Will a propeller work in a superfluid?

Question

Will a propeller work in a superfluid? Opinions differ.

Answer 1

No.

I actually tried this in an undergraduate physics lab long ago. We put two fan blades right up against each other in a glass dewar. One was driven and the other was free to spin. We filled the dewar with liquid He and spun the fan. The other spun just fine.

We pumped on the LHe until it transitioned to a superfluid. We spun the fan. The other just sat there and then slowly slowly started to turn.

Edit as requested

So yes, it worked a little. But so poorly that the best answer is no. As Does every superfluid have a normal and a superfluid component? says, it has two components. The normal component was responsible for the residual viscosity. If we had reduced the temperature, there would be a smaller fraction of normal component, and less viscosity. It would work even more poorly.

And I must include the obligitory XKCD.

Answer 2

Interesting question. First of all, on the more general question, it is certainly possible to design devices that provide thrust in a superfluid, and some examples were given in the thread you have linked to. A simple propeller, on the other hand, will not work, if we can assume that the viscosity is exactly zero. If it's just very small, then the propeller should still work fine.

Now, we could still ask if there was a way to make a propeller work in a zero-viscosity fluid, using some auxiliary devices. The critical issue is that we need to create circulation in the fluid, without the help of viscosity. There are ways to create such circulation (using tangential wall jets, say), but I am not sure if effective devices that create forces in such fluids can be designed this way. My gut feeling is that simply using jet thrusters driven by some type of positive-displacement pumps might be the way to go.

To make this argument more rigorous, we will consider the fundamental equations describing this fluid flow problem, which are the incompressible Navier-Stokes equations for an incompressible fluid,

$$\rho\left(\frac{\partial\mathbf u}{\partial t} +{\mathbf u}\cdot{\mathbf\nabla}{\mathbf u}\right)=-{\mathbf\nabla}p+\nu\Delta{\mathbf u},$$

where we are ignoring potential volume forces (such as gravitational forces), plus the divergence-free condition,

$$\nabla\cdot {\mathbf u}=0,$$

within some closed two-dimensional domain $\Omega$ (in general multiply connected; see comments below for the 3-D case).

We will consider problems characterized by boundary conditions that prescribe the flow as being either normal to the boundary, ${\mathbf u}\cdot{\mathbf t}=0$ on part of the boundary $\partial\Omega$ (at inflow/outflow boundaries), or tangential to the boundary, ${\mathbf u}\cdot{\mathbf n}=0$ on other parts of the boundary (along solid walls). Here $\mathbf n$ and $\mathbf t$ are the unit normal and tangent vectors on the boundary, respectively. As our second boundary condition we prescribe a "traction-free condition", $\nu\,\Delta{\mathbf u}\cdot {\mathbf t}=0$, meaning the fluid slides across the surface without friction.

We choose an initial condition for a velocity field ${\mathbf u}_0={\mathbf u}(t=0)$ which satisfies $\nabla\cdot{\mathbf u}=0$ (incompressibility condition) as well as $\nabla\times{\mathbf u}=0$ (irrotationality) in the interior of $\Omega$. Finally, we require that the initial velocity field ${\mathbf u}_0$ is at least $C^0$-continuous on the closed domain $\Omega$. We note in passing that this condition is non-trivial and often violated in many practical problems (meaning, numerical solutions). It plays a crucial role in the mathematics of the Navier-Stokes problem, however: If this condition is violated then the Navier-Stokes problem is in fact ill-posed. To avoid unnecessary complications I will require the slightly stronger condition of $C^1$-continuity below. The difference is mathematically non-trivial, but should be of no physical consequence.

Now, with these preparations it is possible to prove that the space of potential flow solutions of this problem is an invariant subspace of the Navier-Stokes equations, meaning that if our initial condition represent an irrotational, incompressible flow, the flow must remain so at all future times. One specific consequence of this situation is that the circulation $\Gamma$ satisfies

$$\Gamma=\oint_C {\mathbf u}\cdot\mbox{d}{\mathbf s}=\iint_\Omega \nabla\times{\mathbf u}\,\mbox{d}{\mathbf x}=0$$

at all times. I will note that I'm leaving out the proof. The technical details are somewhat severe; for details see Ladyzhenskaya's Mathematical Theory of Viscous Incompressible Flow.

At this point the only additional ingredient we need is the Joukowski theorem which says that forces in potential flows around closed contours are proportional to the circulation $\Gamma$ around the contour. Since we have shown above that the circulation remains zero, there can be no forces.

I remind the reader that the above argument assumes a two-dimensional domain. I'll just state without proof that it can be extended to the three-dimensional case, at the cost of considerable mathematical complexity...

Finally, it is worth pointing out that the above argument addresses the mathematics of this problem for the case of an ideal inviscid fluid. If you preform an experiment in such a flow, a number of real-world effects may change the outcome significantly. For example, the potential flow around sharp trailing edges of the propeller's airfoils requires extremely strong pressure gradients. I am fairly certain that any real fluid would be subject to cavitation under these conditions, and who knows what might happen then. Most certainly the model of the ideal incompressible fluid will not apply to the situation anymore.

Answer 3

Fluids with viscosity zero (superfluids) are unable to exchange energy with a moving object submerged in it. So if the object is a rotating propeller, the fluid can't transfer energy to it, which is the same as saying that if the propeller is attached to a submarine, the submarine's kinetic energy can't increase, so the propeller is of no use. Even a square piece of metal moving through a superfluid doesn't experience a change in velocity (which is rather counter-intuitive).

Answer 4

Yes, it will work. Design of propellers is often done by assuming inviscid and incompressible fluid dynamics models and simulations. Viscosity changes things quantitatively but not qualitatively. Since a propeller works at high Re number, viscosity is often safely neglected. A simple Google search will show you a lot of technical studies of propeller design in inviscid (zero viscosity) fluids. A superfluid at zero temperature and with no quantum vortices is the best representation of such inviscid fluid. Adding quantum vortices and normal component goes in the direction of having also a viscous component and still the propeller will work.

Answer 5

Studies on propellers in inviscid fluids: Euler equations are often used to study propellers, e.g. A review of propeller modelling techniques based on Euler methods (1998). In the post How can I calculate the thrust of a propeller?, you will see that the thrust does not depend on the viscosity of the fluid. This is consistent with the basic description of how a propeller works, where viscosity or friction is not even mentioned: the fundamental principle behind propellers is Bernoulli, that is valid also in superfluids.

Furthermore, let's make the assumption that a propeller does not work in a perfect fluid. Then, it would not work also in the high Reynolds number regime of any Navier-Stokes fluid, where it is a good approximation to neglect viscosity (note that this is the regime where most propellers are designed to work, in the high Reynolds number limit). See e.g. this paper. Hence, the belief that a propeller needs viscosity to work leads us to a strong contradiction.

Therefore, while other answers claim "no", I would say "yes, it works". In fact, It is well known that the hydrodynamic properties of a zero-temperature superfluid (no viscous normal component) are those of an inviscid fluid ruled by the usual Euler equation (plus the quantization of circulation).

Other kinds of propeller: There is a kind of propeller that works for sure, as it is based on plain action-reaction, i.e. total momentum conservation: consider a propeller that works like a "syringe" filled with superfluid. To "recharge" it, you may close the main nozzle and open valves on the sides of the syringe, so the superfluid is sucked from the sides (and not along the direction of motion). Then, you close the lateral valves and squeeze the superfluid out, from the main nozzle.

How about other ways of "swimming" in a superfluid? Will an "anguilliform" or "thunniform" (see fish locomotion) kind of motion will work in a superfluid? Again, I would say yes, even though the most efficient one may not be the one that is most efficient in water. In fact, there are "swimming strategies" that can work in incompressible perfect (i.e. zero viscosity) fluids, see this paper and references therein, or this one (the other extreme case, viscosity dominated, is described in the paper Life at low Reynolds number). Also, this answer, while a comparison between the low Reynolds number case and the inviscid one is outlined here.