# Viscosity

• ... To describe the motion of a fluid, we must give it properties at every point ... We will write the force density as the sum of three terms. We have already considered the pressure force per unit volume, –${\displaystyle \nabla }$p. Then there are the “external” forces which act at a distance—like gravity or electricity. When they are conservative forces with a potential per unit mass, ${\displaystyle \phi }$, they give a force density –${\displaystyle \rho \nabla \phi }$. (If the external forces are not conservative, we would have to write ƒext for the external force per unit volume.) Then there is another “internal” force per unit volume, which is due to the fact that in a flowing fluid there can also be a shearing stress. This is called the viscous force, which we will write ƒvisc. Our equation of motion is ${\displaystyle \rho \times }$(acceleration) = –${\displaystyle \nabla }$p –${\displaystyle \rho \nabla \phi }$+ƒvisc ... When we drop the viscosity term, we will be making an approximation which describes some ideal stuff rather than real water. John von Neumann was well aware of the tremendous difference between what happens when you don’t have the viscous terms and when you do, and he was also aware that, during most of the development of hydrodynamics until about 1900, almost the main interest was in solving beautiful mathematical problems with this approximation which had almost nothing to do with real fluids. He characterized the theorist who made such analyses as a man who studied “dry water.” Such analyses leave out an essential property of the fluid.