Controlling the unsteady analogue of saddle stagnation points
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In: SIAM Journal on Applied Mathematics, Vol. 73, No. 2, 23.04.2013, p. 1038-1057.
Research output: Journal contributions › Journal articles › Research › peer-review
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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 -