Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/111

Rh to the network boundary system and the wheels with the fluid in the meshes.

§ 70. Irrotational Motion in its Relation to Velocity Potential.—We have above defined irrotational motion as follows:—

The motion of a fluid is irrotational when the sum of the circulation round a complete circuit of the boundary of its each small element is zero.

Assuming this definition, it can be shown that fluid in irrotational motion has a velocity potential.

Let (Fig. 38) the cell $$a\ b\ c\ d$$ be any small element of the fluid in which $$a\ b$$ and $$c\ d$$ are lines of flow and $$a\ c$$ and $$b\ d$$ are normals thereto. Then since the motion in the line of $$a\ c$$ and $$b\ d$$ is nil, the circulation round the circuit is the sum of the circulations along $$a\ b$$ and $$c\ d$$, and since that motion is irrotational, this quantity is zero.

(For the sake of simplicity the axis of $$x$$ has been chosen in the direction of the flow.)

Then let us take two columns of the fluid along the lines $$a\ b$$ and $$c\ d$$ respectively, whose section is defined as $$\delta y \times \delta z$$, then if $$\rho =$$ density, we have masses of the two columns $$x_1\ \rho\ \delta y\ \delta z$$ and $$x_2\ \rho\ \delta y\ \delta z$$ respectively. But their velocities $$u_1$$ and $$u_2$$ are connected by the relationship $$u_1\ x_1 = u_2\ x_2$$, or $$\frac{u_1}{u_2} = \frac{x_2}{x_1}$$. The momenta of the two columns are therefore in the relation $$x_2\ x_1\ \rho\ \delta y\ \delta z$$ is to $$x_1\ x_2 \circ \delta y\ \delta z$$, which are equal; consequently, if a certain force applied to any column for a time $$t$$ will bring it to rest, the same force applied for the same time to the other column will bring that to rest also. But the areas of the columns are equal; therefore to stop or to reproduce the motion