Nondimensionalization

The primitive dependent variables are reduced to dimensionless form by:

$$m_* = m_{*r} + m\Delta m_*;$$ (2.26)

$$\mathbf{u}_* = \frac{g(\beta\Delta T_*+\zeta\Delta m_*)b^2}{\nu} \mathbf{u};$$ (2.27)

$$p_* = \rho g(\beta\Delta T_* + \zeta\Delta m_*)b\mbox{$\mathcal A$}p-\rho gy_*; \;\;\;\mbox{and}$$ (2.28)

$$T_* = T_{*r} + T\Delta T_*;$$ (2.29)

where $b$ is a characteristic length of the domain and $\Delta m_*$ and $\Delta T_*$ are characteristic differences of vapour mass fraction and temperature, to be introduced by the boundary conditions, respectively. Also,

$$\mbox{\boldmath$\nabla$}_* = \frac{1}{b}\mbox{\boldmath$\nabla$}\;\;\;\mbox{and}$$ (2.30)

$$\mathbf{r}_* = b\mathbf{r},$$ (2.31)

where $\mathbf{r}_*$ is the position vector.

The scales in this scheme were chosen to ensure that the variables have values of order unity in the solutions later obtained (ch. 4, 7, 8). In general, these are characterized by a dominance of the diffusive (including conduction and viscous diffusion of momentum) as opposed to the advective terms in the field equations.