= Steady circular Couette profile
{title2=$\Omega=A+B/r^2$}
For a homogeneous <Newtonian fluid> with constant <kinematic viscosity>, a steady axisymmetric purely azimuthal flow obeys $v_\phi''+v_\phi'/r-v_\phi/r^2=0$. Writing $v_\phi=r\Omega$ gives $(r^3\Omega')'=0$, so $\Omega=A+B/r^2$. A domain containing the rotation axis requires $B=0$ for regularity; rotating walls in an annulus can sustain $B\ne0$. An arbitrary time-dependent profile instead evolves by $\partial_t\Omega=\nu(\Omega''+3\Omega'/r)$ within this ansatz.
Back to article page
