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

TL;DR

This paper extends Washburn's capillary rise model by including slip boundary conditions, proving well-posedness, analyzing long-term behavior, and characterizing the basin of attraction for solutions.

Contribution

It introduces a slip condition into Washburn's equation, establishes global existence and uniqueness, and analyzes the solution's long-time dynamics and stability.

Findings

Proved global-in-time existence and uniqueness of solutions.

Demonstrated solutions reach equilibrium monotonically or oscillatory.

Identified the basin of attraction for the equilibrium state.

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.