said to constitute fluidity, and which distinguishes fluid from solid bodies, is inability to sustain stress in shear. A fluid in which this property is perfect is said to be inviscid, and in such a fluid a shearing strain, i.e., distortion, may take place without being accompanied by any corresponding stress: such a fluid must be regarded as hypothetical. All actual fluids possess viscosity; in a viscous fluid a stress in shear may exist, but is accompanied by a continually increasing strain; "stress in shear in a viscous fluid bears, in fact, the same relation to the rate of change of strain that stress in a perfectly elastic solid bears to the strain itself.
The remaining physical properties of a fluid are identical with those of a solid body, and comprise density and elasticity (volumetric). These two quantities are related to a third quantity—pressure—in so much that the density is a function of the pressure, the nature of which function is defined by the law of elasticity; thus in a perfect gas under isothermal conditions we have Pρ = constant where ρ is density, and P pressure.
If we take the two extreme cases in the relation of ρ and P, so that, firstly, the elasticity be supposed zero, we shall have any finite pressure, however small, produce an infinite density, and the fluid becomes identical with the medium of Newton. If, secondly, we suppose the elasticity to be infinite, so that a change in P, however great, produces no change in the density, we have the case of an incompressible fluid. The latter assumption is that of our present hypothesis.
§ 59. Basis of Mathematical Investigation.—The Equations of Motion may be said to constitute the starting point of all analytical investigation; these are:—
(1) The Equation of Continuity, expressing the relation between the density of the fluid and the linear rate of change of flow in each of the co-ordinate directions of space; or, under the restriction that density is constant, the relation between the
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