Porous medium equation

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

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

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]

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 has been found to have a number of applications in gas flow, heat transfer, and groundwater flow.^[5]

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]

• Diffusion equation
• Porous medium

• The Porous Medium Equation: Mathematical theory