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{{lowercase|d'Alembert-Euler condition}} | |||
In [[mathematics]] and [[physics]], especially the study of [[mechanics]] and [[fluid dynamics]], the '''d'Alembert-Euler condition''' is a requirement that the [[Streamlines, streaklines and pathlines|streaklines]] of a flow are [[irrotational]]. Let '''x''' = '''x'''('''X''',''t'') be the coordinates of the point '''x''' into which '''X''' is carried at time ''t'' by a (fluid) flow. Let <math>\ddot{\mathbf{x}}=\frac{D^2\mathbf{x}}{Dt}</math> be the second [[material derivative]] of '''x'''. Then the d'Alembert-Euler condition is: | |||
:<math>\mathrm{curl}\ \mathbf{x}=\mathbf{0}. \, </math> | |||
The d'Alembert-Euler condition is named for [[Jean le Rond d'Alembert]] and [[Leonhard Euler]] who independently first described its use in the mid-18th century. It is not to be confused with the [[Cauchy-Riemann equations|Cauchy-Riemann conditions]]. | |||
==References== | |||
*{{cite book |last=Truesdell |first=Clifford A. |authorlink=Clifford Truesdell |title=The Kinematics of Vorticity |year=1954 |publisher=Indiana University Press |location=Bloomington, IN}} See sections 45–48. | |||
*[http://eom.springer.de/c/c020970.htm d'Alembert–Euler conditions] on the Springer Encyclopedia of Mathematics | |||
{{DEFAULTSORT:D'alembert-Euler Condition}} | |||
[[Category:Fluid mechanics]] | |||
[[Category:Mechanical engineering]] | |||
[[Category:Vector calculus]] | |||
Revision as of 14:27, 9 January 2014
In mathematics and physics, especially the study of mechanics and fluid dynamics, the d'Alembert-Euler condition is a requirement that the streaklines of a flow are irrotational. Let x = x(X,t) be the coordinates of the point x into which X is carried at time t by a (fluid) flow. Let be the second material derivative of x. Then the d'Alembert-Euler condition is:
The d'Alembert-Euler condition is named for Jean le Rond d'Alembert and Leonhard Euler who independently first described its use in the mid-18th century. It is not to be confused with the Cauchy-Riemann conditions.
References
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My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 See sections 45–48. - d'Alembert–Euler conditions on the Springer Encyclopedia of Mathematics