[math-fun] Dumping water
Say I have a jar full of water covered by a thin plate. Then I turn the jar upside down, holding the plate firmly against the lip of the jar to prevent water from spilling. Then I whisk away the plate so that instead of the plate pushing against the water, only the air beneath the jar is pushing upward. Of course, the water will leave the jar. But what will the geometry of the process be? The water can't leave as a cylindrical slug; intuitively, it seems that the process has instability, so that spontaneous fingering in the air-water interface will break the initial cylindrical symmetry. Maybe Doc Edgerton made high-speed pictures of this? Jim Propp PS: The author of this post disclaims responsibility for all property damage caused by experiments prompted by this question, along with any matrimonial rifts resulting therefrom.
Good question. I wonder if with ideal physics, with zero viscosity and no surface tension, with the water were a perfect fluid, then in a perfectly symmetrical situation maybe the water wouldn't fall at all, since that would require symmetry-breaking. --Dan On Aug 11, 2014, at 9:14 PM, James Propp <jamespropp@gmail.com> wrote:
Say I have a jar full of water covered by a thin plate. Then I turn the jar upside down, holding the plate firmly against the lip of the jar to prevent water from spilling. Then I whisk away the plate so that instead of the plate pushing against the water, only the air beneath the jar is pushing upward.
Of course, the water will leave the jar. But what will the geometry of the process be? The water can't leave as a cylindrical slug; intuitively, it seems that the process has instability, so that spontaneous fingering in the air-water interface will break the initial cylindrical symmetry.
Maybe Doc Edgerton made high-speed pictures of this?
Jim Propp
PS: The author of this post disclaims responsibility for all property damage caused by experiments prompted by this question, along with any matrimonial rifts resulting therefrom.
* Dan Asimov <dasimov@earthlink.net> [Aug 12. 2014 07:12]:
Good question.
I wonder if with ideal physics, with zero viscosity and no surface tension, with the water were a perfect fluid, then in a perfectly symmetrical situation maybe the water wouldn't fall at all, since that would require symmetry-breaking.
--Dan
Even if you teach the water that symmetry-breaking ist verboten, there are perfectly symmetric solutions of this problem. Regards, jj
[...]
On 12/08/2014 05:14, James Propp wrote:
Say I have a jar full of water covered by a thin plate. Then I turn the jar upside down, holding the plate firmly against the lip of the jar to prevent water from spilling. Then I whisk away the plate so that instead of the plate pushing against the water, only the air beneath the jar is pushing upward.
Of course, the water will leave the jar. But what will the geometry of the process be? The water can't leave as a cylindrical slug; intuitively, it seems that the process has instability, so that spontaneous fingering in the air-water interface will break the initial cylindrical symmetry.
http://en.wikipedia.org/wiki/Rayleigh%E2%80%93Taylor_instability (I think). -- g
Yes indeed, it is the Rayleigh-Taylor instability, which arises when a low density material pushes against a high density material. This issue is of practical importance in the design of nuclear explosives. A low density chemical explosive is pushing against a high density plutonium ball, and it is essential to maintain accurate spherical symmetry as the ball is being compressed to yet higher density. At the point of maximum compression, and maximum criticality, the chain reaction is initiated by a burst of neutrons. -- Gene
________________________________ From: Gareth McCaughan <gareth.mccaughan@pobox.com> To: math-fun@mailman.xmission.com Sent: Tuesday, August 12, 2014 1:39 AM Subject: Re: [math-fun] Dumping water
On 12/08/2014 05:14, James Propp wrote:
Say I have a jar full of water covered by a thin plate. Then I turn the jar upside down, holding the plate firmly against the lip of the jar to prevent water from spilling. Then I whisk away the plate so that instead of the plate pushing against the water, only the air beneath the jar is pushing upward.
Of course, the water will leave the jar. But what will the geometry of the process be? The water can't leave as a cylindrical slug; intuitively, it seems that the process has instability, so that spontaneous fingering in the air-water interface will break the initial cylindrical symmetry.
http://en.wikipedia.org/wiki/Rayleigh%E2%80%93Taylor_instability (I think).
-- g
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On 8/12/2014 1:39 AM, Gareth McCaughan wrote:
On 12/08/2014 05:14, James Propp wrote:
Say I have a jar full of water covered by a thin plate. Then I turn the jar upside down, holding the plate firmly against the lip of the jar to prevent water from spilling. Then I whisk away the plate so that instead of the plate pushing against the water, only the air beneath the jar is pushing upward.
Of course, the water will leave the jar. But what will the geometry of the process be? The water can't leave as a cylindrical slug; intuitively, it seems that the process has instability, so that spontaneous fingering in the air-water interface will break the initial cylindrical symmetry.
http://en.wikipedia.org/wiki/Rayleigh%E2%80%93Taylor_instability (I think).
Things are made of atoms. Atoms obey quantum mechanics. Quantum mechanics includes randomness. So symmetries get broken. Brent Meeker
Interesting things happen as the diameter of the opening decreases. The plate might be thought of as artificially increasing surface tension. -tom On Tue, Aug 12, 2014 at 9:20 AM, meekerdb <meekerdb@verizon.net> wrote:
On 8/12/2014 1:39 AM, Gareth McCaughan wrote:
On 12/08/2014 05:14, James Propp wrote:
Say I have a jar full of water covered by a thin plate. Then I turn the jar upside down, holding the plate firmly against the lip of the jar to prevent water from spilling. Then I whisk away the plate so that instead of the plate pushing against the water, only the air beneath the jar is pushing upward.
Of course, the water will leave the jar. But what will the geometry of the process be? The water can't leave as a cylindrical slug; intuitively, it seems that the process has instability, so that spontaneous fingering in the air-water interface will break the initial cylindrical symmetry.
http://en.wikipedia.org/wiki/Rayleigh%E2%80%93Taylor_instability (I think).
Things are made of atoms. Atoms obey quantum mechanics. Quantum mechanics includes randomness. So symmetries get broken.
Brent Meeker
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Perhaps I should confess here the background of my question: I'm thinking that my facetious way of explaining megafauna mass extinction via "spontaneous reversal of Earth's gravitational field" would be a good candidate for a BAHFest talk at some point. A BAHFest talk is supposed to include some real science along with the bogus kind, so, in addition to citing dimensional scaling laws to explain why falling up and then falling down would be much more injurious to big animals than to small ones, I'll want to discuss R-T instability and its relevance to the question "What happened to the oceans during this postulated event?" See bahfest. <http://bahfest.org>com if you don't know what BAHFest is. Jim Propp On Tuesday, August 12, 2014, Tom Rokicki <rokicki@gmail.com <javascript:_e(%7B%7D,'cvml','rokicki@gmail.com');>> wrote:
Interesting things happen as the diameter of the opening decreases. The plate might be thought of as artificially increasing surface tension.
-tom
On Tue, Aug 12, 2014 at 9:20 AM, meekerdb <meekerdb@verizon.net> wrote:
On 8/12/2014 1:39 AM, Gareth McCaughan wrote:
On 12/08/2014 05:14 <x-apple-data-detectors://5>, James Propp wrote:
Say I have a jar full of water covered by a thin plate. Then I turn the jar upside down, holding the plate firmly against the lip of the jar to prevent water from spilling. Then I whisk away the plate so that
instead
of the plate pushing against the water, only the air beneath the jar is pushing upward.
Of course, the water will leave the jar. But what will the geometry of the process be? The water can't leave as a cylindrical slug; intuitively, it seems that the process has instability, so that spontaneous fingering in the air-water interface will break the initial cylindrical symmetry.
http://en.wikipedia.org/wiki/Rayleigh%E2%80%93Taylor_instability (I think).
Things are made of atoms. Atoms obey quantum mechanics. Quantum mechanics includes randomness. So symmetries get broken.
Brent Meeker
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-- -- http://cube20.org/ -- http://golly.sf.net/ --
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The pressure difference P across a surface of radius of curvature r with surface tension γis P = 2γ/r. The hydrostatic pressure of a fluid of density ρ with column height h in gravity g is P = ρgh. Suppose there is a circular hole of radius R in the bottom of the jar. A bead of fluid extends beneath the hole, and if R << h, we may neglect the variation of hydrostatic pressure within the bead, so that the bead is a spherical cap of radius of curvature r. Setting the pressures equal, rh = 2γ/ρg. For the bead to be stable, r >= R, so there is a maximum column height H with RH = 2γ/ρg. On Earth g = 9.8 m/s^2, and for water, ρ = 1000 kg/m^3, γ = 0.073 N/m. Thus RH = 1.5e-5 m^2 = 15 mm^2. For example, a 1 mm radius hole should support a 15 mm column height. Note that one must use a jar, not a tube, in order to avoid the attractive capillary force. -- Gene
________________________________ From: James Propp <jamespropp@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Sent: Monday, August 11, 2014 9:14 PM Subject: [math-fun] Dumping water
Say I have a jar full of water covered by a thin plate. Then I turn the jar upside down, holding the plate firmly against the lip of the jar to prevent water from spilling. Then I whisk away the plate so that instead of the plate pushing against the water, only the air beneath the jar is pushing upward.
Of course, the water will leave the jar. But what will the geometry of the process be? The water can't leave as a cylindrical slug; intuitively, it seems that the process has instability, so that spontaneous fingering in the air-water interface will break the initial cylindrical symmetry.
Maybe Doc Edgerton made high-speed pictures of this?
Jim Propp
participants (7)
-
Dan Asimov -
Eugene Salamin -
Gareth McCaughan -
James Propp -
Joerg Arndt -
meekerdb -
Tom Rokicki