The model

The title of this chapter refers to the idealised model discussed in chapter 1, on which Euler and Bernoulli based their contributions to fluid dynamics. An Euler fluid by definition has zero viscosity and zero compressibility. A fluid without viscosity cannot sustain shear stress, and the pressure p within it is therefore isotropic at all points. A fluid without compressibility has a density ρ which is unaffected by variations of p from place to place. The model need not exclude small variations of density due to thermal expansion if the temperature is nonuniform, but such variations are normally irrelevant except in so far as they may drive thermal convection currents in the fluid. Consideration of the topic of convection is deferred to chapter 8. For the time being we may regard temperature as something which has no influence on the flow behaviour of our model fluid and which may therefore be ignored.

Some of the conditions which need to be satisfied if the model is to match the behaviour of real fluids have been discussed in chapter 1. The reader may wish to refer back to that, and to the summary in § 1.16 in particular.

The continuity condition

It is usually safe to assume that fluids remain continuous, and in that case the mass of fluid which occupies any volume V whose boundaries are fixed in space is just the integral over this volume of ρdx, where dx is a volume element.