Diffusion from cylinder without flow

In this example, we will investigate a system consisting of a cylindrical cavity of radius a filled with a stationary liquid. The control equation is $$\frac{\partial}{\partial t} C(r, t) = D \frac{1}{r} \frac{\partial}{\partial r} r \frac{\partial}{\partial r} C(r, t)$$

with the initial and boundary condition $ C(r < a, t = 0) = C_0 $, $ C(r = a, t) = C_c $. The analytical solution is $$\frac{C(r, t)-C_c}{C_0-C_c} = \sum_{n=1}^{\inf} \frac{2}{\mu_n J_1(\mu_n)} exp(-\mu_n^2 \frac{Dt}{a^2}) J_0(\mu_n \frac{r}{a})$$

$ J_0(x) $ and $ J_1(x) $ are 0th and 1st order Bessel functions. $ \mu_n $ is the n-th root of $ J_0(x) $. The first 5 roots are 2.2048, 5.5201, 8.6537, 11.7915, 14.9309 respectively. By taking initial value $ C_0 = 0 $, the analytical solution becomes $$C(r, t) = C_c (1 - \sum_{n=1}^{\inf} \frac{2}{\mu_n J_1(\mu_n)} exp(-\mu_n^2 \frac{Dt}{a^2}) J_0(\mu_n \frac{r}{a}))$$

In our simulation, the cylinder diameter $a=40$ is used as a default value. The D2Q9 layout is used. The pressure anti-bounce back boundary condition is used to compute and compare with the analytical solutions.

Simply executing run.sh to get the above figure. Please try to use different initial condition by setting different values of $ tIni $ in musubi.lua. Test how the initial condition impacts the result.