[This section is excerpted from Fluid Flow: A First Course in Fluid Mechanics, Macmillan Publishing Company, 1989.]
The momentum theorem developed in Chapter 10 gives the force acting on a fixed volume in terms of linear momentum flux through the surface of the volume. In many situations we are interested in the moment or torque on the volume. For this purpose we may adapt the angular momentum law of mechanics to the flow of fluids. Our starting point is the familiar law
where $ m$ , $ \vec{F}$ , and $ \vec{v}$ refer to a single particle. The torque exerted by the force $ \vec{F}$ about a fixed point is where $ \vec{r}$ is the radius vector from the fixed point to the point of application of $ \vec{F}$ . The symbol, $ \times$ , signifies, as usual, that the vector cross-product shall be taken. Then, from Newton's law of motion, We now define a vector $ \vec{H}$ as the vector product of the radius vector to the particle and the linear momentum, that is, The quantity $ \vec{H}$ is called angular momentum. Upon differentiating $ \vec{H}$ with respect to time, we find that However, [画像:$ dr/dt =\vec{v}$] and the cross-product of a vector parallel to itself is zero. The first term in the right-hand side therefore vanishes and we have the result thatWe now seek to modify the law as expressed by Equation (12.1) to be suitable for a fixed volume. The torque on a material volume $ V'$ is
This is readily transformed into a control volume integral. We have, therefore, where
