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### Course: Physics archive > Unit 9

Lesson 3: Fluid Dynamics- Volume flow rate and equation of continuity
- What is volume flow rate?
- Bernoulli's equation derivation part 1
- Bernoulli's equation derivation part 2
- Finding fluid speed exiting hole
- More on finding fluid speed from hole
- Finding flow rate from Bernoulli's equation
- What is Bernoulli's equation?
- Viscosity and Poiseuille flow
- Turbulence at high velocities and Reynold's number
- Venturi effect and Pitot tubes
- Surface Tension and Adhesion

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# Viscosity and Poiseuille flow

Understand the concept of viscosity and how it affects the motion of fluids. Learn about the velocity gradient, the role of adhesive forces, and the impact of fluid depth. Discover the units of viscosity and real-life examples of different viscosities. Get introduced to Newtonian and non-Newtonian fluids and the practical application of Poiseuille's Law.

## Want to join the conversation?

- viscosity depends on temperature but I don't see temperature anywhere in the formula for viscosity.(43 votes)
- Great question. I had the same question myself. I've learned that temperature is accounted for in the calculation of the coefficient of viscosity, by checking out the following link: http://en.wikipedia.org/wiki/Viscosity. Good Luck. Hope this helps.(48 votes)

- difference between density and viscosity please. density to me is how 'thick' something is... i would refer to a 'thicker' liquid as a 'denser' liquid , instead of saying that it is a more 'viscous' liquid.(8 votes)
- Nice point.

In scientific language, you need to think of dense as being 'more mass per unit volume'. And that can refer to liquids, solids and gas.

Viscosity, however, only refers to things that flow. Its is a measure of how badly the substance flows.

it would be worth chewing over your thoughts relating now to materials that are viscous but not dense or visa versa....

For example; honey; mercury; glass;(11 votes)

- I don't understand why the flow rate decreases with an increase in L (length of tube). Is the flow slowing down due to the viscous force?(8 votes)
- Great question, and I think you're on the right track. Length being inversely proportional to the flow rate describes how there is a cumulative effect of friction the longer a fluid must travel down a pipe because it interacts with longer stretches of material. Therefore, as a fluid increases the distance it moves in contact with a pipe, we expect there to be losses of "forward" motion of its particles, which will register as a loss of flow rate of our incompressible fluid.(11 votes)

- If the fluid depth goes to infinity, the derivation of viscosity implies that zero force would be required to slide the lid over the top of the liquid, which is not reasonable. What's the problem? Are there some hidden assumptions in this derivation?(7 votes)
- I think if the depth were really infinity, there would be no friction, because we could perfectly well drag all that water with us. There is no force preventing that because there is no "ocean floor" to provide the friction, and we are assuming there is no other resistance. However, by dragging the water with us we need to conquer its inertia. If the board is 1 kg, the water is 2 kg, and the force pushing the board is 6 N, then acceleration is not 6 N / 1 kg = 6 m/s^2, as is the case when there is no water, but 6 N / (1 kg + 2 kg) = 2 m/s^2, with a = F/m. So the water increases inertia, but doesn't provide a resistance force.

Except with a depth of infinity the mass would probably be infinity too, so it would be impossible to drag the water (a = F/m = F/∞ = 0). But still, you can't blame the mathematics.

Besides, if the board and water are already moving, there sure is no resistance. The "inertial drag" has nothing to do with velocity, only acceleration, and therefore is*not*viscosity.

It may still be hard to believe there is zero resistance, but even if there was another resistance force, by definition it is not the viscous force, but some other force we had assumed didn't exist.

I like your question too. It really started me thinking.(10 votes)

- Is it safe to assume that A1v1=A2v2=poiseuille law ?(3 votes)
- No. Equation of continuity gives you the flow rate when the liquid has no viscosity. Poiseuilles law, simply put, gives you the flow rate when the fluid has viscosity.(15 votes)

- So blood is "thicker" than water because it has a greater viscosity?(7 votes)
- Well almost. Blood flow is better described by viscosity and elasticity, (viscoelasticity).(1 vote)

- How is the viscous force different from the frictional force exerted by the water? If the two are the same, and the coefficient of friction (whether static or kinetic) is independent of both the mass floating above it and the contact surface area, how can the viscous force be dependent on the contact area?(5 votes)
- I think they are the same thing. It's just that usually when you're talking about friction (such as air resistance) you don't usually think about layers of fluid but instead simplify the force as proportional to an object's velocity, such as F = k v or F = k v^2.

F = μ N is just an approximation and only applies to dry friction. The force is really caused by quantum phenomena, the electromagnetic force. Basically electrons from different surfaces interact and create resistance. You can't expect a simple equation to describe all instances of that behavior.

I got this off Wikipedia:

Dry friction resists relative lateral motion of two solid surfaces in contact. Dry friction is subdivided into static friction ("stiction") between non-moving surfaces, and kinetic friction between moving surfaces.

Fluid friction describes the friction between layers of a viscous fluid that are moving relative to each other.

Lubricated friction is a case of fluid friction where a lubricant fluid separates two solid surfaces.

Skin friction is a component of drag, the force resisting the motion of a fluid across the surface of a body.

Internal friction is the force resisting motion between the elements making up a solid material while it undergoes deformation.

Which shows how naïve we all are--what in the world is skin friction!?(4 votes)

- Well, he said Viscosity depends on the "area of contact" of the lid with the surface of the liquid at2:03. My question is, shouldn't Viscosity also depend on the "material" of the lid ? What if we push lid made of rough material like wood then it may take greater force ! Or, if we take smooth plastic lid, then it may take comparatively lesser force ??

I hope my question is clear !!(5 votes)- your question is clear and very interesting (I like quesitons that make us think)

I am thinking honey or syrup... very viscous..

In my view, if the liquid is 'stuck' to the lid as it flows 'over' the lid then it does not matter what the lid is made of.... stuck is stuck. The material is not sliding on the lid; not slipping. so friction does not play a part beyond 'sticking'

Your thoughts??(2 votes)

- If viscosity plays a big role in the velocity of the fluid, how come it's not included in Bernoulli's equation?(3 votes)
- Bernouilli's equation assumes that fluid is ideal so which means that the fluid is nt viscous but in case of non ideal fluid : change in work =change in mechanical energy + energy consumed by viscosity(3 votes)

- I've got a bunch of questions.

First, how can a fluid be Newtonian? Isn't the coefficient of viscosity always affected by the velocity at which an object slides on the fuid's surface? If not, what's an example of Newtonian fluid?

Also, I haven't quite got Poiseuille Law. Is it just valid for these so-called Newtonian fluids? And also, the equation of continuity says that volume flow rate is just influenced by the cross sectional area and by the velocity at which the fluid is flowing. So, why does Poiseuille law state that volume flow rate is inversely proportional to the lenght of the entire pipe?(2 votes)

## Video transcript

- [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.