Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/100

§ 60 direction of the lines of force being everywhere that of the initial acceleration of the particles.

When such a system is initiated in a fluid from rest, at the instant the force is applied the surfaces of equal pressure are everywhere normal to the lines of force. This is not necessarily the case when the fluid is in motion, for we have then superposed pressure differences due to the change in velocity and direction of the particles which modify the pressure distribution.

Let us suppose that the applied force is impulsive, i.e., let it be considered to be an infinite force applied for an infinitely short time; then the form of flow generated will be that due to the initial application of the force, that is to say the field of flow will coincide with the field of force.

Now it does not obviously follow that this form of flow will be stable or permanent. In actual fluids, such as water or air, we know in fact that it is not so. It would, however, appear that in the ideal fluid of hypothesis any form of motion generated by an impulse in this manner will persist without change of form, and therefore the field of force and system of pressure by which the flow is generated may be taken as defining the form of flow for the steady state.

Under these circumstances it is evident that the motion will be the same whether generated by an impulse or by a finite force, since the continued application of the force to the body in motion will accelerate the field everywhere in the line of flow.

If we now examine the initial pressure system, then the velocity produced on the fluid from rest along any line of force after a brief interval of time will be, for any small difference of pressure, inversely as the mass per unit section, that is, inversely as the distance separating, the points at which the said pressure difference exists. Or if $$\delta p$$ is the pressure increment, and $$\delta l$$ the distance along the line of force, the velocity after a certain brief interval of time will be everywhere proportional to $$\frac{\delta p}{\delta l}$$ or, when the increments are taken as evanescent