In this video we are going back in time to 1738 and to the mind and writings of one Daniel
Bernoulli.
By the end of this video you should be able to see how the famous Bernoulli equation is
derived and what assumptions are being made in being able to use it.
Since we have used small differential elements thus far in fluids to explain certain phenomena
let's go ahead and plop an element down on our sheet of paper that follows a stream line.
Here is our fluid element that follows a stream line.
We are going to give it dimensions.
We are going to say in the direction of the stream line it has a dimension of delta s
and normal to the stream line it has a dimension of delta n.
We are going to give this a thickness of delta y.
These coordinates allow us to designate them to regardless of what coordinate system this
element may fall into thus its volume is going to be the following.
delta s times delta n times delta y.
If we go back and visit with Newton to get his second law which states that the sum of
all forces acting in the stream line direction on the elements is going to equal its differential
mass times the acceleration in the stream line direction.
Assume that there are no viscous forces or surface tension acting on the particle and
we do know that acceleration is the change in velocity with respect to time.
If we use the chain rule we could rewrite this as the following.
The reason we do this is so that we can remove the dt term and you can see that this term
right here is just the velocity in the stream line direction.
We can rewrite acceleration as a velocity in the stream line direction times the change
in velocity per change in s.
Now Newton second law looks like the following where we have rewritten the acceleration as
the velocity times the differential.
We can pull the density out of the mass term and we get our rewritten Newton second law.
This equation is valid for both compressible and incompressible fluids.
As you will see as we move along this one of the main assumptions in using Bernoulli
equation is that we assume that it is incompressible.
We need to rewrite the left side of the equation, the sum of all the forces acting on the particle,
by collecting all the different forces that are acting in the stream line directions.
The first force that we are going to start with is gravity which gives the particles
a weight.
In our schematic here the differential weight for the particle does not act in either the
direction of the stream line nor that normal to the stream line.
We draw a triangle so that we can calculate what those contributions will be.
Differential weight can be rewritten as the specific weight times the differential volume
and the contribution in the stream line direction is just going to be sine of theta where theta
marks this angle.
We are also going to put a negative sign here just because of the way we have drawn our
axis.
We have this contribution acting in the negative direction thus if the stream line is horizontal
and theta is 0 the weight component in the stream line direction is also 0.
Term 1 the weight term can be rewritten as such.
The second and last term, and important term of course, acting on the fluid is that of
pressure.
Recall in previous knowledge on pressure fields acting on an element pressure is not constant
throughout the fluid because of its weight.
We can say that pressure will be a function of both s and n.
Let's label the center pressure of our element as P. Pressure is going to act on the right
side of the face and on the left side of the face.
Again we are only concerned in the direction of the stream line.
We are going to say that the pressure here is going to be P plus some delta P in the
stream line direction and here it is going to be P minus some delta s in the stream line
direction.
We could write out a long Taylor series for determining what the Ps is but for a very
small fluid element we can use the first term and obtain delta Ps is equal to the change
in pressure with change in s times the distance over 2.
The reason for that is we are interested in this distance right here which is delta s
over 2 and we are multiplying that times the change in pressure with respect to s.
The total pressure force, which we will designate Fps acting on the fluid element is going to
be on the left side P minus delta Ps times the area delta n times delta y minus pressure
acting on the right side P plus delta Ps times the area delta n times delta y.
You can see quickly that this is going to be equal to the following and plugging what
we know for Ps in we get our force die to pressure.
For those of you keeping track we are going to sum up our delta forces in the stream line
direction as the weight force plus the pressure force.
This looks like the following.
I have taken out the differential volume and we are going to plug in what we know for the
other half of this equation.
We have density times the differential volume times the velocity and the differential velocity
with respect to s.
This allows us to cancel out both terms of the differential volume and rewrite the Bernoulli
equation.
That is not the Bernoulli equation that I showed at the beginning so what are we missing
at this point?
I do not recall a sine theta term above so let's figure out how we can rewrite sine theta.
Referring back to our fluid element we know that this angle theta.
This is delta s and using typically Cartesian coordinates we can say that this is delta
z, that sine theta is just delta z over delta z.
We can rewrite this as dz/ds.
Now we can change the sine theta to our dz/ds and approach our pressure term.
We know that the differential pressure dP is going to be a function of differential
pressure in the s direction and the differential pressure in the normal direction however along
a stream line dn is going to be equal to 0 thus we can rewrite our differential pressure
as just a function of s.
We will go ahead and plug that back in.
Lastly our velocity term can be rewritten so that we have an ordinary differential equation
and we will take that term on the right side and plug that back in to the middle of our
equation and it will look like the following.
This equation simplifies to look like this and we can integrate this.
At this point we have a pretty simplified equation that equals some constant and the
only assumptions we have made so far is that it follows a stream line, that it is inviscid
and that the flow is at steady state.
We do not know if density is going to change or not with pressure so we cannot integrate
that out but if we assume that it is incompressible then density and the specific weight will
stay constant and we can rewrite this equation as the following.
The pressure term plus the velocity term plus the height term along a stream line is going
to be constant.
We have our Bernoulli equation.
This basically implies that given the assumption we have made we can calculate along a stream
line the conditions at one point knowing the conditions at another.
Check out some of the example screencasts to see how we apply Bernoulli's equation to
certain systems.