4. THE EQUATION OF MOTION Department of （4所示。）.pdfVIP

4. THE EQUATION OF MOTION Newton’s law for a fluid element is dV = F, (4. 1) dt where is the mass density (mass per unit volume), V is the velocity of the fluid element, and F is the force per unit volume acting on the element. The latter is composed of two types: volumetric forces and surf ace forces. Volumetric forces act throughout the volume of the fluid element. They can be thought of as acting at the centroid. Examples are: 1. Gravity: F = g , where g is the gravitational acceleration. If the force is a g centralf orce, then F = , where is the gravitational pot ential. g 2. Electromagnetic forces: Since the fluid can conduct electricity, it can have a current density J = n q V (with the sum being over all species of ions and electrons, and n , q , and V the number density, electric charge, and velocity of species ), and, in principle, a net electric charge per unit volume, q . The electromagnetic force (per unit volume) are then the electric f orce, F = E ( E is the electric field), and the Lorentz f orce, q q F = J B ( B is the magnetic field). L Surface forces are more complicat