Viscous Heating

On Wed, 2 May 2007, John Schutkeker wrote:

"Timo A. Nieminen" wrote:
On Wed, 2 May 2007, John Schutkeker wrote:

Has the theory of viscous heating of an ordinary fluid been
developed?

I was recently reading about free convection driven by isothermal
spheres. Heating due to viscosity was mentioned in passed, and in the
case being considered, dismissed as negligible.

I'm wondering if they used the right viscosities. If the core and
mantle are so extremely thick, I'm wondering if they might not be
negligible. Something tells me that nobody's ever measured viscosities
of liquids under the extreme pressures of a planetary interior.

Of course, I could be wrong, in which case I'm wasting time. But I'm
more interested in the math and the model as much as the answers, so
this is as much for my own edification as it is for a result.

Seismic wave propagation would tell us something about viscosity.

As for the core, I recently saw some stuff about turbulent flow in the
core and the generation of the geomagnetic field.

I forgot about that, but sonic flows do generate a lot of friction.
What's the title of Anderson?

John D. Anderson, and there are a couple of his books that might be of
interest: Modern compressible flow, and Hypersonic and high temperature
gas dynamics

--
Timo Nieminen - Home page: http://www.physics.uq.edu.au/people/nieminen/
E-prints: http://eprint.uq.edu.au/view/person/...,_Timo_A..html
Shrine to Spirits: http://www.users.bigpond.com/timo_nieminen/spirits.html

John Schutkeker wrote:

Has the theory of viscous heating of an ordinary fluid been developed?

Brenner and Edward's "Macrotransport Processes" has two chapters on
nonmaterial dispersion (energy and momentum)in a variety of material
systems. Is that what you mean?

--
Andrew Resnick, Ph.D.
Department of Physiology and Biophysics
Case Western Reserve University

John Schutkeker wrote:

Bruce Scott TOK ] wrote in
:


John Schutkeker wrote:

Has the theory of viscous heating of an ordinary fluid been developed?

Are you interested in Navier Stokes fluids (i.e., gasdynamics) or
actual liquids where the quantum physics determines the
microproperties?


AFAIK, Navier-Stokes (NS) is just a momentum balance equation, making me
ask, since when don't liquids obey the same force balances on a
differential fluid element as gasses? If that's true, what momentum
equation replaces NS, in the incompressible liquid case you mentioned?
There should be only one equation, and it's NS, although the viscosity
may be a complicted function, rather than a constant. But it should
still be NS, shouldn't it?

The NS equation*s* are for the *conservation* of momentum, and are a
simplification of Cauchy's first law of motion. To solve the general
flows you describe, one also needs the conservation of mass equations
and the conservation of energy equations.


I'm interested in a fluid whose properties are hardly even known:
planetary mantles and cores, like Earth and Enceladus. Nobody
knows exactly what are those fluid properties, raising a whole 'nother
theory question that I plan to gloss over.

But the fluid properties are intimately tied into the resultant flows.
Especially if there are magnetic effects.

I'm thinking that under such high pressures, Enceladus' "mantle" may be
a highly viscous liquid, which might be something like a solution of
liquids like N2, NH3, and CH4, etc. Unfortunately, it may also be the
mixture of solid/liquid phases that we colloquially know as "slush."

Right- that's why it's sometimes better to stick with Cauchy's law
rather than the NS- viscoelastic (or viscoplastic, or any other
constitutive relation you can dream up) materials can be handled in one,
but not the other.

Whichever it is, I'm betting that it's a highly viscous liquid, more
like a paste or a putty, than what we're used to. Since nobody knows
anything about it, I'll have to just say that it seems obvious enough
that quantum effects will dominate the viscosity, and not hard-body
collisions, like a compressible gas.

Pastes are not viscous fluids. Is there a yield stress? And forget
quantum effects- for planetary-scale motions, quantum effects are
useless unless the temperature is near 0 K.

snip
--
Andrew Resnick, Ph.D.
Department of Physiology and Biophysics
Case Western Reserve University

Timo A. Nieminen wrote:

On Wed, 2 May 2007, John Schutkeker wrote:

Has the theory of viscous heating of an ordinary fluid been developed?


I was recently reading about free convection driven by isothermal
spheres. Heating due to viscosity was mentioned in passed, and in the
case being considered, dismissed as negligible. There's probably some
number that can sit alongside the Reynolds number, the Grashoff number,
and the Nusselt number that tells you whether you can ignore it.

Good point- there's a bunch of dimensionless groups that are relevant
for planetary-scale gravito-magneto-viscoelastic (!) convective flows:
Ekman, Froude, Hartmann, Joule, Kronig, Lewis, Peclet, Prandtl, Taylor,
Weissenberg, and more (yes, I just looked through the CRC list).

There's some recent experimentation on the geomagnetic dynamo using
liquid sodium, Physics Today has had a few blurbs about it.

snip

--
Andrew Resnick, Ph.D.
Department of Physiology and Biophysics
Case Western Reserve University

John Schutkeker wrote:

"Timo A. Nieminen" wrote in
:


On Wed, 2 May 2007, John Schutkeker wrote:


Has the theory of viscous heating of an ordinary fluid been
developed?

I was recently reading about free convection driven by isothermal
spheres. Heating due to viscosity was mentioned in passed, and in the
case being considered, dismissed as negligible.


I'm wondering if they used the right viscosities. If the core and
mantle are so extremely thick, I'm wondering if they might not be
negligible. Something tells me that nobody's ever measured viscosities
of liquids under the extreme pressures of a planetary interior.

Viscosity is a function of temperature and not of pressure, AFAIK. I
suspect that the viscosity of some materials at superhigh temperatures
has been measured by someone.


Of course, I could be wrong, in which case I'm wasting time. But I'm
more interested in the math and the model as much as the answers, so
this is as much for my own edification as it is for a result.

It's not clear what you are trying to model: planetary scale flows?
there hasn't been any mention of the boundary conditions: heat loss,
obviously, but also surface waves and gravity waves. Coriolus forces.
etc. etc.


snip

--
Andrew Resnick, Ph.D.
Department of Physiology and Biophysics
Case Western Reserve University

Andy Resnick wrote in
:

John Schutkeker wrote:

AFAIK, Navier-Stokes (NS) is just a momentum balance equation, making
me ask, since when don't liquids obey the same force balances on a
differential fluid element as gasses? If that's true, what momentum
equation replaces NS, in the incompressible liquid case you
mentioned? There should be only one equation, and it's NS, although
the viscosity may be a complicted function, rather than a constant.
But it should still be NS, shouldn't it?

The NS equation*s* are for the *conservation* of momentum, and are a
simplification of Cauchy's first law of motion. To solve the general
flows you describe, one also needs the conservation of mass equations
and the conservation of energy equations.

I'm not familiar with Cauchy's first law of motion. Is it east enough
to wrote down here, or can you give me a link to a page that explains
it?

Whichever it is, I'm betting that it's a highly viscous liquid, more
like a paste or a putty, than what we're used to. Since nobody knows
anything about it, I'll have to just say that it seems obvious enough
that quantum effects will dominate the viscosity, and not hard-body
collisions, like a compressible gas.

Pastes are not viscous fluids. Is there a yield stress? And forget
quantum effects- for planetary-scale motions, quantum effects are
useless unless the temperature is near 0 K.

I believe Scott was saying that viscosity os due to intermolecular
interactions, whose physics is very complex. That complex physics
exists at all temperatures, not just near absolute zero.

Andy Resnick wrote in news:f1co8d$n8q$1
@eeyore.INS.cwru.edu:

It's not clear what you are trying to model: planetary scale flows?

That's right.

there hasn't been any mention of the boundary conditions:

So far it looks like boundary conditions are easy.

heat loss, obviously,

I'm only interested in the heating term, not the losses.

but also surface waves and gravity waves.

I don't know what you mean by gravity waves, but I think I will neglect
surface waves. If by gravity waves, you mean volume modes, that will be
the whole project.

Coriolis forces.

Thanks for reminding me about that.

Timo Nieminen wrote in
news:Pine.LNX.4.50.0705031113230.14713-100000@localhost:

On Wed, 2 May 2007, John Schutkeker wrote:

"Timo A. Nieminen" wrote:
On Wed, 2 May 2007, John Schutkeker wrote:

Has the theory of viscous heating of an ordinary fluid been
developed?

I was recently reading about free convection driven by isothermal
spheres. Heating due to viscosity was mentioned in passed, and in
the case being considered, dismissed as negligible.

I'm wondering if they used the right viscosities. If the core and
mantle are so extremely thick, I'm wondering if they might not be
negligible. Something tells me that nobody's ever measured
viscosities of liquids under the extreme pressures of a planetary
interior.

Of course, I could be wrong, in which case I'm wasting time. But I'm
more interested in the math and the model as much as the answers, so
this is as much for my own edification as it is for a result.

Seismic wave propagation would tell us something about viscosity.

Not on Enceladus. ?

"N:dlzc D:aol T:com \(dlzc\)" wrote in news:Vea_h.233935
:

"John Schutkeker" wrote in
message
. 33.102...

But thanks for the red spot insight. These planets aren't gas
giants,
but I don't know if that makes the issue go away. I wonder if
the
presence of a surface crust would be enough to suppress that.

If you are requiring an entirely fluid surface (???), then you
must have some vortex... if not two. One would expect them at /
near the poles. Unlike Jupiter.

The boundary condition between the mantle and crust makes the vortex
problem go away, but if you'd still be willing to point my way to a page
that works the math for a free fluid surface, I'd be very grateful. A
man can never read too much math. ?

Dear John Schutkeker:

"John Schutkeker" wrote in
message
. 33.102...
"N:dlzc D:aol T:com \(dlzc\)" wrote in
news:Vea_h.233935
:

"John Schutkeker" wrote in
message
. 33.102...

But thanks for the red spot insight. These planets
aren't gas giants, but I don't know if that makes the
issue go away. I wonder if the presence of a surface
crust would be enough to suppress that.

If you are requiring an entirely fluid surface (???), then
you must have some vortex... if not two. One would
expect them at / near the poles. Unlike Jupiter.

The boundary condition between the mantle and crust
makes the vortex problem go away,

Actually, I think it does not. It would tend to "rotate" the
vortex "neutral axis" to be parallel to any differential rotation
between the core (if any) and the crust.

but if you'd still be willing to point my way to a page
that works the math for a free fluid surface, I'd be very
grateful. A man can never read too much math. ?

I have the text that brings this up at work ("5 Golden Rules").
I'll try and remember to post the necessary keywords to see if
you agree with my take on it.

David A. Smith