4Inviscid irrotational flow
IB Fluid Dynamics
4 Inviscid irrotational flow
From now on, we are going to make a further simplifying assumption. We are
going to assume that our flow is is incompressible, inviscid, and irrotational.
We first check this is a sensible assumption, in that if we start of with an
irrotational flow, then the flow will continue being irrotational. Suppose we have
∇×u
=
0
at
t
= 0, and the fluid is inviscid and homogeneous (i.e.
ρ
is constant).
Then by the vorticity equation
Dω
Dt
= ω · ∇u = 0.
So the vorticity will keep on being zero.
In this case, we can write u = ∇φ for some φ.
Definition
(Velocity potential)
.
The velocity potential of a velocity
u
is a scalar
function φ such that u = ∇φ.
Note that since we are lazy, we don’t not have a negative sign in front of
∇φ
,
unlike, say, gravity.
If we are incompressible, then ∇ · u = 0 implies
∇
2
φ = 0.
So the potential satisfies Laplace’s equation.
Definition
(Potential flow)
.
A potential flow is a flow whose velocity potential
satisfies Laplace’s equation.
A key property of Laplace’s equation is that it is linear. So we can add two
solutions up to get a third solution.