# Does a fluid with zero viscosity cause a body moving through it to decrease it's velocity?

If a massive body (suppose it's a square piece of metal, perpendicular to its velocity) is moving through a fluid with zero viscosity, does the metal feel a force decreasing its velocity?

I would say yes because the metal makes some part of the fluid change momentum, the effect of which is to change the momentum of the piece of metal.And because the fluid has zero viscosity (inner friction) the fluid can bounce back, without affecting other parts of the fluid, and won't create vortices.

• If the fluid has a free surface (or there is a density gradient in the fluid) then the answer is certainly yes through the creation of waves! See, for instance, the Cauchy-Poisson process. – Nick P Jan 31 '17 at 20:01

I believe this is a known paradox in hydrodynamics known as D'Alembert's paradox. The TL;DR on that is if you assume zero viscosity and a potential flow, you don't get any drag force, though this is counterintuitive.

The effect you're describing in the second paragraph reminds me of the phenomenon of induced or added mass - though that's relevant to accelerating bodies, and not to bodies with constant velocity moving through a steady-state fluid.

• D'Alembert's paradox seems to only be a paradox if you don't understand the properties of viscosity as well as we do now. It was a paradox because they didn't account for viscosity but were dealing with viscous fluids. This is the opposite situation. It's hard to really picture an inviscid fluid. I'm not sure it even would impart momentum into the fluid though. There would be no mechanism to transmit any resistance onto the solid object that I can think of. Inviscid fluids are really weird, they can climb walls to exit containers and do a lot of other strange things. – JMac Jan 31 '17 at 11:48
• @Jmac-Why can't a fluid without viscosity don't impart a momentum to the plate we are talking about here? Although the fluid has no internal friction, it can transfer momentum to an object in the fluid, because there's no inner fluid friction involved in the momentum transfer. I changed my answer in a "yes", because in d'Alembert's Paradox an incompressible fluid is considered. But now I think about it, I'll change my answer again... – descheleschilder Jan 31 '17 at 20:31
• @descheleschilder I still don't really understand the method with which the momentum would transfer. Have you seen how inviscid fluids behave? It isn't like regular fluids. I'm still not convinced that the inviscid fluid wouldn't just creep around the plate and keep going as if nothing happened. – JMac Jan 31 '17 at 23:39

The perpendicular impact on the metal causes the fluid to bounce back and change momentum. Without any influence on the surrounding fluid due to friction forces in the fluid, which has, after all, zero viscosity. So no turbulence develops and the fluid on the side of the metal plate onto which the fluid bumps, is build up in two regions, which don't influence each other: a fluid layer with a velocity in the direction of the velocity of the metal plate (due to the change in momentum), and a layer which isn't affected by the plate (see picture). The fluid moving "outside" the plate flows around the edge of the plates towards the lower fluid pressure (created by the moving plate) without forming turbulence and the fluid behind the plate moves along with the same velocity, but lower density, and thus lesser momentum. This compensates for the change momentum of fluid in front of the plate, so no net force is created to slow the plate down. But it stays counter-intuitive. See picture below.(By the way, by now it must be clear that I don't agree anymore with me answering "yes" to the answer in my question).

If I make the plate bigger, the low pressure behind the plate gets higher, so more frictionless fluid is "sucked" in, with the result that the momentum change (due to a greater area of the plate) is compensated for by an adjusted decreased fluid density behind the plate.

the momentum change due to the bouncing back of fluid from the plate is compensated by an equal, but opposite momentum change caused by a lower density of the fluid behind the plate. And thus is the net force (=dP/dt) on the plate zero. The difference with d'Alembert's Paradox is that in this case, the fluid is not incompressible.

So Of course, you can apply this reasoning to masses of any form. EDIT1 I see now that although the fluid density behind the plate becomes lesser, the total momentum stays the same because of the greater area. The momentum of the two layers of fluid above and below the layer with a changed momentum due to the collision with the plate must of course stay the same, so there is indeed (as was my first impression), a net force that slows down the plate. I think that's because, in contrast to d'Alemberts Paradox, the fluid is not incompressible.

EDIT2 The density of the fluid behind the plate doesn't change, but the fluid velocity will be lower (like cars driving on a road with two lawns will decrease their velocity if they go over to a road with four lanes), so the total momentum to the right will be the same as the total momentum of the two layers with velocity to the right.

EDIT3

Last edit! See second picture. The fluid behind the plate is divided into three layers. The layer above and beneath have the same momentum as in the two layers on the left side, but the middle layer has a momentum that is less than the main momentum (this right middle layer doesn't interact with the two layers above and beneath it because of the lack of internal friction), due the low pressure behind the plate. The tube with the fluid is infinite long. So the plate is stopped to the point where it has the same velocity as the fluid, and the whole fluid is streaming to the right again with equal momentum everywhere. So as far as I can see, the motion of a fluid without viscosity will stop an object moving in it.

One last last last remark. Of course, the fluid without viscosity can transfer energy to the plate. So it stays in motion. The question is how the fluid flows. This can be tested in an experiment with a superfluid in motion and placing a metal platelet in it, perpendicular to the direction of the velocity of the fluid. When you make the streamlines visible you can see if the second picture is a good representation of the real fluid. Of course, the momentum increase left to the plate is compensated by a decrease in momentum on the right side. Over and out!

• This post is 11 hours old and you have already edited it 12 times. Please try to make edits substantial - if you know that you're going to look over this post again and again, don't make an individual edit for every minor change, but collect them and do one large edit instead of a dozen small ones. Also, please do not let posts look like revision histories - marking added context with "EDIT" is superfluous since the revision history is accessible to everyone. – ACuriousMind Jan 31 '17 at 20:47
• @ACuriousMind-Last edit! Doesn't it make clear how trying to solve a problem proceeds? Next time I'll make all revisions at once! – descheleschilder Jan 31 '17 at 22:08
• I would submit that the above material mostly consists of the author imagining things and fantasizing. Not surprisingly then the post contains next to no useful/valid information. You cannot do physics by speculating in empty space, with no background and based on no rigorous principles of any sort. Not in this millennium, anyway. The above material would have been o.k. in Aristotle's time, I guess. – Pirx Jan 31 '17 at 23:09
• His point is that you seem to flip flop between proposed solutions without a concrete system. These are all speculations for what happens behind the plate; when in reality there are rigorous methods to determine that. – JMac Jan 31 '17 at 23:47
• @descheleschilder Well, I'm not really surprised to see that you feel there's no difference between physical theories and fantasies. But no, scientists in general do not fantasize. Your latest comment, on the other hand, is just more of the same: pure fantasy. Potential flow around this plate is exactly symmetric, and the net force is therefore zero. As JMac said, the theory of inviscid incompressible flow (also known as potential flow theory) provides a complete description of such flows, that is well understood in every last detail for well over a century now. There's no need to speculate. – Pirx Feb 1 '17 at 1:44