{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T13:58:05Z","timestamp":1753883885673,"version":"3.41.2"},"reference-count":23,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2012,9,12]],"date-time":"2012-09-12T00:00:00Z","timestamp":1347408000000},"content-version":"vor","delay-in-days":255,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2012,1]]},"abstract":"<jats:p>Control of fluid flow is an important, underutilized process possessing potential benefits ranging from avoidance of separation and stall on aircraft wings to reduction of friction in oil and gas pipelines to mitigation of noise from wind turbines. But the Navier\u2010Stokes (N.\u2010S.) equations, whose solutions describe such flows, consist of a system of time\u2010dependent, multidimensional, nonlinear partial differential equations (PDEs) which cannot be solved in real time using current computing hardware. The poor man\u2032s Navier\u2010Stokes (PMNS) equations comprise a discrete dynamical system that is algebraic\u2014hence, easily (and rapidly) solved\u2014and yet which retains many (possibly all) of the temporal behaviors of the PDE N.\u2010S. system at specific spatial locations. Herein, we outline derivation of these equations and discuss their basic properties. We consider application of these equations to the control problem by adding a control force. We examine the range of behaviors that can be achieved by changing this control force and, in particular, consider controllability of this (nonlinear) system <jats:italic>via<\/jats:italic> numerical experiments. Moreover, we observe that the derivation leading to the PMNS equations is very general and may be applied to a wide variety of problems governed by PDEs and (possibly) time\u2010delay ordinary differential equations such as, for example, models of machining processes.<\/jats:p>","DOI":"10.1155\/2012\/746752","type":"journal-article","created":{"date-parts":[[2012,9,12]],"date-time":"2012-09-12T22:03:22Z","timestamp":1347487402000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Application of the Poor Man\u2032s Navier\u2010Stokes Equations to Real\u2010Time Control of Fluid Flow"],"prefix":"10.1155","volume":"2012","author":[{"given":"James B.","family":"Polly","sequence":"first","affiliation":[]},{"given":"J. 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