A hyperbolic model for two-layer thin film flow with a perfectly soluble anti-surfactant

TL;DR

This work studies the dynamics of a two-layer thin-film flow under the influence of a perfectly soluble anti-surfactant ($K=0$), modeling surface-tension variations that drive Marangoni-type flows. By a lubrication limit and neglecting capillarity and diffusion, it yields a first-order hyperbolic system for $(f,b,g,q)$ and proves strict hyperbolicity on the physically relevant set $fb<gq$; an entropy/entropy-flux framework is developed, including a strictly convex entropy that yields local well-posedness via Friedrichs-symmetrization. Using Riemann invariants, the authors construct an exact Riemann solver and a Godunov-type scheme, leveraging a Temple-like property of the fourth field to obtain explicit wave curves. Numerical experiments illustrate accurate Riemann-solution-based simulations and smooth-profile evolution, demonstrating the practical viability and potential for extensions to more complex multi-layer flows.

Abstract

We consider the motion of a two-phase thin film that consists of two immiscible viscous fluids and is endowed with an anti-surfactant solute. The presence of such solute particles induces variations of the surface tension and interfacial stress and can drive a Marangoni-type flow. We first analyze a lubrication limit and derive one-dimensional evolution equations of film heights and solute concentrations. Then, under the assumption that the capillarity and diffusion effects are negligible and the solute is perfectly soluble, we obtain a conservative first-order system in terms of film heights and concentration gradients. This reduced system is found to be strictly hyperbolic for a certain set of states and to admit an entire class of entropy/entropy-flux pairs. We also provide a strictly convex entropy for the hyperbolic system. Thus, the well-posedness for the Cauchy problem is given. Moreover, the system is almost a Temple-class system which allows to provide explicit solutions of the Riemann problem. The paper concludes with some numerical experiments using a Godunov-type finite volume method which relies on the exact Riemann solver.

Figures (7)

• Figure 1: Two-phase thin film flow under the influence of an anti-surfactant.
• Figure 2: Evolution of primitive variables at $t=1.00$ using Godunov solver and Lax-Friedrichs scheme with $\Delta x=4.37\times 10^{-2}$.
• Figure 3: Evolution of primitive variables at $t=1.00$ using the Godunov scheme with $\Delta x= 4.37\times 10^{-2}$.
• Figure :
• Figure :

Theorems & Definitions (9)

• Remark 2.1
• Remark 3.1
• Lemma 4.1
• proof
• Theorem 4.2: Entropy/entropy-flux pairs
• proof
• Theorem 4.3: Convex entropy
• proof
• Remark 4.4