Integral modelling of weakly evaporating 3D liquid film with variable substrate heating

TL;DR

This work develops and validates a three-dimensional Weighted Integral Boundary-Layer (WIBL) reduced-order model for weakly evaporating/condensing liquid films on inclined substrates with spatially and temporally varying heating. Built on a second-order long-wave expansion, the model yields three coupled equations for film thickness, streamwise/spanwise flow rates, and free-surface temperature, and incorporates phase-change with vapour recoil and thermocapillary effects. Compared with full Navier–Stokes and Orr–Sommerfeld analyses, the WIBL framework provides accurate linear stability predictions (up to moderate Re) and nonlinear dynamics within a few percent, while reducing computational cost by orders of magnitude; nonlinear 3D simulations reveal how substrate heating modulates dry-out and wave structures. The approach enables extensive parametric studies and control-oriented design of substrate-temperature strategies to enhance heat transfer and delay dry-out in practical thermal-management systems.

Abstract

Analysing the dynamics of phase-changing liquid films is essential for enhancing the performance of thermal management systems. Still, direct simulation of the full governing equations is computationally expensive. To circumvent this limitation, I derived a weighted-integral boundary-layer (WIBL) model under long-wave assumptions, weak evaporation, and strong surface tension, also accounting for variable substrate heating. In the linear regime, the WIBL reproduces growth rates and the cutoff wavenumber of unstable modes with significantly higher accuracy than commonly used Benney-type models for Re<40, as compared to the Orr-Sommerfeld equations. The linear analysis further reveals a threshold separating streamwise- and spanwise-dominated instabilities in hanging films, arising from the competition between Kapitza and Rayleigh-Taylor mechanisms; the WIBL predicts this threshold accurately for small Re and inclination angles. In the nonlinear regime, with substrate heating that varies in both space and time, the WIBL model captures the evolution of free-surface thickness and temperature within approximately 6% of the original Navier-Stokes equations. Three-dimensional simulations show that a condensing film undergoes dry-out due to Kapitza instability, whereas unsteady substrate heating promotes spanwise momentum spreading, modifies wave dynamics, and prevents dry-out. The WIBL model provides a good level of accuracy at a low computational cost, enabling extensive parametric studies, nonlinear stability analyses, and the design of optimal substrate-heating control strategies.

Figures (22)

• Figure 1: Scheme of the 3D evaporating/condensing falling liquid film flowing over an inclined substrate with temperature $\eta(x,z,t)$ in contact with its pure vapour phase at saturating pressure and temperature.
• Figure 2: Long-wave eigenfunctions $\nu_{k=0}^1$ (red contentious line) and $\nu_{k=0}^2$ (blue contentious line) and their polynomial approximations $\hat{\nu}_1$ (red dashed line with squares) and $\hat{\nu}_2$ (blue dashed line with circles) for $\overline{H}/\hbox{K}$ equal to (a) $10^{-3}$, (b) $16$ and (c) $10^{5}$.
• Figure 3: Critical Marangoni Reynolds number $\hbox{Re}_{M_c}$ for spanwise perturbations as a function of $\overline{H}$ and $\hbox{Ct}$ for $\hbox{Vr}=0.5$ in (a) evaporating ($\hat{\eta}=1$) and (b) condensing conditions ($\hat{\eta}=-1$).
• Figure 4: (a) Neutral Curve in the plane ($k_x,\hbox{Re}$) and (b) growth rate $\omega_i$ as a function of $k_x$ for $\hbox{Re}=20,40,60,80$ given by the WIBL equations (red dashed line with squares) and the Orr-Sommerfeld eigenvalue problem solved with long-wave expansion (blue dash-dotted line) and numerically (black continuos line) for vertical film with 2D streamwise perturbations with $\overline{H}=1$.
• Figure 5: Neutral curves in ($\hbox{Re},k_x)$ with $k_z=0$ obtained solving the linearized WIBL model (red dashed line with squares) and the Navier-Stokes equation numerically (continuos black line) with the long wave expansion (dash-dotted line with triangles) for $\hbox{Ma}=10$ and $\hbox{Vr}=4$, numerical solution of Navier-Stokes for $\hbox{Ma}=\hbox{Vr}=0$ for inclination angle $\beta$ equals to (a) 90° and (b) 15°.