Quaternions and particle dynamics in the Euler fluid equations

Abstract

Vorticity dynamics of the three-dimensional incompressible Euler equations is 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. 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.

14 pages, 3 figures, revised abstract, To appear in Nonlinearity

References in corpus (2)

• On the Collapse of Tubes Carried by 3D Incompressible Flows
• On a new scale of regularity spaces with applications to Euler's equations

Cited by in corpus (12)

• Modeling the pressure Hessian and viscous Laplacian in Turbulence: comparisons with DNS and implications on velocity gradient dynamics
• Solve the linear quaternion-valued differential equations having multiple eigenvalues
• Global structure of quaternion polynomial differential equations
• Lagrangian particle paths and ortho-normal quaternion frames
• A Riccati-type solution of 3D Euler equations for incompressible flow
• An Algorithm for Solving Linear Nonhomogeneous Quaternion-valued Differential Equations
• Ortho-normal quaternion frames, Lagrangian evolution equations and the three-dimensional Euler equations
• Floquet Theory for Quaternion-valued Differential Equations
• A Unified Analysis of Linear Quaternion Dynamic Equations on Time Scales
• Lagrangian analysis of alignment dynamics for isentropic compressible magnetohydrodynamics
• The gradient of potential vorticity, quaternions and an orthonormal frame for fluid particles
• A Semi-Lagrangian Spherical Essentially Non-Oscillatory (SENO) Scheme for Advection Equations of S2-valued Functions