Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/76

§ 39 and adapted for motion under water, will, at a velocity proportioned to the square root of its linear dimension, that is $$\tfrac{1}{2.4}$$th the full scale velocity, give rise to a geometrically similar disturbance in the fluid, and will itself undergo geometrically similar disturbance, and density for density the resistance will be proportional to the cube of the linear dimension—that is to say, in the ratio of $$\tfrac{1}{196}$$ of the full scale model; or, taking count of the relative density of air and water, the resistance of the smaller model will be approximately four times that of the greater.

§ 40. Energy Relation.—In all cases of purely viscous resistance the law of viscosity requires that the resistance shall vary directly as the velocity; and the whole of the energy expended disappears at once into the thermodynamic system. In cases where the resistance is dynamic—that is to say, where it is due to the continuous setting of new masses of the fluid in motion—the whole of the energy expended remains in the fluid in the kinetic form (being only subsequently frittered away), and the resistance varies as the square of the velocity. Where the resistance is due to both causes combined, as in the case of skin friction, the portions of the total resistance varying directly, and as the square, are respectively proportional to the energy expended in the two directions.

Now for any particular velocity, the total resistance—that is, the sum of the viscous and dynamic resistances—may be expressed as varying as the $$n$$ th power of the velocity; it is not necessary that the value of $$n$$ should be constant over the whole range of the $$R\ V$$ curve; it may be a quantity varying as a function of $$V ,$$ the form of which is unknown; but, for the particular value of $$V$$ chosen we have

$$\frac{dR}{dV} = n\ V^{(n-1)}, \quad\mbox{or}\quad = n\ \frac{R}{V} .$$