# What is a diffusionless fluid?

I'm taking a course in astrophysical fluid dynamics and have come across a problem involving "small diffusionless disturbances" of a fluid. Based on the nature of the course I expect the examiner to have an exact mathematical definition of this in mind (i.e. if later in the question they say the flow is solenoidal, which I know to interpret as $\nabla \cdot \mathbf{v}=0$).

In general what is a diffusionless disturbance and do you have any insight how this definition might translate into a simple mathematical relation?

(my best guess so far is that they mean that $\delta \rho$, the pertubations in the density, can be taken as 0, but I'm far from sure about this)

• I suspect they mean $\rho=\rho+\delta\rho$ such that $\nabla\delta\rho\neq\nabla^2\rho$ Commented May 22, 2015 at 11:32
• Could you expand a little, I suspect you mean $\rho =\rho_0 + \delta \rho$, but surely $\nabla \delta \rho$ never equals $\nabla^2 \rho$?, for one thing the spatial dimensions are inconsistent? Also what is the physical meaning of this? Commented May 22, 2015 at 11:37
• I should have written $\nabla\cdot\delta\rho$, rather than $\nabla\delta\rho$ (matches the fluid equations the former way & not the latter). If the variation of the medium, $\delta\rho$, is proportional to the gradient, $\nabla\rho$, then you'd clearly have $\nabla\cdot\delta\rho=\nabla^2\rho$. Commented May 22, 2015 at 14:58