40 ON THE MOTION OF SOLID BODIES THROUGH VISCOUS LIQUID [35
But a closer consideration will show, I think, that there is a substantis foundation for the idea at the basis of Lanchester's argument. If we suppos that the viscosity is so small that the layer of fluid affected by the passag of the blade is very small compared with the width (6) of the latter, it wi appear that the communication of motion at any stage takes place muc as if the blade formed part of an infinite plane moving as a whole. W know that if such a plane starts from rest wibh a velocity V afterward uniformly maintained, the force acting upon it at time t is per unit of are; see (12),
.................................. (50)
The supposition now to be made is that we may apply this formula to th element of width dy, taking t equal to yj V, where y is the distance of th element from the leading edge. Thus
(51)
which agrees with (48) if we take in the latter c = 2/vV.
The formula (51) would seem to be justified when v is small enough, £ representing a possible state of things ; and, as will be seen, it affords a absolutely definite value for the resistance. There is no difficulty in extendin it under similar restrictions to a lamina of any shape. If b, no longe constant, is the width of the lamina in the direction of motion at level . we have
(52)
It will be seen that the result is nob expressible in terms of the area, of th lamina. In (49) c is not constant, unless the lamina remains always simik in shape.
The fundamental condition as to the smallness of v would seem to I realized in numerous practical cases; but any one who has looked over tli side of a steamer will know that the motion is not usually of the kin supposed in the theory. It would appear that the theoretical motion : subject to instabilities which prevent the motion from maintaining its simp] stratified character. The resistance is then doubtless more nearly as tl: square of the velocity and independent of the value of v.
"When in the case of bodies moving through air or water we exprei V, a, and v in a consistent system of units, we find that in all ordinary cast v/Va is so very small a quantity that it is reasonable to identify f(v/V( with/(0). The influence of linear scale upon the character of the motic then disappears. This seems to be the explanation of a difficulty raised I Mr Lanchester (loc. cit. § 56).e argument as rigorous is that complete similarity cannot be secured so long as b is constant as has boon supposed. If, as ia necessary to this end, we take /; proportional to n, it is bV/n, or V (and not V/n), which varies as nVa, or &7a. The conclusion is then simply that bV must be constant (v being given). This is merely the UBiial condition of dynamical similarity, and no conclusion as to the law of velocity follows.