3Dynamics

IB Fluid Dynamics


3.2 Pressure
In the Navier-Stokes equation, we have a pressure term. In general, we classify
the pressure into two categories. If there is gravity, then we will get pressure in
the fluid due to the weight of fluid above it. This is what we call hydrostatic
pressure. Technically, this is the pressure in a fluid at rest, i.e. when u = 0.
We denote the hydrostatic pressure as
p
H
. To find this, we put in
u
= 0 into
the Navier-Stokes equation to get
∇p
H
= f = ρg.
We can integrate this to obtain
p
H
= pg · x + p
0
,
where p
0
is some arbitrary constant. Usually, we have g = (0, 0, −g). Then
p
H
= p
0
− gρz.
This exactly says that the hydrostatic pressure is the weight of the fluid above
you.
What can we infer from this? Suppose we have a body
D
with boundary
∂D
and outward normal n. Then the force due to the pressure is
F = −
Z
∂D
p
H
n · dS
= −
Z
D
∇p
H
dV
= −
Z
D
gρ dV
= −g
Z
D
ρ dV
= −Mg,
where
M
is the mass of fluid displaced. This is known as Archimedes’ principle.
In particular, if the body is less dense than the fluid, it will float; if the body
is denser than the fluid, it will sink; if the density is the same, then it does not
move, and we say it is neutrally buoyant.
This is valid only when nothing is moving, since that was our assumption.
Things can be very different when things are moving, which is why planes can
fly.
In general, when there is motion, we might expect some other pressure
gradient. It can either be some external pressure gradient driving the motion
(e.g. in the case of Poiseuille flow), or a pressure gradient caused by the flow
itself. In either case, we can write
p = p
H
+ p
0
,
where
p
H
is the hydrostatic pressure, and
p
0
is what caused/results from motion.
We substitute this into the Navier-Stokes equation to obtain
ρ
Du
Dt
= −∇p
0
+ µ∇
2
u.
So the hydrostatic pressure term cancels with the gravitational term. What we
usually do is drop the “prime”, and just look at the deviation from hydrostatic
pressure. What this means is that gravity no longer plays a role, and we can
ignore gravity in any flow in which the density is constant. Then all fluid particles
are neutrally buoyant. This is the case in most of the course, except when we
consider motion of water waves, since there is a difference in air density and
water density.