View source for Porous medium equation

The '''porous medium equation''', also called the '''nonlinear heat equation''', is a [[Nonlinear partial differential equation|nonlinear]] [[Partial differential equation|partial differential equation]] taking the form:<ref>{{Cite web |last1=Wathen |first1=A |last2=Qian |first2=L. |title=Porous medium equation |url=https://people.maths.ox.ac.uk/trefethen/pdectb/porous2.pdf |publisher=University of Oxford}}</ref>{{Equation box 1|cellpadding|border|indent=:|equation=<math> \frac{\partial u}{\partial t} = \Delta \left(u^{m}\right), \quad m > 1 </math>|border colour=#0073CF|background colour=#F5FFFA}}where <math>\Delta</math> is the [[Laplace operator]]. It may also be put into its equivalent divergence form:<math display="block">{\partial u\over{\partial t}} = \nabla \cdot \left[ D(u)\nabla u \right]</math>where <math>D(u) = mu^{m-1}</math> may be interpreted as a [[Physics:Mass diffusivity|diffusion coefficient]] and <math>\nabla\cdot(\cdot)</math> is the [[Divergence|divergence]] operator.
== Solutions ==
Despite being a nonlinear equation, the porous medium equation may be solved exactly using [[Separation of variables|separation of variables]] or a [[Self-similar solution|similarity solution]]. However, the separation of variables solution is known to blow up to infinity at a finite time.<ref>{{Cite book |last=Evans |first=Lawrence C. |title=Partial Differential Equations |publisher=American Mathematical Society |year=2010 |isbn=9780821849743 |edition=2nd |series=Graduate Studies in Mathematics |volume=19 |pages=170–171}}</ref>

=== Barenblatt-Kompaneets-Zeldovich similarity solution ===
The similarity approach to solving the porous medium equation was taken by [[Biography:Grigory Barenblatt|Barenblatt]]<ref>{{Cite journal |last=Barenblatt |first=G.I. |date=1952 |title=On some unsteady fluid and gas motions in a porous medium |journal=Prikladnaya Matematika i Mekhanika |language=Russian |volume=10 |issue=1 |pages=67–78}}</ref> and Kompaneets/[[Biography:Yakov Zeldovich|Zeldovich]],<ref>{{Cite journal |last1=Zeldovich |first1=Y.B. |last2=Kompaneets |first2=A.S. |date=1950 |title=Towards a theory of heat conduction with thermal conductivity depending on the temperature |journal=Collection of Papers Dedicated to 70th Anniversary of A. F. Ioffe |publisher=Izd. Akad. Nauk SSSR |pages=61–72}}</ref> which for  <math>x \in \mathbb{R}^{n}</math> was to find a solution satisfying:<math display="block">u(t,x) = {1\over{t^{\alpha}}}v\left( {x\over{t^{\beta}}} \right), \quad t > 0</math>for some unknown function <math>v</math> and unknown constants <math>\alpha,\beta</math>. The final solution to the porous medium equation under these scalings is:<math display="block">u(t,x) = {1\over{t^{\alpha}}}\left( b - {m-1\over{2m}} \beta {\|x\|^{2}\over{t^{2\beta}}} \right)_{+}^{1\over{m-1}}</math>where <math>\|\cdot\|^{2}</math> is the <math>\ell^{2}</math>-[[Lp space#The p-norm in infinite dimensions and ℓp spaces|norm]], <math>(\cdot)_{+}</math> is the positive part, and the coefficients are given by:<math display="block">\alpha = {n\over{n(m-1) + 2}}, \quad \beta = {1\over{n(m-1) + 2}}</math>

== Applications ==
The porous medium equation has been found to have a number of applications in gas flow, heat transfer, and groundwater flow.<ref>{{Cite journal |last=Boussinesq |first=J. |date=1904 |title=Recherches théoriques sur l'écoulement des nappes d'eau infiltrées dans le sol et sur le débit des sources |url=https://eudml.org/doc/235283 |journal=Journal de Mathématiques Pures et Appliquées |volume=10 |pages=5–78 |issn=}}</ref>

=== Gas flow ===
The porous medium equation name originates from its use in describing the flow of an [[Physics:Ideal gas|ideal gas]] in a homogeneous porous medium.<ref>{{Cite book |last=Muskat |first=M. |title=The Flow of Homogeneous Fluids Through Porous Media |publisher=McGraw-Hill |year=1937 |isbn=9780934634168 |location=New York}}</ref> We require three equations to completely specify the medium's density <math>\rho</math>, flow velocity field <math>{\bf v}</math>, and pressure <math>p</math>: the [[Continuity equation|continuity equation]] for [[Physics:Conservation of mass|conservation of mass]]; [[Physics:Darcy's law|Darcy's law]] for flow in a porous medium; and the ideal gas [[Physics:Equation of state|equation of state]].  These equations are summarized below:<math display="block">\begin{aligned}
\varepsilon {\partial \rho\over{\partial t}} &= -\nabla \cdot (\rho {\bf v}) & (\text{Conservation of mass}) \\
{\bf v} &= -{k\over{\mu}}\nabla p & (\text{Darcy's law}) \\
p &= p_{0}\rho^{\gamma} & (\text{Equation of state})
\end{aligned}</math>where <math>\varepsilon</math> is the [[Earth:Porosity|porosity]], <math>k</math> is the [[Earth:Permeability (Earth sciences)|permeability]] of the medium, <math>\mu</math> is the [[Physics:Dynamic viscosity|dynamic viscosity]], and <math>\gamma</math> is the [[Physics:Polytropic process|polytropic exponent]] (equal to the [[Physics:Heat capacity ratio|heat capacity ratio]] for [[Physics:Isentropic process|isentropic process]]es). Assuming constant porosity, permeability, and dynamic viscosity, the partial differential equation for the density is:<math display="block">{\partial \rho\over{\partial t}} = c\Delta \left( \rho^{m} \right)</math>where <math>m = \gamma + 1</math> and <math>c = \gamma k p_{0}/(\gamma+1)\varepsilon\mu</math>.

=== Heat transfer ===
Using Fourier's law of heat conduction, the general equation for temperature change in a medium through conduction is:<math display="block">\rho c_{p} {\partial T\over{\partial t}} = \nabla \cdot (\kappa \nabla T)</math>where <math>\rho</math> is the medium's density, <math>c_{p}</math> is the [[Physics:Heat capacity|heat capacity at constant pressure]], and <math>\kappa</math> is the [[Physics:Thermal conductivity|thermal conductivity]].  If the thermal conductivity depends on temperature according to the power law:<math display="block">\kappa = \alpha T^{n}</math>Then the heat transfer equation may be written as the porous medium equation:<math display="block">{\partial T\over{\partial t}} = \lambda\Delta \left(T^{m}\right)</math>with <math>m=n+1</math> and <math>\lambda = \alpha/\rho c_{p}m</math>.  The thermal conductivity of high-temperature [[Physics:Plasma|plasmas]] seems to follow a power law.<ref>{{Cite book |last1=Zeldovich |first1=Y.B. |title=Physics of Shock Waves and High Temperature Hydrodynamic Phenomena |last2=Raizer |first2=Y.P. |publisher=Academic Press |year=1966 |isbn=9780127787015 |edition=1st |pages=652–684}}</ref>

== See also ==

* [[Diffusion equation]]
* [[Chemistry:Porous medium|Porous medium]]

== References ==
<references />

== External links ==

* [https://verso.mat.uam.es/~juanluis.vazquez/BKPME2006six.pdf The Porous Medium Equation: Mathematical theory]

[[Category:Partial differential equations]]
[[Category:Diffusion]]
[[Category:Hydrogeology]]
[[Category:Heat transfer]]
[[Category:Transport phenomena]]
[[Category:Exactly solvable models]]

{{Sourceattribution|Porous medium equation}}