Viscosity and Poiseuille flow Video Lecture - NEET

- [Voiceover] Check out this empty box.
Now, check out this lid
that I can put on the empty box.
Let's say there is no resistance
so that if I gave this lid a little nudge,
it just keep moving
with the constant speed.
Let's say there's no air resistance.
No resist to forces at all.
Now what I'm gonna do,
I'm gonna stick a fluid.
Let's say water.
And I'm gonna fill it to the brim
all the way to the top.
This thing was overflowing.
This goes all the way to the top.
Now what happens is you take this lid,
you go to slide it across again.
You give it a little nudge,
it doesn't keep going,
it slows down and stops.
You give it another nudge,
it slows down and stops.
The fact that this fluid's in here now
is resisting the motion
that's because it's viscous.
What do I mean by that?
I mean that when this lid
was moving across the top,
the fact that this fluid
was in contact with the lid
caused this top most layer to start moving
with the same speed as the lid.
There's adhesive forces
between this fluid and the lid
on an atomic and molecular level,
this fluid gets pulled with it.
And so it resist the motion
but it's worse than that
because if this top most
layer gets pulled this way
then the layer right below it
gets pulled by the top most layer
and the second most layer
pulls this third most layer
and this keeps going down the line,
and what you get is a velocity gradient
is the fancy name for it.
Which just means that this
velocity of this fluid
gets smaller and smaller.
Once you get down to the bottom
well, now the fluid is in contact
with this surface at the bottom
and this surface at the
bottom is not moving.
This fluid is at the lowest
most point doesn't move at all.
All right, so that's
why this lid slowed down
when we gave it a nudge,
it was dragging that fluid along with it.
And if it exerts a force
to the right on the fluid,
then the fluid is gonna exert a force
by doing its third law
to the left on the lid
and I'm gonna call this a viscous force.
I'm gonna call it Fv.
So, what does this depend on?
What does this viscous force depend on?
One thing it depends on is the area
and not the whole area of the lid.
It's just the area of the lid
that's actually in contact with the fluid.
So if you imagine the
dimensions of this box
would only expand here.
So it's only that part of the lid
that's actually in contact with the fluid.
So that area and it's
proportional to that area.
The bigger that area,
the more fluid you're gonna be dragging,
the larger the viscous
force that makes sense
so it's this area here.
And something else that it depends on
is the speed with which you drag the lid.
So the faster I pull the lid,
well, the faster I'm gonna
be pulling this water,
the bigger that force
which means the bigger the viscous force,
which was proportional
to the speed as well.
It's inversely proportional
to the depth of the fluid.
I'll call that D.
And then it depends on one more thing.
It depends on the viscosity of the fluid.
Maybe the most important factor
in this whole discussion.
Eta is gonna be called
the viscosity of the fluid
or the coefficient of viscosity.
And what this number tells you
is how viscous, how thick
essentially the fluid is.
How much it resist flow.
And so, coefficient of viscosity.
So to give you an idea honey
or corn syrup would
have a large viscosity.
Water would have that
smaller viscosity coefficient
and gasses would have a
coefficient of viscosity even less.
So what are the units of this
coefficient of viscosity?
Well, if we solved.
If we were to solve for the eta,
what would we get?
We'd get force divided by area.
So this would be force divided by area,
and multiplied by the
distance divided by the speed.
What units do these have?
Force is newtons, distance is in meters,
area is in meters squared,
speed is in meters per second.
So I bring that second up top
because it was divided in the denominator.
So it goes up top.
And what am I left with?
Meters cancels meters
and I'm left with the units of viscosity
as being newtons per meter
squared times a second,
but a newton per meter squared is a pascal
so this is pascal seconds.
So these units are a little strange
but the units of eta, the
coefficient of viscosity
is a pressure times a
time, pascal seconds.
But some people use the unit poise
and one poise
is equal to 1/10 of a pascal second.
Or in other words 10 poise
and it's abbreviated capital P is
one pascal second.
And so, you'll often hear this unit poise
as a measure of viscosity.
So what are some real life
values for the viscosity?
Well, the viscosity of water
at zero degree Celsius is,
and I'm not talking about ice
but water that's actually at zero degrees
but not frozen yet is about 1.8.
But 1.8 millipascal seconds.
And another way to say that,
look at millipascal seconds,
that would be a centi, a centipoise, cP.
Because a poise is already
1/10 of a pascal second.
Remember one poise is 1/10.
And so a centipoise is
really a millipascal second
or water at 20 degrees Celsius is
1.0 millipascal seconds or centipoise.
Now you can see there's a huge
dependency on temperature.
The viscosity is highly
dependent on temperature.
The colder it gets,
the more viscous a fluid typically gets
which you know, because
if you start your car
and it's too cold outside,
that car is not gonna want to start.
That oil inside's gonna be more viscous
than it's prepared for,
and that engine might
not start very easily.
Blood typically has a viscosity
between three to four millipascal
seconds or centipoise.
And then engine oil can have
viscosities in the hundreds.
Around 200 centipoise.
And then gasses, gasses would have
viscosities that are even less.
Air has a viscosity of around
0.018 centipoise.
Now it's important to note,
if a fluid follows this
rule for the viscous force
and the coefficient of
viscosity does not depend
on the speed with which
this fluid is flowing
or with which you pull
this lid over the top
does not depend on that.
Then we call this a Newtonian Fluid.
Then it's a Newtonian Fluid.
But if the coefficient of viscosity
does depend on the speed
with which the fluid is flowing
or the speed with which
you pulled this lid,
then it would be a non-Newtonian fluid.
So if this coefficient of viscosity
stays the same regardless
of what the speed is,
it's a Newtonian fluid.
If that's not the case,
it would be a non-Newtonian fluid.
Now, you might be thinking
well, this is kind of stupid.
I mean, how many cases are we gonna have
where you're trying to
pull a lid over a box,
you've probably never tried
to do that in real life.
But this is just a handy way
to determine the viscosity.
Once you know the viscosity
you can apply this number.
This is a constant of the fluid.
Now, anywhere that this fluid is flowing
now that you measured it carefully,
you could determine what kind
of flow rate you would get.
So imagine, let's get rid
of all these stuff here.
So you had a stationary tube or a pipe
and there was a fluid flowing through it,
maybe it's a vein or a vessel
and it's blood flowing through it.
Anyways, now it's stationary though.
Both the top and the bottom are stationary
so that means the fluid near the top
and the fluid near the
bottom aren't really moving
but it's the fluid in the middle
that's moving fastest
and then slightly less fast,
and you get this somewhat parabolic type
velocity gradient where
it gets bigger and bigger
and then it gets smaller and smaller,
and so, the velocity profile
might look something like this.
And, if we wanted to know how much volume
of fluid, how many meters cube of fluid
passed by a certain point per time,
we can figure that out.
There's a formula for this.
The volume per time, the
meters cube per time.
Now the formula is
called Poiseuille's Law.
And it says this.
It says the volume that will flow per time
is dependent on delta P
times pi, times R to the fourth,
divided by eight eta, times L.
Now this is a crazy equation.
Let's break this thing down
and see what it's really talking about.
So here's Poiseuille's Law.
So, this delta P is referring
to the pressure differential.
So it will be pressure on this side.
We'll call it P one.
There'll be pressure on this side, P two.
If those were the same,
this fluid's not gonna
be flowing very long.
There's gonna be a difference.
If we want the fluid to flow to the right,
P one has to be bigger than P two,
and the greater the difference,
the greater this difference
P one minus P two,
the more volume that's gonna flow per time
and that makes sense.
And then pi is a geometric factor.
R to the fourth.
This R is referring to
the radius of the tube.
So that's R.
And then eight and eta we know.
Eta is the viscosity.
So this is the viscosity of the fluid.
And the volume flow rate's
inversely proportional to the viscosity
because the more viscous the fluid,
the more it resists flowing,
and the less meters cube
you'll get per second.
And it's inversely proportional also
to the length of the tube.
The more tube this fluid's
got to flow through,
the smaller the volume flow rate.
So this is called Poiseuille's Law.
It's useful on a lot of medical
and engineering applications.
For whenever you want to determine
the volume flow rate,
now you've got to be careful.
We're assuming this is a Newtonian fluid,
that means eta's not a function
of the speed of the fluid.
We're also assuming you have
nice streamlined laminar flow.
So laminar flow means
these layers of fluid stay
in their lane basically.
They do not crossover.
Once you start getting this,
you'll start getting turbulence
and once you get turbulence,
you'll need a much more complicated
equation to describe the dynamics of this.
So we're assuming no turbulence
and a nice Newtonian fluid.
And if that's the case,
Poiseuille's law gives you the volume
that flows through a pipe per time.