  We study the vanishing viscosity/diffusivity limit for the transport equation
of a passive scalar $f(x,t)\in\mathbb{R}$ along a divergence-free vector field
$u(x,t)\in\mathbb{R}^2$, given by $\frac{\partial f}{\partial t} + \nabla\cdot
(u f) = 0$; and the associated advection-diffusion equation of $f$ along $u$
for positive viscosity/diffusivity parameter $\nu>0$, expressed by
$\frac{\partial f}{\partial t} + \nabla\cdot (u f) -\nu\Delta f = 0$. We
demonstrate failure of the vanishing viscosity limit of the advection-diffusion
equation to select unique solutions, or to select entropy-admissible solutions,
to transport along $u$.
  First, we construct a bounded divergence-free vector field $u$ which admits,
for each (non-constant) initial datum, two weak solutions to the initial value
problem for the transport equation. Moreover, we show that both these solutions
are renormalised weak solutions, and are strong limits along different
subsequences of vanishing viscosity of solutions to the corresponding
advection-diffusion equation.
  Second, we construct a second bounded divergence-free vector field $u$
admitting, for any initial datum, a weak solution to the transport equation
which is perfectly mixed to its spatial average, and after some delay in time,
it unmixes to its initial state. Moreover, we show that this
entropy-inadmissible unmixing is the unique weak vanishing viscosity limit of
the corresponding advection-diffusion equation.
