Quaternions and particle dynamics in the Euler fluid equations

Abstract

Vorticity dynamics of the three-dimensional incompressible Euler equations are cast into a quaternionic representation governed by the Lagrangian evolution of the tetrad consisting of the growth rate and rotation rate of the vorticity. In turn, the Lagrangian evolution of this tetrad is governed by another that depends on the pressure Hessian. Together these form the basis for a direction of vorticity theorem on Lagrangian trajectories. Moreover, in this representation, fluid particles carry ortho-normal frames whose Lagrangian evolution in time are shown to be directly related to the Frenet-Serret equations for a vortex line. The frame dynamics suggest an elegant Lagrangian relation regarding the pressure Hessian tetrad. The equations for ideal MHD are similarly considered.

Dynamic stability of the three‐dimensional axisymmetric Navier‐Stokes equations with swirl

“…Moreover, they show that the vorticity does not grow faster than double exponential in time, and the velocity field remains bounded up to T D 19, beyond the singularity time alleged in [16]. Finally, we mention the recent work of Gibbon et al (see [13] and the references therein), where they reveal some interesting geometric properties of the Euler equations in quaternion frames.…”

The three-dimensional Euler equations: Where do we stand?

“…Fluid particles not only take complicated trajectories but they also rotate in motion. Recent work has shown that Hamilton's quaternions are applicable to this type of problem [78,[100][101][102][103]. In his lifetime Hamilton's ideas did not meet with the approval of his contemporaries [104][105][106] but in the context of modern-day problems the crucial property that quaternions possess -that they represent a composition of rotations -has made them the technical foundation of modern inertial guidance systems in the aerospace industry where tracking the paths and the orientation of satellites and aircraft is critical [107].…”

“…Another example that could be cast into this format are the equations of ideal MHD in Elsasser form (see [78,100,101] although the existence of two material derivatives requires some generalization. In Section 5.3 it will be shown how the quartet in (35), based upon the pair of Lagrangian evolution equations (33) and (34), can determine the evolution of an ortho-normal frame for a fluid particle in a trajectory.…”

“…The variables α p , χ p corresponding (53) (with a change of sign) are defined in the same manner [101,102] …”

Quaternionic-valued ordinary differential equations. The Riccati equation

“…We present results for the Riccati equation. The special interest in this equation comes from the fact that it appears in the Euler vorticity dynamics (see [10]). …”