# The 2D annulus benchmark#

This section was contributed by C. Thieulot and E. G. Puckett.

This benchmark is based on a manufactured solution in which an analytical solution to the isoviscous incompressible Stokes equations is derived in an annulus geometry. The velocity and pressure fields are as follows:

\begin{split}\begin{aligned} v_r(r,\theta) &=& g(r) k \sin(k\theta), \\ v_\theta(r,\theta)&=& f(r) \cos(k \theta), \\ p(r,\theta) &=& k h(r) \sin(k \theta), \\ \rho (r,\theta) &=& \aleph(r) k \sin (k \theta),\end{aligned} with \begin{aligned} f(r)&=&Ar+B/r, \\ g(r) &=& \frac{A}{2}r + \frac{B}{r} \ln r + \frac{C}{r}, \\ h(r)&=& \frac{2g(r)-f(r)}{r}, \\ \aleph(r) &=& g'' - \frac{g'}{r} - \frac{g}{r^2} (k^2 - 1) + \frac{f}{r^2} + \frac{f'}{r}, \\ A &=& -C\frac{2(\ln R_1 - \ln R_2)} { R_2^2 \ln R_1 - R_1^2 \ln R_2}, \\ B &=& -C \frac{R_2^2-R_1^2}{R_2^2 \ln R_1 - R_1^2 \ln R_2}.\end{aligned}\end{split}

The parameters $$A$$ and $$B$$ are chosen so that $$v_r(R_1)=v_r(R_2)=0$$, i.e. the velocity is tangential to both inner and outer surfaces. The gravity vector is radial and of unit length.

The parameter $$k$$ controls the number of convection cells present in the domain, as shown in Fig. 3.

In the present case, we set $$R_1=1$$, $$R_2=2$$ and $$C=-1$$. Fig. 4 shows the velocity and pressure errors in the $$L_2$$-norm as a function of the mesh size $$h$$ (taken in this case as the radial extent of the elements). As expected we recover a third-order convergence rate for the velocity and a second-order convergence rate for the pressure.