[Poimsm]

what does it mean that a passive scalar will not vary alongside a streamline when the flow is incompressible?

[FMDenaro]

I think you are a student learning the basic of fluid mechanics.

Consider the equation in a slight different form:


dpsi/dt+u.Grad psi = psi Div u


or introducing the Lagrangian derivative



Dpsi/Dt = psi Div u

for a divergence-free velocity field you see that Dpsi/Dt=0, that is the variable psi is constant along the path-line (trajectory) dx/dt=u. For steady flow (dpsi/dt=0) the trajectory coincides with the streamline, thus Psi is constant along the line having the velocity field tangent at each point.



This discussion can be applied directly to the continuity equation, that is the equation for the density. What textbooks are you reading?