TY - JOUR

T1 - Controlling the unsteady analogue of saddle stagnation points

AU - Balasuriya, Sanjeeva

AU - Padberg-Gehle, Kathrin

PY - 2013/4/23

Y1 - 2013/4/23

N2 - It is well known that saddle stagnation points are crucial flow organizers in steady (autonomous) flows due to their accompanying stable and unstable manifolds. These have been extensively investigated experimentally, numerically, and theoretically in situations related to macroand micromixers in order to either restrict or enhance mixing. Saddle points are also important players in the dynamics of mechanical oscillators, in which such points and their associated invariant manifolds form boundaries of basins of attraction corresponding to qualitatively different types of behavior. The entity analogous to a saddle point in an unsteady (nonautonomous) flow is a timevarying hyperbolic trajectory with accompanying stable and unstable manifolds which move in time. Within the context of nearly steady flows, the unsteady velocity perturbation required to ensure that such a hyperbolic (saddle) trajectory follows a specified trajectory in space is derived and shown to be equivalent to that which can be obtained via a heuristic approach. An expression for the error in the hyperbolic trajectory's motion is also derived. This provides a new tool for the control of both fluid transport and mechanical oscillators. The method is applied to two examples-a four-roll mill and a Duffing oscillator-and the performance of the control strategy is shown to be excellent in both instances.

AB - It is well known that saddle stagnation points are crucial flow organizers in steady (autonomous) flows due to their accompanying stable and unstable manifolds. These have been extensively investigated experimentally, numerically, and theoretically in situations related to macroand micromixers in order to either restrict or enhance mixing. Saddle points are also important players in the dynamics of mechanical oscillators, in which such points and their associated invariant manifolds form boundaries of basins of attraction corresponding to qualitatively different types of behavior. The entity analogous to a saddle point in an unsteady (nonautonomous) flow is a timevarying hyperbolic trajectory with accompanying stable and unstable manifolds which move in time. Within the context of nearly steady flows, the unsteady velocity perturbation required to ensure that such a hyperbolic (saddle) trajectory follows a specified trajectory in space is derived and shown to be equivalent to that which can be obtained via a heuristic approach. An expression for the error in the hyperbolic trajectory's motion is also derived. This provides a new tool for the control of both fluid transport and mechanical oscillators. The method is applied to two examples-a four-roll mill and a Duffing oscillator-and the performance of the control strategy is shown to be excellent in both instances.

KW - Flow control

KW - Hyperbolic trajectory

KW - Nonautonomous flows

KW - Oscillator control

KW - Saddle point

KW - Stagnation point

KW - Mathematics

UR - http://www.scopus.com/inward/record.url?scp=84878031609&partnerID=8YFLogxK

U2 - 10.1137/120886042

DO - 10.1137/120886042

M3 - Journal articles

AN - SCOPUS:84878031609

VL - 73

SP - 1038

EP - 1057

JO - SIAM Journal on Applied Mathematics

JF - SIAM Journal on Applied Mathematics

SN - 0036-1399

IS - 2

ER -