Modelling Capillary Rise with a Slip Boundary Condition: Well-posedness and Long-time Dynamics of Solutions to Washburn's Equation

This paper extends Washburn's capillary rise equation by incorporating a slip boundary condition, proving the global well-posedness of the resulting initial-value problem and characterizing the long-time dynamics, including monotonic or oscillatory convergence to equilibrium, for a wide range of slip parameters and initial fluid heights.

The Gist

Imagine you have a very thin straw dipped into a glass of water. You know that the water doesn't just sit there; it climbs up the straw on its own, defying gravity for a while. This is called capillary rise, and it's the same principle that allows trees to pull water from their roots to their leaves or how a paper towel soaks up a spill.

For over a century, scientists have used a famous formula called Washburn's Equation to predict exactly how fast and how high that liquid will climb. However, this old formula had a few problems:

1. It assumed the liquid sticks perfectly to the walls of the pipe (like glue).
2. In the very first split second of the climb, the math broke down and gave impossible answers (like dividing by zero).
3. No one could mathematically prove that the liquid would always settle down at a specific height without going crazy or oscillating forever.

This paper by Rapajić, Simić, and Süli fixes those problems. Here is a simple breakdown of what they did, using some everyday analogies.

1. The "Slippery" Wall (The New Physics)

The old model assumed the water molecules at the very edge of the pipe were stuck in place, like a car with its brakes locked. But in reality, especially in tiny pipes, the liquid can "slip" a little bit along the wall, like a car on a patch of ice.

The authors added a "slip parameter" to the equation. Think of this as a "slipperiness dial."

• Dial at 0: The wall is sticky (no slip).
• Dial turned up: The wall is slippery.
• Result: When the wall is slippery, the liquid climbs faster because there is less friction holding it back. The authors proved that even with this new "slippery" factor, the physics still makes sense.

2. Fixing the "Zero Second" Problem (The Math)

The biggest headache with the old equation was what happens at Time = 0. If you start with the liquid at the very bottom (height 0) and velocity 0, the math tries to calculate the acceleration and screams, "I can't do this!" It's like trying to calculate the speed of a car the exact moment it starts moving from a dead stop without any engine power.

The authors realized the old proof that the equation works was "leaky." They patched it up by:

• Smoothing the start: They allowed the liquid to start at a tiny, non-zero height (like a tiny drop already sitting at the bottom) rather than a perfect mathematical point.
• The "Regularization" Trick: They temporarily added a tiny bit of "mathematical glue" to the equation to make it behave nicely, solved it, and then slowly removed the glue. This proved that a solution exists and is unique—meaning the liquid behaves predictably, not chaotically.

3. The Race to the Finish Line (Stability)

Once the liquid starts climbing, does it stop? Or does it overshoot, fall back, bounce up, and bounce down forever like a bouncy ball?

The authors asked: Will the liquid eventually stop at a specific height?

• The Destination: There is a "comfort zone" height (called the equilibrium height) where the pull of the surface tension exactly balances the weight of the water column.
• The Journey: Depending on how "heavy" the liquid is (inertia) and how "thick" it is (viscosity), the liquid might:
  • Climb smoothly like a snail reaching a leaf.
  • Overshoot and wobble like a pendulum swinging back and forth before settling.
• The Proof: The authors built a mathematical "energy trap" (called a Lyapunov function). Imagine a bowl with a marble inside. No matter where you drop the marble (as long as it's inside the bowl), it will eventually roll to the very bottom. They proved that for any starting height (as long as it's not absurdly high), the liquid is like that marble, and it will eventually settle at the correct height. It won't escape the bowl.

4. The "Slippery" Effect on the Journey

One interesting finding is that while the "slippery wall" makes the liquid climb faster, it doesn't change the destination. The liquid still stops at the same final height, regardless of how slippery the pipe is. It just gets there with a different style of movement (maybe a faster wobble or a smoother glide).

Summary: Why This Matters

Think of this paper as the "User Manual Update" for how we understand liquids in tiny tubes.

• Before: The manual had a warning that said, "If you start from zero, the math breaks," and "We aren't 100% sure the liquid stops."
• After: The new manual says, "We fixed the math so it works from the very first second. We proved the liquid will always stop at the right height, whether the pipe is sticky or slippery."

This is crucial for engineers designing micro-fluidic devices (tiny lab-on-a-chip computers), medical devices that draw blood, or understanding how plants drink water. It gives us the confidence that our models are solid and won't fail when we build real-world things.

Technical Summary

1. Problem Statement

The paper addresses the mathematical modeling of spontaneous capillary rise of a liquid in a vertical cylindrical pipe. The classical Washburn's equation (1921) describes this phenomenon but traditionally assumes a no-slip boundary condition at the pipe wall. This assumption leads to the Huh–Scriven paradox, where the fluid velocity at the wall is zero, yet the contact line (the interface between liquid, solid, and gas) is moving.

To resolve this, the authors incorporate a slip boundary condition (characterized by a slip length $L$ or dimensionless parameter $\beta$). Furthermore, the paper aims to rigorously address gaps in previous mathematical proofs regarding the global existence, uniqueness, and well-posedness of solutions to Washburn's equation, particularly for non-zero initial data and over the infinite time interval $t \in [0, \infty)$.

2. Methodology

A. Physical Derivation

The authors derive Washburn's equation from first principles (continuum mechanics) rather than simply applying Newton's second law to a control volume.

• Assumptions: Incompressible Newtonian fluid, constant density $\rho$ and viscosity $\mu$, constant surface tension $\gamma$, and contact angle $\theta$.
• Flow Profile: They assume an axially symmetric, parallel flow with a Poiseuille velocity profile modified by a slip parameter.
• Momentum Balance: They utilize the global (integral) form of the momentum balance law to derive the governing ordinary differential equation (ODE) for the meniscus height $h(t)$.
• Resulting Equation:
  $$\frac{d}{dt}\left[\rho h(t) \frac{dh}{dt}\right] + \frac{8\mu\beta}{R^2} h(t) \frac{dh}{dt} + \rho g h(t) = \frac{2\gamma \cos \theta}{R}$$
  where $\beta = (1 + 4L/R)^{-1}$ is the slip parameter ($\beta=1$ for no-slip).

B. Dimensional Analysis and Model Reduction

The equation is non-dimensionalized using the equilibrium height $h_e$ and a characteristic time scale $\tau$. This yields the dimensionless equation:
$$\omega(HH')' + \beta HH' + H = 1$$
where $H$ is the dimensionless height, $T$ is dimensionless time, and $\omega$ is a parameter related to the Ohnesorge and Bond numbers.

• The authors perform a systematic model reduction analysis by introducing power-law scalings for time and height.
• They identify four distinct flow regimes based on the relative magnitude of $\omega$ (inertia vs. viscosity vs. gravity):
  1. Negligible gravity (Inertia + Viscosity).
  2. Negligible inertia (Viscosity + Gravity).
  3. Negligible gravity and inertia (Viscous dominated, recovering the classical $\sqrt{t}$ Washburn law).
  4. Negligible viscosity (Inertia + Gravity, leading to oscillatory behavior).

C. Mathematical Analysis (Well-posedness)

The core mathematical contribution involves proving the existence and uniqueness of solutions for the transformed equation:
$$u'' + \frac{\beta}{\sqrt{\omega}} u' + \sqrt{2u} = 1$$
where $u = \frac{1}{2}H^2$.

• Transformation: The nonlinearity in the leading term is removed via the substitution $u = H^2/2$.
• Existence: The authors use a regularization approach. They solve a regularized version of the equation (adding a small $\epsilon$ to the nonlinear term) to ensure the Lipschitz condition holds. They then apply the Picard–Lindelöf theorem for local existence and use Arzelà–Ascoli compactness arguments to pass to the limit $\epsilon \to 0$, establishing global existence.
• Uniqueness: They prove uniqueness by analyzing the energy functional and utilizing Grönwall's lemma for positive initial data, and a fixed-point theorem (Precup's Theorem 11.3) for zero initial data, carefully addressing the singularity at $t=0$ where the Lipschitz condition fails.
• Well-posedness: They demonstrate continuous dependence on initial data in the maximum norm, satisfying Hadamard's criteria.

D. Stability Analysis

• Linear Stability: The system is linearized around the equilibrium point $(u_e, v_e) = (1/2, 0)$. The authors determine that the equilibrium is asymptotically stable. The type of convergence (monotonic vs. oscillatory) depends on a critical value $\omega^* = \beta^2/4$.
• Global Stability: To prove global convergence from the specific initial data ($H(0)=\alpha, H'(0)=0$), they construct a Lyapunov function:
  $$V(u, v) = \frac{1}{2}v^2 - u + \frac{2\sqrt{2}}{3}u^{3/2} + \frac{1}{6}$$
• Using LaSalle's invariance principle, they identify the basin of attraction $\Omega_0$ which explicitly includes the physically relevant initial states ($0 \le \alpha \le 3/2$).

3. Key Contributions

1. Rigorous Derivation: Derivation of Washburn's equation with slip from fundamental conservation laws, providing a solid basis for generalizations.
2. Correction of Previous Proofs: The paper identifies and corrects flaws in a previous proof of global existence (Ref [9]), specifically regarding the failure to verify the self-mapping property of the fixed-point operator and the incorrect application of monotonicity theorems at $t=0$.
3. General Initial Data: The proofs hold for a broader range of initial conditions ($0 \le \alpha \le 3/2$), not just zero initial height.
4. Slip Parameter Analysis: Demonstration that while the slip parameter $\beta$ affects the rate of convergence and the critical threshold for oscillatory behavior, it does not alter the qualitative long-time behavior (convergence to equilibrium) or the equilibrium height itself.
5. Global Stability Proof: Construction of a Lyapunov function that rigorously proves the system converges to the equilibrium state from the specific initial data used in capillary rise experiments, a result previously lacking a rigorous mathematical proof.

4. Key Results

• Existence & Uniqueness: For any $\beta > 0, \omega > 0$ and initial height ratio $\alpha \in [0, 3/2]$, there exists a unique, bounded, positive solution $H(T)$ defined for all $T \in [0, \infty)$.
• Well-posedness: The initial value problem is well-posed in the sense of Hadamard (existence, uniqueness, and continuous dependence on data).
• Equilibrium: The unique equilibrium height is $H_e = 1$ (dimensionless), which is independent of the slip parameter $\beta$.
• Dynamics:
  • If $\omega < \beta^2/4$: The solution approaches equilibrium monotonically (stable node).
  • If $\omega > \beta^2/4$: The solution approaches equilibrium oscillatorily (stable spiral).
  • The slip parameter $\beta$ shifts the critical threshold $\omega^*$, effectively damping the oscillations more strongly as slip increases (lower $\beta$).
• Convergence: All solutions starting from rest with $0 \le H(0) \le 1.5 H_e$ converge to the equilibrium height $H_e$ as $t \to \infty$.

5. Significance

This work bridges the gap between physical modeling and rigorous mathematical analysis for capillary flow.

• Physical Relevance: By incorporating the slip condition, the model resolves the Huh–Scriven paradox, making the equation more physically accurate for the initial phase of flow where the contact line moves.
• Mathematical Rigor: It provides the first complete and rigorous proof of global well-posedness for Washburn's equation with slip, correcting incomplete arguments in prior literature.
• Predictive Power: The identification of the basin of attraction ensures that the model is robust for experimental conditions where the fluid starts from rest at a small height.
• Future Directions: The paper sets the stage for analyzing variable slip parameters, negative slip (contact angle hysteresis), and the development of stable numerical methods for long-time simulations.