Possible Flow-Dependent Temperature Gradients in He Films

Various experiments, especially recent experiments by Keller, clearly demonstrate that the thickness of a helium film is independent of the flow rate $Q$. This is consistent with our current picture of the hydrodynamics of a helium film if there are temperature gradients in the film which produce a thermodynamic potential ${\varphi }_T=\int \mathrm{SdT}$ that cancels the $\frac{{\rho }_S{v_S}^2}{2\rho }$ term in the hydrodynamic potential $\mu$. If such temperature gradients exist, they would produce unacceptably large evaporation rates unless the saturated vapor pressure $p_{\mathrm{vap}}$ is a function of $\mu$ rather than $T$. If $p_{\mathrm{vap}}=p_{\mathrm{vap}}\left(\mu \right)$, then there should be no thinning of the film with changes in $Q$, and the theory is in complete agreement with Keller's results.