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

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?

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.

• How can $r$ be independent of the flow $\boldsymbol{u}_v$ if by definition $r=Q\nabla\cdot\boldsymbol{u}_v$? Unless I misunderstood what you meant. By the way, $\boldsymbol{u}_v$ is independent of the material flow $\boldsymbol{u}$ (well, technically it is not, but only because $\boldsymbol{u}_v$ depends in a complicated manner on $Q$). Jan 4, 2022 at 16:25
• What I meant by saying that $r$ can be independent of the flow field is that, in most but not all cases, transport variables are defined such that the creation/dissipation of the transported scalar is not implicitly defined in terms of the flow (but there is no hard and fast rule about that—just convenient convention). If it turns out that $\boldsymbol{u}_v$ is “just some variable” and $Q$ is actually being transported, then the relationship between $\boldsymbol{u}_v$ and the “true” transport parameters $\boldsymbol{u}$ and $r$ are what’s written above. Jan 4, 2022 at 18:18

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.

• Ok thanks, this is pretty clear! But since in my case the flow is not a material flow, I suppose it makes more sense to simply view this term as some kind of source/sink term? I find it hard to understand conceptually what these sources or sinks could physically mean, in combination with the advection (I understand I didn't write any detail with regards to the particular problem) And what about a non-conservative diffusion term? It would mean there is a heat source/sink somewhere? Jan 4, 2022 at 10:54
• Source and sink are just "pictures" for a mathematically clear concept related to the divergence of any field, see this answer math.stackexchange.com/a/3793151/532409 Jan 5, 2022 at 18:21
• The fact that your flow is not a material one is not important to the present discussion. You have a vector field $u_v$ and a scalar field $Q$, and an equation of motion ("continuity" or "advection"): regardless of the real interpretation of $u_v$ or $Q$ you will see that the $Q$ field is conserved or advected. Being conserved or advected are mathematical concepts. Jan 5, 2022 at 18:27
• Finally, there is no diffusion in your equations (a Laplacian should pop out for that). Jan 5, 2022 at 18:28

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

• Isn't that just viscous dissipation? This flow I'm interested in is not a material flow in the sense that it is not driven by a fluid, rather by peculiar magnetic effects. So that the scalar quantity (a component of the magnetic field) is advected nonetheless (but in a non-conservative way, whatever that means). Jan 3, 2022 at 21:05