# What is the theoretical justification for a fluid flow's being irrotational?

I am not a fluid dynamicist, and I really just began thinking about this problem as my curiousity drew me into building an answer for the question What really allows airplanes to fly?.

It is very clear from the answers to the following questions:

When is a flow vortex free?

Does a wing in a potential flow have lift?

that viscous flows cannot be everywhere irrotational. Moreover, some handwaving justification is given in answer to the first question that low viscosity fluid can be irrotational.

Now, the spin angular momentum per unit volume of a fluid is the vector $\rho\,\nabla \wedge \vec{v}$. So the assumption of irrotational flow is the assumption of lack of spin angular momentum.

I can accept that it is reasonable in some cases to accept this to be true. But is there a deeper theoretical justification as to why and when a fluid's spin angular momentum should be nought?

In electromagnetism, we have specific equations - the Ampère and Faraday laws i.e. $\nabla\wedge\vec{H} = \vec{J}+\partial_t\vec{D}$ and $\nabla\wedge\vec{E} = -\partial_t\vec{B}$ - for the "source" of such vorticity in the fields. At an elementary level, we can see that in electrostatics conservation of energy behests that $\oint\vec{E}\cdot \mathrm{d} \vec{r} = 0$, so we immediately grasp irrotationalhood for the electrostatic field.

Is there any such analogue in fluid dynamics?

The Navier-Stokes equation doesn't seem "split" the velocity field up into curl and divergence terms like the Maxwell equations do, or to put it more pithily, the Maxwell equations say that the source distribution must be an exact form $\mathrm{d} \vec{F} = \mu_0 \vec{J}$ and we can thus "invert" $\mathrm{d}$, to within a constant (which we can set to nought the grounds that it would have infinite energy).

Are there any other ways to "split" the Navier-Stokes equation like Maxwell's equations are to shed intuitive light on the nature of an irrotational flow?

It's not just a simplifying assumption. There are good reasons as to why the approximation is used for incompressible flows: Vorticity is introduced into incompressible flows only through surface boundaries. A flow that is initially irrotational and and does not interact with a surface will remain irrotational. You can see this from the equation for the evolution of vorticity $$\frac{D\vec{\omega}}{Dt}=\nu\nabla^2\vec{\omega} + \vec{\omega}\cdot\nabla\vec{U}$$ Almost all flows interact with surface boundaries, however, because flows that start out irrotational remain so, outside of the boundary layer the irrotational approximation remains a good one.

• Many thanks, very helpful. I dimly recall this now that you raise it: this is Kelvin's vorticity theorem, right? Oct 31, 2013 at 22:59

it's just a simplifying assumption. It's in fact rarely true. The main problem is that the Navier-Stokes equations are just tremendously more complex than the Maxwell equations--they have chaotic solutions, they couple nonlinearly, they behave wildly differently in the viscous and inviscid limits, and most strikingly, they have no proven existence and uniqueness theorem.

So, when learning them, it's best to work out simplified, specialized cases, like the steady flow, irrrotational case (which can then be solved using methods more analogous to E&M), and then to work from there.

• I guess the N-S existence and smoothness problem doesn't have a million dollar Clay Mathematics Institute prize for its solution for nothing then! Sep 23, 2013 at 5:06

You flow field is irrotational when the differential parcels of fluids are not rotating about itself. That is why a free vortex is irrotational, even if its streamlines have a circular path. A solid body can translate or rotate or both. It is the same for fluids. They are irrotational if there is only translation on the fluid particles but for fluids translation is given by pressure forces. You do not need incompressible viscous fluids to have irrotational flows. You also can have a irrotational flows with newtonian, compressible, viscous fluids. For example the flow: U = 1j + 1k +1z is irrotational for whatever fluid. But of course general flows will have vorticity since in a general case they depend on space, time and pressure.