Page 113 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

5.1 Mathematical Background of Chemical Dissolution Front Instability Problems 101

along the positive x direction, it has a negative algebraic value (i.e. p < 0) in this
fx
analysis.
If the propagation speed of the planar dissolution front is denoted by v front , then
it is possible to transform a moving boundary problem of the dissolution front (in an
x-t coordinate system) into a steady-state boundary problem of the dissolution front
(in an ξ − t coordinate system) using the following coordinate mapping:

ξ = x − v front t. (5.20)

It is necessary to relate partial derivatives with respect to ξ and t to those with
respect to x and t (Turcotte and Schubert 1982).

∂ ∂ ∂ ∂x ∂ ∂
= + = + v front , (5.21)
∂t ∂t ∂x ∂t ∂t ∂x
ξ x x
∂ ∂

= , (5.22)
∂ξ ∂x
t t
where derivatives are taken with x or t held constant as appropriate.
Since the transformed system in the ξ − t coordinate system is in a steady state,
the following equations can be derived from Eqs. (5.21) and (5.22).

∂ ∂

=−v front , (5.23)
∂t ∂ξ
x

∂ ∂
= . (5.24)
∂ξ t ∂x t
Substituting Eqs. (5.23) and (5.24) into Eqs. (5.15), (5.16) and (5.17) yields the
following equations:

∂ ∂p
ψ(φ) + v front φ = 0, (5.25)
∂ξ ∂ξ

∂ ∂C ∂p
φD(φ) + Cψ(φ) + v front (C − ρ s )φ = 0, (5.26)
∂ξ ∂ξ ∂ξ

∂φ A p
v front − k chemical (φ f − φ)(C − C eq ) = 0. (5.27)
∂ξ V p
Integrating Eqs. (5.25) and (5.26) from negative infinite to positive infinite and
using the boundary conditions (i.e. Eqs. (5.18) and (5.19)) yields the following
equations: