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fluid dynamics In physics, physical chemistry and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids – liquids and gases. It has several subdisciplines, including (the study of air and other gases in motion ...
, momentum theory or disk actuator theory is a theory describing a
mathematical model A mathematical model is an abstract and concrete, abstract description of a concrete system using mathematics, mathematical concepts and language of mathematics, language. The process of developing a mathematical model is termed ''mathematical m ...
of an ideal actuator disk, such as a
propeller A propeller (often called a screw if on a ship or an airscrew if on an aircraft) is a device with a rotating hub and radiating blades that are set at a pitch to form a helical spiral which, when rotated, exerts linear thrust upon a working flu ...
or
helicopter rotor On a helicopter, the main rotor or rotor system is the combination of several rotary wings (rotor blades) with a control system, that generates the aerodynamic lift (force), lift force that supports the weight of the helicopter, and the thrust ...
, by W.J.M. Rankine (1865),
Alfred George Greenhill Sir Alfred George Greenhill (29 November 1847 in London – 10 February 1927 in London), was a British mathematician. George Greenhill was educated at Christ's Hospital School and from there he went to St John's College, Cambridge in 1866. In ...
(1888) and Robert Edmund Froude (1889). 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 In physics and mechanics, torque is the rotational analogue of linear force. It is also referred to as the moment of force (also abbreviated to moment). The symbol for torque is typically \boldsymbol\tau, the lowercase Greek letter ''tau''. Wh ...
and induced velocity.
Friction Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. Types of friction include dry, fluid, lubricated, skin, and internal -- an incomplete list. The study of t ...
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: :P = \sqrt where: * T is the
thrust Thrust is a reaction force described quantitatively by Newton's third law. When a system expels or accelerates mass in one direction, the accelerated mass will cause a force of equal magnitude but opposite direction to be applied to that ...
* \rho is the
density of air The density of air or atmospheric density, denoted '' ρ'', is the mass per unit volume of Earth's atmosphere at a given point and time. Air density, like air pressure, decreases with increasing altitude. It also changes with variations in atmosph ...
(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
propeller A propeller (often called a screw if on a ship or an airscrew if on an aircraft) is a device with a rotating hub and radiating blades that are set at a pitch to form a helical spiral which, when rotated, exerts linear thrust upon a working flu ...
s,
windmill A windmill is a machine operated by the force of wind acting on vanes or sails to mill grain (gristmills), pump water, generate electricity, or drive other machinery. Windmills were used throughout the high medieval and early modern period ...
s,
helicopter rotor On a helicopter, the main rotor or rotor system is the combination of several rotary wings (rotor blades) with a control system, that generates the aerodynamic lift (force), lift force that supports the weight of the helicopter, and the thrust ...
s,
centrifugal pump Centrifugal pumps are used to transport fluids by the Energy transformation, conversion of rotational kinetic energy to the hydrodynamic energy of the fluid flow. The rotational energy typically comes from an engine or electric motor. They are ...
s,
wind turbine A wind turbine is a device that wind power, converts the kinetic energy of wind into electrical energy. , hundreds of thousands of list of most powerful wind turbines, large turbines, in installations known as wind farms, were generating over ...
s,
turbocharger In an internal combustion engine, a turbocharger (also known as a turbo or a turbosupercharger) is a forced induction device that is powered by the flow of exhaust gases. It uses this energy to compress the intake air, forcing more air into th ...
s and chemical agitators.


See also

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Blade element theory Blade element theory (BET) is a Mathematics, mathematical process originally designed by William Froude (1878), David W. Taylor (1893) and Stefan Drzewiecki (1885) to determine the behavior of propellers. It involves breaking a blade down into sev ...
*
Circulation (fluid dynamics) In physics, circulation is the line integral of a vector field around a closed curve embedded in the field. In fluid dynamics, the field is the fluid velocity field. In electrodynamics, it can be the electric or the magnetic field. In aerodyn ...
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Disk loading In fluid dynamics, disk loading or disc loading is the average pressure change across an actuator disk, such as an airscrew. Airscrews with a relatively low disk loading are typically called rotors, including helicopter Helicopter rotor, main ro ...
*
Kutta–Joukowski theorem The Kutta–Joukowski theorem is a fundamental theorem in aerodynamics used for the calculation of lift of an airfoil (and any two-dimensional body including circular cylinders) translating in a uniform fluid at a constant speed so large that th ...


References

{{reflist Fluid dynamics Propellers Momentum Aircraft aerodynamics