in detail in EFE. Below, we concentrate on its extension to superfluids with an emphasis on the new effects of mutual friction and mutual gravitational attraction of the superfluid and normal fluid.

First order virial equations
----------------------------

On taking the zeroth moment of Eq. (\[eq:AS:euler\]) which amounts to integrating over $V_\alpha$ we obtain the “first order ‘virial’ equation”[^2] $$\begin{aligned}
{d\over dt}\biggl(\int_{V_\alpha}{d^3x\rho_\alpha u_{\alpha,i}}
\biggr)&=&2\epsilon_{ilm}\Omega_m\int_{V_\alpha}{d^3x\rho_\alpha
u_{\alpha,l}}+(\Omega^2\delta_{ij}-\Omega_i\Omega_j)
\int_{V_\alpha}{d^3x\rho_\alpha x_j}
\nonumber\\& & 
-(1-\delta_{\alpha\beta})
\int_{V_\alpha}{d^3x\rho_\alpha{\partial\phi_\beta\over\partial x_i}}
+\int_{V_\alpha}{d^3xF_{\alpha\beta,i}}.
\label{eq:AS:v1}\end{aligned}$$ Note that we impose the boundary condition $p_\alpha=0$ for each fluid on the bounding surface of $V_\alpha$. The fluids are not restricted to occupy the same volume. Apart from simple doubling of the number of the inertial forces, which do not couple the two fluids, there are two forces that do couple them: gravity and friction. The net mutual gravitational force between the fluids vanishes only if they (i) occupy the same volume and (ii) have densities that are proportional to one another (i.e. $\rho_S\propto
\rho_N$). The mutual friction force is nonzero as long as the fluids move relative to one another. Although the mutual friction force is nonzero only in the [*overlap*]{} volume of the two fluids - a restriction which is necessary to derive conservation 