  We study the behaviour of the Laplacian flow evolving closed G$_2$-structures
on warped products of the form $M^6\times{\mathbb S}^1$, where the base $M^6$
is a compact 6-manifold endowed with an SU(3)-structure. In the general case,
we reinterpret the flow as a set of evolution equations on $M^6$ for the
differential forms defining the SU(3)-structure and the warping function. When
the latter is constant, we find sufficient conditions for the existence of
solutions of the corresponding coupled flow. This provides a method to
construct immortal solutions of the Laplacian flow on the product manifolds
$M^6\times{\mathbb S}^1$. The application of our results to explicit cases
allows us to obtain new examples of expanding Laplacian solitons.
