Circulation (physics)

In physics, circulation is the line integral of a vector field around a closed curve. In fluid dynamics, the field is the fluid velocity field. In electrodynamics, it can be the electric or the magnetic field.

Circulation was first used independently by Frederick Lanchester, Martin Kutta and Nikolay Zhukovsky.^{[citation needed]} It is usually denoted $Γ$ (Greek uppercase gamma).

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Here's a diagram of the four chambers of the heart. So let's name them to get started. So we've got the right atrium up here. We've got the right ventricle down here. We've got the left atrium and the left ventricle. So these are the four chambers. And blood is going to flow through all of them and then get out to the body. So to do this and to do this right, the heart has got to coordinate how it squeezes. And we know that the way that it kind of squeezes down is, you have a cell. And that cell is usually negatively charged. And it will, at some point, become more positively charged. And we call that process depolarization. So depolarization is the idea of going from a negative membrane potential to something much more positive. And when you depolarize is when the muscle cell can squeeze down. So where does that begin? Let's actually draw it into our diagram. So if you were to look, there's actually an area here where little cells can actually depolarize by themselves. And that's actually quite unique, because most of the cells in the body are going to be polarized, when that neighboring cell depolarizes. So these are actually really unique cells, because they're depolarizing all by themselves. And we call that area the sinoatrial node, sometimes called the SA node. And the fact that they can actually depolarize by themselves, we have a word for that too. We call it automaticity. So it just means that they can kind of automatically depolarize, without having a neighbor do it first. So once they depolarize, what happens after that? Well, when these cells depolarize, immediately they're connected through little gap junctions to the neighboring muscle cells. And so they're going to start sending out waves of depolarization in all directions. And so it's almost like going to a football game and watching the wave start. It just goes on and on and on. And so all the neighboring cells are going to start depolarizing as well. And that orange arrow is moving kind of slowly. That depolarization wave is moving kind of slowly, relative to how fast it could be moving if it went, through a specialized band of tissue. So this band of tissue that I'm drawing, this blue band is almost like a highway, compared to that orange arrow, which is like a little road. And that highway is going to take that same depolarization wave over to the other side, over to the left atrium. And all of these cells begin to do the same thing. They start depolarizing as well. See you've go depolarization happening both in the right atrium and the left atrium in a coordinated way. So it's happening all very evenly. And this band or bundle is called Bachmann's bundle. So it's like a little bundle of tissue. So it's called Bachmann's bundle. So now we've named two things-- the sinoatrial node and Bachmann's bundle. And actually, just like Bachmann's bundle, there are actually a few more little bands of tissue, almost like little highways that take that signal down to another node, called the atrial ventricular node. So this right here is the atrial ventricular node. And the atrial ventricular node is really the only major connection-- I shouldn't even say, only major-- only connection in most of us, between the atria and the ventricles. The atrial Ventricular node, and this is actually sometimes called the AV node. So the AV node is going to get this signal. And actually, I didn't even tell you what that signal came through. It cam through-- this kind of a generic name-- internodal, mean between two nodes, tracks. And that's kind of the name for all three of them. So the signal went from the SA node through the internodal tracks down to the AV node. And there, kind of an interesting thing happens. So if you actually take a step back and look at the AV node, let's imagine we're now kind of focused in on exactly what's happening there. And to figure out what's happening there, I'm going to give you a little scenario. So let's say that you've got-- I don't know, let's say, a little timeline here. And that timeline is, let's say, one, two, three seconds-- three seconds. And your job is just to watch the atria and see how they contract. So you just watch the atria. And you say, wow, I saw one contraction that happened right there and one contraction that happened right there and one contraction that happened right there. So the atria, as they get their wave of depolarization, are contracting now three times in three seconds. So for the atria, you saw three contractions. Now you do the exact same thing, but you do it for the ventricles. So for the ventricles, you kind of just keep and eye, and you watch exactly what happens. And you notice that there's a contraction in the ventricles there and again there, and one more there. So both the atria and the ventricles are both contracting the same number of times. But the unique thing is that there's this little delay between the two. They're not actually contracting at the same moment in time. There's this tiny delay. And if you measured it, it would be about 0.1 seconds, so just a tiny little fraction of a second. But the reason that there's that delay is due to the AV node. So one of the kind of interesting things about the AV node is that it creates a delay, between the atria and the ventricles. And the reason that's really important is, that if the atria and the ventricles were actually contracting simultaneously, then they would actually be squeezing blood against each other. They would be actually doing work that wouldn't actually move the blood in the right direction. So by creating the delay, the atria can squeeze. The blood can move from the atria to the ventricles. And then, a tenth of a second later, the ventricles can squeeze. And then the ventricles can move that blood onward. So the reason for the delay is actually to make sure the blood moves in a coordinated way through the heart. So now this signal has delayed by a tenth of a second. But then it continues on. It continues on, and it goes to little area right there. And this is called the bundle of His. Kind of funny names, I know-- bundle of His. And even those it's spelled H-I-S, you don't say, his. It's hiss, almost like what a snake does. And then it continues from the bundle of His through one track down here. And this is considered the right bundle. And then it goes through the left bundle. And actually, the left bundle splits. There's like a forward part that goes up to the front and a part that goes to the back. And I'm going to draw the back part kind of dashed like that. So this is called the left posterior-- because posterior means back posterior fascicle. And this is called the left anterior-- because it's coming forward-- anterior fascicle. And you got to kind of imagine that it's going forward and back, because obviously, in two dimensions it's hard to show that. And then this is just called the right bundle. And actually, just so you're not ever mistaken, this part right here is called the left bundle, where it's still combined and it's not broken into the two fascicles. So you have the left and the right bundle. And then, the left bundle splits again. And then all of the fibers get really kind of split up here at the end. And these are called the Purkinje fibers. And it happens on both sides, the Purkinje fibers. And from this point, basically, the electrical signal can kind of dash out in all directions. So you can finally get all the muscle cells involved. So up until now, it's been part of the electrical conduction system, meaning these are all the highways. But now you have all the waves of depolarization going through all the little tiny roads. And I'm using the idea of roads and highways, just to point out the idea, that through the electrical conduction system, the signal moves really fast. And when you get down to the muscle itself, then the signal moves slightly slower. But you can see that's important, because that's the only way to get all the muscle cells on the same page. So this is how the electrical signal moves from the SA node, all the way through the electrical conduction system, so that the atria beating together, and then goes into the AV node, where there's a little delay and then down into the ventricles storage, where again, the ventricles are going to beat together.

Definition and properties

If $V$ is a vector field and $d l$ is a vector representing the differential length of a small element of a defined curve, the contribution of that differential length to circulation is $dΓ$ :

\mathrm {d} \Gamma =\mathbf {V} \cdot \mathrm {d} \mathbf {l} =\left|\mathbf {V} \right|\left|\mathrm {d} \mathbf {l} \right|\cos \theta .

Here, $θ$ is the angle between the vectors $V$ and $d l$ .

The circulation $Γ$ of a vector field $V$ around a closed curve $C$ is the line integral:^[1]^[2]

\Gamma =\oint _{C}\mathbf {V} \cdot \mathrm {d} \mathbf {l} .

In a conservative vector field this integral evaluates to zero for every closed curve. That means that a line integral between any two points in the field is independent of the path taken. It also implies that the vector field can be expressed as the gradient of a scalar function, which is called a potential.^[2]

Relation to vorticity and curl

Circulation can be related to curl of a vector field $V$ and, more specifically, to vorticity if the field is a fluid velocity field,

{\boldsymbol {\omega }}=\nabla \times \mathbf {V} .

By Stokes' theorem, the flux of curl or vorticity vectors through a surface S is equal to the circulation around its perimeter,^[2]

\Gamma =\oint _{\partial S}\mathbf {V} \cdot \mathrm {d} \mathbf {l} =\iint _{S}\nabla \times \mathbf {V} \cdot \mathrm {d} \mathbf {S} =\iint _{S}{\boldsymbol {\omega }}\cdot \mathrm {d} \mathbf {S}

Here, the closed integration path $\partialS$ is the boundary or perimeter of an open surface $S$ , whose infinitesimal element normal $d S = n dS$ is oriented according to the right-hand rule. Thus curl and vorticity are the circulation per unit area, taken around a local infinitesimal loop.

In potential flow of a fluid with a region of vorticity, all closed curves that enclose the vorticity have the same value for circulation.^[3]

Uses

Kutta–Joukowski theorem in fluid dynamics

In fluid dynamics, the lift per unit span (L') acting on a body in a two-dimensional flow field is directly proportional to the circulation, i.e. it can be expressed as the product of the circulation Γ about the body, the fluid density $\rho$ , and the speed of the body relative to the free-stream $v_{\infty }$ :

L'=\rho v_{\infty }\Gamma

This is known as the Kutta–Joukowski theorem.^[4]

This equation applies around airfoils, where the circulation is generated by airfoil action; and around spinning objects experiencing the Magnus effect where the circulation is induced mechanically. In airfoil action, the magnitude of the circulation is determined by the Kutta condition.^[4]

The circulation on every closed curve around the airfoil has the same value, and is related to the lift generated by each unit length of span. Provided the closed curve encloses the airfoil, the choice of curve is arbitrary.^[3]

Circulation is often used in computational fluid dynamics as an intermediate variable to calculate forces on an airfoil or other body.

Fundamental equations of electromagnetism

In electrodynamics, the Maxwell-Faraday law of induction can be stated in two equivalent forms:^[5] that the curl of the electric field is equal to the negative rate of change of the magnetic field,

\nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}

or that the circulation of the electric field around a loop is equal to the negative rate of change of the magnetic field flux through any surface spanned by the loop, by Stokes' theorem

\oint _{\partial S}\mathbf {E} \cdot \mathrm {d} \mathbf {l} =\iint _{S}\nabla \times \mathbf {E} \cdot \mathrm {d} \mathbf {S} =-{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{S}\mathbf {B} \cdot \mathrm {d} \mathbf {S} .

Circulation of a static magnetic field is, by Ampère's law, proportional to the total current enclosed by the loop

\oint _{\partial S}\mathbf {B} \cdot \mathrm {d} \mathbf {l} =\mu _{0}\iint _{S}\mathbf {J} \cdot \mathrm {d} \mathbf {S} =\mu _{0}I_{\text{enc}}.

For systems with electric fields that change over time, the law must be modified to include a term known as Maxwell's correction.

References

^ Robert W. Fox; Alan T. McDonald; Philip J. Pritchard (2003). Introduction to Fluid Mechanics (6 ed.). Wiley. ISBN 978-0-471-20231-8.
^ ^a ^b ^c "The Feynman Lectures on Physics Vol. II Ch. 3: Vector Integral Calculus". feynmanlectures.caltech.edu. Retrieved 2020-11-02.
^ ^a ^b Anderson, John D. (1984), Fundamentals of Aerodynamics, section 3.16. McGraw-Hill. ISBN 0-07-001656-9
^ ^a ^b A.M. Kuethe; J.D. Schetzer (1959). Foundations of Aerodynamics (2 ed.). John Wiley & Sons. §4.11. ISBN 978-0-471-50952-3.
^ "The Feynman Lectures on Physics Vol. II Ch. 17: The Laws of Induction". feynmanlectures.caltech.edu. Retrieved 2020-11-02.

This page was last edited on 10 December 2023, at 01:25

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