  A 3-Sasakian structure on a 7-manifold may be used to define two distinct
Einstein metrics: the 3-Sasakian metric and the squashed Einstein metric. Both
metrics are induced by nearly parallel $G_2$-structures which may also be
expressed in terms of the 3-Sasakian structure. Just as Einstein metrics are
critical points for the Ricci flow up to rescaling, nearly parallel
$G_2$-structures provide natural critical points of the (rescaled) geometric
flows of $G_2$-structures known as the Laplacian flow and Laplacian coflow. We
study each of these flows in the 3-Sasakian setting and see that their
behaviour is markedly different, particularly regarding the stability of the
nearly parallel $G_2$-structures. We also compare the behaviour of the flows of
$G_2$-structures with the (rescaled) Ricci flow.
