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Consider these two cases. There is fluid flowing over a sphere and the fluid can either flow
at a slow speed, or it can flow at a fast speed, and in the slow speed it is nice laminar
flow profile. As a high speed there is turbulent regime in the back. Lets say you wanted to
calculate the force of drag in both cases. Now there is no way to characterize the force
of drag for this turbulent flow. There is no way to integrate the Novier-Stokes equations
and calculate the drag force. What you could do is actually run some experiments and measure
to force of drag in both cases. Thus you might have some way to measure the fluid flow, the
velocity from left to right, and you might note that things like the density and the
viscosity of the fluid. In that case you can just measure the forces of drag as a function
of these other parameters. What you could do is make a graph of the drag force as a
function of velocity. So you would run the experiment at a low velocity and run it at
a high velocity. Pleasurably you would get some curve force of 0 at a low velocity. You
might see some increasing curve like this. What you would do is fit a line to this data
or curve of some sort and you might do this with a sphere with a diameter of 1 cm. Then
what you would want to do is explore what would happen if I change it to a 10 cm sphere
and you would get another curve. So in this case these data points represent 10 cm, and
you fit a line to these, and you might will want to do something at 1 mm. Here is what
the data might look like for a smaller sphere. So presumably you might get a bigger sphere
and you would expect to have a higher drag force. Then what you might be interested in
is not the force as a function of velocity. You are looking at a bunch of different fluids.
Lets see how the force varies with the viscosity of the fluid. So we will do an experiment.
Lets say we have a fixed diameter, which we are going to say 1 cm sphere, and we are going
to collect data as a function of viscosity. We are going to do it for a couple of different
experiments. What you might find is some trend, and use a velocity of 10 cm/s. You draw a
curve and so here is a force as a function of viscosity. So what happens if I have a
higher velocity or a lower velocity in the same diameter. So presumably what we would
see. Is at a small velocity the drag force would be small and again you increase the
velocity and if we increase the viscosity I would anticipate that the force of drag
would be greater. So we get more experimental data. Notice though that all of these experimental
points are only for 1 diameter sphere. In this case it was 1 cm. So we are doing a whole
lot of experiments, and we want to figure out if what happens if the diameter now is
10 cm. Well in this case you have to go back and measure all of these velocities again
using a sphere that was 10 times bigger. You would calculate the force in that case as
a function of viscosity. So we are doing a whole lot of experiments just to get the drag
force around a simple geometry like a sphere. Then you might say well not only is the viscosity
significant. What happens if we look at the force as a function of the density of the
fluid. It is hard to say of the back what this relationship might be. Does a denser
fluid have a bigger drag, a smaller density. If we look at this now how does the density
compare of we had given fluid with a certain density and a higher viscosity or high density
and a low viscosity. You can imagine you would have 100s and 100s of experiments that you
could possibly run to explore the interactions between all of these variables. As it terms
out we can combine all of those data based on viscosity, Re number, and diameter of the
sphere, and the density of the fluid. We could make a graph of what is known as the coefficient
of drag as a function of what is known as the Re number. Both of these are dimensionless
parameters they don't have dimensions like meters or kilograms. The coefficient of drag
is equal to the force of drag divided by 1/2 times the area of the sphere times the density
times the velocity of the fluid squared. So this area of the sphere is the cross sectional
area that is exposed to the fluid. So the area in this case is pi over 4 times the diameter
squared. In the Re number is equal to the density of the fluid, times the velocity,
times the diameter of the sphere, divided by the viscosity of the fluid. So lets say
you did a whole lot of experiments and in each experiment you measured the force of
drag and you look at a particular diameter sphere, particular cross-sectional area, particular
density of the fluid, and a particular velocity. So if you know all of these parameters you
can calculate the coefficient of drag. You also calculate for a given experiment you
know the density of the fluid, its velocity, its diameter of the sphere, the viscosity
of the fluid. You could calculate what is known as the Re number. We make a graph of
this and it turns out for a sphere. What you would absorb is some decrease in the coefficient
of drag and in some independent region where the coefficient of drag does not change much
with the Re numbers. Then it decreases due to the changes in the fluid flow behavior
around the sphere, and then it increases again. So lets say we did that, and say that the
blue would indicate a certain diameter, and if you do this again we will double the diameter.
So now we are looking at a bigger sphere. We are going to vary the velocity the density
of the fluids, and different types pf fluids. If you graph it this way. What it turns out
these data points fall right on top of each other. They are very, very close to one another.
You see this exact same trend. This might be a smaller sphere or it might have a more
viscous fluid what have you, but you calculate the coefficient of drag and the Re number,
and you do it again. We will say alright here an even bigger sphere and in this case we
are going to do it in a very thick fluid, and in this case it turns out the data points
all fall on top of each other. What you are able to do is collapse all of these different
experiments into a single curve. What that allows you to do is extrapolate now. Lets
say we have a diameter sphere that we have not tested, and we have tested 1 cm, 10 cm,
100 cm, well lets say we have a 50 cm diameter sphere. Some strange fluid with a different
viscosity. You haven't actually done any experiments. You just know the diameter of the sphere,
the velocity, the density, and the viscosity of the fluid. What we ultimately want to know
is the drag force on this sphere. Lets say the Re number is in this region. You might
be something like 100,000 for this Re number. So I will draw a line. What that means is
in this case there is a particular coefficient of drag associated with that, and this coefficient
of drag might be 0.6. So if you know the velocity of all these other fluid diameters. You have
not even done the experiment yet, but you can calculate the force of drag.