滑移模型（简称滑移模型）。 我们通过渐近分析和变分分析两种不同的方法得到相同的控制方程。在第一种方法中，我们从两相流不可压缩Navier-Stokes方程出发，引入基于热力学原理的边界条件来形成原始的控制系统； 然后进行渐近分析，通过薄膜近似、适当选择参数尺度，成功匹配了不同阶下的方程组和边界条件，得到了首阶控制方程。在第二种方法中，我们的研究基于系统总自由能和能量耗散的分析，通过体上和界面上的首阶Rayleighian泛函的变分分析，得到了相同的控制方程。

为了说明滑移长度的变化如何影响流体动力学，并证明模型的有效性和可行性，我们考虑了模型的几种应用。第一个应用是液滴在光滑平坦基底上的蔓延，在圆形初始形状下，我们通过Onsager变分原理进行模型简化，推导出一个简化控制方程，这个简化方程（ODE）是在有限参数空间中对原方程（PDE）的一个近似，并且仍然能捕捉液体的演化特征；同时，通过简化方程能够得到，在长时间意义下，接触点 $R$ 和演化时间 $t$ 满足幂律 $R \propto t^{1/6}$。此外，当液滴具有矩形初始形状时，我们观察到时间尺度分离现象，并使用量纲分析对其进行解释。

第二个应用探讨了液滴在倾斜基底上的滑动，而第三个应用则重点关注涂层问题。在这两个应用中，滑移模型出现了不同于经典模型的新现象。对于液滴在斜坡上滑落的问题，我们展示了液滴形状如何随Bond数变化。对于涂层问题，我们逐渐增大无量纲基板速度$U_r$（被定义为基板速度和特征速度的比值）。对于滑移模型，当$U_r$小于临界值时，液滴最终演化至稳态；当$U_r$超过临界值时，在演化过程中形成了厚度渐近恒定的薄膜，并且在$U_r$相对小时（$O(10^{-2})$），数值测试发现薄膜厚度$h_f$和无量纲基板速度$U_r$满足幂律$h_f \propto U_r^2$，同时，整个液滴在演化过程中相对于基底向后移动。对于经典模型，对于任意小的拉板速度，在演化过程中都会形成厚度渐近恒定的薄膜，而不是演化至稳态，并且在$U_r$相对小时（$O(10^{-3})$），有着经典结论：薄膜厚度$h_f$和无量纲基板速度$U_r$满足幂律$h_f \propto U_r^{2/3}$，同时，整个液滴在演化过程中相对于基底向前移动。可以看到，两个模型在涂层问题中的演化规律完全不同。通过三个应用，我们揭示了滑移长度的变化如何影响流体动力学，并揭示了滑移模型的特征。为了数值求解上述问题，我们设计了滑移模型的数值格式，对于所有应用，该格式的数值收敛性都已得到验证。

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