17.1: F.1 Local Equilibrium

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Recall the famous story of Galileo dropping cannon balls from a tower to demonstrate that balls of different sizes fall at the same rate. This showed that gravity pulls more strongly on a big ball than on a small one. Suppose this was not so, and the force acting on both balls was in fact the same. In that case, by Newton’s second law \(a = F/m\), the smaller ball would accelerate faster. If it were light enough, it could zip to the ground like a bullet! (Conversely, a sufficiently massive ball would seem to hang in mid-air, a prisoner of its own inertia.)

In our universe that is not how it works: as the size of an object goes to zero, the force acting on it must go to zero also, so that the acceleration remains finite. This is true not only of gravity but of any force. It is also true of rotational motion: as an object’s size shrinks to zero, the net torque acting on its surface must also vanish, or else it would spin infinitely fast.

These observations support the principle of local equilibrium, which goes a long way towards specifying how force is transmitted within a fluid. Picture an imaginary closed surface within the fluid, with intermolecular forces and torques acting on it, and now imagine that the closed surface shrinks to infinitesimal size. Both the net force and the net torque acting on the surface must go to zero. In the next two sections we will see how these requirements lead to the three statements about the stress tensor listed at the beginning of this appendix.

Figure \(\PageIndex{1}\): If the gravitational force was independent of mass.