{{Short description|Mathematical model in fluid dynamics}} thumb|An actuator disk accelerating a fluid flow from right to left
In fluid dynamics, '''momentum theory''' or '''disk actuator theory''' is a theory describing a mathematical model of an ideal actuator disk, such as a propeller or helicopter rotor, by W.J.M. Rankine (1865),<ref>{{Cite journal |last=Rankine |first=W.J. Macquorn |date=6 April 1865 |title=On the Mechanical Principals of the Action of Propellers |url=https://hdl.handle.net/2027/mdp.39015022700325?urlappend=%3Bseq=47%3Bownerid=13510798885881399-61 |journal=Transactions of the Royal Institution of Naval Architects |volume=6 |pages=13 |hdl=2027/mdp.39015022700325?urlappend=%3Bseq=47 |via=HathiTrust}}</ref> Alfred George Greenhill (1888) and Robert Edmund Froude (1889).<ref>{{Cite journal |last=Froude |first=Robert |date=12 April 1889 |title=On the Part Played in Propulsion by Differences in Fluid Pressure |url=https://babel.hathitrust.org/cgi/pt?id=hvd.hnhtv1&seq=424 |journal=Transactions of the Royal Institution of Naval Architects |volume=30 |pages=390 |via=HathiTrust}}</ref>
The rotor is modeled as an infinitely thin disc, inducing a constant velocity along the axis of rotation. The basic state of a helicopter is hovering. This disc creates a flow around the rotor. Under certain mathematical premises of the fluid, there can be extracted a mathematical connection between power, radius of the rotor, torque and induced velocity. Friction is not included.
For a stationary open rotor with no outer duct, such as a helicopter in hover, the power required to produce a given thrust is:
:<math>P = \sqrt{\frac{T^3}{2 \rho A}}</math>
where: * T is the thrust * <math>\rho</math> is the density of air (or other medium) * A is the area of the rotor disc * P is power
A device which converts the translational energy of the fluid into rotational energy of the axis or vice versa is called a '''Rankine disk actuator'''. The real life implementations of such devices include marine and aviation propellers, windmills, helicopter rotors, centrifugal pumps, wind turbines, turbochargers and chemical agitators.
==See also== * Blade element theory * Circulation (fluid dynamics) * Disk loading * Kutta–Joukowski theorem {{Commons|Category:Actuator disk theory|Actuator disk theory}}
==References== {{reflist}}
Category:Fluid dynamics Category:Propellers Category:Momentum Category:Aircraft aerodynamics