In this paper we investigate the optimal control approach for the active control and drag optimization of incompressible viscous flow past circular cylinders. The control function is the time angular velocity of the rotating cylinder. The wake flow is solved in the laminar regime (Re = 200) with a finite element method. Due to the CPU and memory costs related to the optimal control theory, a Proper Orthogonal Decomposition (POD) Reduced Order Model (ROM) is used as the state equation. The key enablers to an accurate and robust POD ROM are the introduction of a time dependent eddy-viscosity estimated for each POD mode as the solution of an auxiliary optimization problem and the use of a snapshot ensemble for POD based on chirp-forced transients. Since the POD basis represents only velocities, we minimize a drag-related cost functional characteristic of the wake unsteadiness. The optimization problem is solved using Lagrange multipliers to enforce the constraints. 25% of relative drag reduction is found when the Navier-Stokes equations are controlled using an harmonic control function deduced from the optimal solution determined with the POD ROM. Earlier numerical studies concerning mean drag reduction are confirmed: it is shown in particular that without a sufficient penalization of the control input, our approach is energetically inefficient. The main result is that a cost reduction factor of one hundred and 760 is obtained for the CPU time and the memory respectively. Finally, limits of the performance of our approach are discussed.

A viscosity stratification is considered as a possible mean to postpone the onset of transition to turbulence in channel flow. As a prototype problem, we focus on the linear stability of shear-thinning fluids modelled by the Carreau rheological law. To assess whether there is stabilization and by how much, it is important both to account for a viscosity disturbance in the perturbation equations, and to employ an appropriate viscosity scale in the definition of the Reynolds number. Failure to do so can yield qualitatively and quantitatively incorrect conclusions. Results are obtained for both exponentially and algebraically growing disturbances, demonstrating that a viscous stratification is a viable approach to maintain laminarity.

The electromagnetic forces in a ferrofluid depend on the domain occupied by the fluid. We study here the equilibrium positions of a ferrofluid drop with a boundary which is partially or totally free. The method used is based on the minimization of the energy with respect to the shape of the drop. We show bifurcations of the solutions and hysteresis phenomena when the parameters vary.