Optimal mixing and optimal stirring for fixed energy, fixed power, or fixed palenstrophy flows

Abstract

A new instrument for dynamic helical squeeze flow which superposes oscillatory shear and oscillatory squeeze flow Rev. Sci. Instrum. 83, 085105 (2012) The effects of hydrodynamic interaction and inertia in determining the high-frequency dynamic modulus of a viscoelastic fluid with two-point passive microrheology Phys. Fluids 24, 073103 (2012) MHD free convection flow of a visco-elastic (Kuvshiniski type) dusty gas through a semi infinite plate moving with velocity decreasing exponentially with time and radiative heat transfer AIP Advances 1, 022132 (2011) Transitional flow of a non-Newtonian fluid in a pipe: Experimental evidence of weak turbulence induced by shearthinning behavior Phys. Fluids 22, 101701 (2010) Effects of viscoelasticity on the probability density functions in turbulent channel flowWe consider passive scalar mixing by a prescribed divergence-free velocity vector field in a periodic box and address the following question: Starting from a given initial inhomogeneous distribution of passive tracers, and given a certain energy budget, power budget, or finite palenstrophy budget, what incompressible flow field best mixes the scalar quantity? We focus on the optimal stirring strategy recently proposed by Lin et al. ["Optimal stirring strategies for passive scalar mixing," J. Fluid Mech. 675, 465 (2011)] that determines the flow field that instantaneously maximizes the depletion of the H − 1 mix-norm. In this work, we bridge some of the gap between the best available a priori analysis and simulation results. After recalling some previous analysis, we present an explicit example demonstrating finite-time perfect mixing with a finite energy constraint on the stirring flow. On the other hand, using a recent result by Wirosoetisno et al. ["Long time stability of a classical efficient scheme for two dimensional Navier-Stokes equations," SIAM J. Numer. Anal. 50(1), 126-150 (2012)] we establish that the H − 1 mix-norm decays at most exponentially in time if the two-dimensional incompressible flow is constrained to have constant palenstrophy. Finite-time perfect mixing is thus ruled out when too much cost is incurred by small scale structures in the stirring. Direct numerical simulations in two dimensions suggest the impossibility of finite-time perfect mixing for flows with fixed power constraint and we conjecture an exponential lower bound on the H − 1 mixnorm in this case. We also discuss some related problems from other areas of analysis that are similarly suggestive of an exponential lower bound for the H − 1 mix-norm. C 2012 American Institute of Physics. [http://dx.

Optimal mixing in two-dimensional plane Poiseuille flow at finite Péclet number

“…In order to keep the time period constant and equal to one, we have to choose the velocity magnitude during the nth stage to be u(n) = 1/2 n−1 . We here stress that we depart from the fixed-power (constant velocity u(n)) approach adopted in Lunasin et al (2012). The steps corresponding to n < t < n + 1/2 can be seen as a folding stage whereas a stretching motion is observed when n + 1/2 < t < n + 1.…”

“…This flow is considered in a doubly periodic unit square. We consider a finite-Péclet-number version of the toy model flow presented in Lunasin et al (2012). This model will give insight into how different mixing measures behave in the presence of diffusion, which is essential for effective mixing.…”

Accurate control of hyperbolic trajectories in any dimension

“…We remark that we have achieved global transport by applying localized controls at the two hyperbolic trajectories, highlighting the role hyperbolic trajectories have on the phase space. By targeting energy towards regions where it has the most impact in this fashion, we offer a new approach towards energy-constrained transport maximization [39,44,49,50]. A refinement of this idea by additionally being able to control the directions in which the stable and unstable manifolds emanate from the hyperbolic trajectory is underway.…”

“…Thus, if hyperbolic trajectories with a heteroclinic manifold connecting them are made to move in a judiciously chosen manner, it will be possible to break apart the heteroclinic manifold into intersecting stable and unstable manifolds, thereby causing complicated (i.e., chaotic) mixing. Thus, controlling hyperbolic trajectories is a yet-unexplored avenue in the topic of controlling and optimizing mixing which is eliciting much recent interest [23,24,[38][39][40][41][42][43][44][45][46][47][48][49][50].…”

Stable mixing estimates in the infinite Péclet number limit

“…It is therefore not surprising that a way to quantify this is through the decay of a negative Sobolev norm, as in (1.12). In the context of passive scalars, this point of view was introduced in [31], and it is deeply connected with the regularity of transport equations [12,18,26,45], the quantification and lower bounds on mixing rates [1,6,19,25,32,36,46], and the inviscid damping in the two-dimensional Euler equations linearized around shear flows [8,17,23,40,42,43,[47][48][49].…”