User:Tohline/PGE/RotatingFrame

NOTE to Eric Hirschmann & David Neilsen... I have move the earlier contents of this page to a new Wiki location called Compressible Riemann Ellipsoids.

Rotating Reference Frame

At times, it can be useful to view the motion of a fluid from a frame of reference that is rotating with a uniform (i.e., time-independent) angular velocity <math>\Omega_f</math>. In order to transform any one of the principal governing equations from the inertial reference frame to such a rotating reference frame, we must specify the orientation as well as the magnitude of the angular velocity vector about which the frame is spinning, <math>{\vec\Omega}_f</math>; and the <math>d/dt</math> operator, which denotes Lagrangian time-differentiation in the interial frame, must everywhere be replaced as follows:

<math> \biggl[\frac{d}{dt} \biggr]_{inertial} \rightarrow \biggl[\frac{d}{dt} \biggr]_{rot} + {\vec{\Omega}}_f \times . </math>

Performing this transformation implies, for example, that

<math> \vec{v}_{inertial} = \vec{v}_{rot} + {\vec{\Omega}}_f \times \vec{x} , </math>

and,

<math> \biggl[ \frac{d\vec{v}}{dt}\biggr]_{inertial} = \biggl[ \frac{d\vec{v}}{dt}\biggr]_{rot} + 2{\vec{\Omega}}_f \times {\vec{v}}_{rot} + {\vec{\Omega}}_f \times ({\vec{\Omega}}_f \times \vec{x}) </math>

<math> = \biggl[ \frac{d\vec{v}}{dt}\biggr]_{rot} + 2{\vec{\Omega}}_f \times {\vec{v}}_{rot} - \frac{1}{2} \nabla \biggl[ |{\vec{\Omega}}_f \times \vec{x}|^2 \biggr] </math>

(If we were to allow <math>{\vec\Omega}_f</math> to be a function of time, an additional term involving the time-derivative of <math>{\vec\Omega}_f</math> also would appear on the right-hand-side of these last expressions; see, for example, Eq.~1D-42 in BT87.) Note as well that the relationship between the fluid vorticity in the two frames is,

<math> [\vec\zeta]_{inertial} = [\vec\zeta]_{rot} + 2{\vec\Omega}_f . </math>

Continuity Equation (rotating frame)

Applying these transformations to the standard, inertial-frame representations of the continuity equation presented elsewhere, we obtain the:

Lagrangian Representation
of the Continuity Equation
as viewed from a Rotating Reference Frame

<math>\biggl[ \frac{d\rho}{dt} \biggr]_{rot} + \rho \nabla \cdot {\vec{v}}_{rot} = 0</math> ;

Eulerian Representation
of the Continuity Equation
as viewed from a Rotating Reference Frame

<math>\biggl[ \frac{\partial\rho}{\partial t} \biggr]_{rot} + \nabla \cdot (\rho {\vec{v}}_{rot}) = 0</math> .

Euler Equation (rotating frame)

Applying these transformations to the standard, inertial-frame representations of the Euler equation presented elsewhere, we obtain the:

Lagrangian Representation
of the Euler Equation
as viewed from a Rotating Reference Frame

<math>\biggl[ \frac{d\vec{v}}{dt}\biggr]_{rot} = - \frac{1}{\rho} \nabla P - \nabla \Phi - 2{\vec{\Omega}}_f \times {\vec{v}}_{rot} - {\vec{\Omega}}_f \times ({\vec{\Omega}}_f \times \vec{x})</math> ;

Eulerian Representation
of the Euler Equation
as viewed from a Rotating Reference Frame

<math>\biggl[\frac{\partial\vec{v}}{\partial t}\biggr]_{rot} + ({\vec{v}}_{rot}\cdot \nabla) {\vec{v}}_{rot}= - \frac{1}{\rho} \nabla P - \nabla \biggl[\Phi - \frac{1}{2}|{\vec{\Omega}}_f \times \vec{x}|^2 \biggr] - 2{\vec{\Omega}}_f \times {\vec{v}}_{rot} </math> ;

Euler Equation
written in terms of the Vorticity and
as viewed from a Rotating Reference Frame

<math>\biggl[\frac{\partial\vec{v}}{\partial t}\biggr]_{rot} + ({\vec{\zeta}}_{rot}+2{\vec\Omega}_f) \times {\vec{v}}_{rot}= - \frac{1}{\rho} \nabla P - \nabla \biggl[\Phi + \frac{1}{2}v_{rot}^2 - \frac{1}{2}|{\vec{\Omega}}_f \times \vec{x}|^2 \biggr]</math> .

Centrifugal and Coriolis Accelerations

Following along the lines of the discussion presented in Appendix 1.D, §3 of BT87, in a rotating reference frame the Lagrangian representation of the Euler equation may be written in the form,

<math>\biggl[ \frac{d\vec{v}}{dt}\biggr]_{rot} = - \frac{1}{\rho} \nabla P - \nabla \Phi + {\vec{a}}_{fict} </math>,

where,

<math> {\vec{a}}_{fict} \equiv - 2{\vec{\Omega}}_f \times {\vec{v}}_{rot} - {\vec{\Omega}}_f \times ({\vec{\Omega}}_f \times \vec{x}) . </math>

So, as viewed from a rotating frame of reference, material moves as if it were subject to two fictitious accelerations which traditionally are referred to as the,

Coriolis Acceleration

<math> {\vec{a}}_{Coriolis} \equiv - 2{\vec{\Omega}}_f \times {\vec{v}}_{rot} , </math>

(see the related Wikipedia discussion) and the

Centrifugal Acceleration

<math> {\vec{a}}_{Centrifugal} \equiv - {\vec{\Omega}}_f \times ({\vec{\Omega}}_f \times \vec{x}) = \frac{1}{2} \nabla\biggl[ |{\vec{\Omega}}_f \times \vec{x}|^2 \biggr] </math>

(see the related Wikipedia discussion).

Nonlinear Velocity Cross-Product

In some contexts —for example, our discussion of Riemann ellipsoids or the analysis by Korycansky & Papaloizou (1996) of nonaxisymmetric disk structures — it proves useful to isolate and analyze the term in the "vorticity formulation" of the Euler equation that involves a nonlinear cross-product of the rotating-frame velocity vector, namely,

<math> \vec{A} \equiv ({\vec{\zeta}}_{rot}+2{\vec\Omega}_f) \times {\vec{v}}_{rot} . </math>

Without lose of generality we can set <math>{\vec\Omega}_f = \hat{k}\Omega_f</math>, that is, we can align the frame rotation axis with the z-axis of a Cartesian coordinate system. Also, to simplify notation, for most of the remainder of this subsection we will drop the subscript "rot" on both the velocity and vorticity vectors. The Cartesian components of <math>{\vec{A}}</math> are then,

<math> \hat{i}: ~~~~~~ </math>

<math> \hat{j}: ~~~~~~ </math>

<math> \hat{k}: ~~~~~~ </math>

© 2014 - 2021 by Joel E. Tohline |   H_Book Home   |   YouTube   | Appendices: | Equations | Variables | References | Ramblings | Images | myphys.lsu | ADS | Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation