The steady state of gravity-capillary problem with inclined walls

1. Problem Statement

The paper investigates the steady-state configurations of a two-dimensional, static, viscous, incompressible fluid flowing over an open-top vessel with inclined walls. The vessel is modeled as a triangular domain $\Gamma \subset \mathbb{R}^2$ defined by two walls with inclination angles $\theta_1$ and $\theta_2$ relative to the horizontal plane.

The physical system is governed by the balance of:

• Gravity: Acting vertically.
• Surface Tension: Acting along the free surface.
• Contact Angle: The interaction between the fluid, the solid wall, and the vapor, characterized by a prescribed contact angle condition at the walls.

The primary mathematical challenge is to determine the shape of the free surface (the interface between the fluid and air) for a prescribed fluid volume $V$, given arbitrary inclination angles $\theta_1, \theta_2$ and arbitrary contact angles. The problem is formulated as finding the minimizer of a specific energy functional subject to a volume constraint, which leads to a nonlinear Euler-Lagrange equation with free boundary conditions.

2. Mathematical Formulation

The steady state is characterized as a solution to the Euler-Lagrange equation derived from the energy functional $E(\rho)$ in polar coordinates $(\rho, \theta)$, where the origin is the intersection of the two walls:
$$E(\rho) = \frac{1}{3}g \int_{\theta_2}^{\pi-\theta_1} \rho^3 \sin \theta \, d\theta + \int_{\theta_2}^{\pi-\theta_1} \sigma \sqrt{\rho^2 + (\rho')^2} \, d\theta - [[\gamma]](\rho(\theta_1) + \rho(\theta_2))$$
Subject to the volume constraint:
$$V(\rho) = \frac{1}{2} \int_{\theta_2}^{\pi-\theta_1} \rho^2(\theta) \, d\theta = V$$
Here, $g$ is gravity, $\sigma$ is surface tension, and $[[\gamma]]$ represents the difference in surface energies (solid-vapor vs. solid-fluid).

The resulting Euler-Lagrange equation is a second-order nonlinear ODE involving a Lagrange multiplier $P_0$ (related to pressure). A key difficulty is that $P_0$ is unknown and depends on the volume $V$.

3. Methodology

The author employs a shooting method combined with geometric analysis and coordinate transformations to construct solutions. The methodology is divided into two main cases based on the signs of the contact angles at the walls ($\psi_1$ and $\psi_2$).

Key Technical Tools:

1. Coordinate Shift: To eliminate the unknown pressure term $P_0$, the author introduces a vertical shift in Cartesian coordinates: $(x, y) \to (x, y - P_0/g)$. This transforms the system into a form where the pressure term vanishes, simplifying the ODE analysis.
2. Intrinsic Variable $\psi$: Instead of using the polar angle $\theta$ or the Cartesian coordinate $x$ directly, the author introduces an angle $\psi$ representing the slope of the free surface in Cartesian coordinates. This variable serves as a natural parameterization for the curve.
3. Shooting Parameter: The construction relies on identifying a "distinguished point" on the free surface (e.g., the point of maximum height or maximum slope) and shooting the solution from this point to the walls. The volume constraint is then used to determine the specific value of the shooting parameter.

Case Analysis:

The paper splits the problem into two distinct geometric scenarios:

Case 1: Opposite Signs ($\psi_1 \cdot \psi_2 < 0$)

• Geometry: The contact angles have opposite signs, implying the free surface has a unique maximum height within the domain.
• Parametrization: The slope angle $\psi$ provides a global one-to-one parametrization of the free boundary curve.
• Construction: The solution is constructed by shooting from the point of maximum height $(x_m, u_m)$ where $\psi=0$.
• Result: The volume $V$ is expressed as a function of the maximum height parameter $u_m$. The author proves that $V(u_m)$ is a strictly increasing function (under certain conditions on surface tension), mapping $(-\infty, 0)$ to $(0, \infty)$. This guarantees the existence and uniqueness of the solution for any prescribed volume $V$.

Case 2: Same Signs ($\psi_1 \cdot \psi_2 \geq 0$)

• Geometry: The contact angles have the same sign. The free surface may not have a unique maximum in the interior, or the curve may not be representable as a single-valued function of the slope angle $\psi$ globally.
• Sub-cases:
  • Small Volume: The curve can be parameterized by $\psi$, and the solution is constructed similarly to Case 1 but from a boundary point.
  • Large Volume: The curve attains a maximum slope angle $\psi_m$ in the interior. The curve is split into two segments (left and right of the maximum), each parameterized by $\psi$.
• Construction: The author constructs solutions by shooting from the interior maximum point $(x_m, 0)$ with a specific slope angle $\psi_m$.
• Result: The volume function $V(\psi_m)$ is shown to diverge to $+\infty$ as $\psi_m \to 0$ and has a positive lower bound $V_m$.
  • If $V \geq V_m$, a solution exists where the maximum is in the interior.
  • If $V \leq V_1$ (an upper bound derived from boundary-shooting solutions), a solution exists where the maximum is at the boundary.
  • Non-uniqueness: In the overlap region where $V_m < V < V_1$, multiple steady-state solutions may exist (one with an interior maximum and one with a boundary maximum).

4. Key Contributions

1. General Existence Theorem: The paper establishes the existence of steady-state solutions for arbitrary inclination angles $\theta_1, \theta_2$ and arbitrary contact angles, extending previous work which was often limited to symmetric or vertical wall configurations.
2. Resolution of Non-Uniqueness: The author rigorously identifies conditions under which the solution is unique (Case 1) and conditions under which non-uniqueness arises (Case 2, large volumes). This provides a complete picture of the solution landscape.
3. Novel Parametrization Strategy: The use of the slope angle $\psi$ as the primary parameter, combined with the vertical coordinate shift to eliminate the Lagrange multiplier, offers a robust analytical framework for handling the nonlinear free boundary problem.
4. Asymptotic Analysis: Detailed asymptotic analysis of the volume functional as the shooting parameters approach critical limits (e.g., $u_m \to 0$ or $\psi_m \to 0$) is provided to prove the surjectivity of the volume map.

5. Main Results

• Theorem 1.2 & 2.15 (Case 1): For opposite contact angles, for any prescribed volume $V > 0$, there exists a unique steady-state solution.
• Theorem 1.4 & 3.11 (Case 2): For same-sign contact angles, there exist constants $V_m$ and $V_1$ such that:
  • Solutions exist for all $V \geq 0$.
  • If $V_1 > V_m$, the solution is not unique for volumes in the interval $(V_m, V_1)$.
• Theorem 2.16 & 3.15: The constructed solutions can be represented as graphs $\rho(\theta)$ in polar coordinates, satisfying the volume constraint.

6. Significance

This work significantly advances the mathematical theory of capillary surfaces in non-symmetric, inclined geometries.

• Theoretical Impact: It resolves the long-standing challenge of proving existence for general contact line problems with two distinct inclined walls, moving beyond the axisymmetric or vertical wall assumptions common in literature (e.g., Finn's work).
• Practical Relevance: The results are directly applicable to engineering problems involving fluid transport in channels, spillways, and microfluidic devices where walls are inclined and surface tension effects are dominant. The identification of non-uniqueness regimes is crucial for predicting hysteresis and stability in such systems.
• Methodological Innovation: The shooting method combined with the specific coordinate shift and intrinsic angle parametrization provides a template for solving other free-boundary problems involving Lagrange multipliers.

In summary, the paper provides a comprehensive existence and uniqueness theory for the steady gravity-capillary problem with inclined walls, characterizing the solution space and identifying regimes of multiplicity based on fluid volume and geometric parameters.