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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'''&nbsp;=&nbsp;'''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 15:27, 9 January 2014

Template:Lowercase

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