User:Tohline/Cylindrical 3D/Linearization
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Linearized Equations in Cylindrical Coordinates
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Eulerian Formulation of Nonlinear Governing Equations
From our more detailed, accompanying discussion we pull the Eulerian representation of the set of principal governing equations written in cylindrical coordinates.
Component of Euler Equation
Component of Euler Equation
z Component of Euler Equation
Equation of Continuity
These match, for example, equations (3.1)  (3.4) of Papaloizou & Pringle (1984, MNRAS, 208, 721750), hereafter, PPI.
Linearization
If we assume that the initial equilibrium configuration is axisymmetric with no radial or vertical velocity, the linearized equations become:
Linearizing Radial Component of Euler Equation












This last expression has been obtained by recognizing that, in the nexttolast expression: (1) The terms inside the curly braces on the righthand side collectively provide a statement of equilibrium (in the radialcoordinate direction) in the initial, unperturbed configuration and therefore the terms sum to zero; and (2) the terms inside square brackets on the lefthand side can be rewritten in a more compact form because we have adopted a polytropic equation of state to build the unperturbed initial equilibrium configuration and are examining only adiabatic perturbations with , in which case,

and 

Linearizing Azimuthal Component of Euler Equation
Keeping in mind that the initial equilibrium configuration is axisymmetric — that is, equilibrium parameters exhibit no variation in the azimuthal direction — and, in addition, exhibits no variation in the vertical direction, we have,






Linearizing Vertical Component of Euler Equation












where the logic followed in deriving the last expression from the nexttolast one is directly analogous to the logic used, above, in obtaining the final expression for the radial component of the linearized Euler equation.
Linearizing Continuity Equation






Summary
Set of Linearized Principal Governing Equations in Cylindrical Coordinates 



See Also
© 2014  2020 by Joel E. Tohline 