Porous medium equation

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

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.

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

External links

• The Porous Medium Equation: Mathematical theory

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