Sections other than rectangular

In deriving the solutions for the horizontal velocity components (7.13) and vapour mass fraction and temperature (7.16), no reference was made to the shape of the section, since the solutions satisfy the boundary conditions (7.1), (7.2) and (7.5) for all $X$ and $Z$. Thus the problem for a general section can be written:

Find $v=v_f+v_n$ in $\Omega$, the domain, where $v_f$ and $v_n$ satisfy (7.19) and (7.20), respectively, in $\Omega$ and (7.21) on $\upartial \Omega$, its boundary.