Porous medium equation

The porous medium equation, also called the nonlinear heat equation, is a nonlinear partial differential equation taking the form:[1]

[math]\displaystyle{ \frac{\partial u}{\partial t} = \Delta \left(u^{m}\right), \quad m \gt 1 }[/math]

where [math]\displaystyle{ \Delta }[/math] is the Laplace operator. It may also be put into its equivalent divergence form:[math]\displaystyle{ {\partial u\over{\partial t}} = \nabla \cdot \left[ D(u)\nabla u \right] }[/math]where [math]\displaystyle{ D(u) = mu^{m-1} }[/math] may be interpreted as a diffusion coefficient and [math]\displaystyle{ \nabla\cdot(\cdot) }[/math] is the divergence operator.

## Contents

* 1 Solutions
* 1.1 Barenblatt-Kompaneets-Zeldovich similarity solution
* 2 Applications
* 2.1 Gas flow
* 2.2 Heat transfer
* 3 See also
* 4 References
* 5 External links

## Solutions

Despite being a nonlinear equation, the porous medium equation may be solved exactly using separation of variables or a similarity solution. However, the separation of variables solution is known to blow up to infinity at a finite time.[2]

### Barenblatt-Kompaneets-Zeldovich similarity solution

The similarity approach to solving the porous medium equation was taken by Barenblatt[3] and Kompaneets/Zeldovich,[4] which for [math]\displaystyle{ x \in \mathbb{R}^{n} }[/math] was to find a solution satisfying:[math]\displaystyle{ u(t,x) = {1\over{t^{\alpha}}}v\left( {x\over{t^{\beta}}} \right), \quad t \gt 0 }[/math]for some unknown function [math]\displaystyle{ v }[/math] and unknown constants [math]\displaystyle{ \alpha,\beta }[/math]. The final solution to the porous medium equation under these scalings is:[math]\displaystyle{ 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]\displaystyle{ \|\cdot\|^{2} }[/math] is the [math]\displaystyle{ \ell^{2} }[/math]-norm, [math]\displaystyle{ (\cdot)_{+} }[/math] is the positive part, and the coefficients are given by:[math]\displaystyle{ \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.[5]

### Gas flow

The porous medium equation name originates from its use in describing the flow of an ideal gas in a homogeneous porous medium.[6] We require three equations to completely specify the medium's density [math]\displaystyle{ \rho }[/math], flow velocity field [math]\displaystyle{ {\bf v} }[/math], and pressure [math]\displaystyle{ p }[/math]: the continuity equation for conservation of mass; Darcy's law for flow in a porous medium; and the ideal gas equation of state. These equations are summarized below:[math]\displaystyle{ \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]\displaystyle{ \varepsilon }[/math] is the porosity, [math]\displaystyle{ k }[/math] is the permeability of the medium, [math]\displaystyle{ \mu }[/math] is the dynamic viscosity, and [math]\displaystyle{ \gamma }[/math] is the polytropic exponent (equal to the heat capacity ratio for isentropic processes). Assuming constant porosity, permeability, and dynamic viscosity, the partial differential equation for the density is:[math]\displaystyle{ {\partial \rho\over{\partial t}} = c\Delta \left( \rho^{m} \right) }[/math]where [math]\displaystyle{ m = \gamma + 1 }[/math] and [math]\displaystyle{ 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]\displaystyle{ \rho c_{p} {\partial T\over{\partial t}} = \nabla \cdot (\kappa \nabla T) }[/math]where [math]\displaystyle{ \rho }[/math] is the medium's density, [math]\displaystyle{ c_{p} }[/math] is the heat capacity at constant pressure, and [math]\displaystyle{ \kappa }[/math] is the thermal conductivity. If the thermal conductivity depends on temperature according to the power law:[math]\displaystyle{ \kappa = \alpha T^{n} }[/math]Then the heat transfer equation may be written as the porous medium equation:[math]\displaystyle{ {\partial T\over{\partial t}} = \lambda\Delta \left(T^{m}\right) }[/math]with [math]\displaystyle{ m=n+1 }[/math] and [math]\displaystyle{ \lambda = \alpha/\rho c_{p}m }[/math]. The thermal conductivity of high-temperature plasmas seems to follow a power law.[7]

## See also

* Diffusion equation
* Porous medium

## References

1. ↑ Wathen, A; Qian, L.. "Porous medium equation". University of Oxford. https://people.maths.ox.ac.uk/trefethen/pdectb/porous2.pdf.
2. ↑ Evans, Lawrence C. (2010). Partial Differential Equations. Graduate Studies in Mathematics. 19 (2nd ed.). American Mathematical Society. pp. 170–171. ISBN 9780821849743.
3. ↑ Barenblatt, G.I. (1952). "On some unsteady fluid and gas motions in a porous medium" (in Russian). Prikladnaya Matematika i Mekhanika 10 (1): 67–78.
4. ↑ Zeldovich, Y.B.; Kompaneets, A.S. (1950). "Towards a theory of heat conduction with thermal conductivity depending on the temperature". Collection of papers dedicated to 70th Anniversary of A. F. Ioffe (Izd. Akad. Nauk SSSR): 61–72.
5. ↑ Boussinesq, J. (1904). "Recherches théoriques sur l'écoulement des nappes d'eau infiltrées dans le sol et sur le débit des sources". Journal de Mathématiques Pures et Appliquées 10: 5–78. https://eudml.org/doc/235283.
6. ↑ Muskat, M. (1937). The Flow of Homogeneous Fluids Through Porous Media. New York: McGraw-Hill. ISBN 9780934634168.
7. ↑ Zeldovich, Y.B.; Raizer, Y.P. (1966). Physics of Shock Waves and High Temperature Hydrodynamic Phenomena (1st ed.). Academic Press. pp. 652–684. ISBN 9780127787015.

## External links

* The Porous Medium Equation: Mathematical theory

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