The continuity equation

We study the flux of quantity $\xi$, where $\xi$ gives the amount of a physical quantity per volume (its concentration). The flux-vector associated with quantity $\xi$ will be called  $\vec{J}(\xi)$, and gives the amount of $\xi$, which passes per unit of time through a unit area. An important contribution to the total flux of quantity $\xi$ is convective flux  $\vec{J}_{c}(\xi)$. The convective contribution gives the amount of $\xi$ that is transported along with a mass flux. The instantaneous value of the convective contribution  $\vec{J}_{c}(\xi)$ to the flux of $\xi$ is given by the product of the physical quantity with velocity vector $\vec{u}$ (The components of velocity $\vec{u}$ will be called $(u,v,w)$):

$$\vec{J}_{c}(\xi) \equiv \xi \: \vec{u}$$ (2.7)

This is seen as follows: The velocity does not only give the pace with which the air is traveling (unit: m/s), it also reflects the amount of volume, which crosses per unit of time through a unit surface (unit: $($m$^{3}/$m$^{2})/$s). In convective flux  $\vec{J}_{c}$ we count both convection and molecular diffusion (which is a mass-bound flux contribution). In the study of fluxes of chemicals, the convective flux vector is the only flux vector involved:

$$\vec{J}( ) = \vec{J}_{c}( )$$ (2.8)

In the study of the transfer of heat or momentum there are more contributions to the total flux than just the convective term. Heat and momentum may also be transferred by conduction or friction. Furthermore heat may be transferred by radiation. We will use symbols  $\vec{J}_{i}$ to denote other contributions to the flux vector of $\xi$ than the convective term (such as radiation and the flow of potential energy in heat exchange processes), where index $i$ counts the contributions, and symbols $S_{i}$ for sources and sinks of $\xi$.

The sources of quantity $\xi$ will be called  $S_{i}(\xi)$. Sources of chemicals can be found in chemical reactions and in the evaporation of liquid chemicals. Sources of heat can be chemistry, viscous dissipation and phase transitions.

With the above given notation for flux-vector $\vec{J}$ and sources $S_{i}$ the general form for a continuity equation is:

$$\hspace{1cm} 0 = \frac{\partial \xi}{\partial t} + \: \vec{J}(\xi) - \sum_{i} S_{i}(\xi)$$ (2.9)

$$= \frac{\partial \xi}{\partial t} + \left\lgroup \xi \vec{u} + \sum_{j} \vec{J}_{j} \right \rgroup -\sum_{i} S_{i}$$ (2.10)

When we combine this relation with the following (usual) definition of the convective derivative:

$$\frac{\mbox{d}}{\mbox{d}t} \equiv \frac{\partial}{\partial t} + \vec{u} \cdot \nabla$$ (2.11)

then we find the following form for the continuity relation:

$$0 = \frac{\mbox{d} \xi}{\mbox{d} t} - \sum_{i} S_{i} + \xi \: \... ...iv} \: \vec{u} + \mbox{div} \left\lgroup \sum_{j} \vec{J}_{j} \right \rgroup$$ (2.12)

A quantity is called conserved when an outflux of that quantity implies that there will be accordingly less of it left in the region where it came from. In other words: there are no sources and sinks of $\xi$:

$$\hspace{1cm} 0 = \frac{\partial \xi}{\partial t} + \left\lgroup \xi \vec{u} + \sum_{j} \vec{J}_{j} \right \rgroup$$ (2.13)

An example of a conserved quantity: Molecules of a certain kind (concentration  $\varrho_{\xi}$) can only be exchanged via either convection or via molecular diffusion (both represented by flux vector  $\vec{J}_{c}(\varrho_{\xi})$). Molecular diffusion plays an important role close to boundaries, where e.g. water molecules evaporate into a layer of air, which has very low speed relative to the ground. The budget equation for a single chemical is therefore:

$$0 = \frac{\mbox{d} \varrho_{\xi}}{\mbox{d} t} + \varrho_{\xi} \: \mbox{div} \: \vec{u}_{\xi}$$ (2.14)

where  $\vec{u}_{\xi}$ represents the fractional velocity of the $\xi$-molecules. For the mixture of air as a whole, we can use mean velocity $\vec{u}$ in relation 2.12.

For quantity  $q_{\xi} \equiv \xi/\varrho$, which gives the amount per unit mass of the same quantity of which $\xi$ gives the amount per unit volume ($\varrho$ is air density), the following budget equation can be derived:

$$\frac{\mbox{d}q_{\xi}}{\mbox{d}t} = \frac{1}{\varrho} \frac{\mbo... ...e{0.5cm} \mbox{with} \hspace{0.5cm} \tilde{S}_{i} \equiv \frac{S_{i}}{\varrho}$$ (2.15)

In the derivation of relation 2.13 we have used conservation relation 2.12 for $\varrho$. Quantity $q_{\xi}$ is called a 'specific quantity'. Source strengths  $\tilde{S}_{i}$ give the creation per unit time of $\xi$ per unit mass, while $S$ is per unit volume. Specific quantities can sometimes provide better insight into exchange processes than absolute quantities. When a quantity $\xi$ is conserved, the continuity equation of the associated specific quantity  $q_{\xi} = \xi/\varrho$ is:

$$\frac{\mbox{d} q_{\xi}}{\mbox{d}t} = 0$$ (2.16)

even in situations with non-stationary density. This relation, which expresses that nothing of interest happens to a conserved quantity, is much simpler than relation 2.12, which includes density effects.