There are several physical conservation laws that we use to predict the behavior of a
fluid.
In Chapter 1 we discussed Newtons's Second Law ( F = ma ) which is really a
statement of the conservation of momentum, and the Law of Conservation of Energy.
In
addition, we use the physical law describing the conservation of mass.
This law is
typically described for a fluid in terms of an equation called the "continuity" equation.
Before we derive the continuity equation, we must first define three terms regarding fluid
flow fields.
The first is
steady state
.
Steady state means that at any given point, the flow
does not fluctuate with time.
Imagine looking at a single point in a flow field.
If, at that
point, the direction and speed (velocity) of the flow remains constant, the flow is
steady
.
The second term is the
streamline
.
Consider a small fluid element moving along a path
in steady state.
This moving fluid element traces out a fixed path in space.
The fixed
path is called a streamline.
The fluid flow is always parallel to the streamline since that is
how the streamline is defined.
There can be no flow across a streamline.
The third term is the
stream tube
.
A stream tube consists of all the streamlines
surrounding a volume of flowing fluid.
A hose or a pipe can be thought of as a stream
tube.
The flow can only go through the stream tube.
It cannot go through the walls of
the stream tube since the walls are streamlines and there can be no flow across a
streamline.
Now we can derive the continuity equation.
We start the the phyical principle
"conservation of mass" that says mass can be neither created nor destroyed.
To apply this
principle to a flowing gas, let's look at the figure below, and consider an imaginary
arbitrary shape drawn perpendicular to the flow direction.
The long dotted lines represent
a stream tube, thus, the fluid flows within these lines.
Flow enters the stream tube on the
left with a speed of
V
1
through an arbitrarily shaped area
A
1
.
The flow exits on the right
with a speed of
V
2
through an arbitrarily shaped area
A
2
.
(Note:
A
2
is in general not the
same shape as
A
1
.
)
If the flow is steady, then the mass that flows through the cross
section at Point 1 is the same as the mass that flow through the crosssection at Point 2.
The mass flowing through the stream tube is confined by the streamlines of the boundary,
just as the flow in a garden hose is confined by the wall of the hose.
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 Spring '09
 Abbitt
 Fluid Dynamics, stream tube, imcompressible flow

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