aerodynamics Aerodynamics () is the study of the motion of atmosphere of Earth, air, particularly when affected by a solid object, such as an airplane wing. It involves topics covered in the field of fluid dynamics and its subfield of gas dynamics, and is an ...

, Crocco's theorem relates the

flow velocity In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the f ...

, vorticity, and

stagnation pressure In fluid dynamics, stagnation pressure, also referred to as total pressure, is what the pressure would be if all the kinetic energy of the fluid were to be converted into pressure in a reversable manner.; it is defined as the sum of the free-strea ...

(or

entropy Entropy is a scientific concept, most commonly associated with states of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynamics, where it was first recognized, to the micros ...

) of a

potential flow In fluid dynamics, potential flow or irrotational flow refers to a description of a fluid flow with no vorticity in it. Such a description typically arises in the limit of vanishing viscosity, i.e., for an inviscid fluid and with no vorticity pre ...

. This

theorem In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to esta ...

gives the relation between the thermodynamics and fluid kinematics. The theorem was first enunciated by Alexander Friedmann for the particular case of a

perfect gas In physics, engineering, and physical chemistry, a perfect gas is a theoretical gas model that differs from real gases in specific ways that makes certain calculations easier to handle. In all perfect gas models, intermolecular forces are neglecte ...

and published in 1922: :

\frac=T \nabla s-\nabla h

However, usually this theorem is connected with the name of Italian scientist ,Crocco L
Eine neue Stromfunktion für die Erforschung der Bewegung der Gase mit Rotation
ZAMM, Vol. 17, Issue 1, pp. 1–7, 1937. DOI: 10.1002/zamm.19370170103. Crocco writes the theorem in the form

\scriptstyle\mathrm\,\mathbf u\times\mathbf u=T\mathrm\,S

for

(the last formula on page 2). a son of Gaetano Crocco. Consider an element of fluid in the flow field subjected to translational and rotational motion: because stagnation pressure loss and entropy generation can be viewed as essentially the same thing, there are three popular forms for writing Crocco's theorem: # Stagnation pressure:

\mathbf u \times \boldsymbol \omega =v \nabla p_0

Shapiro, Ascher H. "National Committee for Fluid Mechanics Films Film Notes for 'Vorticity,'" 1969. Encyclopædia Britannica Educational Corporation, Chicago, Illinois. (retrieved from http://web.mit.edu/hml/ncfmf/09VOR.pdf (5/29/11) # Entropy (the following form holds for plane steady flows):

T \frac = \frac +u \omega

Liepmann, H. W. and Roshko, A. "Elements of Gasdynamics" 2001. Dover Publications, Mineola, NY (eq. (7.33)). # Momentum:

\frac + \nabla \left(\frac + h \right) = \mathbf u \times \boldsymbol \omega + T \nabla s + \mathbf,

In the above equations,

\mathbf u

is the flow velocity vector,

\omega

is the vorticity,

v

is the specific volume,

p_0

is the

T

temperature Temperature is a physical quantity that quantitatively expresses the attribute of hotness or coldness. Temperature is measurement, measured with a thermometer. It reflects the average kinetic energy of the vibrating and colliding atoms making ...

s

is specific

h

is specific

enthalpy Enthalpy () is the sum of a thermodynamic system's internal energy and the product of its pressure and volume. It is a state function in thermodynamics used in many measurements in chemical, biological, and physical systems at a constant extern ...

\mathbf

is specific

body force In physics, a body force is a force that acts throughout the volume of a body.Springer site - Book 'Solid mechanics'preview paragraph 'Body forces'./ref> Forces due to gravity, electric fields and magnetic fields are examples of body forces. Bod ...

, and

n

is the direction normal to the streamlines. All quantities considered (entropy, enthalpy, and body force) are ''specific'', in the sense of "per unit mass".

References

{{reflist * * * * Fluid dynamics Aerodynamics