Which are the governing equations for heat convection in porous media? [Archive]

Andrea

May17-04, 06:21 PM

tessel@tum.bot

Jun3-04, 04:19 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>On Mon, 17 May 2004, Andrea wrote:\n\n> I would like to numerically simulate a unsteady natural convection in a\n> saturated porous medium, assuming equality of fluid and solid\n> temperature. First of all, I need to find the governing equations of the\n> problem. I found many differences in the same balance equations on three\n> different references: Bear, (book,1972), Bear and Bachmat (book,1991),\n> and Haji-Sheikh A., and Vafai,A. (paper, 2004). So I ask you: which are\n> considered today to be the correct governing equations for this problem?\n\nDid anyone ever answer this?\n\nI know very little about porous media, so I can\'t answer it properly\nmyself, but I -can- say something which might at least help explain the\nterminology.\n\nIn various books on Lie analysis of partial differential equations, I too\nhave come across various equations labelled "porous media equation".\n(References below.) The usages I have seen can be reconciled by saying\nthat the most general governing equation which authors consider a\ncandidate for the governing equation for diffusion in a porous medium is\nthis "nonlinear diffusion equation":\n\nu_t = (K(u) u_x)_x (*)\n\nwhere x,t are the independent variables and u is the dependent variables,\nsubscripts denote partial derivatives, and K is an arbitrary but\nsufficiently smooth function. (Higher dimensional generalizations are\nobvious.)\n\nThe "point symmetry group" of (*) depends, in general, on the form of the\nfunction K. In general, the Lie algebra of this group (sometimes called\nthe Lie algebra of "isovectors") is only three dimensional and is\ngenerated by the following flows:\n\nX1 = @/@x X2 = @/@t X3 = x @/@x + 2 t @/@t\n\nThere are some special forms of K which admit additional symmetries,\nincluding K(u) = 1/u^2, and some authors call\n\nu_t = (u_x/u^2)_x (**)\n\nthe porous media equation. Its Lie algebra suggests that this\n(nonlinear!) PDE can be transformed into the heat equation (a linear\nPDE!), and this is in fact the case.\n\n> Which book or review paper would you suggest for further informations?\n\nFor Lie symmetries (with mention of "porous media" only as examples):\n\nauthor = {Bluman, George W., and Kumei, Sukeyuki},\ntitle = {Symmetries and Differential Equations},\nseries = {Applied mathematical sciences},\nvolume =81,\npublisher = {Springer-Verlag},\nyear = {1989}}\n\nauthor = {Peter J. Olver},\ntitle = {Applications of {L}ie Groups to Differential Equations},\nseries = {Graduate Texts in Mathematics},\nvolume = 107,\npublisher = {Springer-Verlag},\nyear = 1993}\n\nauthor = {L. V. Ovsiannikov},\ntitle = {Group analysis of differential equations},\nnote = {translation edited by W.F. Ames},\npublisher = {Academic Press},\nyear = 1982}\n\nThe textbook\n\nauthor ={J. D. Murray},\ntitle = {Mathematical Biology},\npublisher = {Springer-Verlag},\nseries = {Biomathematics},\nvolume = {19}\nedition = {second},\nyear = 2002}\n\nhas some discussion of diffusion equations, including (d) above and the\nFisher equation. The collection of expository essays\n\neditor = {A. H. Taub},\ntitle = {Studies in Applied Mathematics},\npublisher = {Mathematical Association of America},\nseries = {Studies in Mathematics},\nvolume = 7,\nyear = 1971}\n\nincludes a nice paper on nonlinear diffusion equations.\n\n"T. Essel" (hiding somewhere in cyberspace)\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Mon, 17 May 2004, Andrea wrote:

> I would like to numerically simulate a unsteady natural convection in a
> saturated porous medium, assuming equality of fluid and solid
> temperature. First of all, I need to find the governing equations of the
> problem. I found many differences in the same balance equations on three
> different references: Bear, (book,1972), Bear and Bachmat (book,1991),
> and Haji-Sheikh A., and Vafai,A. (paper, 2004). So I ask you: which are
> considered today to be the correct governing equations for this problem?

Did anyone ever answer this?

I know very little about porous media, so I can't answer it properly
myself, but I -can- say something which might at least help explain the
terminology.

In various books on Lie analysis of partial differential equations, I too
have come across various equations labelled "porous media equation".
(References below.) The usages I have seen can be reconciled by saying
that the most general governing equation which authors consider a
candidate for the governing equation for diffusion in a porous medium is
this "nonlinear diffusion equation":

u_t = (K(u) u_x)_x (*)

where x,t are the independent variables and u is the dependent variables,
subscripts denote partial derivatives, and K is an arbitrary but
sufficiently smooth function. (Higher dimensional generalizations are
obvious.)

The "point symmetry group" of (*) depends, in general, on the form of the
function K. In general, the Lie algebra of this group (sometimes called
the Lie algebra of "isovectors") is only three dimensional and is
generated by the following flows:

X1 = @/@x X2 = @/@t X3 = x @/@x + 2 t @/@t

There are some special forms of K which admit additional symmetries,
including K(u) = 1/u^2, and some authors call

u_t = (u_x/u^2)_x (**)

the porous media equation. Its Lie algebra suggests that this
(nonlinear!) PDE can be transformed into the heat equation (a linear
PDE!), and this is in fact the case.

> Which book or review paper would you suggest for further informations?

For Lie symmetries (with mention of "porous media" only as examples):

author = {Bluman, George W., and Kumei, Sukeyuki},
title = {Symmetries and Differential Equations},
series = {Applied mathematical sciences},
volume =81,
publisher = {Springer-Verlag},
year = {1989}}

author = {Peter J. Olver},
title = {Applications of {L}ie Groups to Differential Equations},
series = {Graduate Texts in Mathematics},
volume = 107,
publisher = {Springer-Verlag},
year = 1993}

author = {L. V. Ovsiannikov},
title = {Group analysis of differential equations},
note = {translation edited by W.F. Ames},
publisher = {Academic Press},
year = 1982}

The textbook

author ={J. D. Murray},
title = {Mathematical Biology},
publisher = {Springer-Verlag},
series = {Biomathematics},
volume = {19}
edition = {second},
year = 2002}

has some discussion of diffusion equations, including (d) above and the
Fisher equation. The collection of expository essays

editor = {A. H. Taub},
title = {Studies in Applied Mathematics},
publisher = {Mathematical Association of America},
series = {Studies in Mathematics},
volume = 7,
year = 1971}

includes a nice paper on nonlinear diffusion equations.

"T. Essel" (hiding somewhere in cyberspace)