Volume forces, work rate, heating

The fluid can be subjected to a force field f (r, t), the most common example being gravitational acceleration g. In a non-inertial frame this would also include d’Alembert, centrifugal, and Coriolis forces,

f (r, t) = g — U — О xr — О x (flxr) — 20xV (1.19)

where U(t) is the inertial velocity of the frame’s reference point, O(t) is the frame’s rotation, r is the position vector relative to the reference point, and V(r, t) is the velocity within the non-inertial frame. These quantities are diagrammed in Figure 7.1, in which V is denoted by Vrei. Flow-Field description in non-inertial frames will not be performed here, so that a constant f = g will be assumed in the most general case.

For application of f to the equations of fluid motion, the actual relevant quantity is p f, which has units of force per unit volume. When acting on fluid moving with local velocity V, this volume force will impart a work rate wv(r, t) equal to

Wv = p f ■ V (1.20)

which has units of power per unit volume. Possibly adding to this mechanical power is a thermal heating rate, quantified by some imposed body heating source density

qv = qv (r, f)

which also has units of power per unit volume. This might be from absorbed radiation or combustion. Outside a turbomachine combustor, and for the vast majority of external aerodynamic flows, qv is zero.