A model for the temporal evolution of two-component systems on an infinite graph ${G=(V,E)}$ is constructed [cf. O. Häggström and R.~A. Pemantle, J. Appl. Probab. {\bf 35} (1998), no.~3, 683--692; MR1659548]. In physical terms it can be seen as an elementary example of classical, non-reversible fluid flow. Suppose that the first component consists of infinitely many light particles, which, at time $t=0$, start out reaching the vacant sites of $G$ from a single vertex-reservoir $r_0\in V$, and assume, in addition, that the dynamics are defined by a rate-1 first-passage percolation process [cf. G.~R. Grimmett, {\it Percolation}, second edition, Grundlehren Math. Wiss., 321, Springer, Berlin, 1999; MR1707339]. Let the second subsystem be an assembly of infinitely many heavy particles, and let at ${t=0}$ a countable collection of reservoirs of heavy particles be distributed on ${V\setminus\{r_0\}}$ according to the product Bernoulli measure with parameter $p\in(0,1)$. If a (light or a heavy) particle hits a reservoir of the heavy fluid, then this reservoir starts wetting the empty vertices of $G$, where the time-evolution is governed through first-passage percolation with rate $q\in\Bbb{R}_+$ [cf. V. Sidoravicius and A.~O. Stauffer, Invent. Math. {\bf 218} (2019), no.~2, 491--571; MR4011705]. \par The present paper establishes the limiting behavior of this model, provided that $G$ is hyperbolic and non-amenable. In particular, the authors prove that if $G$ is a bounded degree non-amenable, hyperbolic, vertex-transitive graph, then for any $q>0$, there exists a $p_q(G)>0$ such that, for all $p\in(0,p_q)$, the two processes coexist with strictly positive probability as $t\to\infty$. \par Key to their approach is the use of a multi-scale analysis, based on a theorem of I.~Benjamini and O. Schramm [Geom. Funct. Anal. {\bf 7} (1997), no.~3, 403--419; MR1466332].