Permeability Models for Magma Flow through the Earth's Mantle: A Lie Group Analysis

Table 1

The permeability 𝐾(𝜙) and the corresponding Lie point symmetries of (1).

Permeability	Lie point symmetry
	𝑋1𝜕=(2−𝑛)𝑡𝜕𝜕𝑡+𝑛𝑧𝜕𝜕𝑧+2𝜙𝜕𝜙
𝐾(𝜙)=𝜙^𝑛	𝑋2=𝜕𝜕𝑧
	𝑋3=𝜕𝜕𝑡
	𝑋1𝜕=−𝑛𝑡𝜕𝜕𝑡+𝑛𝑧𝜕𝜕𝑧+2𝜕𝜙
𝐾(𝜙)=exp[𝑛(𝜙−1)]	𝑋2=𝜕𝜕𝑧
	𝑋3=𝜕𝜕𝑡
	𝑋1𝜕=𝑓(𝑡)𝜕𝑡
	𝑋2𝜕=𝜙𝜕𝜙
	𝑋3=𝜕𝜕𝑧
𝐾(𝜙)=1	𝑋4𝜕=cosh(2𝑧)𝜕𝜕𝑧+𝜙sinh(2𝑧)𝜕𝜙
	𝑋5𝜕=sinh(2𝑧)𝜕𝜕𝑧+𝜙cosh(2𝑧)𝜕𝜙
	𝑋6𝜕=𝐵(𝑡,𝑧)𝜕𝜙, where 𝐵(𝑡,𝑧) satisfies
	𝜕𝐵−𝜕𝜕𝑡^3𝐵𝜕𝑡𝜕𝑧^2=0