Neoclassical Viscosity Matrices

The $2\times 2$ dimensionless species-$s$ neoclassical viscosity matrix, $[\mu_s](r)$ (see Section 2.17), is defined to have the following elements [7]:

$\displaystyle \mu_{s\,11}$ $\displaystyle = K_{s\,11},$ (A.53)
$\displaystyle \mu_{s\,12}$ $\displaystyle = \frac{5}{2}\,K_{s\,11}- K_{s\,12},$ (A.54)
$\displaystyle \mu_{s\,21}$ $\displaystyle = \frac{5}{2}\,K_{s\,11}- K_{s\,12},$ (A.55)
$\displaystyle \mu_{s\,22}$ $\displaystyle = K_{s\,22} - 5\,K_{s\,12}+\frac{25}{4}\,K_{s\,11}.$ (A.56)

(Note that the viscosity matrix elements defined here differ from those defined in Section 2.17 by a factor $g_t$.) Here,

$\displaystyle K_{e\,jk}(r)$ $\displaystyle = g_t\,\frac{8}{3\!\sqrt{\pi}}\int_0^\infty
\frac{e^{-x^{2}}\,x^{...
...D\,e}(x)]\,[x^{4}+(5\pi/8)\,(\omega_{t\,e}\,\tau_{ee})^{\,-1}\,\nu_{T\,e}(x)]},$ (A.57)
$\displaystyle \nu_{D\,e}(x)$ $\displaystyle = \frac{3\!\sqrt{\pi}}{4}\left[\left(1-\frac{1}{2\,x^{2}}\right)Y(x)+Y'(x)\right]+\frac{3\!\sqrt{\pi}}{4}\,Z_{\rm eff},$ (A.58)
$\displaystyle \nu_{\epsilon\,e}(x)$ $\displaystyle = \frac{3\!\sqrt{\pi}}{2}\left[Y(x)-Y'(x)\right],$ (A.59)
$\displaystyle \nu_{T\,s}(x)$ $\displaystyle = 3\,\nu_{D\,s}(x)+\nu_{\epsilon\,s}(x),$ (A.60)

and

$\displaystyle Y(x)$ $\displaystyle = \frac{2}{\sqrt{\pi}}\int_0^{x^2}\!\sqrt{t}\,{\rm e}^{-t}\,dt,$ (A.61)
$\displaystyle Y'(x)$ $\displaystyle = \frac{2}{\sqrt{\pi}}\,x\,{\rm e}^{-x^{2}}.$ (A.62)

Furthermore,

$\displaystyle K_{i\,jk}(r)$ $\displaystyle =g_t\,\frac{8}{3\!\sqrt{\pi}}\int_0^\infty
\frac{e^{-x^{2}}\,x^{4...
...D\,i}(x)]\,[x^{3}+(5\pi/8)\,(\omega_{t\,i}\,\tau_{ii})^{\,-1}\,\nu_{T\,i}(x)]},$ (A.63)
$\displaystyle \nu_{D\,i}(x)$ $\displaystyle = \frac{3\!\sqrt{\pi}}{4}\left[\left(1-\frac{1}{2\,x^{2}}\right)Y(x)+Y'(x)\right]\frac{1}{x}$    
$\displaystyle \phantom{===}$ $\displaystyle \phantom{=}+\frac{3\!\sqrt{\pi}}{4}\,\alpha_I\left[\left(1-\frac{...
...left(\frac{x}{x_{iI}}\right)
+Y\left(\frac{x}{x_{iI}}\right)\right]\frac{1}{x},$ (A.64)
$\displaystyle \nu_{\epsilon\,i}(x)$ $\displaystyle =\frac{3\!\sqrt{\pi}}{2}\left[Y(x)-Y'(x)\right]\frac{1}{x}+\frac{...
...frac{x}{x_{iI}}\right)
-\psi'\!\left(\frac{x}{x_{iI}}\right)\right]\frac{1}{x},$ (A.65)

and, finally,

$\displaystyle K_{I\,jk}(r)$ $\displaystyle = g_t\,\frac{8}{3\!\sqrt{\pi}}\int_0^\infty
\frac{e^{-x^{2}}\,x^{...
...D\,I}(x)]\,[x^{3}+(5\pi/8)\,(\omega_{t\,I}\,\tau_{II})^{\,-1}\,\nu_{T\,I}(x)]},$ (A.66)
$\displaystyle \nu_{D\,I}(x)$ $\displaystyle = \frac{3\!\sqrt{\pi}}{4}\left[\left(1-\frac{1}{2\,x^{2}}\right)Y(x)+Y'(x)\right]\frac{1}{x}$    
$\displaystyle \phantom{===}$ $\displaystyle \phantom{=}+\frac{3\!\sqrt{\pi}}{4}\,\frac{1}{\alpha_I}\left[\lef...
...eft(\frac{x}{x_{Ii}}\right)
+Y'\left(\frac{x}{x_{Ii}}\right)\right]\frac{1}{x},$ (A.67)
$\displaystyle \nu_{\epsilon\,I}(x)$ $\displaystyle = \frac{3\!\sqrt{\pi}}{2}\left[Y(x)-Y'(x)\right]\frac{1}{x}+\frac...
...eft(\frac{x}{x_{Ii}}\right)
-Y'\left(\frac{x}{x_{Ii}}\right)\right]\frac{1}{x}.$ (A.68)

Note that our expressions for the neoclassical viscosity matrices interpolate in the most accurate manner possible between the three standard neoclassical collsionality regimes (i.e., the banana, plateau, and Pfirsch-Schl├╝ter regimes [8]) [7].