of the plane is continually engaging with new masses of the fluid, and setting them in motion by virtue of the viscous stress exerted. But the conditions under which any given mass of fluid is acted on are not those of the previous hypothesis; the force resisting the motion of the plane is that of the inertia of the fluid itself; and if we confine our attention to any one portion of the fluid, its condition is not that of a steady state, but one of acceleration. Now it is evident that when the leading edge first enters the undisturbed region the stratum of fluid affected will be quite thin; and as the following portions of the plane successively traverse the same region the thickness of the stratum set in
Fig. 27.
motion continuously increases, and the velocity gradient will correspondingly diminish. This is illustrated in Fig. 27, in which the line a, a, a, represents the original position of a series of particles of the fluid at some given instant, and b, b, b, the position assumed after a short time has elapsed.
§ 34. Skin-friction.—Basis of Investigation.—Owing to the considerations dealt with in the preceding section, it is evident that we cannot regard skin-friction as of necessity amenable to the ordinary viscous law, i.e., F ∝ V where l and µ are constant; it is in fact easy to prove that this law will not apply.
Let us suppose for example that the motion of the fluid is strictly homomorphous in respect of changes of V, that is to say, if u, v, w, be the velocity of the fluid particles at any instant of
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