The equations of continuity

For steady-state laminar flow, the equation for the continuity of the mixture is (Bird et al. 1960, p. 556):

$$\mbox{\boldmath$\nabla$}_*\cdot(\rho \mathbf{u}_*)=0.$$ (2.1)

The mass flux of vapour with respect to a frame of reference fixed in space is (Bird et al. 1960, p. 561):

$$\mathbf{n}_* = \rho m_*\mathbf{u}_* + \mathbf{j}_*$$ (2.2)

where $\rho$ is the mixture density, $m_*$ is the mass fraction of the vapour in the gas-vapour mixture, $\mathbf{u}_*$ is the mass average velocity and $\mathbf{j}_*$ is the mass flux of the vapour relative to the mass average velocity.

Considering only Fickian diffusion (Bird et al. 1960, p. 502; see also `Omissions', below),

$$\mathbf{j}_*=-\rho D\mbox{\boldmath$\nabla$}_* m_*,$$ (2.3)

where $D$ is the binary diffusivity, so that

$$\mathbf{n}_* = \rho \left(m_* \mathbf{u}_* - D\mbox{\boldmath$\nabla$}_* m_*\right).$$ (2.4)

For steady-state vapour transport in a laminar flow, with no destruction or creation of vapour (see `Omissions', below), conservation of vapour requires that its flux (2.4) is solenoidal (Bird et al. 1960, p. 561); i.e.

$$\mbox{\boldmath$\nabla$}_*\cdot\mathbf{n}_* = 0.$$ (2.5)

If the spatial variations of $\rho$ and $D$ are neglected in the mass fluxes $(\rho\mathbf{u}_*)$ and $\mathbf{n}_*$, the equations of continuity reduce to

$$\mbox{\boldmath$\nabla$}_* \cdot\mathbf{u}_* = 0$$ (2.6)

$$\mathbf{u}_*\cdot\mbox{\boldmath$\nabla$}_* m_* = D\nabla^2_* m_*$$ (2.7)

It is obvious from (2.7) that the velocity field, $\mathbf{u}_*$, must be known for the distribution of vapour to be calculable. To this end, an equation of motion is obtained in §2.1.2.