Description
Capillary-driven flow is a key transport mechanism in many microfluidic systems, graphic products, porous medium, and biological microchannels. Classical models of capillary rise, most notably Washburn’s equation, have been widely used to describe the movement of the liquid column in narrow tubes. This nonlinear second-order ordinary differential equation predicts that the equilibrium height (Jurin’s height) can be approached either monotonically or through damped oscillations, a behavior that has been confirmed by experiments.
In this work, we extend the classical Washburn’s model by introducing a slip boundary condition at the channel wall. The introduction of a slip parameter is motivated by experimental observations, where the no-slip assumption may fail, particularly during the initial stages of liquid motion. Starting from the fundamental conservation laws of mass and momentum, we derive a physically consistent model that includes inertial effects in addition to capillary, viscous, and gravitational forces. The explicit presence of inertia represents a key difference from simplified models that are commonly used in microfluidic applications.
After appropriate scaling, the governing equation relies on a single dimensionless parameter. This formulation allows systematic study of different flow regimes and provides a basis for stability analysis. Using linearization around the equilibrium height, we show that, in the presence of slip, the system reaches the equilibrium state either monotonically or oscillatory, in analogy with the classical no-slip case. These results, however, are valid only locally, as the analysis assumes initial conditions close to equilibrium.
To establish convergence to the equilibrium height for a broader range of initial conditions, we perform a nonlinear stability analysis by introducing a Lyapunov function based on energy estimate. This approach allows us to rigorously prove asymptotic stability and to determine the basin of attraction of the equilibrium state. The results provide insight into the global dynamics of capillary rise with slip and are relevant for the design and interpretation of microfluidic experiments where wall slip and inertial effects cannot be neglected.
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.
I. Rapajić, S. Simić, E. Süli, Modeling capillary rise with a slip boundary condition: Well-posedness and long-time dynamics of solutions to Washburn’s equation, Physica D: Nonlinear Phenomena 481 (2025) 134842.