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 ${\displaystyle {\ddot {\mathbf {x} }}={\frac {D^{2}\mathbf {x} }{Dt}}}$ be the second material derivative of x. Then the d'Alembert-Euler condition is:

${\displaystyle \mathrm {curl} \ \mathbf {x} =\mathbf {0} .\,}$

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