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Law of Continuity, Bernoulli, Contradiction
Name: Dil
Status: student
Grade: 9-12
Location: CA
Country: N/A
Date: 11/29/2005
Question:
According to the Law of Continuity as the velocity of a
fluid increases, the area decreases. So, if area decreases, pressure
must increase.
That would mean that as velocity increases, pressure also increases, which
contradicts Bernoulli's Principle.
There is obviously something wrong with this reasoning. What would that
be?
Replies:
Hi Dil,
What is wrong with this reasoning is that what seems so common-sense
reasonable (to me too) is just wrong. As the area decreases, the velocity
increases and, as you correctly state, by the equation of continuity the
cross sectional area of the fluid decreases. However, Bernoulli is right!
When that happens the pressure decreases!
Some arguments to guide your common sense:
In order for the liquid to flow faster, a force must be exerted on it to
accelerate it. This force can only come from the pressure in the fluid.
For the fluid to accelerate as its area decreases, the pressure upstream of
that point MUST be higher that the pressure where the velocity is greater.
Clear?
Also, the energy MUST be conserved if no external work is done on the
fluid. When the fluid moves faster, its kinetic energy increases. If it
is moving horizontally, its gravitational potential energy does not change.
The only other energy in the liquid is due to the pressure, so to
compensate for the increase in kinetic energy, the pressure must decrease.
OK?
Best, Dick Plano
Professor of Physics emeritus, Rutgers University...
Your statement of pressure increasing is only slightly correct, the only
questions is "where is the pressure increasing?" Let me explain a little
better. Say we have a pipe that is 2 inches in diameter and reduces down to
one inch. Like you said, continuity says that I will have the same flow
through the 2 inch pipe that I have through the 1 inch pipe. Therefore,
with the reduction in area, I have an increase in the velocity. Where is
this pressure increase you wonder? Well, pressure is the resistance to
flow. Where does the resistance start to build? It builds in the 2 inch
pipe, not the 1 inch pipe. That is proven by Bernoulli's equation and the
continuity equation:
P1/rho + V1^2/2 + gZ1= P2/rho + V2^2/2 + gZ2
Q=V1A1=V2A2 -> V1=Q/A1 & V2=Q/A2
Then P1/rho + Q^2/2A1^2 + gZ1=P2/rho + Q^2/2A2^2 + gZ2
Assuming the pipe is on the same line i.e. z1=z2
Then
P1/rho=P2/rho + (Q^2/2)*(1/A2^2 - 1/A1^2)
Substituting our 2 inch and 1 inch pipe with A1=Pi*2^2/4=Pi and
A2=Pi*1^2/2=Pi/2
P1/rho=P2/rho+Q^2/2*(4/Pi^2 - 1/Pi^2)=P2/rho+(Q^2/2)*(3/Pi^2)
Therefore P1=P2+(Q^2*rho)*(3/2Pi^2)
So you can see that my pressure will increase in the area BEFORE the
reduction in the pipe, not AT the pipe reduction. Say P2 remains constant,
like going out to open atmosphere, you can see that increasing the flow rate
increases the velocity in both sections and therefore increases the pressure
at P1. Continuity and Bernoulli's principles still intact.
Hopefully this helped clear things up some. If you still have questions,
feel free to write back with your questions. Thanks for using NEWTON.
Christopher Murphy, P.E.
Mechanical Engineer
Air Force Research Laboratory
Dil-
Do not take the "Law of Continuity" as a law of physics.
It is more like an intuitive principle good in many common circumstances.
I think it means all this:
When the airflows around an object are subsonic, the
pressure-differences are less than an atmosphere,
and the gas molecules (moving about the speed of sound) can move around in
all directions to even out the pressure, so
there are no sharp density changes (aka "shock-waves").
This means that the volume of a particular 3-D patch of gas is
approximately conserved as it flows by the object.
For this kind of thinking, it's important to define for yourself a "piece"
of gas,
the "patch" I just referred to. "Packet" might be another adequate word
for it.
Imagine starting with an imaginary cubic outline or soap-bubble box,
small compared with the twists and turns of the airflow.
The walls are completely flexible, but the volume is conserved.
Molecules diffusing across the imaginary boundaries average equal and
opposite, so they can be ignored.
This starting cube can stretch longer and skinny, like a rod flying end-wards.
or squash shorter and wider, like a square plate flying face-wards.
It can also shear (slant-deformation) into parallelogram-prism shapes.
But usually we try to imagine a cube of air so small that
rather little shear deformation happens within a single cube.
Adjacent cube-patches are glued face-to-face and corner-to-corner,
permanently:
the back end of one patch has to stay the same width and shape as the
front end of the patch behind it.
In the long run turbulent flow will shred this tidy imaginary pattern,
but we are trying to think clear, quantifiable thoughts about short-run
changes,
and we can always resort to imagining smaller cubes that are not yet shredded.
The "area" you referred to is the cross-section of that small patch of gas
as seen
from the axis of motion or flow (from the front or back).
As the velocity of a patch increases, its volume and density stay the same.
Patches pass sequentially, so the rate of passing patches per second on a
given flow line
stays the same all the way past the object (such as a wing).
Together these mean that as a patch speeds up it must be stretched out in
the axis of motion,
and get narrower in the two perpendicular directions.
And conversely for slowing.
All this thinking is embedded in your "Law of Continuity".
"Continuity" of what? Of volume and density, during the time the
air-patch flows past the object, I suppose.
This principle does not predict a pressure increase in higher-speed parts
of the airflow.
On the contrary, each patch of air has mass, so it will accelerate a
little as it goes from high-pressure places to lower-pressure places,
and conversely the faster places must have slightly lower pressure to have
given rise to their increased speed.
That relation (speed up, pressure down) is the same polarity as the
Bernoulli effect,
but it may not be as big.
There is more such change, in the same direction, caused by the finiteness
of the thermal velocity of air molecules.
As a patch of air launches itself from a high-pressure zone into low, it
smoothly expands and cools by a small amount.
Some of its random thermal kinetic energy in all directions gets re-directed
into being the communal kinetic energy of forwards air-flow.
Then it is no longer available to make as much pressure sideways, on the
wing or on adjacent air-patches.
I'm not sure I understand Bernoulli exactly or am explaining the cause right,
but it's a mental starting-point you might be able to fix up for yourself.
There must be many explanations of the Bernoulli effect around the Internet.
Jim Swenson
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