MECHANICS OF LIQUIDS AND GASES

158. Fluids.

PART I.-FLUIDS AT REST

PRESSURE IN LIQUIDS AND GASES.

Certain substances, such as air, water, glycerin, etc., are characterized by great mobility, changing their shapes and flowing under the smallest forces. They are known as fluids. Fluids are divided into two classes, liquids and gases.

Liquids change but slightly in volume when subjected to great pressure and may have a free surface.

Gases are far more compressible than liquids and fill all parts. of the containing vessel. Water is a type of liquid, and air

of gas.

159. Density.-The mass of any substance contained in unit volume is known as its density. In the C. G. S. system of units density is expressed in grams per cubic centimeter, while in the foot-pound-second system it is expressed in pounds per cubic

foot.

Thus the density of water is 1.0 on the first system, while it is 62.5 on the latter system.

A table showing the densities of some substances will be found on page 150.

160. Viscosity.-Fluids differ greatly in mobility. If a dish. of water is tilted, the flow is so rapid that it gives rise to waves that surge to and fro, while in case of glycerin or syrup the flow is slow and the liquid only gradually settles to the new level. This difference in mobility is due to viscosity or internal friction ($245). Substances like pitch or tar are very viscous, while water, alcohol, and ether are but slightly so.

A perfect fluid is one that has no viscosity and is an ideal. All known fluids, even gases, have some viscosity.

161. Force in Fluid at Rest.-The force exerted by a fluid at rest against any surface is perpendicular to that surface. Otherwise, owing to the mobility of the fluid, flow must take

place along the surface, which of course cannot be in a liquid

at rest.

This law is true of all fluids, even those which are very viscous, after they have settled into equilibrium.

162. Pressure.-Let a very small flat surface be imagined at some point in a fluid. The fluid on one side of that surface exerts a force perpendicular to the surface against the fluid on the opposite side. This force is proportional to the surface, and the force per unit surface is called the pressure.

In C. G. S. units pressure is measured in dynes per square centimeter; it may also be measured in grams per square centimeter, pounds per square inch, etc.

163. Hydrostatic Pressure. At any point in a fluid at rest the pressure is the same in every direction. This is a direct consequence of the mobility of fluids, for a little sphere of liquid at the given point could not be in equilibrium if the pressure against its surface were not the same in

every direction. 164. Pressures on Same Level.-In a liquid at rest the pressure is the same at all points on the same level. For a horizontal cylindrical column of liquid reaching from A to B is in equilibrium under the pressure of the surrounding liquid. The pressure against its sides is perpendicular to the line A B, and therefore has no influence to move the column toward A or B. And since it is level it has no tendency to slide

FIG. 80.

B

toward A or B by reason of its weight. The force against the end at A must therefore be balanced by the force against the end at B. These forces are due to the pressures at A and B, and since the ends have equal areas the pressure at A must be equal to the pressure at B.

165. Pressures at Different Depths.-The difference in pressure between two points at different levels in a mass of fluid at rest under gravity is equal to the weight of a column of the fluid of unit cross section reaching vertically from one level to the other. For a vertical cylindrical column of the fluid of unit cross section. reaching from B to C is in equilibrium under the pressure of the

surrounding fluid. The pressure against the sides of the vertical column is horizontal and has no power to support its weight, consequently the upward force at C must balance the weight of the column in addition to the downward force at B. Hence, since the force against the end of a unit column is equal to the pressure, the pressure at C is greater

than the pressure at B by the weight of the column of fluid of unit cross section reaching from B to C.

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If h is the height of the column in centimeters and d is the weight of one cubic centimeter of the fluid in grams, then hd is the weight of the column and is thus the difference in pressure between B and C in grams per sq. cm. The difference in pressure expressed in dynes per sq. cm. is hdg where g is the acceleration of gravity in cm.-seconds. The total pressure at a point h centimeters below the surface is therefore as follows:

FIG. 81.

Pressure in grams per sq. cm. =hd + pressure on surface in grams per sq. cm.

Pressure in dynes per sq. cm. =hdg+pressure on surface in dynes per sq. cm.

Note as to Units.-In calculating pressure by the use of the formula hd, it must be remembered that if the pressure is to be found in pounds per square inch, then h must be expressed in inches and d is the weight of one cubic inch of the liquid in pounds. The student is advised, however, to compute directly the weight of a column of the substance of unit cross section without thinking of any formula.

In gases the density is so small that the pressure is practically the same everywhere throughout a small volume.

166. Pascal's Principle.-Pressure is transmitted equally in all directions throughout a mass of fluid at rest, or if the pressure at any point is increased, it is increased everywhere throughout the fluid mass by the same amount.

167. Hydraulic or Hydrostatic Press. An important mechanical device known as the hydraulic press is a good illustration of the application of the laws of fluid pressure. If was first constructed by Bramah in 1796, and is sometimes known as Bramah's press.

It consists of a strong cylinder in which works a cylindrical piston or ram of large diameter. A collar of oiled leather or copper surrounds the piston in such a way that the greater the pressure of the liquid filling the cylinder, the more closely does the collar fit the piston. By means of a small pump, oil or water is forced into the large cylinder, a check-valve preventing its In consequence of the law of pressure just enunciated,

return.

FIG. 82.-Hydrostatic press.

whatever pressure is communicated to the liquid by the pump will be exerted everywhere equally against the walls of the containing cylinders. So that if the large piston has 100 times the area of the other it will exert a force 100 times as great as that applied to the pump piston.

Hydraulic jacks act on this principle: they contain a reservoir of oil which may be pumped into the main cylinder, thus forcing up the ram; opening a small stopcock permits the flow of oil back to the reservoir.

Oil

is used as it keeps the machine lubricated and does not freeze. It is to be observed that when the liquid in the hydraulic press is incompressible as much work is done by the large piston as is expended upon the smaller one.

168. Pressure Independent of Shape of Vessel.-It has been shown that the pressure at any point in a liquid under gravity depends only on the depth of the point below the surface, on the density of the liquid, and on the pressure on its surface.

The total force exerted against the bottom of a vessel by the pressure of the liquid which it contains is the product of the pressure at the bottom by its area, and may therefore be very different from the actual weight of liquid which the vessel contains; and when a vessel is filled with water to a given height the force against its bottom is the same whether the upper part of the vessel is flaring, cylindrical, or narrow. The reasonable

ness of this result will be evident from the following considerations.

In the case of the vessel with flaring sides we may think of a cylindrical column resting on the bottom and pressed upon by the surrounding water as shown in the figure (Fig. 83). This pressure is necessarily perpendicular to the surface of the cylindrical column and therefore can have no effect in either supporting it or pressing it down. The whole weight of the cylindrical column is therefore supported by the bottom plate. In case of the vessel which is narrow at the top, the liquid exerts a downward force on the bottom greater than its weight because

the sides of the vessel press the liquid down.

Just as a man in a

box may brace himself against the top and press against the bottom with a force far greater than his own weight.

This fact that the force exerted on the bottom of a vessel may be greater than the weight of all the liquid in the vessel has been called the hydrostatic paradox.

Pascal succeeded in bursting a strong cask by the pressure produced by a column of water in a narrow pipe forty feet high.

169. Center of Pressure. The center of pressure of a surface is the point of application of the resultant force due to the pressure against the surface. The pressure is so distributed that the surface will just balance if supported at that point.

In case of a tank having rectangular sides and filled with water, the center of pressure on a side will evidently be nearer the bottom than the top, because the pressure increases with the depth. Suppose the side to be divided into narrow horizontal strips of equal widths, the force exerted on each strip by the liquid pressure may be represented by an arrow as in the diagram, and it is clear that each of these forces