2Kinematics

IB Fluid Dynamics


2.1 Material time derivative
We first want to consider the problem of how we can measure the change in a
quantity, say
f
. This might be pressure, velocity, temperature, or anything else
you can imagine.
The obvious thing to do would be the consider the time derivative
∂f
∂t
.
In physical terms, this would be equivalent to fixing our measurement instrument
at a point and measure the quantity over time. This is known as the Eulerian
picture.
However, often we want to consider something else. We pretend we are a
fluid particle, and move along with the flow. We then measure the change in
f
along our trajectory. This is known as the Lagrangian picture.
Let’s look at these two pictures, and see how they relate to each other.
Consider a time-dependent field
f
(
x, t
). For example, it might be the pressure
of the system, or the temperature of the fluid. Consider a path
x
(
t
) through the
field, and we want to know how the field varies as we move along the path.
Along the path x(t), the chain rule gives
df
dt
(x(t), t) =
∂f
∂x
dx
dt
+
∂f
∂y
dy
dt
+
∂f
∂z
dz
dt
+
∂f
∂t
= ∇f ·
˙
x +
∂f
∂t
.
Definition
(Material derivative)
.
If
x
(
t
) is the (Lagrangian) path followed by a
fluid particle, then necessarily
˙
x(t) = u by definition. In this case, we write
df
dt
=
Df
Dt
.
This is the material derivative.
In other words, we have
Df
Dt
= u · ∇f +
∂f
∂t
.
On the left of this equation, we have the Lagrangian derivative, which is the
change in
f
as we follow the path. On the right of this equation, the first term
is the advective derivative, which is the change due to change in position. The
second term is
∂f
∂t
, the Eulerian time derivative, which is the change at a fixed
point.
For example, consider a river that gets wider as we go downstream. We know
(from, say, experience) that the flow is faster upstream than downstream. If the
motion is steady, then the Eulerian time derivative vanishes, but the Lagrangian
derivative does not, since as the fluid goes down the stream, the fluid slows
down, and there is a spacial variation.
Often, it is the Lagrangian derivative that is relevant, but the Eulerian time
derivative is what we can usually put into differential equations. So we will need
both of them, and relate them by the formula above.