Conservation Laws for a Barotropic Fluid. in a Conservative Body Force Field

Conservation Laws for a Barotropic Fluid. in a Conservative Body Force Field Подпись: S2 — Подпись: dp P. Подпись: (1-11)

Under the limitations of the present section, it is easily seen that the law of conservation of momentum, (1-3), can be written

The term “barotropic ” implies a unique pressure-density relation through­out the entire flow field; adiabatic-reversible or isentropic flow is the most important special case. As we shall see, (1-11) can often be integrated to yield a useful relation among the quantities pressure, velocity, density, etc., that holds throughout the entire flow.

Подпись: where Подпись: DT Dt Подпись: (, dp c P Подпись: c T ds’ Подпись: (1-12)

Another consequence of barotropy is a simplification of Kelvin’s theorem of the rate of change of circulation around a path C always composed of the same set of fluid particles. As shown in elementary textbooks, it is a consequence of the equations of motion for inviscid fluid in a conservative body force field that


is the circulation or closed line integral of the tangential component of the velocity vector. Under the present limitations, we see that the middle member of (1-12) is the integral of a single-valued perfect differential and therefore must vanish. Hence we have the result DT/Dt = 0 for all
such fluid paths, which means that the circulation is preserved. In par­ticular, if the circulation around a path is initially zero, it will always remain so. The same result holds in a constant-density fluid where the quantity p in the denominator can be taken outside, leaving once more a perfect differential; this is true regardless of what assumptions are made about the thermodynamic behavior of the fluid.