I stumbled upon a continuity equation with a $\nabla^2$ term on the right-hand side:

$$ \partial_t \rho + \nabla (\vec b \rho) = D \nabla^2 \rho , $$

where $b$ denotes the forward velocity and $D$ is a constant.

What's the meaning of such a diffusion equation?

Some background:

Since, we have particle number conservation, we have

$$ \partial_t \rho + \nabla (\vec v \rho) = 0 , $$

where $v$ denotes the ordinary flux velocity. Moreover, if there are sources, we have

$$ \partial_t \rho + \nabla (\vec v \rho) = \sigma . $$

  • $\begingroup$ Can you elaborate what the operator '$D$' refers to here? $\endgroup$ – Lelouch Apr 12 at 14:34
  • $\begingroup$ I edited the question. Thanks! $\endgroup$ – jak Apr 12 at 14:36
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    $\begingroup$ Can you define what the $\Delta$ operator is in the $\Delta\rho$ term in the 1st equation? $\endgroup$ – Lelouch Apr 12 at 14:41
  • $\begingroup$ $\Delta = \nabla^2$. I edited the question again. $\endgroup$ – jak Apr 12 at 14:45
  • $\begingroup$ I asked a similar question a few years ago. Could the problem be that you have started with a form that is not conservative (well not in general), fundamental unit of transport of the flux, see this answer scicomp.stackexchange.com/a/7269/3691 $\endgroup$ – boyfarrell Apr 12 at 15:39

It is the so called convection–diffusion equation (but it is also known under other names). You may find more information in the above linked wikipedia page, but, in brief, it is an equation which combines a convective cause of time variation at one point (the $\nabla (\vec b \rho)$ term), with a diffusive process, controlled by the Laplacian term.

The convective term in the continuity equation captures the information about the flux of the quantity $\rho$ which locally moves with velocity $\vec b$. When integrated over a closed surface, the net time variation of $\int \rho$ over the volume is due to the surface integral of the current $\rho \vec b $.

The diffusive term with the Laplacian, provides a different mechanism for the net time variation of $\int \rho$ over the volume: the presence of a diffusive flux, not accompanied by a macroscopic field of velocity, proportional to the local gradient of $\rho$, according to a Fick's-like law ( current = $-D \nabla \rho$).

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  • $\begingroup$ Is there any intuitive way to understand why the convective flux is described by a $\nabla (\vec b \rho)$ term, while the diffusive flow is described by the laplacian term? $\endgroup$ – jak Apr 13 at 6:15
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    $\begingroup$ @jak added two paragraphs to explain the meaning of convective and diffusive flux. $\endgroup$ – GiorgioP Apr 13 at 7:52
  • $\begingroup$ thanks a lot! What I still find somewhat confusing is that people use the same symbol $\vec J$ for both currents, i.e. $\vec J \equiv \rho \vec b$ and $\vec J \equiv -D \nabla \rho$. However, your answers seems to suggest that there are really two currents and different symbols should be used, i.e. $\vec J_c \equiv \rho \vec b$ and $\vec J_d \equiv \nabla \rho$?! A second small question: is it correct that while there is no macroscopic field for the diffusive flux, there is a microscopic field (sometimes called osmotic velocity $u$)? $\endgroup$ – jak Apr 13 at 9:29
  • $\begingroup$ @jak I would say that it is matter of taste. Either one is stressing a unique flux made by two physical mechanisms, or one is stressing the convective term as the responsible of the macroscopic flux. But certainly it is possible to speak about two different fluxes. About the diffusive flux, yes, its physical origin is due to the presence of microscopic velocities disappearing at the coarse grained level of usual hydrodynamics. $\endgroup$ – GiorgioP Apr 13 at 9:42
  • $\begingroup$ Thanks! But how can there be "a unique flux made by two physical mechanisms"? Isn't the total flux the sum of the two fluxes: $\vec J \equiv \vec{J}_c + \vec{J}_d $? In particular, if we use the same symbol this seems to suggest that we can equate the two equations to find $\rho \vec b = -D \nabla \rho$. However, as far as I know, this formula is only valid for the osmotic velocity $\vec u$, which we can introduce by defining $ \vec{J}_d \equiv \rho \vec u $. $\endgroup$ – jak Apr 13 at 10:11

It looks like a diffusion equation with advection. Such an equation would be relevent for heat transport in a fluid moving at velocty ${\bf b}$. If $D=0$ the LHS says that the stuff whose density is $\rho$ is being moved about the flow. If ${\bf b}=0$ the suff is just diffusing. With both ${\bf b}$ and $D$ non zero, you have a combination of both processes.

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  • $\begingroup$ Thanks! But isn't diffusion also some kind of flow? I suppose there is a different vector field that describes it. And wouldn't there also be a continuity equation for this diffusion flow? (After all it should be continuous?!) Is there any way to drape words around the term on the right-hand side? $\endgroup$ – jak Apr 13 at 6:14

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