Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/101

velocity varies as ${\frac {\delta p}{\delta l}}$, or, resolving into its three co-ordinate components, we have—

$u,\ v,\ w,=c\ {\frac {\delta p}{\delta x}},\ c\ {\frac {\delta p}{\delta y}},\ c\ {\frac {\delta p}{\delta z}},$

where $c$ is a constant.

In the above expression the density of the fluid and the magnitude of the applied force are involved in the constant $c.$ It is, however, evident that we may regard the form of flow as a matter of pure kinematics, since the existence of the flow is not dependent upon the pressure system by which it is generated. Consequently we may substitute for $p$ a function $\phi ,$ which has no dynamic import, and which is termed velocity potential, and we may write the expression—

$u,\ v,\ w,={\frac {\delta \phi }{\delta x}},\ {\frac {\delta \phi }{\delta y}},\ {\frac {\delta \phi }{\delta z}},$

the terms on the right-hand side of this equation being sometimes written with a minus sign.

In the foregoing illustration $\phi$ is a single-valued function, inasmuch as it can have a definite value assigned for every point in the field of flow.

 

§ 61. Flux ($\psi$ Function), $\phi$ and $\psi$ interchangeable.—In cases of fluid motion in which a velocity potential exists the lines of flow are, as pointed out, everywhere normal to the equipotentials, that is to say the surfaces of $\phi =$ constant. It can be shown analytically that if the curves of flow be plotted for equal increments of flux (that is, so that the amount of fluid that flows per unit time past any point and between two adjacent lines is constant), and the curves $\phi =$ constant be plotted over the same field, the two series of lines will divide the field into a number of similar elements whose ultimate form in the case of motion in two dimensions, when the units employed are sufficiently small, becomes square within any desired degree of approximation. Thus (Fig. 33) let $e\ e$ be two lines of flow, and $f\ f$ be two lines