The Dynamical Equations
Olivier Darrigol · 2005
Abstract Modern derivations of the fundamental equations for non-viscous fluids have an air of evidence. The fluid is divided into volume elements, and the acceleration of a volume element is equated to a force divided by a mass. The force on the element dτ is the sum of an external action fdτ (e.g. gravity) and of the resultant -( ▽P)dt of the pressures exerted on the surface of the element by the surrounding fluid. If v (r, t) denotes the velocity of the fluid at the point r and time t, and r (t) is the position of the element at time t, then the acceleration of the element is the time derivative of v [r(t), t], that is, ∂v= ∂tþ (v • ∂)v. The mass of the element is the product of its density ρ and its volume dτ. Hence Euler’s equation follows: