the velocity $$\mathbf{V}_p + (\mathbf{V}_g-\mathbf{V})$$ and in the linear regime this will be entirely radial. The advection velocity is the sum of two parts: the particle velocity and a secondary contribution issuing from the relative drag (the bracketed term). The first term should be familiar, and simply describes the Lagrangian advection of particles by the particle velocity itself. The second term is novel. Physically, it represents the fact that particle density maxima induce a *decrease* in the relative velocity - because the reciprocal drag of the two fluids is greater at that location (cf. the first term in equation \[\[Euler-DeltaV\]\]). This decrease leads to an additional mass flux, which is anticorrelated with $\rho_p'$. An outward effective advection is the result, which is in addition (and opposition) to that of $\mathbf{V}_p$. Thus a density peak will tend to be both (a) pushed inward simply by the primary particle velocity $\mathbf{V}_p$, and (b) pushed outward by the second drag-induced mass flux that it has itself excited. It is easy to show that for $f_g=f_p$ the sum of the two velocities is precisely zero, i.e. the two drift velocities cancel perfectly. Conversely, when there are very few particles, i.e. $f_p\to 0$, the advection 