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

Isidora Rapajić, Srboljub Simić, Endre Süli

TL;DR

The paper derives a Washburn-type capillary-rise equation that incorporates a slip boundary condition via a slip parameter $\beta$, starting from first principles and a cylindrical geometry. Through nondimensionalization, it obtains the dimensionless model $\omega(HH')' + \beta HH' + H = 1$ with equilibrium height $H_e=1$, and provides a rigorous analysis establishing global existence, uniqueness, and continuous dependence of solutions for initial heights $H(0)=\alpha$ with $\alpha\in[0,3/2]$. It further analyzes long-time behavior by recasting the problem as a 2D autonomous system, identifying a critical threshold $\omega_*=\beta^2/4$ that distinguishes monotone and oscillatory approaches, and proving convergence to the equilibrium via a Lyapunov function and LaSalle’s invariance principle. The results show that while the slip parameter shapes transient dynamics, the unique equilibrium height and global convergence are preserved for all positive $\beta$ and $\omega$, providing a solid mathematical foundation for capillary rise with slip and guiding future numerical and physical investigations.

Abstract

The aim of this paper is to extend Washburn's capillary rise equation by incorporating a slip condition at the pipe wall. The governing equation is derived using fundamental principles from continuum mechanics. A new scaling is introduced, allowing for a systematic analysis of different flow regimes. We prove the global-in-time existence and uniqueness of a bounded positive solution to Washburn's equation that includes the slip parameter, as well as the continuous dependence of the solution in the maximum norm on the initial data. Thus, the initial-value problem for Washburn's equation is shown to be well-posed in the sense of Hadamard. Additionally, we show that the unique equilibrium solution may be reached either monotonically or in an oscillatory fashion, similarly to the no-slip case. Finally, we determine the basin of attraction for the system, ensuring that the equilibrium state will be reached from the initial data we impose. These results hold for any positive value of the nondimensional slip parameter in the model, and for all values of the ratio $h_0/h_e$ in the range $[0,3/2]$, where $h_0$ is the initial height of the fluid column and $h_e$ is its equilibrium height.

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

TL;DR

The paper derives a Washburn-type capillary-rise equation that incorporates a slip boundary condition via a slip parameter , starting from first principles and a cylindrical geometry. Through nondimensionalization, it obtains the dimensionless model with equilibrium height , and provides a rigorous analysis establishing global existence, uniqueness, and continuous dependence of solutions for initial heights with . It further analyzes long-time behavior by recasting the problem as a 2D autonomous system, identifying a critical threshold that distinguishes monotone and oscillatory approaches, and proving convergence to the equilibrium via a Lyapunov function and LaSalle’s invariance principle. The results show that while the slip parameter shapes transient dynamics, the unique equilibrium height and global convergence are preserved for all positive and , providing a solid mathematical foundation for capillary rise with slip and guiding future numerical and physical investigations.

Abstract

The aim of this paper is to extend Washburn's capillary rise equation by incorporating a slip condition at the pipe wall. The governing equation is derived using fundamental principles from continuum mechanics. A new scaling is introduced, allowing for a systematic analysis of different flow regimes. We prove the global-in-time existence and uniqueness of a bounded positive solution to Washburn's equation that includes the slip parameter, as well as the continuous dependence of the solution in the maximum norm on the initial data. Thus, the initial-value problem for Washburn's equation is shown to be well-posed in the sense of Hadamard. Additionally, we show that the unique equilibrium solution may be reached either monotonically or in an oscillatory fashion, similarly to the no-slip case. Finally, we determine the basin of attraction for the system, ensuring that the equilibrium state will be reached from the initial data we impose. These results hold for any positive value of the nondimensional slip parameter in the model, and for all values of the ratio in the range , where is the initial height of the fluid column and is its equilibrium height.

Paper Structure

This paper contains 18 sections, 7 theorems, 136 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Derivation of the model
  3. Modelling assumptions
  4. Local form of the balance laws
  5. Flow field and mean velocity
  6. Global form of the momentum balance law
  7. Dimensional analysis and model reduction
  8. Global existence and uniqueness
  9. Stability and convergence to the equilibrium solution
  10. Conclusions
  11. Local forms of balance laws
  12. Global forms of balance laws and Washburn's equation
  13. Domain
  14. Mean velocity
  15. Momentum balance law in global form
  16. ...and 3 more sections

Key Result

Lemma 4.1

The initial-value problem can be transformed via $u(s) := \frac{1}{2}H(T)^{2}$ with $s := \frac{T}{\sqrt{\omega}}$ into an initial-value problem with $\beta := \left(1 + \frac{4L}{R}\right)^{-1}$, $\alpha \geq 0$, and the independent variable $s \in [0,\infty)$.

Figures (2)

  • Figure 1: Capillary pipe with Poiseuille velocity profile and column height $h(t)$ at time $t$.
  • Figure 2: Flow regimes for different values of the slip parameter.

Theorems & Definitions (12)

  • Lemma 4.1
  • Theorem 4.2
  • proof
  • Theorem 5.1
  • proof
  • Corollary 5.2
  • proof
  • Lemma 5.3
  • proof
  • Theorem 5.4
  • ...and 2 more