Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/127

velocity. Applying this to the case of stream motion, let $b$ be the distance between stream lines demarcating some definite increment of $\psi$, then we know that $v\propto {\frac {1}{b}}$, or $v^{2}\propto {\frac {1}{b^{2}}}$, that is to say, the energy per unit mass, is, in two-dimensional motion, inversely as the area of the square elements cut off by the $\psi$, $\phi$ lines. But the mass of fluid contained in these elements is directly as their area, or varies as $b^{2}$, consequently the kinetic energy in each element is proportional to $b^{2}\times {\frac {1}{b^{2}}}$, which is constant; therefore:

The kinetic energy contained in each element cut off by lines of equal increment of $\psi$ and $\phi$ is constant.

In a $\psi$, $\phi$ diagram, such as Fig. 48, the total kinetic energy is thus measured by the total number of squares, and the kinetic energy in any circumscribed region is equal to the number of squares in that region. In order to give an absolute value to the energy on this basis it is necessary that the quantity of energy in some particular square element should be known.

The kinetic energy in the field of flow round a body in motion is imparted to the fluid when the body is started from rest, and is given up when the motion is arrested. The effect of the fluid motion is thus to add to the apparent inertia of the body, so that a given force requires to act through a greater distance to impart a given velocity than for the same body in vacuo. Not only has a force to act for a greater distance, but also for a longer time, which means that the body possesses in effect a greater store of momentum for a given velocity. In reality, however, such increase of momentum is only apparent; the momentum of the body and fluid system combined is actually less than that of the body at the same velocity in vacuo by the amount due to its fluid displacement. That is to say, if the body be of the same specific gravity as the fluid the total dynamic system possesses no momentum at all, whatever the velocity. This apparent paradox is accounted for by the fact that during the period of application of force

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