HAGEN POISEUILLE EQUATION DERIVATION PDF
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.
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. 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. 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. If the pipe is too short, the Hagen—Poiseuille equation may result in unphysically high flow rates; the flow is bounded by Bernoulli's principle , under less restrictive conditions, by Relation to Darcy—Weisbach Normally, Hagen-Poiseuille flow implies not just the relation for the pressure drop, above, but also the full solution for the laminar flow profile, which is parabolic.
However, the result for the pressure drop can be extended to turbulent flow by inferring an effective turbulent viscosity in the case of turbulent flow, even though the flow profile in turbulent flow is strictly speaking not actually parabolic. In both cases, laminar or turbulent, the pressure drop is related to the stress at the wall, which determines the so-called friction factor.
Viscosity and Poiseuille flow
The wall stress can be determined phenomenologically by the Darcy—Weisbach equation in the field of hydraulics , given a relationship for the friction factor in terms of the Reynolds number. It proves more useful to define the Reynolds number in terms of the mean flow velocity because this quantity remains well defined even in the case of turbulent flow, whereas the maximal flow velocity may not be, or in any case, it may be difficult to infer.
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.
Poiseuilles Law Formula
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:.
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.
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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.Viscosity effects will pull from the faster lamina immediately closer to the center of the tube.
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:. 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. Or in other words 10 poise and it's abbreviated capital P is one pascal second.
So this is the viscosity of the fluid. Low viscosity or a wide pipe may result in turbulent flow, making it necessary to use more complex models, such as Darcy—Weisbach equation. And then engine oil can have viscosities in the hundreds.
And if that's the case, Poiseuille's law gives you the volume that flows through a pipe per time. The differential equation can be integrated to: To find A and B, we use the boundary conditions.
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