Fig. 18. Automatic relief valve.
The
stiffness of spring 2
required
for an upward displacement of the piston
=
10 mm is
CH APT E R III
Relative rest of a liquid
10. Basic concepts
In the previous chapter we considered the equilibrium of a liquid subjected to the action of only one body force, gravity. This is the case of a liquid at rest in a vessel that is motionless or is moving uniformly in a straight line with respect to the earth.
If a vessel containing a liquid is in nonuniform or nonrectilinear motion, all the particles of the liquid are subjected, in addition to gravity, to the forces developed by accelerated motion. These forces tend to displace the fluid in its vessel in some way, and if they are uniform in time the fluid occupies a new equilibrium position, i. e., it comes to rest relative to the walls of the vessel. This case of fluid equilibrium is called relative rest.
The free surface, as well as the other surfaces of equal pressure (see Sec. 6) of a liquid relatively at rest may differ substantially from the surfaces of equal pressure of a liquid in a motionless vessel, which are horizontal planes. In determining the shape and position of the free surface of a liquid in relative rest conditions, the basic property of any surface of equal pressure must be taken into account, namely, that the resultant body force is always normal to the surface of equal pressure.
If the resultant body force were directed at an angle to the surface of equal pressure its tangential component would displace the liquid particles along that surface. But in relative rest the particles of a liquid are at rest with respect to the walls of the vessel and to each other. Henoe, the only possible direction of the resultant body force is normal to the free surface and the other surfaces of equal pressure.
Furthermore, no two surfaces of equal pressure can intersect, as otherwise points on the line of intersection would be simultaneously subjected to two different pressures, which is impossible.
We shall consider two characteristic cases of relative rest of a liquid: (i) the vessel moves with uniform acceleration in a straight line, and (ii) the vessel rotates uniformly about a vertical axis.
11. Liquid in a vessel moving with uniform acceleration in a straight line
Let a vessel containing a liquid, say an aircraft fuel tank, be mov->g in a straight line with a uniform acceleration a. The resultant ody force acting on the liquid is found as the vector sum of the eleration force, which is directed opposite the acceleration a, ii J the force of gravity (Fig. 19).
Denoting the resultant body force referred to unit mass by the symbol j, we have
The resultant body forces of all the particles of the liquid volume under consideration are parallel, and the surfaces of equal pressure are perpendicular to the forces. Hence, all the surfaces of equal pressure, including the free surface, are parallel planes. Their angle of inclination to the horizontal is determined from the condition of their perpendicularity to force j.
I
n
order to determine the position of a liquid free surface in avessel
moving with uniform acceleration in a straight line, the foregoing
condition must be supplemented by the equation of volumes, i.
e.,
we must know the volume of the liquid in the vessel in terms of
the dimensions B
and
H
of
the vessel and the initial level h
of
the liquid.
The equation for determining the pressure at any point of the fluid volume can be developed in the same way as in Sec. 6.
Take at a point M an area dS parallel to the free surface and erect normal to the free surface a cylindrical volume with dS as its face. The condition of equilibrium of this volume is
where the last term is the total body force acting on the liquid volume and l is the distance from point M to the free surface. Cancelling out dS, we obtain
(3.1)
In the special case when a = 0, j = g, and Eq. (3.1) turns into he basic hydrostatic equation (2.2).
In investigating the action of acceleration forces in aviation ractice the concept of ^-loading is used. Hie g-load may be tangen-ial (nx) or normal (ny). In straight flight the g-load is equal to
Normal g-loading develops when an airplane flies in a curved path living, nosing up, banking) and it is given by the equation
here v speed of flight;
R radius of curvature.
The example considered above referred to a tangential g-load, it in view of the stnall dimensions of an aircraft fuel tank compared ith the radius R, the reasoning set forth can be applied to normal loading, the acceleration-a being calculated from the equation
It should be noted that in airplane flight the normal g-loading usually much greater than the tangential g-loading (eight to tenmes).
When the g-load is great and the amount of fuel in a tank is small, displacement of the fuel may lay bare the fuel intake pipe and it out the fuel supply. To prevent this special devices are provided around the intake.