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2 The Navier–Stokes and Euler Equations – Fluid and Gas Dynamics
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Under the assumption of incompressibility of the fluid the Navier–Stokes
equations, determining the fluid velocity u and the fluid pressure p, read:

∂u
+(u · grad)u +grad p = νΔu + f
∂t
div u = 0

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3
Here x denotes the space variable in R or R depending on whether 2 or 3
dimensional flows are to be modeled and t> 0 is the time variable. The velocity
3
3
2
2
field u = u(x, t) (vector field on R or, resp., R )isin R or R ,resp.,and the
pressure p = p(x, t) is a scalar function. f = f (x, t) is the (given) external force
field (again two and, resp. three-dimensional) acting on the fluid and ν > 0
the kinematic viscosity parameter. The functions u and p are the solutions of
the PDE system, the fluid density is assumed to be constant (say, 1) here as
consistent with the incompressibility assumption. The nonlinear Navier–Stokes
system has to be supplemented by an initial condition for the velocity field and
by boundary conditions if spatially confined fluid flows are considered (or by
decay conditions on whole space). A typical boundary condition is the so-called
no-slip condition which reads

u = 0
on the boundary of the fluid domain.
The constraint div u = 0enforcesthe incompressibilityofthe fluid andserves
to determine the pressure p from the evolution equation for the fluid velocity u.
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If ν = 0 then the so called incompressible Euler equations, valid for very
small viscosity flows (ideal fluids), are obtained. Note that the viscous Navier–
Stokes equations form a parabolic system while the Euler equations (inviscid
case) are hyperbolic. The Navier–Stokes and Euler equations are based on New-
ton’s celebrated second law: force equals mass times acceleration. They are
consistent with the basic physical requirements of mass, momentum and energy
conservation.
The incompressible Navier–Stokes and Euler equations allow an interesting
simple interpretation, when they are written in terms of the fluid vorticity,
defined by
ω := curl u .

Clearly, the advantage of applying the curl operator to the velocity equation
is the elimination of the pressure. In the two-dimensional case (when vorticity
can be regarded as a scalar since it points into the x 3 direction when u 3 is zero)
we obtain
Dω
= νΔω +curl f ,
Dt
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http://gap-system.org/∼history/Mathematicians/Euler.html