304 THE PRINCIPLE OF SIMILITUDE [392
[Nature, Vol. xcv. p. 644, Aug. 1915.]
The question raised by Dr Riabouchinsky (Nature, July 29, p. 105)* belongs rather to the logic than to the use of the principle of similitude, with which I was mainly concerned. Ib would be well worthy of further discussion. The conclusion that I gave follows on the basis of the usual Fourier equation for the conduction of heat, in which heat and temperature are regarded as sui generis. It would indeed be a paradox if further knowledge of the nature of heat afforded by molecular theory put us in a worse position than before in dealing with a particular problem. The solution would seem to be that the Fourier equations embody something as to the nature of heat and temperature which is ignored in the alternative argument of Dr Riabouchinsky.
[19IT. Reference may be made also to a letter signed J. L. in the same number of Nature, and to Nature, April 22, 1915. See further Buckingham, Nature, Vol. xcvi. p. 396, Dec. 1915. Mr Buckingham had at an earlier date (Oct. 1914) given a valuable discussion of the whole theory (Physical Review, Vol. iv. p. 345), and further questions have been raised in the same Review by Tolman.
As a variation of the last example, we may consider the passage of heat between two infinite parallel plane surfaces maintained at fixed temperatures differing by 0, when the intervening space is occupied by a stream of incompressible viscous fluid (e.g. water) of mean velocity v. In a uniform regime the heat passing acr.oss is proportional to the time and to the area considered ; but in many cases the uniformity is not absolute and it is necessary to take the mean passage over either a large enough area or a long enough time. On this understanding there is a definite quantity h', representing the passage of heat per unit area and per unit time.
If there be no stream (v= 0), or in any case if the kinematic viscosity (v) is infinite, we have
h' = tc6/a,
a being the distance between the surfaces, since then the motion, if any, takes place in plane strata. But when the velocity is high enough, or the viscosity low enough, the motion becomes turbulent, and the flow of heat may be greatly augmented. With the same reasoning and with the same notation as before we have
,_K0 -pfavc cv\ i . U { , 1 , a \ K K }
* "In Nature of March 18, Lord Kayleigh gives this formula h = Ka0 .F (avc/ic), considering heat, temperature, length, and time as four ' independent ' units. If we suppose that only three of these quantities are really independent, we obtain a different result. For example, if the temperature is defined as the mean kinetic energy of the molecules, the principle of similarity allows us only to affirm that 7i=/ca0 . F (v/Ğa2, ca3)."1892); Scientific Papm, Vol. iv. j>. Ifl. t Ann. der Plnjsik, Vol. xx. p. 848 (1906).