In
mathematics and
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which rel ...
, especially the study of
mechanics
Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objects ...
and
fluid dynamics
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including '' aerodynamics'' (the study of air and other gases in motion) ...
, the d'Alembert-Euler condition is a requirement that the
streaklines of a flow are
irrotational
In vector calculus, a conservative vector field is a vector field that is the gradient of some function (mathematics), function. A conservative vector field has the property that its line integral is path independent; the choice of any path betwee ...
. 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
In continuum mechanics, the material derivative describes the time rate of change of some physical quantity (like heat or momentum) of a material element that is subjected to a space-and-time-dependent macroscopic velocity field. The material de ...
of x. Then the d'Alembert-Euler condition is:
:
The d'Alembert-Euler condition is named for
Jean le Rond d'Alembert
Jean-Baptiste le Rond d'Alembert (; ; 16 November 1717 – 29 October 1783) was a French mathematician, mechanician, physicist, philosopher, and music theorist. Until 1759 he was, together with Denis Diderot, a co-editor of the '' Encyclop� ...
and
Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
who independently first described its use in the mid-18th century. It is not to be confused with the
Cauchy–Riemann conditions.
References
* See sections 45–48.
d'Alembert–Euler conditionson the Springer Encyclopedia of Mathematics
Fluid mechanics
Mechanical engineering
Vector calculus
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