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

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.

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

(1-13)

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.