Torsional warping ($u_z=w(x,y)$)

In uniform torsion about the $z$-axis, the axial displacement $w(x,y)$ describes departure of a cross-section from a plane. It is determined up to an additive axial translation by $w_x=\Omega y+{\varphi }_y/\mu$, $w_y=-\Omega x-{\varphi }_x/\mu$, where $\varphi$ is the torsion stress function for a simply connected bar. For a circular section these derivatives vanish. For an ellipse they give $w=\Omega (b^2-a^2)xy/(a^2+b^2)+\text{constant}$, so a noncircular section generally warps. The stress-function Poisson equation is exactly the mixed-derivative compatibility condition for this axial displacement.