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The Theorems of Hydrostatics are thus true for all stagnant fluids, however viscous they may be; it is only when we come to Hydrodynamics, the Science of the Motion of Fluids, that the effect of viscosity will make itself felt, and modify the phenomena; unless we begin by postulating perfect fluids, that is, fluids devoid of viscosity.
We have used the word STRESS in the Definition of a Fluid above; a stress is defined as composed of two equal and opposite balancing forces, acting between two bodies or two parts of the same body.
These two forces constitute the "Action and Reaction" of Newton's Third Law of Motion, which acccording to this law "are equal and opposite." (Maxwell, Matter
and Motion, p. 46.)
The Stress between two parts of a body is either (i.) of the nature of a PULL or TENSION, tending to prevent separation of the parts, or (ii.) of the nature of a THRUST or PRESSURE, tending to prevent approach, or (iii.) of the nature of a SHEARING STRESS, tending to prevent the parts from sliding on each other.
In a Solid Substance all three kinds of Stress can exist, but in a Fluid at rest the stress can only be a normal Thrust or Pressure; a tensional stress would overcome the cohesion of the fluid particles.
Nevertheless a column of mercury, many times the barometric height, may be supported in a vertical tube by its adhesion to the top of the tube, in which case the hydrostatic pressure is negative above the barometric height, or the mercury is in a state of tension; and Mr. Worthington has measured experimentally in ethyl
THE PRESSURE IN A FLUID.
alcohol enclosed in a glass vessel a tension up to 17 atmospheres, or 255 pounds per square inch. (Phil. Trans., 1892.)
The Stress across a dividing plane in a Solid can be resolved into two components, one perpendicular to the plane, of the nature of a tension or pressure, and the other component tangential to the plane; and it is this tangential stress which is absent in a Fluid at rest.
7. The Measurement of Fluid Pressure.
If we consider a fluid at rest on one side of any imaginary dividing plane, the fluid is in equilibrium under the forces acting upon it and of the stress across the plane, which is of the nature of a THRUST (poussée), perpendicular to the plane.
Definition. "The PRESSURE (pression) at any point of the plane is the intensity of the Thrust estimated per unit of area of the plane."
Thus if a thrust of P pounds is uniformly distributed over a plane area of A square feet, as on the horizontal bottom of the sea or of any reservoir, the pressure at any point of the plane is P/A pounds per square foot, (but P/144A pounds per square inch).
If the thrust P is not uniformly distributed over the area A, as for instance on the vertical or inclined face of a wall of a reservoir, then P/A represents the average pressure over the area, in pounds per square foot; and the actual pressure at any point is the average pressure over a small area enclosing the point.
Thus if AP pounds denotes the thrust on a small plane area AA square feet enclosing the point, the pressure there is the limit of AP/AA (=dP/dA, in the notation of the Differential Calculus) pounds per square foot.
8. Units of Length, Weight, and Force.
As we are dealing with a Statical subject, we shall employ the statical gravitation unit of force, which is generally defined as the Attraction of the Earth on the Unit of Weight; but more strictly it is the tension of the plumb line when supporting the Unit of Weight, thus allowing for the discount in the Attraction of the Earth due to its rotation.
The British Unit of Weight is the Pound, defined by Act of Parliament, so that our unit of force is the force which is equal to the tension of a thread or plumb line supporting a Pound Weight; and we shall call this force the FORCE OF A POUND.
With a foot as Unit of Length, our pressures will be measured in pounds per square foot; this may be written as lb per foot, or, or ft2, or as lb/ft2.
The Metric Units of Length and Weight are the Metre and Kilogramme, or the Centimetre and the Gramme; and with these units, pressure will be given in kilogrammes per square metre, or grammes per square centimetre.
According to the Act of Parliament, 8th August, 1878, Schedule III.,
Therefore a pressure of one lb/ft2 is equivalent to a pressure of 4536 × (3·2809)2 = 4.8826 kilogrammes per square metre (kg/m2); and a pressure of one kg/m2 is equivalent to a pressure of
2.2046 × (3048)2=0.2048 lb/ft2.
UNITS OF MEASUREMENT.
A pressure of one kg/cm2 is thus 2048 lb/ft2, or 14-2 lb/in2; so that the normal atmospheric pressure, called an atmosphere, being taken as 143 or 147 lb/in2, is the same as 1033 kg/cm2; and therefore, for practical purposes, the atmosphere may be taken as one kg/cm2.
With the Gravitation Unit of Force, the weight of a body is at once the measure of the quantity of matter in the body, and also of the force with which it is apparently attracted by the Earth; and the word Weight may be used in either sense without ambiguity or confusion, when dealing with hydrostatical problems on the surface of the Earth.
We must notice however that, in consequence of the variation of g, this unit of force will vary slightly in magnitude at different points of the Earth; but the variation is so small that it makes no practical difference in engineering problems; the variation is only important when we consider tidal or astronomical phenomena, covering the Earth and extending to the Moon, Sun, and planets.
9. The Safety Valve.
To measure the pressure of a fluid in a vessel, and to prevent the pressure from exceeding a certain amount, the Safety Valve was invented by Papin, 1681.
It consists essentially of a spherical or conical plug C', fitting accurately into a circular orifice in the vessel, and kept closed against the pressure of the fluid by a lever AB, with fulcrum at A; carrying either a sliding weight W lb, when used on a steady fixed vessel; or else held down at the end B by a spiral spring S, which can be screwed to any desired pull of T lb, when the vessel is subject to shock and oscillation (fig. 1).
Then if the pressure of the fluid on the seat of the valve is p lb/in2, and the orifice is d inches in diameter, the thrust on the valve is d2p lb; so that, taking moments about the fulcrum A of the lever AB,
πd2р × AC= W × AE or T× AB,
when the valve is on the point of lifting.
Sometimes the valve is held down by a weight (fig. 2) or by a spiral spring, superposed directly without the intervention of a lever as in fig. 3, the form used in steamers and hydraulic machinery.
The danger of the sticking of the valve in the seat is obviated in Ramsbottom's safety valve (fig. 4), consisting of two equal conical valves, held down by a bar and a spring midway between them; then one or the other valve, or both valves, will open when the thrust of the fluid on it is half the pull of the spring.