What does it mean when a flow is non-conservative?

Question

From a physical point of view (not numerical), does a non-conservative flow have any meaning? I'm solving this scalar advection-diffusion equation with source and sink terms. I want to add another term which I interpret as an additional source term. But curiously, rearranging the term and using symmetry properties, I end up with this equation for the scalar quantity $Q$ if I ignore all the other terms: $$ \frac{\partial Q}{\partial t}=-\boldsymbol{u}_v\cdot\nabla Q, $$ where $\boldsymbol{u}_v$ is a complicated expression depending on other terms (it does depend indirectly on $Q$ though - note that this is not a real material flow). The RHS is an advection term. However, we also have that: $$ \nabla\cdot\boldsymbol{u}_v\neq 0, $$ and this "flow" is not divergenceless, which means we can't rewrite the equation in conservation form, or: $$ \frac{\partial Q}{\partial t}\neq-\nabla\cdot(\boldsymbol{u}_v Q). $$ Now this confuses me a bit because usually, we get such an equation (the first one) from the conservation form in the case the flow is divergenceless, which is not the case here. However, we can rewrite the first equation as: $$ \frac{D_vQ}{D_vt}=0, $$ and $Q$ is conserved along the streamlines (definition of advection without compression I suppose). But if $Q$ is conserved along the stream lines, doesn't that mean that it is in general a conserved quantity? Or maybe we can interpret this flow as having open stream lines? Are there more possibilities?

Answer 1

You stated that the fact that $\frac{\partial Q}{\partial t}\neq-\nabla\cdot(\boldsymbol{u}_v Q)$ means that you can't write the equation in conservation form; but really, all it means is that you would have to write your transport equation as:

$$\frac{\partial Q}{\partial t} +\nabla\cdot(\boldsymbol{u}_v Q) = r$$

where $r$ and $\boldsymbol{u}_v$ are defined appropriately (i.e. $\boldsymbol{u}_v$ properly describes the "flow field" of the scalar $Q$, and $r$ represents the creation/dissipation of $Q$ independent of flow.)

Notice that, if $r = 0$ and $\nabla \cdot \boldsymbol{u}_v = 0$, then you would get the first equation you wrote; $ \frac{\partial Q}{\partial t}=-\boldsymbol{u}_v\cdot\nabla Q. $ However, it may also be the case that you have defined a $\boldsymbol{u}_v$ that does not represent only the advective changes in $Q$, but also includes some creation/destruction of $Q$. For that to be the case and for the equation you obtain to be correct, $\boldsymbol{u}_v$ has to be related to the "true" flow field $\boldsymbol{u}$ and the source/sink term $r$ such that:

$$-\boldsymbol{u}_v\cdot\nabla Q = r -\nabla\cdot(\boldsymbol{u} Q)$$

Check and see if that's the case for you. Depending on the properties of $\boldsymbol{u}$, the connection may simplify.

Answer 2

If you have the equation (continuity equation)

$$\frac{\partial Q}{\partial t}+\nabla \cdot (\boldsymbol{u}_v Q)=0$$

then it means that $Q$ is a conserved quantity, in the sense that the integral of $Q$ over a volume changes in time only according to the flow of the associated current $\boldsymbol{u}_v Q$ across the volume boundary (the total inside the volume can change only if stuff leaves or enters the volume, nothing disappears or is created).

If you have :

$$\frac{\partial Q}{\partial t}+\boldsymbol{u}_v\cdot\nabla Q=0$$

then it means that $Q$ is advected by the velocity field. However, it is in general not conserved in the sense provided above. Imagine a sink (i.e. a part of the flow where the divergence of the velocity is negative): $Q$ is transported by the velocity that goes into the sink and disappears (despite suff enters the volume containing the sink, the volume integral of $Q$ may not necessarily increase accordingly). This should convince you that if $Q$ is conserved along the stream lines, then it doesn't necessarily mean that it is a conserved quantity.

The two are mathematically equivalent only if the velocity field is solenoidal.

Answer 3

It means that there are internal forces between the molecules of the liquid which convert some mechanical energy to heat.