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 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]\displaystyle{ \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]\displaystyle{ \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 conditions.

References

• Truesdell, Clifford A. (1954). The Kinematics of Vorticity. Bloomington, IN: Indiana University Press.  See sections 45–48.
• d'Alembert–Euler conditions on the Springer Encyclopedia of Mathematics