J.C. Maxwell's 1879 Paper on Thermal Transpiration and Its Relevance to Contemporary PDE

In his 1879 PRSL paper on thermal transpiration J.C.MAXWELL addressed the problem of steady flow of a dilute gas over a flat boundary. The experiments of KUNDT and WARBURG had demonstrated that if the boundary is heated with a temperature gradient , say increasing from left to right, the gas will flow from left to right. On the other hand MAXWELL using the continuum mechanics of his (and indeed our) day solved the ( compressible) NAVIER- STOKES- FOURIER equations for balance of mass, momentum, and energy and found a solution: the gas has velocity equal zero. The Japanese fluid mechanist Y. SONE has termed this the incompleteness of fluid mechanics. In this talk I will sketch MAXWELL's program and how it suggests KORTEWEG's 1904 theory of capillarity to be a reasonable “ completion” of fluid mechanics. Then to push matters in the analytical direction I will suggest that these results show that HILBERT's 1900 goal expressed in his 6th problem of passage from the BOLTZMANN equation to the EULER equations as the KNUDSEN number tends to zero in unattainable.