Porous medium equation

The similarity approach to solving the porous medium equation was taken by Barenblatt[3] and Kompaneets/Zeldovich,[4] which for ${\displaystyle x\in \mathbb {R} ^{n}}$ was to find a solution satisfying:

$${\displaystyle u(t,x)={1 \over {t^{\alpha }}}v\left({x \over {t^{\beta }}}\right),\quad t>0}$$

for some unknown function ${\displaystyle v}$ and unknown constants ${\displaystyle \alpha ,\beta }$. The final solution to the porous medium equation under these scalings is:

$${\displaystyle u(t,x)={1 \over {t^{\alpha }}}\left(b-{m-1 \over {2m}}\beta {\|x\|^{2} \over {t^{2\beta }}}\right)_{+}^{1 \over {m-1}}}$$

where ${\displaystyle \|\cdot \|^{2}}$ is the ${\displaystyle \ell ^{2}}$-norm, ${\displaystyle (\cdot )_{+}}$ is the positive part, and the coefficients are given by:

$${\displaystyle \alpha ={n \over {n(m-1)+2}},\quad \beta ={1 \over {n(m-1)+2}}}$$

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 ${\displaystyle \rho }$, flow velocity field ${\displaystyle {\bf {v}}}$, and pressure ${\displaystyle p}$: 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:

$${\displaystyle {\begin{aligned}\varepsilon {\partial \rho \over {\partial t}}&=-\nabla \cdot (\rho {\bf {v}})&({\text{Conservation of mass}})\\{\bf {v}}&=-{k \over {\mu }}\nabla p&({\text{Darcy's law}})\\p&=p_{0}\rho ^{\gamma }&({\text{Equation of state}})\end{aligned}}}$$

where ${\displaystyle \varepsilon }$ is the porosity, ${\displaystyle k}$ is the permeability of the medium, ${\displaystyle \mu }$ is the dynamic viscosity, and ${\displaystyle \gamma }$ 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:

$${\displaystyle {\partial \rho \over {\partial t}}=c\Delta \left(\rho ^{m}\right)}$$

where ${\displaystyle m=\gamma +1}$ and ${\displaystyle c=\gamma kp_{0}/(\gamma +1)\varepsilon \mu }$.

Using Fourier's law of heat conduction, the general equation for temperature change in a medium through conduction is:

$${\displaystyle \rho c_{p}{\partial T \over {\partial t}}=\nabla \cdot (\kappa \nabla T)}$$

where ${\displaystyle \rho }$ is the medium's density, ${\displaystyle c_{p}}$ is the heat capacity at constant pressure, and ${\displaystyle \kappa }$ is the thermal conductivity. If the thermal conductivity depends on temperature according to the power law:

$${\displaystyle \kappa =\alpha T^{n}}$$

Then the heat transfer equation may be written as the porous medium equation:

$${\displaystyle {\partial T \over {\partial t}}=\lambda \Delta \left(T^{m}\right)}$$

with ${\displaystyle m=n+1}$ and ${\displaystyle \lambda =\alpha /\rho c_{p}m}$. The thermal conductivity of high-temperature plasmas seems to follow a power law.[7]