  We consider the Laplacian coflow of a $\mathrm{G}_2$-structure on warped
products of the form $M^7= M^6 \times_f S^1$ with $M^6$ a compact 6-manifold
endowed with an $\mathrm{SU}(3)$-structure. We give an explicit
reinterpretation of this flow as a set of evolution equations of the
differential forms defining the $\mathrm{SU}(3)$-structure on $M^6$ and the
warping function $f$. Necessary and sufficient conditions for the existence of
solution for this flow are given. Finally we describe new long time solutions
for this flow where the $\mathrm{SU}(3)$-structure on $M^6$ is nearly K\"ahler,
symplectic half-flat or balanced.
