Measurement of Viscosity. Coefficient of Viscosity.—Suppose the plane ab, fig. 1 of area ω, to move with the velocity V relatively to the surface cd and parallel to it. Let the space between be filled with liquid. The layers of liquid in contact with ab and cd adhere to them. The intermediate layers all offering an equal resistance to shearing or distortion, the rectangle of fluid abcd will take the form of the parallelogram a′b′cd. Further, the resistance to the motion of ab may be expressed in the formR = κωV,(1)where κ is a coefficient the nature of which remains to be determined.If we suppose the liquid between ab and cd divided into layers as shown in fig. 2, it will be clear that the stress R acts, at each dividing face, forwards in the direction of motion if we consider the upper layer, backwards if we consider the lower layer. Now suppose the original thickness of the layer T increased to nT; if the bounding plane in its new position has the velocity nV, the shearing at each dividing face will be exactly the same as before, and the resistance must therefore be the same. Hence,R = κ′ω (nV).(2)But equations (1) and (2) may both be expressed in one equation if κ and κ′ are replaced by a constant varying inversely as the thickness of the layer. Putting κ = μ/T, κ′ = μ/nT,R = μωV/T;or, for an indefinitely thin layer,R = μωdV/dt,(3)an expression first proposed by L. M. H. Navier. The coefficient μ is termed the coefficient of viscosity.According to J. Clerk Maxwell, the value of μ for air at θ° Fahr. in pounds, when the velocities are expressed in feet per second, isμ = 0.000 000 025 6 (461° + θ);that is, the coefficient of viscosity is proportional to the absolute temperature and independent of the pressure.The value of μ for water at 77° Fahr. is, according to H. von Helmholtz and G. Piotrowski,μ = 0.000 018 8,the units being the same as before. For water μ decreases rapidly with increase of temperature.
Measurement of Viscosity. Coefficient of Viscosity.—Suppose the plane ab, fig. 1 of area ω, to move with the velocity V relatively to the surface cd and parallel to it. Let the space between be filled with liquid. The layers of liquid in contact with ab and cd adhere to them. The intermediate layers all offering an equal resistance to shearing or distortion, the rectangle of fluid abcd will take the form of the parallelogram a′b′cd. Further, the resistance to the motion of ab may be expressed in the form
R = κωV,
(1)
where κ is a coefficient the nature of which remains to be determined.
If we suppose the liquid between ab and cd divided into layers as shown in fig. 2, it will be clear that the stress R acts, at each dividing face, forwards in the direction of motion if we consider the upper layer, backwards if we consider the lower layer. Now suppose the original thickness of the layer T increased to nT; if the bounding plane in its new position has the velocity nV, the shearing at each dividing face will be exactly the same as before, and the resistance must therefore be the same. Hence,
R = κ′ω (nV).
(2)
But equations (1) and (2) may both be expressed in one equation if κ and κ′ are replaced by a constant varying inversely as the thickness of the layer. Putting κ = μ/T, κ′ = μ/nT,
R = μωV/T;
or, for an indefinitely thin layer,
R = μωdV/dt,
(3)
an expression first proposed by L. M. H. Navier. The coefficient μ is termed the coefficient of viscosity.
According to J. Clerk Maxwell, the value of μ for air at θ° Fahr. in pounds, when the velocities are expressed in feet per second, is
μ = 0.000 000 025 6 (461° + θ);
that is, the coefficient of viscosity is proportional to the absolute temperature and independent of the pressure.
The value of μ for water at 77° Fahr. is, according to H. von Helmholtz and G. Piotrowski,
μ = 0.000 018 8,
the units being the same as before. For water μ decreases rapidly with increase of temperature.
§ 4. When a fluid flows in a very regular manner, as for instance when It flows in a capillary tube, the velocities vary gradually at any moment from one point of the fluid to a neighbouring point. The layer adjacent to the sides of the tube adheres to it and is at rest. The layers more interior than this slide on each other. But the resistance developed by these regular movements is very small. If in large pipes and open channels there were a similar regularity of movement, the neighbouring filaments would acquire, especially near the sides, very great relative velocities. V. J. Boussinesq has shown that the central filament in a semicircular canal of 1 metre radius, and inclined at a slope of only 0.0001, would have a velocity of 187 metres per second,2the layer next the boundary remaining at rest. But before such a difference of velocity can arise, the motion of the fluid becomes much more complicated. Volumes of fluid are detached continually from the boundaries, and, revolving, form eddies traversing the fluid in all directions, and sliding with finite relative velocities against those surrounding them. These slidings develop resistances incomparably greater than the viscous resistance due to movements varying continuously from point to point. The movements which produce the phenomena commonly ascribed to fluid friction must be regarded as rapidly or even suddenly varying from one point to another. The internal resistances to the motion of the fluid do not depend merely on the general velocities of translation at different points of the fluid (or what Boussinesq terms the mean local velocities), but rather on the intensity at each point of the eddying agitation. The problems of hydraulics are therefore much more complicated than problems in which a regular motion of the fluid is assumed, hindered by the viscosity of the fluid.
Relation of Pressure, Density, and Temperature of Liquids§ 5.Units of Volume.—In practical calculations the cubic foot and gallon are largely used, and in metric countries the litre and cubic metre (= 1000 litres). The imperial gallon is now exclusively used in England, but the United States have retained the old English wine gallon.1 cub. ft.= 6.236 imp. gallons= 7.481 U.S. gallons.1 imp. gallon= 0.1605 cub. ft.= 1.200 U.S. gallons.1 U.S. gallon= 0.1337 cub. ft.= 0.8333 imp. gallon.1 litre= 0.2201 imp. gallon= 0.2641 U.S. gallon.Density of Water.—Water at 53° F. and ordinary pressure contains 62.4 ℔ per cub. ft., or 10 ℔ per imperial gallon at 62° F. The litre contains one kilogram of water at 4° C. or 1000 kilograms per cubic metre. River and spring water is not sensibly denser than pure water. But average sea water weighs 64 ℔ per cub. ft. at 53° F. The weight of water per cubic unit will be denoted by G. Ice free from air weighs 57.28 ℔ per cub. ft. (Leduc).§ 6.Compressibility of Liquids.—The most accurate experiments show that liquids are sensibly compressed by very great pressures, and that up to a pressure of 65 atmospheres, or about 1000 ℔ per sq. in., the compression is proportional to the pressure. The chief results of experiment are given in the following table. Let V1be the volume of a liquid in cubic feet under a pressure p1℔ per sq. ft., and V2its volume under a pressure p2. Then the cubical compression is (V2− V1)/V1, and the ratio of the increase of pressure p2− p1to the cubical compression is sensibly constant. That is, k = (p2− p1)V1/(V2− V1) is constant. This constant is termed the elasticity of volume. With the notation of the differential calculus,k = dp/ (−dV)= − Vdp.VdVElasticity of Volume of Liquids.Canton.Oersted.Colladonand Sturm.Regnault.Water45,990,00045,900,00042,660,00044,000,000Sea water52,900,000······Mercury705,300,000··626,100,000604,500,000Oil44,090,000······Alcohol32,060,000··23,100,000··According to the experiments of Grassi, the compressibility of water diminishes as the temperature increases, while that of ether, alcohol and chloroform is increased.§ 7.Change of Volume and Density of Water with Change of Temperature.—Although the change of volume of water with change of temperature is so small that it may generally be neglected in ordinary hydraulic calculations, yet it should be noted that there is a change of volume which should be allowed for in very exact calculations. The values of ρ in the following short table, which gives data enough for hydraulic purposes, are taken from Professor Everett’sSystem of Units.Density of Water at Different Temperatures.Temperature.ρDensity ofWater.GWeight of1 cub. ft.in ℔.Cent.Fahr.032.0.99988462.417133.8.99994162.420235.6.99998262.423337.41.00000462.424439.21.00001362.425541.01.00000362.424642.8.99998362.423744.6.99994662.421846.4.99989962.418948.2.99983762.4141050.0.99976062.4091151.8.99966862.4031253.6.99956262.3971355.4.99944362.3891457.2.99931262.3811559.0.99917362.3731660.8.99901562.3631762.6.99885462.3531864.4.99866762.3411966.2.99847362.3292068.0.99827262.3162271.6.99783962.2892475.2.99738062.2612678.8.99687962.2292882.4.99634462.1963086.99577862.1613595.9946962.09340104.9923661.94745113.9903861.82350122.9882161.68855131.9858361.54060140.9833961.38765149.9807561.22270158.9779561.04875167.9749960.86380176.9719560.67485185.9688060.47790194.9655760.275100212.9586659.844The weight per cubic foot has been calculated from the values of ρ, on the assumption that 1 cub. ft. of water at 39.2° Fahr. is 62.425 ℔. For ordinary calculations in hydraulics, the density of water (which will in future be designated by the symbol G) will be taken at 62.4 ℔ per cub. ft., which is its density at 53° Fahr. It may be noted also that ice at 32° Fahr. contains 57.3 ℔ per cub. ft. The values of ρ are the densities in grammes per cubic centimetre.§ 8.Pressure Column. Free Surface Level.—Suppose a small vertical pipe introduced into a liquid at any point P (fig. 3). Then the liquid will rise in the pipe to a level OO, such that the pressure due to the column in the pipe exactly balances the pressure on its mouth. If the fluid is in motion the mouth of the pipe must be supposed accurately parallel to the direction of motion, or the impact of the liquid at the mouth of the pipe will have an influence on the height of the column. If this condition is complied with, the height h of the column is a measure of the pressure at the point P. Let ω be the area of section of the pipe, h the height of the pressure column, p the intensity of pressure at P; thenpω = Ghω ℔,p/G = h;that is, h is the height due to the pressure at p. The level OO will be termed the free surface level corresponding to the pressure at P.Relation of Pressure, Temperature, and Density of GasesFig. 3.§ 9.Relation of Pressure, Volume, Temperature and Density in Compressible Fluids.—Certain problems on the flow of air and steam are so similar to those relating to the flow of water that they are conveniently treated together. It is necessary, therefore, to state as briefly as possible the properties of compressible fluids so far as knowledge of them is requisite in the solution of these problems. Air may be taken as a type of these fluids, and the numerical data here given will relate to air.Relation of Pressure and Volume at Constant Temperature.—At constant temperature the product of the pressure p and volume V of a given quantity of air is a constant (Boyle’s law).Let p0be mean atmospheric pressure (2116.8 ℔ per sq. ft.), V0the volume of 1 ℔ of air at 32° Fahr. under the pressure p0. Thenp0V0= 26214.(1)If G0is the weight per cubic foot of air in the same conditions,G0= 1/V0= 2116.8/26214 = .08075.(2)For any other pressure p, at which the volume of 1 ℔ is V and the weight per cubic foot is G, the temperature being 32° Fahr.,pV = p/G = 26214; or G = p/26214.(3)Change of Pressure or Volume by Change of Temperature.—Let p0, V0, G0, as before be the pressure, the volume of a pound in cubic feet, and the weight of a cubic foot in pounds, at 32° Fahr. Let p, V, G be the same quantities at a temperature t (measured strictly by the air thermometer, the degrees of which differ a little from those of a mercurial thermometer). Then, by experiment,pV = p0V0(460.6 + t) / (460.6 + 32) = p0V0τ/τ0,(4)where τ, τ0are the temperatures t and 32° reckoned from the absolute zero, which is −460.6° Fahr.;p/G = p0τ/G0τ0;(4a)G = pτ0G0/p0τ.(5)If p0= 2116.8, G0= .08075, τ0= 460.6 + 32 = 492.6, thenp/G = 53.2τ.(5a)Or quite generally p/G = Rτ for all gases, if R is a constant varying inversely as the density of the gas at 32° F. For steam R = 85.5.
Relation of Pressure, Density, and Temperature of Liquids
§ 5.Units of Volume.—In practical calculations the cubic foot and gallon are largely used, and in metric countries the litre and cubic metre (= 1000 litres). The imperial gallon is now exclusively used in England, but the United States have retained the old English wine gallon.
Density of Water.—Water at 53° F. and ordinary pressure contains 62.4 ℔ per cub. ft., or 10 ℔ per imperial gallon at 62° F. The litre contains one kilogram of water at 4° C. or 1000 kilograms per cubic metre. River and spring water is not sensibly denser than pure water. But average sea water weighs 64 ℔ per cub. ft. at 53° F. The weight of water per cubic unit will be denoted by G. Ice free from air weighs 57.28 ℔ per cub. ft. (Leduc).
§ 6.Compressibility of Liquids.—The most accurate experiments show that liquids are sensibly compressed by very great pressures, and that up to a pressure of 65 atmospheres, or about 1000 ℔ per sq. in., the compression is proportional to the pressure. The chief results of experiment are given in the following table. Let V1be the volume of a liquid in cubic feet under a pressure p1℔ per sq. ft., and V2its volume under a pressure p2. Then the cubical compression is (V2− V1)/V1, and the ratio of the increase of pressure p2− p1to the cubical compression is sensibly constant. That is, k = (p2− p1)V1/(V2− V1) is constant. This constant is termed the elasticity of volume. With the notation of the differential calculus,
Elasticity of Volume of Liquids.
According to the experiments of Grassi, the compressibility of water diminishes as the temperature increases, while that of ether, alcohol and chloroform is increased.
§ 7.Change of Volume and Density of Water with Change of Temperature.—Although the change of volume of water with change of temperature is so small that it may generally be neglected in ordinary hydraulic calculations, yet it should be noted that there is a change of volume which should be allowed for in very exact calculations. The values of ρ in the following short table, which gives data enough for hydraulic purposes, are taken from Professor Everett’sSystem of Units.
Density of Water at Different Temperatures.
The weight per cubic foot has been calculated from the values of ρ, on the assumption that 1 cub. ft. of water at 39.2° Fahr. is 62.425 ℔. For ordinary calculations in hydraulics, the density of water (which will in future be designated by the symbol G) will be taken at 62.4 ℔ per cub. ft., which is its density at 53° Fahr. It may be noted also that ice at 32° Fahr. contains 57.3 ℔ per cub. ft. The values of ρ are the densities in grammes per cubic centimetre.
§ 8.Pressure Column. Free Surface Level.—Suppose a small vertical pipe introduced into a liquid at any point P (fig. 3). Then the liquid will rise in the pipe to a level OO, such that the pressure due to the column in the pipe exactly balances the pressure on its mouth. If the fluid is in motion the mouth of the pipe must be supposed accurately parallel to the direction of motion, or the impact of the liquid at the mouth of the pipe will have an influence on the height of the column. If this condition is complied with, the height h of the column is a measure of the pressure at the point P. Let ω be the area of section of the pipe, h the height of the pressure column, p the intensity of pressure at P; then
pω = Ghω ℔,
p/G = h;
that is, h is the height due to the pressure at p. The level OO will be termed the free surface level corresponding to the pressure at P.
Relation of Pressure, Temperature, and Density of Gases
§ 9.Relation of Pressure, Volume, Temperature and Density in Compressible Fluids.—Certain problems on the flow of air and steam are so similar to those relating to the flow of water that they are conveniently treated together. It is necessary, therefore, to state as briefly as possible the properties of compressible fluids so far as knowledge of them is requisite in the solution of these problems. Air may be taken as a type of these fluids, and the numerical data here given will relate to air.
Relation of Pressure and Volume at Constant Temperature.—At constant temperature the product of the pressure p and volume V of a given quantity of air is a constant (Boyle’s law).
Let p0be mean atmospheric pressure (2116.8 ℔ per sq. ft.), V0the volume of 1 ℔ of air at 32° Fahr. under the pressure p0. Then
p0V0= 26214.
(1)
If G0is the weight per cubic foot of air in the same conditions,
G0= 1/V0= 2116.8/26214 = .08075.
(2)
For any other pressure p, at which the volume of 1 ℔ is V and the weight per cubic foot is G, the temperature being 32° Fahr.,
pV = p/G = 26214; or G = p/26214.
(3)
Change of Pressure or Volume by Change of Temperature.—Let p0, V0, G0, as before be the pressure, the volume of a pound in cubic feet, and the weight of a cubic foot in pounds, at 32° Fahr. Let p, V, G be the same quantities at a temperature t (measured strictly by the air thermometer, the degrees of which differ a little from those of a mercurial thermometer). Then, by experiment,
pV = p0V0(460.6 + t) / (460.6 + 32) = p0V0τ/τ0,
(4)
where τ, τ0are the temperatures t and 32° reckoned from the absolute zero, which is −460.6° Fahr.;
p/G = p0τ/G0τ0;
(4a)
G = pτ0G0/p0τ.
(5)
If p0= 2116.8, G0= .08075, τ0= 460.6 + 32 = 492.6, then
p/G = 53.2τ.
(5a)
Or quite generally p/G = Rτ for all gases, if R is a constant varying inversely as the density of the gas at 32° F. For steam R = 85.5.
II. KINEMATICS OF FLUIDS
§ 10. Moving fluids as commonly observed are conveniently classified thus:
(1)Streamsare moving masses of indefinite length, completely or incompletely bounded laterally by solid boundaries. When the solid boundaries are complete, the flow is said to take place in a pipe. When the solid boundary is incomplete and leaves the upper surface of the fluid free, it is termed a stream bed or channel or canal.
(2) A stream bounded laterally by differently moving fluid of the same kind is termed acurrent.
(3) Ajetis a stream bounded by fluid of a different kind.
(4) Aneddy,vortexorwhirlpoolis a mass of fluid the particles of which are moving circularly or spirally.
(5) In a stream we may often regard the particles as flowing along definite paths in space. A chain of particles following each other along such a constant path may be termed a fluid filament or elementary stream.