Porous medium equation

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

∂u∂t=Δ(um),m>1

where

is the Laplace operator. It may also be put into its equivalent divergence form:

∂u∂t=∇⋅[D(u)∇u]

where

D(u)=mum−1

may be interpreted as a diffusion coefficient and

∇⋅(⋅)

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 x∈ℝn was to find a solution satisfying:u(t,x)=1tαv(xtβ),t>0for some unknown function v and unknown constants α,β. The final solution to the porous medium equation under these scalings is:u(t,x)=1tα(b−m−12mβ‖x‖2t2β)+1m−1where ‖⋅‖2 is the ℓ2-norm, (⋅)+ is the positive part, and the coefficients are given by:α=nn(m−1)+2,β=1n(m−1)+2

## 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 ρ, flow velocity field 𝐯, and pressure p: 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:ε∂ρ∂t=−∇⋅(ρ𝐯)(Conservation of mass)𝐯=−kμ∇p(Darcy's law)p=p0ργ(Equation of state)where ε is the porosity, k is the permeability of the medium, μ is the dynamic viscosity, and γ 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:∂ρ∂t=cΔ(ρm)where m=γ+1 and c=γkp0/(γ+1)εμ.

### Heat transfer

Using Fourier's law of heat conduction, the general equation for temperature change in a medium through conduction is:ρcp∂T∂t=∇⋅(κ∇T)where ρ is the medium's density, cp is the heat capacity at constant pressure, and κ is the thermal conductivity. If the thermal conductivity depends on temperature according to the power law:κ=αTnThen the heat transfer equation may be written as the porous medium equation:∂T∂t=λΔ(Tm)with m=n+1 and λ=α/ρcpm. 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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