[page 4, 238.]

STREAM-FUNCTION

...from which we infer that the expression u dy − v dx is a perfect differential A mathematical condition where a multivariable expression can be integrated to find a single, consistent scalar function.. Calling it dψ original: "dψ"; "ψ" is the Greek letter psi., we have as the equivalent of equation (4):

where ψ is a function of the coordinates which, so far, is completely arbitrary. The function ψ is called the stream-function, since the motion of the fluid is everywhere in the direction of the curves where ψ is constant. When the motion is steady—meaning it is always the same at the same point in space—the curves where ψ is constant mark out a system of pipes or channels in which the fluid may be imagined to flow. Mathematically, the substitution of one function (ψ) for the two functions (u and v) is often a step of great importance.

Another important case occurs when there is symmetry around an axis—for example, the x-axis. Everything is then expressed in terms of x and r, where r = √(y² + z²), and the motion takes place in planes passing through the axis of symmetry. If the velocities parallel and perpendicular to the axis of symmetry are u and q, respectively, the equation of continuity is:

which, as before, is equivalent to:

where ψ is the stream-function.

239.

In almost all the cases we will deal with, the hydrodynamical equations undergo a remarkable simplification because of a principle first stated by Lagrange Joseph-Louis Lagrange (1736–1813), a foundational mathematician in the study of mechanics.. If, for any part of a fluid mass, the expression u dx + v dy + w dz is a perfect differential dφ original: "dφ"; "φ" is the Greek letter phi. at one moment, it will remain so for all subsequent time. In particular, if a fluid is originally at rest and is then set in motion by conservative forces and pressures transmitted from the outside, the quantities:

(which we shall denote by ξ, η, ζ original: "ξ, η, ζ"; these are the Greek letters xi, eta, and zeta.) can never deviate from zero.