Integral modelling of weakly evaporating three-dimensional liquid film with variable-substrate heating

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 the growth rates and cutoff wavenumbers of unstable modes with significantly higher accuracy than commonly used Benney-type models for Reynolds number italic Re less than 40 Re < 40 ${{\\textit{Re}}}\\lt 40$ , and with accuracy comparable to that of 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 the Kapitza and Rayleigh–Taylor mechanisms; the WIBL accurately predicts this threshold for small italic Re Re ${\\textit{Re}}$ and inclination angles. In the nonlinear regime, with space- and time-varying substrate heating, the WIBL model captures the evolution of the free-surface thickness and temperature with an accuracy of approximately 6 % relative to 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.