3.5 Change of coordinates
To simplify the calculations for a domain with time-dependent dimension because of an imposed or self-induced boundary velocity, we introduce a unitless variable, \(y\), defined as:
\begin{equation} \left(9\right)\ y=\frac{z}{H(t)}=\ \frac{z}{H_{0}-\int_{0}^{t}{v_{H}\left(t\right)\text{dt}}}\text{.\ }\nonumber \\ \end{equation}
Here, \(H_{0}\) is the initial thickness of the compacting domain. In the following section, this variable will be substituted in place of the dimensional quantity, \(z\).