Imagine a plane surface covered by water, with the water at rest. If you place a hole, or drain, in the plane, the water doesn’t simply pour down and over the edges of the drain, it begins to rotate.
How does the rotation come about? Indeed, given that there was zero angular momentum when the water was at rest before the drain was made, where does the angular momentum from the rotating water come from as it swirls down the drain? I know that conservation of angular momentum must hold, yet this seems not to be the case here - I must be missing something.
Is it due to the shape of the drain hole (i.e. circular)? Is it simply a topological phenomenon, i.e. when you ‘transform’ a 2D-plane surface (the water at rest) into a 3D-column (the water pouring down), it introduces a curve?
Please note that I am not interested (just now) in why the water rotates in one direction or another, only that it begins to rotate at all (although how the “choice” of rotational direction comes about is mysterious to me as well).
I tried searching for an answer but all the hits seemed to be concerned with whether toilets flush in different directions in the northern and southern hemispheres (and with Bart Simpson and Australia!), but not with my question as to why there is rotation in the first place.
Any asymmetry will induce a rotation, which will propagate and be amplified by the energy of the falling water.
If absolute perfect symmetry is achieved, then Corialis force will kick off the rotation. I am surprised you didn’t run into a discussion of Corialis force when the issue of northern vs southern hemisphere came up. At the scale of bathroom fixtures it is the smallest of factors that determine rotation direction, so risidual angular momentum or asymmetry usually wins out, allowing sinks in both hemispheres to drain either direction.
Actually, the Corialis force was all I kept hitting, and I assumed that it wasn’t relevant on the scale of my everyday life.
You state that ‘any asymmetry will induce a rotation’. I don’t understand what you mean - what asymmetries are you referring to? In my mind, I just imagined a typical bathtub, frankly, and assumed the bottom was planar. I’m not sure what asymmetries you mean and how they effect a rotation.
Thanks!
ETA - even if the plane (or tub) is, say, tilted on one side of the drain and flat on the other, how does that asymmetry cause it to rotate? (assuming that’s the type of asymmetry you’re getting at). Where does the angular momentum come from?
In a typical bathtub the drain is near the center of one end. If it were exactly symmetrical it would be exactly in th middle, but that is very unlikely.
Now you open the drain. Water starts flowing into the drain from all directions. Since the drain is close to one edge, that side empties a little faster, and water flows in from the other side to fill this low spot. If the drain is even slightly off center, then the low spot fills faster from the wider side, and this is rotation.
If there wasn’t any angular momentum before, then any angular momentum in the water must come from a transfer of angular momentum with the objects it’s in contact with. Picture, if you will, a bathtub sitting on a turntable (and no, don’t make the obvious wisecrack). At the start of the experiment, the bathtub and the water in it are both perfectly still (so no angular momentum anywhere), but the shape of the bathtub isn’t quite perfectly symmetric. The plug is pulled, and the water starts flowing out. When it’s about half-drained, you’ll find that the water is circling the drain one way, but that the bathtub and the turntable it’s sitting on are turning the other way, such that the net angular momentum of the whole system is still zero.
Chronos - I am beginning to feel denser than usual - are you saying that the rotation does arise from the Corialis force? If not, then I don’t see the point of the turntable in your analogy.
Kevbo - thanks for your answers. Again, I feel dense, but to make sure I understand, if, say, one side empties faster and that causes water to flow in from the other side, isn’t the flow straight to the low spot? Or, is there a change of direction of the water that occurs (since water from all points of the high side flows to the low spot) and that change in direction of the flow is the origin of the rotation? I guess I’m stuck because the direction change of the flow doesn’t seem to be rotatory or circular to me. Instead, I see it as a tilt of a line of flow towards the low spot.
If the drain were in the center, the flow lines could be strait radials. If the drain is off center, then some of the flow lines will hook around to make use of the side of the drain that would be draining less area if the lines were straight.
You can drain a container without the water rotating. You have to let the water sit for a long time before draining because filling it caused a lot of movement in the water to begin with. For best results use a round container with a drain in the center, and a plug that can be removed from the bottom so you don’t have to stick your hand in the water and cause some motion. This is done at various establishments on the equator as a demonstration of the lack of coriolis force, but it’s been reproduced all over the world, coriolis force has nothing to do with the rotation of water in a draining bathtub.
The spinning was originally described to me as conservation of momentum across vectors. Any small movement in the water becomes greatly magnified as it is pulled in to drain through a small hole.
Well, pretty much anything manufactured that you have is going to have asymmetries due to manufacturing imperfections. Stuff is made to an engineering tolerance that is deemed acceptable for the purpose for which it is built - the wheels of a passenger car don’t need to be exactly, 100% really and truly-o round in a geometric sense for the car to work ok. If the wheel is off balance by .0001 mm it doesn’t really matter and it’s too expensive for GM or Toyota to make it any better at a reasonable price. Tolerances don’t have to be the same either - spaceships and fighter planes probably have a tighter tolerance than passenger cars which probably have a tighter tolerance than a bathtub or fly swatter.
Basically, an engineering tolerance defines how far off a measurement of a manufactured part can be from the number printed on the spec and still be acceptable. For example, a spec might say that a rod must be 10 cm long give or take 1 mm. A rod that is anywhere from 9.9 to 10.1 cm is acceptable. What level of tolerance is acceptable is a fundamental engineering question.
Actually, no geometric asymmetry is required to explain the phenomenon, nor is Coriolis acceleration due to the rotation of the Earth required to initiate the swirling motion, although both may contribute to the exact motion of the fluid. The real reason that water and other viscous fluids tend to rotate when going through a drain is fundamentally simple, albeit somewhat complex to explain, requiring a fairly deep understanding of real world fluid mechanics. I am going to attempt to offer up a straightforward explanation that eliminates the influences of asymmetry of the drain geometry and external mechanics in an intuitive fashion requiring only a basic understanding of Newton’s laws and trigonometry, e.g. not requiring a detailed understanding of non-linear partial differential equations.
First of all, let us eliminate any external influences by simplifying the drain to a volume of revolution of an asymptotic curve like the funnel shape seen here, and assume no external rotations. We will use this boundary condition to impose reaction forces between the fluid and the rest of the world, with the asymptotic geometry allowing the fluid smoothly and continuously go from being completely supported (at the top) to completely unsupported (at the spout) without introducing any rapid change in momentum of the flow due to geometry. The funnel is filled with a Newtonian fluid, i.e. it is viscous, incompressible and isentropic. This is the fluid dynamicist way of saying that the forces arising within the fluid are proportional to the difference in flow velocity between one area and another, that the fluid density doesn’t change, and that it has the same mechanical properties everywhere. (While no real world fluids are truly Newtonian, this is a perfectly reasonable assumption for water at standard temperature and pressure.) The flow velocity at the boundary condition against the funnel is assumed to be zero in all directions which is not quite true, but good enough for our purposes and eliminates any discussions about the influences of surface roughness or adhesion.
Now for the purposes of visualization, we will consider the fluid as being composed of individual particles which are elastically connected to their adjacent particles, and in their natural state all of the forces are balanced per Newton’s third law. This is not, in general, how fluid flow is treated mathematically and in computational fluid simulation (except for a relatively recent method called smoothed particle hydrodynamics or SPH which idealizes the fluid as virtual particles with somewhat arbitrary parameters), but it makes this easier to visualize than flow fields. The actual behavior of fluid mechanics is described completely by what are called the Navier-Stokes equations. These may look complicated to the non-engineer or physicist, but they are actually trivially simple in concept (at least compared to, say, Maxwell’s equations for electrodynamics). Fundamentally they combine Newton’s general laws of mechanics with viscous flow, and at least in linear cases allow one to completely solve fluid flow states at any point in the flow, so that we can break the flow field into individual particles and look at the forces and rotations.
Now if we fill the funnel with fluid but plug the spout we can see that the reaction force upward must support the weight of the fluid (which is mass times the acceleration due to gravity) but the total force at any point on the boundary surface is normal to the tangent line of the surface. So the surface is “pushing hard” upward at the top to support the fluid, but down in the spout the only forces are the hydrostatic pressure pushing inward (except that the bottom where the spout is plugged). This hydrostatic pressure is equal everywhere (technically it changes with depth as the water below supports the weight of the water above, but for a very small funnel we can ignore this). If we suddenly cut away all but a small slice of the spout and water (but keep a condition in which the water cannot move in the circumferential direction out of our slice, called radial symmetry), the water would jump away normal to the boundary at a speed proportional to this pressure. In other words, the water at the top of the funnel would go up, while the water in the spout would go sideways. The natural flow of a particle in the water would be along streamlines that are normal to these forces, so if you drew the streamlines as you move in radially they would look like smaller versions of the funnel.
When we remove the plug at the bottom of the funnel, the fluid immediately begins to flow. This changes the pressure from the hydrostatic state to a dynamic state; obviously, there is no pressure (ignoring the atmosphere) on the bottom of the spout where water is flowing out. However, the pressure radially within the water varies as a function of velocity; near the boundary the flow rate is zero and the pressure is high, but as you get close to the center the flow rate is close to what a free particle falling in a gravity field would be and the pressure is low as the fluid moves faster. The incompressible condition of the fluid prevents all of the fluid from running into the center and falling all out at once, which creates both a pressure and speed difference in the flow. If this difference is small the flow field looks smooth (laminar flow) and our virtual particle moves along a smooth line (called a streamline) but if the difference is too large the flow starts to peel away from itself and make small internal loops (turbulent flow), making the particle spin around and introducing a torque. We can determine this in incompressible fluids by looking at the Reynolds number (the ratio of inertial forces to viscous forces, noted as Re); high Re flows are inherently irregular and unstable; that is, areas of turbulence with form and disappear with time, and in ways that are not easily calculable.
Side note: Most computational fluid dynamic (CFD) simulations handle this by treating it as a stochastic phenomenon that obeys certain aggregate rules based upon the Re of the flow rather than attempt to actually model the local velocity of the turbulent region. These are called Reynolds-averaged Navier Stokes methods (RANS) and require a very keen understanding of both the mechanics and the details of the code to apply correctly, but since most interesting problems in CFD simulation are turbulent, they are critical to being able to get any useful answers to real world problems. There are other methods, such as the SPH methods listed above, to actually treat turbulent motion directly, but they can easily become more computationally intensive than even the largest computing clusters can handle and are still sensitive to the parameters and assumptions that the analyst uses, and so they are really more useful in working with subscale or development models than general simulation in a large flow field with complex boundary geometry although they are becoming more robust and are increasingly used for fluid-structure interaction problems like fluid tank slosh or coupled vibration.
So we’ve seen that if you have a high Re in the flow, you will naturally get turbulence, which creates inherent instabilities in the flow field and makes our virtual particle move in a jagged fashion, and the Re of the flow will become greater as we go down the spout due to the difference in reaction forces and speed differential from the wall to the center. Of course, these are all within our slice, which is orthogonal to the swirling rotation. So how do we go from turbulence in-plane to a swirling, through-plane motion? Well, as noted the turbulent cells are unstable and will not form through this circumferential region in the fluid all at once. This means that when it does occur in one region it has a lower pressure on each side than the surrounding region, and now we have forces on our virtual particle by adjacent particles. If these forces are not perfectly matched, the particle will move sideways and, depending on other nearby motion, may start to spin. Hence, we get flow that is out of plane, even though our simulation is by definition radially symmetric.
Now, we’ve said that this turbulent behavior is stochastic; that is, it follows a statistical distribution but we can’t predict what it will be at any given location and time. However, if enough virtual particles end up all moving or spinning in the same direction, they’ll dominate the rest of the flow just like a conga line at a drunken office party. This rotation will take over and drive the flow to swirl and keep swirling, even in the absence of any obvious asymmetry. The degree to which this occurs and is consistent depends upon the speed and flow differential of the funnel; a very shallow wall funnel with a large opening in the spout will only provide very modest rotation which may well be damped out by hysteresis in the water (loses due to internal heating and acoustics), while a very steep walled funnel or one with a narrow spout that encourages high flow differentials will create very pronounced swirl. This is used in devices like centrifugal pumps and fuel injectors specifically to create turbulence and encourage high flow or mixing.
Now it is demonstrated that water can swirl even without consideration of the spout geometry, surface roughness, external acceleration (Coriolis forces, starting rotation), et cetera, but can these factors influence the direction of the swirling? Absolutely! In fact, they’ll tend to bias the stochastic behavior that starts the flow, accentuating favorable motion and damping out motion that goes in opposition of these influences. The most pronounced influence is typically the geometry of the spout, especially if it has any kind of oriface or jet. Any irregularities in the upper funnel geometry (if it is, say, square instead of round) will also influence swirling flow, often creating separate regions of high and low rotation. And the Coriolis acceleration, while small, can influence very large systems, as we can see with hurricanes and tropical storms (though these are obviously not directly comparable to our idealized incompressible flows). A smooth surface can cause viscous damping which smooths out turbulence and swirl, while a roughened surface can enhance it, which is a critical consideration when making something like a heat exchanger, where you generally want some turbulence to give maximum heat transfer throughout the coolant, but not introduce cavitation or other detrimental conditions.
So you don’t need an external asymmetry to get a swirling behavior; you just need a very rapid flow with high differentials. However, irregular geometry, forces, initial motion in the fluid, et cetera can all influence the degree, direction, and consistency of rotational flow. For the most part this is all contained within the flow field and so angular momentum is conserved, but in any real world condition in which the fluid can transfer load to the boundary via shear (tangential) adhesion, the extra angular momentum is transferred to the outside world, changing (insignificantly) the angular momentum of the planet. So if everyone flushes their toilets all at the same time…
From class years ago so I will keep KISS it. The water in the bathtub is rotating if it is on the planet earth. Though it would be hard to tell because the tub is rotating. On one side of the equator it rotates one way on the other side the opposite.
No, he’s saying that the rotation of the water about the drain is balanced by an equal and opposite rotation of the Earth in the other direction - it’s just that, given the mass differential, the effect on the Earth in terms of revolutions per minute is rather small.
Sure it does. At the poles, it rotates once per day, or 1/1440 RPM. At some other latitude, it rotates at sin(latitude)/1440 RPM. Not a lot, and usually overwhelmed by the motion of water relative to the bathtub or sink or whatever.
What Snnipe 70E said isn’t incorrect, but neither does it simplify anything. When one wishes to simplify a problem, one should discard all of the less-significant aspects of the problem, leaving only the most significant. However, the Coriolis effect (what Snnipe is describing) is a much less significant aspect of this problem than the many things which Snnipe is omitting, and his simplification therefore fails.
Stranger - many thanks, indeed, for taking the time to present such a detailed and thoughtful explanation. Of course, I understood very little of the physics but even so, I believe I’ve begun to develop a bit of intuition about the situation. In particular, I thank you for stating explicitly that, “water can swirl even without consideration of the spout geometry, surface roughness, external acceleration (Coriolis forces, starting rotation), et cetera” and that “you don’t need an external asymmetry to get a swirling behavior; you just need a very rapid flow with high differentials”. Wonderful!
