1917] ANALOGY BETWEEN CONDUCTION OF HEAT AND MOMENTUM 487
with two similar equations, where
D d d d d
TU — ~n + u ~r + v j~ + w T > JJt dt ax dy dz
representing differentiation with respect to time when a particle of the fluid is followed.
In like manner, the equation for the conduction of heat is
Although we identify the values of k and v, and impose the same boundary conditions upon u and 6, we see that the same values will not serve for both u and 6 in the interior of the fluid on account of the term in dp/dx, which is not everywhere zero.
It is to be observed that turbulent motion is not steady in the hydro-dynamical sense, and that a uniform regime can be spoken of only when we contemplate averages of u and 6 for all values of so or for all values of t. It is conceivable that, although there is no equality between the passage of heat and the tangential traction at a particular time and place, yet that the average values of these quantities might still be equal. This question must for the present remain open, but the suggested equality does not seem probable.
The principle of similitude may be applied in the present problem to. find a general form for H, the heat transmitted per unit area and per unit time (compare Nature, Vol. xcv. p. 67, 1915)*. In the same notation as there used, let a be the distance between the planes, v the mean velocity of the stream, 0 the temperature difference between the planes, K the conductivity of the fluid, c the capacity for heat per unit volume, v the kinematic viscosity. Then
fT — -- F faVC a ' \ K '
or, whiph comes to the same,
icO fav cv
JDL — — . JP i ~~~ , —
a \ v • K where F, FI denote arbitrary functions of two variables. When v =
For a given fluid CV/K is constant and may be omitted. Dynamical similarity is attained when av is constant, so that a complete determination of F (experimentally or otherwise) does not require the variation of both a and v. There is advantage in keeping a constant; for if a be varied, geometrical similarity demands that any roughnesses shall be in proportion.
The objection that K, c, v are not constants, but functions of the temperature, may be obviated by supposing that 6 is small.
[* This volume, p. 300.] fi & )