In a 2D or 3D channel flow, the vorticity is created at the physical boundary and diffuses into the body of the flow. Far away from the boundary, the flow can be assumed to be irrotational, i.e. having zero vorticity [1], [2, Chapter 14]. Let be the velocity field. The vorticity is
Here,
is the differential operator with respect to spatial variables only. The assumption that
is equivalent to the assumption that the flow is potential, i.e.
for some scalar function
Again, this assumption/approximation is only good in the body of the flow far away from the boundary. For
to satisfy the Navier-Stokes equations
must be a harmonic function and
J. Serrin [3] pointed out that a weak solution of this kind can have no regularity in time. Solutions of the form
is usually referred to as Serrin’s example.
Near the physical boundary, potential flow is no longer a good approximation for the true flow due to large vorticity. Interestingly, this fact is also confirmed at the mathematical level: one cannot impose a non-slip boundary condition on the solution
without forcing it to be identically zero (static flow).
Proposition: Let
be an open connected subset of
such that
is the closure of an open subset of
. Let
be a function that is continuous on
and harmonic on
If
on
then
is a constant function.
Proof: Because and
is path-connected,
must be constant on
Subtracting a constant from
if necessary, we can assume that
on
By translation, one can assume that
is an interior point of
One can extend
by Schwarz reflection principle to the domain
where
as follows:
With a little abuse of notation, let us denote as
Note that points in the interior of
are also in the interior of
Because
one can prove by induction on
that the
-partial derivative
for any multi-index From the fact that
on
one can show that (1) is true for
Then from the fact that
one can show that (1) is true for any Therefore, all partial derivatives of
vanish at 0. Together with the fact that
is analytic on
which is an open connected subset of
one concludes that
is identically zero.
Remark: for , there is a simple proof using Complex Analysis. Namely,
is the real part of a holomorphic function on
Let
be the imaginary part. The Riemann-Cauchy equations read
and
The condition
and
implies that both
and
must be constant on
Subtracting a complex constant from
if necessary, we can assume that
on
Because the holomorphic function
vanishes on
it must be identically zero.
References:
- A Story of Potential Flow
- Applications of Classical Physics (2013) by Blandford and Thorne.
- On the interior regularity of weak solutions of the Navier-Stokes equations by J. Serrin. Arch. Rational Mech. Anal. 9, 187–195 (1962).