In this screencast we are going to work though an example, in which we determine the viscosity
of the fluid based on some simple force measurements that we might take and an experiment involving
a toothpick and a straw.
So here we have a toothpick that is placed inside a 5 mm diameter straw full of some
fluid, and there is a gap of 1 mm between the toothpick and the straw.
Now if the 50 mm of the toothpick is pulled from the straw with a force of 0.1 N such
that the toothpick moves with a constant velocity of 0.1 m/s.
What is the viscosity of the fluid?
We are making 2 assumptions that the fluid is Newtonian and that the velocity profile
is linear.
We are just going to work with linear coordinates since we are assuming that this velocity profile
is linear.
So you can see that I have drawn cylindrical coordinates.
The radius in the vertical direction, and the z-axis in the longitude direction.
We have some inner radius of our toothpick, and some outer radius of our straw, and what
is happening is we are pulling the toothpick out.
We know that the length that we are pulling out is 50 mm.
We are told that the diameter of the straw is 5 mm and there is a gap of 1 mm on each
side.
So this is probably not drawn to scale here.
We have a 3 mm diameter toothpick.
We know that the velocity that I am going to write as u is equal to 0.1 m/s.
So lets draw what is occurring between in this fluid space between the toothpick and
the straw.
So I am going to draw the space that is highlighted in green.
So if we label the top surface as the straw boundary and the bottom surface as our toothpick.
We are assuming the the velocity profile is linear.
So if we were to draw this.
We know that the velocity at the straw is going to be 0.
Since the straw is not moving.
The toothpick is.
So our maximum velocity is going to be at the toothpick surface.
So if we are pulling the toothpick out this way.
This would not be correct.
It should be the other way, and we can draw our velocity arrows moving to the left, such
that our maximum velocity is at the toothpick surface and out velocity at the straw is 0.
Now we know this maximum velocity is 0.1 m/s.
So at this point we need some constructive equation that is going to relate these forces
that we have calculated for the toothpick, and pulled out of this fluid, and compare
it to this velocity distribution that we have here, and they are related through the propriety
of viscosity.
Newton's Law of viscosity says that the shear stress acting on the fluid is related to the
viscosity of the fluid times a velocity gradient.
Now in this case again since we are assuming a linear relationship.
We are going to work in Cartesian coordinates.
So we have some information about this.
We do not know the viscosity of the fluid, but if we were to look at this differential
velocity with space.
This case the y direction.
We are going from 0.1 m/s to 0, at a distance of 1.5 mm to 2.5 mm.
We also have a way to characterize the shear stress acting on the fluid.
We know that this is the force that is applied over some area, and our force in this case
is negative 0.1N. Now having it at a negative because we are going to the left direction,
based on how I draw the coordinate system earlier the positive goes the the right.
Now our area in this case is going to be the surface area of our toothpick.
So that is going to be pi times our diameter in this case our diameter is 3 mm or 0.003
m times our length.
We said that it was 50 mm of our toothpick being pulled out.
So that is 0.050 m.
We set this equal to the viscosity of the fluid times this ratio that we have calculated.
I am going to convert the 1 mm into m to keep units consistent, and now you can see we can
solve for our viscosity.
When we do this we should get an answer of 2.12 N*s/m^2.
Another system of units used to describe the viscosity of a fluid is that we know that
1 N*s/m^2 is equal to 10000 cP.
So in this case we could write this as 2,120 cP.
To give you an idea water at 20 degree C is a viscosity of 1 cP.
Honey might be somewhere around 10,000 cP.
So this viscosity of the Newtonian fluid is somewhere between water and honey.
Hope this gives you an idea on calculating viscosity based on a velocity gradient, and
the shear stress applied to the fluid.