Continuity equation for two-dimensional real fluids is the same obtained for two- . The derivation of the Hagen-Poiseuille equation for laminar flow in straight. In nonideal fluid dynamics, the Hagen–Poiseuille equation, also known as the . Although more lengthy than directly using the Navier–Stokes equations, an alternative method of deriving the Hagen–Poiseuille equation is as follows. This is known as Hagen-Poiseuille flow, named after the The last equation ( ) balances the pressure force and the viscous damping in our system. Question.
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PDF | The Hagen-Poiseuille equation has been widely applied to the study of fluid feeding by insects that have sucking (haustellate). In nonideal fluid dynamics, the Hagen–Poiseuille equation, also known as the The theoretical derivation of a slightly different form of the law was made. From the velocity gradient equation above, and using the empirical velocity gradient limits, an integration can be made to get an expression for the velocity.
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. 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.
Poiseuille's equation for compressible fluids
But 1. And another way to say that, look at millipascal seconds, that would be a centi, a centipoise, cP. And so a centipoise is really a millipascal second or water at 20 degrees Celsius is 1. 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 centipoise. And then gasses, gasses would have viscosities that are even less. Air has a viscosity of around 0. 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.
The theoretical derivation of a slightly different form of the law was made independently by Wiedman in and Neumann and E. Hagenbach in , Hagenbach was the first who called this law the Poiseuille's law.
The law is also very important in hemorheology and hemodynamics , both fields of physiology. Wilberforce, based on Hagenbach's work.
Although more lengthy than directly using the Navier—Stokes equations , an alternative method of deriving the Hagen—Poiseuille equation is as follows. Liquid flow through a pipe a A tube showing the imaginary lamina. Those closest to the edge of the tube are moving slowly while those near the center are moving quickly. Assume the liquid exhibits laminar flow.
Laminar flow in a round pipe prescribes that there are a bunch of circular layers lamina of liquid, each having a velocity determined only by their radial distance from the center of the tube. Also assume the center is moving fastest while the liquid touching the walls of the tube is stationary due to the no-slip condition. To figure out the motion of the liquid, all forces acting on each lamina must be known: The pressure force pushing the liquid through the tube is the change in pressure multiplied by the area:.
Viscosity and Poiseuille flow
This force is in the direction of the motion of the liquid. The negative sign comes from the conventional way we define. Viscosity effects will pull from the faster lamina immediately closer to the center of the tube. Viscosity effects will drag from the slower lamina immediately closer to the walls of the tube. Viscosity Two fluids moving past each other in the x direction. The liquid on top is moving faster and will be pulled in the negative direction by the bottom liquid while the bottom liquid will be pulled in the positive direction by the top liquid.
When two layers of liquid in contact with each other move at different speeds, there will be a shear force between them. By Newton's third law of motion , the force on the slower liquid is equal and opposite no negative sign to the force on the faster liquid. This equation assumes that the area of contact is so large that we can ignore any effects from the edges and that the fluids behave as Newtonian fluids. Faster lamina Assume that we are figuring out the force on the lamina with radius.
From the equation above, we need to know the area of contact and the velocity gradient. The area of contact between the lamina and the faster one is simply the area of the inside of the cylinder:.
We don't know the exact form for the velocity of the liquid within the tube yet, but we do know from our assumption above that it is dependent on the radius. Therefore, the velocity gradient is the change of the velocity with respect to the change in the radius at the intersection of these two laminae.So if you imagine the dimensions of this box would only expand here. 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.
The laminar flow through a pipe of uniform circular crosssection is known as HagenPoiseuille flow. One thing it depends on is the area and not the whole area of 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.
And so, you'll often hear this unit poise as a measure of viscosity.
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