2. Navier slip regularisation: motivation, challenge, solution

In their recent paper, Luo & Gao (Reference Luo and Gao2025) set out to provide a closed-form theory for moving contact lines, using the Navier slip boundary condition as a regularisation mechanism. The physical reality of Navier slip and the experimental determination of the slip length have been unambiguously established since the early 2000s (Bocquet & Charlaix Reference Bocquet and Charlaix2010). Remarkably, for simple liquids it was found that the continuum description remains valid down to the scale at which slip occurs, which can be as small as a few molecular sizes. These developments provide a solid foundation for slip in wetting flows.

In spite of this long history, an explicit theory of moving contact lines with slip has remained elusive. The difficulty can be traced back to the nature of Huh & Scriven’s wedge flow solution. In a polar coordinate system $(r,\varphi )$ that is co-moving with the contact line, wedge flow is described by a scale-invariant streamfunction:

(2.1) \begin{equation} \psi (r,\varphi ) = \textit{Ur}^a f(\varphi ), \end{equation}

where $a$ and $f(\varphi )$ follow from boundary conditions. Without slip the fluid velocity along the plate is uniform, which fixes the exponent to $a=1$ . The introduction of a slip length $\lambda$ , however, breaks the scale invariance and is not compatible with (2.1). Explicit solutions with the Navier slip boundary conditions are only available for special cases. Figure 1(b) shows the fluid velocity at the wall in the presence of slip for the special case of $\theta =90^\circ$ (Hocking Reference Hocking1976). The wall velocity is not uniform: a stagnant zone appears near the contact line and the no-slip limit is only recovered at distances $r \gg \lambda$ .

The key idea by Luo & Gao (Reference Luo and Gao2025) is to introduce an auxiliary ‘slippery wedge flow’ problem that is enforced to be compatible with (2.1). To achieve this, they introduce a variable slip length $\lambda = \beta r$ , where $\beta$ is a constant. This modified boundary condition indeed admits a family of solutions with $a=1$ , parametrised by $\beta$ . Subsequently, the analysis returns to the true Navier slip boundary condition with constant $\lambda$ : for a given distance to the contact line, the effective value of $\beta$ is estimated such that locally the Navier slip boundary condition is satisfied. Strictly speaking, this local approximation scheme does not provide an exact solution. However, Luo & Gao verified against direct numerical simulations that this scheme provides an extremely accurate approximation of the flow. Here, I further illustrate the excellent quality of the proposed slippery wedge flow by a direct comparison of the wall velocity with the exact solution for $\theta =90^\circ$ (figure 1 b).

Their discovery of an accurate representation of the slip flow enabled Luo & Gao to derive closed-form expressions for $Q$ in (1.2), for arbitrary contact angles and arbitrary viscosity ratios. In addition, they showed how to consistently introduce slip in the generalised lubrication equation. When the outer viscosity is negligible, this amounts to a simple modification of (1.1):

(2.2) \begin{equation} \frac {{\rm d}^2\theta }{{\rm d}s^2} = \frac {\textit{Ca} \, F(\theta )}{h\left [h+c \lambda \right ]}, \end{equation}

a form that was previously hypothesised by Chan et al. (Reference Chan, Srivastava, Marchand, Andreotti, Biferale, Toschi and Snoeijer2013, Reference Chan, Kamal, Snoeijer, Sprittles and Eggers2020). Luo & Gao’s slippery wedge flow gives the simple prediction $c=F(\theta )$ , which compares well with previous numerical determination. Only for very large angles ( $\theta \to 180^\circ$ ) does the slippery wedge flow lose its accuracy; in this limit one can resort to asymptotic expressions (Chan et al. Reference Chan, Kamal, Snoeijer, Sprittles and Eggers2020).

3. Future

The Cox–Voinov theory has been a cornerstone for the hydrodynamics of wetting flows. This formalism can now be used without adjustable parameters owing to the closed-form expressions derived by Luo & Gao. This paves the way for fully quantitative numerical schemes where the expensive nanoscale resolution can be replaced by the asymptotic Cox–Voinov solution as an effective boundary condition, or even resolve the smallest scales down to $h\to 0$ using (2.2).

Looking ahead, there are numerous developments that reach beyond the viscous-wedge description of wetting. From a hydrodynamic perspective, technologies often operate at velocities for which neither the Reynolds number nor the capillary number are small, or can involve non-Newtonian fluids. In the last few decades attention has shifted from the liquid to the substrate, with the aim of designing surfaces with specific functionality (Quéré Reference Quéré2008; Hardt & McHale Reference Hardt and McHale2022). Examples are patterned substrates to tune hydrophobicity, lubricant-infused substrates to enhance slipperyness or adaptive substrates that respond to the presence of the wetting fluid. The latter typically involves polymeric substrates that can deform elastically (Style et al. Reference Style, Jagota, Hui and Dufresne2017; Andreotti & Snoeijer Reference Andreotti and Snoeijer2020), and the resulting contact line speed can be tuned by the degree of cross-linking or swelling (Hourlier-Fargette et al. Reference Hourlier-Fargette, Antkowiak, Chateauminois and Neukirch2017). Additional microscopic phenomena like phase separation (Hauer et al. Reference Hauer, Cai, Skabeev, Vollmer and Pham2023; Qian et al. Reference Qian, Zhao, Qian and Xu2024) or spontaneous charging (Li et al. Reference Li, Ratschow, Hardt and Butt2023) can start to play a dominant role. As such, moving contact lines will continue to serve as macroscopic probes of nanoscale physics.