Numerical confirmation of the infinitely soluble front-scaling law

Confirm numerically that the surfactant-front location for the infinitely soluble surfactant thin-film equations satisfies the late-time scaling r_f proportional to t, and thereby verify the associated exponential thinning behavior over sufficiently long times.

Background

For infinitely soluble surfactants, the paper argues that the bulk concentration remains order one behind the propagating front, producing a stronger Marangoni stress than in the insoluble case. The authors infer from Rankine–Hugoniot jump conditions that the front location should scale as r_f proportional to t, while the minimum film thickness decays exponentially in time.

The numerical simulations available to the authors could not be run for sufficiently long times to test this inferred front-scaling law. Consequently, the late-time linear-in-time front propagation remains numerically unconfirmed in the axisymmetric, diffusion-free formulation studied here.

References

However, we were not able to run the numerical simulations longer in our code to numerically confirm this scaling. A code specifically designed to handle exponential behaviour, such as that developed by , should be able to give numerical solutions at later times.

— The effects of surfactant solubility on inertial Marangoni flow: theory and numerics  (2609.10337 - Eshima et al., 9 Sep 2026) in Section 2.2, subsection “Infinitely soluble surfactants”