3.1 Navier-Stokes equations

So far, we haven’t done anything useful. We have not yet come up with an

equation of motion for fluids, and thus we don’t know how fluids actually behave.

The equations for the parallel viscous flow was a good start, and it turns out

the general equation of motion is of a rather similar form.

Unfortunately, the equation is rather complicated and difficult to derive, and

we will not derive it in this course. Instead, it is left for the Part II course.

However, we will be able to derive some special cases of it later under certain

simplifying assumptions.

times acceleration, and the right is the individual forces — the pressure gradient,

viscosity, and the body forces (per unit volume) respectively.

In general, these are very difficult equations to solve because of non-linearity.

For example, in the material derivative, we have the term u · ∇u.

There are a few things to note:

(i)

The acceleration of a fluid particle is the Lagrangian material derivative of

the velocity.

The derivation of the viscous term is complicated, since for each side of the

cube, there is one normal direction and two tangential directions. Thus

this requires consideration of the stress tensor. This is only done in IID

2

In an incompressible fluid, this reduces to

∇

2

u = −∇ × (∇ × u) = −∇ × ω,

where

ω

=

∇ × u

y

, u

z

), we have

∇

2

u = (∇

2

u

x

, ∇

2

The Navier-Stokes equation reduces to the parallel flow equation in the

special case of parallel flow, i.e. u = (u(y, t), 0, 0). This verification is left

as an exercise for the reader.