2Kinematics
IB Fluid Dynamics
2.2 Conservation of mass
We can start formulating equations. The first one is the conservation of mass.
We fix an arbitrary region of space
D
with boundary
∂D
and outward normal
n. We imagine there is some flow through this volume
D
∂D
n
What we want to say is that the change in the mass inside
D
is equal to the
total flow of fluid through the boundary. We can write this as
d
dt
Z
D
ρ dV = −
Z
∂D
ρu · n dS.
We have the negative sign since we picked the outward normal, and hence the
integral measures the outward flow of fluid.
Since the domain is fixed, we can interchange the derivative and the integral
on the left; on the right, we can use the divergence theorem to get
Z
D
∂ρ
∂t
+ ∇ · (ρu)
dV = 0.
Since D was arbitrary, we must have
∂ρ
∂t
+ ∇ · (ρu) = 0
everywhere in space.
This is the general form of a conservation law — the rate of change of
“stuff” density plus the divergence of the “stuff flux” is constantly zero. Similar
conservation laws appear everywhere in physics.
In the conservation equation, we can expand ∇ · (ρu) to get
∂ρ
∂t
+ u · ∇ρ + ρ∇ · u = 0.
We notice the first term is just the material derivative of ρ. So we get
Dρ
Dt
+ ρ∇ · u = 0.
With the conservation of mass, we can now properly say what incompressibility
is. What exactly happens when we compress a fluid? When we compress mass,
in order to conserve mass, we must increase the density. If we don’t allow
changes in density, then the material derivative
Dρ
Dt
must vanish. So we define
an incompressible fluid as follows:
Definition
(Incompressible fluid)
.
A fluid is incompressible if the density of a
fluid particle does not change. This implies
Dρ
Dt
= 0,
and hence
∇ · u = 0.
This is also known as the continuity equation.
For parallel flow,
u
= (
u,
0
,
0). So if the flow is incompressible, we must have
∂u
∂x
= ∇ · u = 0. So we considered u of the form u = u(y, z, t).
Of course, incompressibility is just an approximation. Since we can hear
things, everything must be compressible. So what really matters is whether
the slow speed is small compared to the speed of sound. If it is relatively
small, then incompressibility is a good approximation. In air, the speed of
sound is approximately
340 m s
−1
. In water, the speed of sound is approximately
1500 m s
−1
.