The analysis assumes that the flow is an incompressible flow, and that the perturbations are governed by the linearized Euler equations and, thus, are inviscid.
42.
To solve the PDE we recognize that it is a Cauchy Euler equation which can be transformed into a diffusion equation by introducing the change-of-variable transformation
43.
We remark that also the Euler equation even when conservative ( no external field, Froude limit ) have "'no Riemann invariants "'in general.
44.
At the dynamic level, the fact that vortex lines are transported by any flow governed by the classical Euler equations implies conservation of any vortical structure within the flow.
45.
This culminated in the three equation new Keynesian model found in the survey by Richard Clarida, Jordi Gali, and dynamic consumption equation ( household's Euler equation ).
46.
The Madelung equations, being quantum Euler equations ( fluid dynamics ), differ philosophically from the de Broglie Bohm mechanics and are the basis of the stochastic interpretation of quantum mechanics.
47.
If they are all distinguished, the system is defined " strictly hyperbolic " ( it will be proved to be the case of one-dimensional Euler equations ).
48.
In the limit of high Reynolds numbers, the Navier Stokes equations approach the inviscid Euler equations, of which the potential-flow solutions considered by d'Alembert are solutions.
49.
The CL vortex force finds its origins in the appearance of the Stokes drift in the convective acceleration terms in the mean momentum equation of the Euler equations or Navier Stokes equations.
50.
From the mathematical point of view, Euler equations are notably hyperbolic conservation equations in the case without external field ( i . e . in the limit of high Froude number ).
