Reynolds, Maxwell, and the Radiometer, Revisited
Authors: Martin Holger 
Published: March 2011

In 1969, S. G. Brush and C. W. F. Everitt published a historical review, that was reprinted as subchapter 5.5 Maxwell, Osborne Reynolds, and the radiometer, in Stephen G. Brush’s famous book The Kind of Motion We Call Heat. This review covers the history of the explanation of the forces acting on the vanes of Crookes radiometer up to the end of the 19th century. The forces moving the vanes in Crookes radiometer (which are not due to radiation pressure, as initially believed by Crookes and Maxwell) have been recognized as thermal effects of the remaining gas by Reynolds — from his experimental and theoretical work on Thermal Transpiration and Impulsion, in 1879 — and by the development of the differential equations describing Thermal Creeping Flow, induced by tangential stresses due to a temperature gradient on a solid surface by Maxwell, earlier in the same year, 1879. These fundamental physical laws have not yet made their way into the majority of textbooks of heat transfer and fluid mechanics so far. A literature research about the terms of Thermal Transpiration and Thermal Creeping Flow, in connection with the radiometer forces, resulted in a large number of interesting papers; not only the original ones as mentioned in subchapter 5.5 of Brush’s book, but many more in the earlier twentieth century, by Martin Knudsen, Wilhelm Westphal, Albert Einstein, Theodor Sexl, Paul Epstein and others. The forces as calculated from free molecular flow (by Knudsen), increase linearly with pressure, while the forces from Maxwell’s Thermal Creeping Flow decrease with pressure. In an intermediate range of pressures, depending on the characteristic geometrical dimensions of flow channels or radiometer vanes, an appropriate interpolation between these two kinds of forces, as suggested by Wilhelm Westphal and later by G. Hettner, goes through a maximum. Albert Einstein’s approximate solution of the problem happens to give the order of magnitude of the forces in the maximum range. A comprehensive formula and a graph of the these forces versus pressure combines all the relevant theories by Knudsen (1910), Einstein (1924), Maxwell (1879) (and Hettner (1926), Sexl (1928), and Epstein (1929) who found mathematical solutions for Maxwells creeping flow equations for non-isothermal spheres and circular discs, which are important for thermophoresis and for the radiometer). The mechanism of Thermal Creeping Flow will become of increasing interest in micro- and submicro-channels in various new applications, so it ought to be known to every graduate student of heat transfer in the future. That’s one of the reasons why some authors have recently questioned the validity of the classical Navier-Stokes, Fourier, and Fick equations: Dieter Straub (1996) published a book on an Alternative Mathematical Theory of Non-equilibrium Phenomena. Howard Brenner (since 2005) wrote a number of papers, like Navier-Stokes, revisited, and Bi-velocity hydrodynamics, explicitly pointing to the forces acting on the vanes of the lightmill, to thermophoresis and related phenomena. Franz Durst (since 2006) also developed modifications of the classical Navier-Stokes equations. So, Reynolds, Maxwell, and the radiometer may finally have initiated a revision of the fundamental equations of thermofluiddynamics and heat- and mass transfer.