The porous medium equation, also called the nonlinear heat equation, is a nonlinear partial differential equation taking the form:
formula_1
where formula_2 is the Laplace operator. It may also be put into its equivalent divergence form:formula_3where formula_4 may be interpreted as a diffusion coefficient and formula_5 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 formula_6 was to find a solution satisfying:formula_7for some unknown function formula_8 and unknown constants formula_9. The final solution to the porous medium equation under these scalings is:formula_10where formula_11 is the formula_12-norm, formula_13 is the positive part, and the coefficients are given by:formula_14
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 formula_15, flow velocity field formula_16, and pressure formula_17: 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:formula_18where formula_19 is the porosity, formula_20 is the permeability of the medium, formula_21 is the dynamic viscosity, and formula_22 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:formula_23where formula_24 and formula_25.
Heat transfer.
Using Fourier's law of heat conduction, the general equation for temperature change in a medium through conduction is:formula_26where formula_15 is the medium's density, formula_28 is the heat capacity at constant pressure, and formula_29 is the thermal conductivity. If the thermal conductivity depends on temperature according to the power law:formula_30Then the heat transfer equation may be written as the porous medium equation:formula_31with formula_32 and formula_33. The thermal conductivity of high-temperature plasmas seems to follow a power law.