= Full-film eccentric journal-bearing rotation
For inner radius $a$, mean gap $\Delta\ll a$, fixed eccentricity parameter $|\alpha|<1$, gap $h=\Delta(1+\alpha\sin\theta)$, and inner <angular velocity> $\Omega$, the constant circumferential flux is
$$
q=\frac{\Omega a\Delta}{2}\frac{1-\alpha^2}{1+\alpha^2/2}.
$$
It follows by integrating the <lubrication theory> relation $p_s=6\mu\Omega a/h^2-12\mu q/h^3$ over a full period. The leading holding <force> is horizontal:
$$
F_x=-\frac{6\pi\mu\Omega a^3}{\Delta^2}\frac{\alpha}{\sqrt{1-\alpha^2}(1+\alpha^2/2)},\qquad F_z=0.
$$
<Integration by parts> computes this <force> without a closed-form <pressure>. The fluid couples about the separate cylinder axes obey $C_i+C_o=\alpha\Delta F_x$; moments about a common axis balance exactly at this order.
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