Homologous spherical flow ($\mathbf{u}=q(t)\mathbf{r}$)

A homologous spherical flow has velocity proportional to radius. Its velocity gradient is $q(t)I$ and its velocity divergence is $3q(t)$.

A self-similar ansatz expresses evolving fields through a dimensionless similarity coordinate such as $x=r/R(t)$, reducing a partial differential equation to ordinary differential equations in $x$.