Energy-consistent thermo-evaporative viscous pressure correction for oscillating droplets

Droplets oscillating in thermal or evaporating environments can lose energy to viscous damping or gain energy from surface-tension gradients, but existing viscous pressure-correction models do not provide a closed energy criterion for when thermocapillary work can amplify a capillary mode. We derive an energy-consistent thermo-evaporative generalised viscous pressure-correction (EC-GVPC) oscillator for a single small-amplitude spherical Legendre mode, coupling EC-GVPC damping, capillary stiffness, thermocapillary work, surface-temperature relaxation and slow D 2 -law shrinkage in one mechanical-energy balance. A surface-tension perturbation acts through two interfacial channels, a tangential Marangoni stress and a capillary-pressure modulation. Projecting both channels onto the modal velocity field gives the net thermocapillary coefficient B M = 4 π R σ T ( n − 1 ) / ( 2 n + 1 ) , which vanishes at n = 1 , as required for the translation mode. Harmonic averaging yields a Rayleigh-type neutral condition: growth requires the cycle-averaged thermocapillary work to exceed the EC-GVPC loss; at optimal thermal-relaxation tuning ( ω n τ T = 1 ) this reduces to the minimum imposed gradient | G R | min = 4 ( 2 n + 1 ) μ ω n / ( n | σ T | ) . The Rayleigh–Lamb frequency spectrum is recovered in the capillary–viscous limit, and the implementation is checked against the Sichani–Mehring two-dimensional elliptical blob benchmark. Against the DNS data of Reutzsch et al. modal frequencies differ by 0.36%–1.55%; the D 2 -law coupling predicts a mode-independent evaporative frequency shift, consistent with DNS values for n = 4 and n = 6 within 0.15 percentage points after calibration of the evaporation constant to the n = 2 mode. The model provides a reduced-order screening tool for thermocapillary damping and amplification in evaporating droplets and sprays, with the amplification branch constituting a closed-form prediction that is not yet validated against simulations or experiments.