# Porous medium equation

The porous medium equation, also called the nonlinear heat equation, is a nonlinear partial differential equation taking the form:
${\frac {\partial u}{\partial t}}=\Delta \left(u^{m}\right),\quad m>1$
where $\Delta$ is the Laplace operator. It may also be put into its equivalent divergence form:${\partial u \over {\partial t}}=\nabla \cdot \left[D(u)\nabla u\right]$where $D(u)=mu^{m-1}$ may be interpreted as a diffusion coefficient and $\nabla \cdot (\cdot )$ is the divergence operator.

## 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.

### Barenblatt-Kompaneets-Zeldovich similarity solution

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

## Applications

The porous medium equation has been found to have a number of applications in gas flow, heat transfer, and groundwater flow.

### Gas flow

The porous medium equation name originates from its use in describing the flow of an ideal gas in a homogeneous porous medium. We require three equations to completely specify the medium's density $\rho$, flow velocity field ${\bf {v}}$, 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:${\begin{aligned}\varepsilon {\partial \rho \over {\partial t}}&=-\nabla \cdot (\rho {\bf {v}})&({\text{Conservation of mass}})\\{\bf {v}}&=-{k \over {\mu }}\nabla p&({\text{Darcy's law}})\\p&=p_{0}\rho ^{\gamma }&({\text{Equation of state}})\end{aligned}}$where $\varepsilon$ is the porosity, $k$ is the permeability of the medium, $\mu$ is the dynamic viscosity, and $\gamma$ 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:${\partial \rho \over {\partial t}}=c\Delta \left(\rho ^{m}\right)$where $m=\gamma +1$ and $c=\gamma kp_{0}/(\gamma +1)\varepsilon \mu$.

### Heat transfer

Using Fourier's law of heat conduction, the general equation for temperature change in a medium through conduction is:$\rho c_{p}{\partial T \over {\partial t}}=\nabla \cdot (\kappa \nabla T)$where $\rho$ is the medium's density, $c_{p}$ is the heat capacity at constant pressure, and $\kappa$ is the thermal conductivity. If the thermal conductivity depends on temperature according to the power law:$\kappa =\alpha T^{n}$Then the heat transfer equation may be written as the porous medium equation:${\partial T \over {\partial t}}=\lambda \Delta \left(T^{m}\right)$with $m=n+1$ and $\lambda =\alpha /\rho c_{p}m$. The thermal conductivity of high-temperature plasmas seems to follow a power law.