Balanced scales for a thermocapillary film (${ϵ}^2=3B{h}_{\ast }/(2{\gamma }_0),\hat{t}=4\mu {\gamma }_0/(3B^2{h}_{\ast })$)

Write the surface tension as ${\gamma }_s={\gamma }_0+Bh$, where $B>0$. The thermocapillary thin-film equation has flux ${\gamma }_0{h}^3{h}_{xxx}/(3\mu )+B{h}^2{h}_x/(2\mu )$. Balancing both terms at thickness scale $h_{\ast }$ gives aspect ratio ${ϵ}^2=3B{h}_{\ast }/(2{\gamma }_0)$ and time scale $4\mu {\gamma }_0/(3B^2{h}_{\ast })$. The fractional surface-tension variation is $B{h}_{\ast }/{\gamma }_0=2{ϵ}^2/3$, so using constant tension in the leading capillary pressure term is consistent with the balanced long-wave limit.