1Parallel viscous flow
IB Fluid Dynamics
1.3 Derived properties of a flow
The velocity and pressure already fully describe the flow. However, there are
some other useful quantities we can compute out of these.
The first thing we consider is how much stuff is being transported by the
flow.
Definition
(Volume flux)
.
The volume flux is the volume of fluid traversing a
cross-section per unit time. This is given by
q =
Z
h
0
u(y) dy
per unit transverse width.
We can calculate this immediately for the two flows.
Example. For the Couette flow, we have
q =
Z
h
0
Uy
h
dy =
Uh
2
.
For the Poiseuille flow, we have
q =
Z
h
0
G
2µ
y(h − y) dy =
Gh
3
12µ
.
We can also ask how much “rotation” there is in the flow.
Definition (Vorticity). The vorticity is defined by
ω = ∇ × u.
In our case, since we have
u = (u(y, t), 0, 0),
we have
ω =
0, 0, −
∂u
∂y
.
Example.
For the case of the Couette flow, the vorticity is
ω
=
0, 0, −
U
h
.
This is a constant, i.e. the vorticity is uniform.
U
ω
For the case of the Poiseuille flow, we have
ω =
0, 0,
G
µ
y −
h
2
.
P
1
P
0
ω
ω
Recall that the tangential stress
τ
s
is the tangential force per unit area
exerted by the fluid on the surface, given by
τ
s
= µ
∂u
∂n
,
with n pointing into the fluid.
Example. For the Couette flow, we have
τ
s
=
(
µ
U
h
y = 0
−µ
U
h
y = h
.
We see that at
y
= 0, the stress is positive, and pulls the surface forward. At
y = h, it is negative, and the surface is pulled backwards.
For the Poiseuille flow, we have
τ
s
=
(
Gh
2
y = 0
Gh
2
y = h
Both surfaces are pulled forward, and this is independent of the viscosity. This
makes sense since the force on the surface is given by the pressure gradient,
which is independent of the fluid.