There's a certain danger in fluid
mechanics to using the mass balance
equation to predict what is going to
happen to the flow. Let me show you what
it is. The mass balance equation in
mechanics tells us that Rho V A is the
mass flow. Rho (density) V (velocity) and a
(cross section area). and so a common
assumption to make is to say that "if I
increase a then V must decrease". And vice-versa:
if I decrease A then V must
increase. It's sometimes true but not
always, and let me show you cases where
this does not this does not work.
Let's take a look first at a jet — jet
engine. And we're looking here at a
Phantom, a jet from the 1960s —not the
latest technology but still the
principles remain in modern engines
today. We're looking at the engine of
that aircraft here, so the
engine built into the fuselage of the
Phantom. It's also the engine that's
built onto the Gulfstream 4, nice jet of
NASA that you see here. And it's a
Rolls-Royce Spey turbofan engine. Not the
most modern engine you can find but
again the principles remain true today.
If we take that engine here off the
airplane,
remove the cover, you would get this
machine here. This is the inlet of
the engine and the outlet would
be on the other side over there.
And if we remove and all these panels here in
the middle then you would find the
combustion chamber. The combustion
chamber from the Rolls-Royce Spey looks like
this. This is a combustion chamber from
that engine over there.
In this combustion chamber, the fuel
comes in into the middle here as a
nozzle that would spray a very small
mass flow of fuel in the center, and the
air would come in through this part here,
would swirl around the fuel, mix with it,
combust, and then here the hot gases would 
outlet into the turbine, where it would
spin the turbine and run the engine over
there. So what happens inside here? The
air comes in here at some temperature
which is already pretty high. But by the
time it has crossed the combustion chamber
and has mixed with the fuel and burned,
the air comes in here has extremely high
temperature and doing so its density
has changed. Its density has decreased by a huge amount
because suddenly it has expanded. And so even
though the area here is much larger than
the area here in here at the inlet, the
velocity has /increased/ because the
density has decreased by a great deal,
yeah? So be careful when you say that
when area decreases velocity must
increase or vice versa. Because this
assumes in the background that the
density is constant which is not always
the case. So the math says this is
only true if rho remains constant, and rho
may remain constant for water flows, but not if there is heat transfer
and not if there is expansion or
contraction of the fluid. So be careful,
it's a classical result in supersonic
flow for example, so in faster-than-sound flow,
that if you decrease the area then you
also decrease the velocity because the
density increases. Rho V A is always the
same, it's the same at the inlet of the
combustion chamber than in the outlet, but all /three/ parameters change, not just two of them.
Another way to get this equation wrong
and to make this assumption wrong
if you increase the area even when density remains constant,
is as follows. Now let's take a water pipe.
Let's take a hose, let's put our thumb
into this yeah? So suppose it's just
summer and your friends are over there
and they’re sun bathing. And what you
have at your disposal is a water hose
and you take the hose out and the water
is running out and what you want to do
of course is to run an experiment to
verify whether this Rho V A thing is
working in the general direction of your
friends who are sunbathing over there.
And so what do you do of course? You reduce the
area and this will increase the velocity
And we know that! You have that feeling,
everybody has played with the hose
before. The question is: until which point
does it work? Because you certainly know
that the more you constrain the outside
area and the faster flow goes, but if you
constrain it too much, then you end up
with just a drip over here. so what's
going on? You reduce the area and you
also reduce the velocity and the answer
is you reduce the area ,you reduce the
velocity, and you reduce the mass flow.
So the
mass flow at this section of the pipe is
still equal to the mass flow out of that
section of the pipe. But the total mass
flow is not the same as before, yeah? So
you have to be careful with how you
manipulate this equation where you say "I
decrease the area therefore I increase
the velocity". It's not always true, yeah?
So let's look at the mathematical
side of this. There is no causal
relationship. Just because you have
increased the area doesn't mean that you
have decreased the velocity and
vice-versa.  A2
decreasing may mean that V2 is
increasing, but there's not a guarantee
that A2×V2 remains a constant. So you may
kill off the flow inside the pipe just
because you have increased the losses to
such a point that it's not possible for
the flow to exit anymore, yeah? So be very
careful with this, those two equations.
The lesson is if you're using the mass
balance to predict velocity, yeah? A mass
balance equation to predict velocity,
then you also must ask yourself what
force is required? What is the pressure
change required to move this fluid around?
And what is the power involved? If you do
not do that, you may find yourself
describing flows that are completely
unrealistic.
