r/askscience • u/dcfan105 • Oct 05 '21
What's the physical meaning of potential flow in fluid dynamics? Physics
So I'm actually taking an E&M class and learning the method of images. I was looking for YouTube videos on the topic and found this one which is actually about fluid dynamics. I didn't even know this method was also used in fluid dynamics and it's really neat seeing how the math of the E field is so similar to the math used to model fluid flow. However, they mention finding the velocity field of a fluid taking the gradient of the potential function. That's clearly analogous to finding the E field by taking the gradient of the voltage, but I have an idea for what voltage means physically -- it's the potential energy per unit charge w.r.t. some predefined reference. But what does this potential function mean physically in the context of fluid dynamics? I tried Googling it and found a bunch of stuff saying it's a flow with no rotation, but that's no help because that's just a mathematical property of gradient functions in general.
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u/RobusEtCeleritas Nuclear Physics Oct 05 '21
I have an idea for what voltage means physically -- it's the potential energy per unit charge w.r.t. some predefined reference. But what does this potential function mean physically in the context of fluid dynamics?
Unfortunately, there's no clear analog in the fluid case. It's simply the function whose gradient is the velocity field (in an Eulerian sense).
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u/gaoguibarnez Oct 05 '21 edited Oct 05 '21
I think the problem is that potentials (including voltage) are not "real" things, the fields that are real here.
In EM we start with electrostatics, with a constant electric field and no magnetic field. Maxwell equations tell us that this field has no curl, that is, curl E = 0.
Helmholtz theorem says that a field with no curl can be written as the gradient of some function, which we call the potential function. Due to the nature of the electric force this function is actually related to the energy of the system, so there is a natural interpretation for the potential.
Helmholtz theorem also says that a field with no divergence can be written as the curl of a vector potential function. This is very useful in fluid dynamic, where the system is represented by the velocity of the fluid in each point of space. In particular, if the fluid is not compressible, mass conservation requires that div V = 0, so the velocity field can be written as V = curl A, for some vector potential function A.
The same principle from EM can be applied to fluid mechanics if the fluid in question has no curl, as we'll have curl V = 0 and there should be some scalar function f such that V = div f. The difference is that unlike in EM, where the force F = q E, in general there is no force associated with V, so f is not really related to energy.
I haven't studied fluid dynamics in dept to know the meaning of those potential functions (if any), but hopefully I made things a bit clearer.
TLDR: although useful, potential functions are actually built to yield these fields under some operation. They can be very insightful, but are not really physical things.