Recent results on mixing optimization and stability of flows in channels with porous walls

The first part of this talk will discuss the development of an optimal mixer. An optimal mixer can be defined as a mixing device able to deliver a uniformly optimal mixing performance over a
wide range of operating and initial conditions. We conceptually design an optimal mixer starting from a reference mixing device,the sine flow. A careful characterization shows that the
time-periodic sine flow performs poorly and erratically over most operating and initial conditions. The optimal mixer is derived in steps modifying the design of the reference mixer
and optimizing its operations. We first optimize the time-sequence of the stirring velocity fields. We obtain a mixer which performs
substantially better than the sine flow, although its performance remains suboptimal due to a deficiency of the actuating system.
Second, we improve the sine flow by implementing a new actuating system that allows optimized shifts of the stirring velocity fields in the cross-flow direction. The performance of the resulting mixer is suboptimal only at low operating conditions due to the use of a periodic stirring protocol. Finally, an optimal mixer is
obtained by coupling the time and shift optimizations. The resulting optimal mixer is able to deliver a uniform optimal performance over the entire operating range and is quite
insensitive to the geometry of the initial conditions.

The second part of this talk will discuss the three-dimensional linear stability analysis of a pressure driven, incompressible, fully developed, laminar flow in a channel delimited by rigid,
homogeneous, isotropic, porous layers. We consider porous materials of small permeability in which the maximum fluid velocity is small compared to the mean velocity in the channel
region and for which inertial effects may be neglected. We solve the fully coupled linear stability problem, arising from the adjacent channel and porous flows, using a spectral collocation technique. We find that very small amounts of wall permeability significantly affect the Orr-Sommerfeld spectrum and can dramatically decrease the stability of the channel flow. Within our assumptions, in channels with two porous walls, permeability destabilizes up to two Orr-Sommerfeld wall modes and introduces two new damped wall modes on the left branch of the spectrum.
Permeability also introduces a new class of damped modes associated with the porous regions. The size of the unstable region delimited by the neutral curve grows substantially, and the critical Reynolds number can decrease to only 10% the corresponding value for a channel flow with impermeable walls.