Axisymmetric Navier–Stokes component equations

Core idea

For an axisymmetric velocity, pressure, and force, define the scalar operators on $r>0$

\mathcal D=\partial_t+u_r\partial_r+u_z\partial_z, \qquad \Delta_0=\partial_r^2+\frac1r\partial_r+\partial_z^2.

Then the Navier–Stokes equations are

\begin{aligned} \mathcal D u_r-\frac{u_\theta^2}{r}+\partial_rp &=\nu(\Delta_0-r^{-2})u_r+f_r,\\ \mathcal D u_\theta+\frac{u_ru_\theta}{r} &=\nu(\Delta_0-r^{-2})u_\theta+f_\theta,\\ \mathcal D u_z+\partial_zp&=\nu\Delta_0u_z+f_z,\\ \partial_r u_r+\frac{u_r}{r}+\partial_z u_z&=0. \end{aligned}

Geometric terms

Differentiating the moving cylindrical basis produces $-u_\theta^2/r$ and $u_ru_\theta/r$ in material acceleration. The vector Laplacian produces the terms $-u_r/r^2$ and $-u_\theta/r^2$ ; applying only the scalar Laplacian to cylindrical components would omit them.

Axis and inviscid versions

These formulas hold directly for $r>0$ . A smooth solution containing the axis must also obey Cartesian axis regularity. The axisymmetric Euler equations follow by setting $\nu=0$ .