5.1.2. Euler's Equation

Since we assume that there is no external perturbation on the fluid and no magnetic fields, the only two forces which are operative within the fluid are due to pressure and gravity. Pressure forces arise directly from any gradient in the fluid density. Gravitational forces arise from gradients in the gravitational field itself. These two forces oppose one another. Where gravity acts to collapse a fluid element, its internal pressure opposes this collapse. If these forces are not in equilibrium there will be a net force which causes an acceleration in the fluid. This change in fluid velocity, as it responds to this net force, is specified by Newton's second law. For the fluid, we express this as

(2)

where P represents the pressure source and is the gravitational source.

The effects of this net force on the fluid can be described in two different ways. We can describe the change in velocity at some fixed point in space or we can identify a particular volume element within the fluid and describe its change in velocity. The former treatment is historically referred to Eulerian where the time derivatives are determined at this fixed point in space. The latter treatment is known as the Lagrangian form where the calculations follow a particular fluid element. In general, Newtonian mechanics is based on the Lagrangian formulation but for our purposes, it is more useful to describe the behavior of the fluid in Eulerian terms. These two forms can be related by the following:

(3)

The second term on the RHS represents changes which are caused by the bulk motion of the fluid. It is this bulk motion that needs to be accounted for if equation 6-2 is to become the fluid analog of momentum conservation. Substituting equation 6-3 into equation 6-2 yields

(4)

This equation was first derived in 1755 and is now referred to as Euler's Equation. The equation describes the net change in velocity of a fluid element due to an imbalance between pressure and gravitational forces. The gravitational force, per unit mass, which acts on a fluid element is a reflection of the gradient in the gravitational potential which is generated by the distribution of the surrounding matter. This is fully specified by Poisson's equation

(5)