Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/70

§ 34 time in the direction of co-ordinates x, y, z, travelling with the body (or plane), we are supposing that any variation in V results in a variation in like ratio of u, v, w, for all values of x, y, z.

In order to simplify the thinking in connection with this problem it is convenient to suppose the body to be a plane travelling in the direction of the axis of x, and confine our attention to the motion of the fluid taking place in like direction. Let y be taken as the axis at right angles to the plane.

The viscous stress at every point will be proportional to, the velocity gradient, that is $du⁄dy$, which on the present supposition varies as V for every point x, y, z, in the region, consequently we shall have F ∝ V, which is the viscous law. Now if the prescribed conditions satisfy the dynamic requirements of the problem, we might conclude that the motion is strictly homomorphous, and that the viscous law obtains, but such is not the case. The momentum communicated per second to any given layer of the fluid, and therefore to the whole fluid, is = mass × the velocity per second imparted, that is ∝ u V ∝V2; so that under strictly homomorphous conditions the viscous stress cannot be satisfied for varying speeds by the inertia of the fluid.

If, when the velocity V increases, we suppose that the layer of fluid affected to any given degree becomes thinner, and vice versa, it is clear that the viscous forces will rise in a greater ratio than directly as V, for the velocity gradient will be steeper, also the inertia forces will be less, for the mass of fluid to be set in motion will be less. It is, therefore, evident that we may suppose the thickness of the affected strata varies with the velocity in the degree necessary to preserve a balance between the viscous and inertia forces.

§ 35. Law of Skin-friction.—Let us suppose that in any two systems, differing only as to velocity V, the whole region be divided into strata by an imaginary series of equidistant planes, so that the thickness of corresponding strata in the two systems,