Chandrasekhar virial equations

Virial equations in rotating frame of reference

The Euler equations in a rotating frame of reference, rotating with an angular velocity ${\displaystyle \mathbf{\Omega }}$ is given by

${\displaystyle \rho \frac{d{u}_i}{dt}=-\frac{\mathrm{\partial }p}{\mathrm{\partial }{x}_i}+\rho \frac{\mathrm{\partial }\mathrm{\Phi }}{\mathrm{\partial }{x}_i}+\frac{1}{2}\rho \frac{\mathrm{\partial }}{\mathrm{\partial }{x}_i}|\mathbf{\Omega }\times \mathbf{x}{|}^2+2\rho {\epsilon }_{i\ell m}{u}_{\ell }{\mathrm{\Omega }}_m}$

where ${\displaystyle {\epsilon }_{i\ell m}}$ is the Levi-Civita symbol, ${\displaystyle \frac{1}{2}|\mathbf{\Omega }\times \mathbf{x}{|}^2}$ is the centrifugal acceleration and ${\displaystyle 2\mathbf{u}\times \mathbf{\Omega }}$ is the Coriolis acceleration.

Steady state second order virial equation

In steady state, the second order virial equation becomes

${\displaystyle 2T_{ij}+W_{ij}+{\mathrm{\Omega }}^2{I}_{ij}-{\mathrm{\Omega }}_i{\mathrm{\Omega }}_k{I}_{kj}+2{ϵ}_{i\ell m}{\mathrm{\Omega }}_m{\int }_V\rho u_{\ell }{x}_{j}d\mathbf{x}=-{\delta }_{ij}\mathrm{\Pi }}$

If the axis of rotation is chosen in ${\displaystyle x_3}$ direction, the equation becomes

${\displaystyle W_{ij}+{\mathrm{\Omega }}^2(I_{ij}-{\delta }_{i3}{I}_{3j})=-{\delta }_{ij}\mathrm{\Pi }}$

and Chandrasekhar shows that in this case, the tensors can take only the following form

${\displaystyle W_{ij}=\left(\begin{array}{ccc}{W}_{11}& W_{12}& 0\\ W_{21}& W_{22}& 0\\ 0& 0& W_{33}\end{array}\right),I_{ij}=\left(\begin{array}{ccc}{I}_{11}& I_{12}& 0\\ I_{21}& I_{22}& 0\\ 0& 0& I_{33}\end{array}\right)}$

Steady state third order virial equation

In steady state, the third order virial equation becomes

${\displaystyle 2(T_{ij;k}+T_{ik;j})+W_{ij;k}+W_{ik;j}+{\mathrm{\Omega }}^2{I}_{ijk}-{\mathrm{\Omega }}_i{\mathrm{\Omega }}_{\ell }{I}_{\ell jk}+2{\epsilon }_{i\ell m}{\mathrm{\Omega }}_m{\int }_V\rho u_{\ell }{x}_j{x}_{k}d\mathbf{x}=-{\delta }_{ij}{\mathrm{\Pi }}_k-{\delta }_{ik}{\mathrm{\Pi }}_j}$

If the axis of rotation is chosen in ${\displaystyle x_3}$ direction, the equation becomes

${\displaystyle W_{ij;k}+W_{ik;j}+{\mathrm{\Omega }}^2(I_{ijk}-{\delta }_{i3}{I}_{3jk})=-({\delta }_{ij}{\mathrm{\Pi }}_k+{\delta }_{ik}{\mathrm{\Pi }}_j)}$

Steady state fourth order virial equation

With ${\displaystyle x_3}$ being the axis of rotation, the steady state fourth order virial equation is also derived by Chandrasekhar in 1968. The equation reads as

${\displaystyle \frac{1}{3}(2W_{ij;kl}+2W_{ik;lj}+2W_{il;jk}+W_{ij;k;l}+W_{ik;l;j}+W_{il;j;k})+{\mathrm{\Omega }}^2(I_{ijkl}-{\delta }_{i3}{I}_{3jkl})=-({\delta }_{ij}{\mathrm{\Pi }}_{kl}+{\delta }_{ik}{\mathrm{\Pi }}_{lj}+{\delta }_{il}{\mathrm{\Pi }}_{jk})}$

Virial equations with viscous stresses

Consider the Navier-Stokes equations instead of Euler equations,

${\displaystyle \rho \frac{d{u}_i}{dt}=-\frac{\mathrm{\partial }p}{\mathrm{\partial }{x}_i}+\rho \frac{\mathrm{\partial }\mathrm{\Phi }}{\mathrm{\partial }{x}_i}+\frac{\mathrm{\partial }{\tau }_{ik}}{\mathrm{\partial }{x}_k},\text{where}{\tau }_{ik}=\rho \nu \left(\frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }{x}_k}+\frac{\mathrm{\partial }{u}_k}{\mathrm{\partial }{x}_i}-\frac{2}{3}\frac{\mathrm{\partial }{u}_l}{\mathrm{\partial }{x}_l}{\delta }_{ik}\right)}$

and we define the shear-energy tensor as

${\displaystyle S_{ij}={\int }_V{\tau }_{ij}d\mathbf{x}.}$

With the condition that the normal component of the total stress on the free surface must vanish, i.e., ${\displaystyle (-p{\delta }_{ik}+{\tau }_{ik})n_k=0}$, where ${\displaystyle \mathbf{n}}$ is the outward unit normal, the second order virial equation then be

${\displaystyle \frac{1}{2}\frac{d^2{I}_{ij}}{d{t}^2}=2T_{ij}+W_{ij}+{\delta }_{ij}\mathrm{\Pi }-S_{ij}.}$

This can be easily extended to rotating frame of references.