User:Tohline/PGE/RotatingFrame
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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.)
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> .
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