Consider an isotropic stochastic flow in Rd(i.e. a simultaneous random, correlated motion of all points in space), whered=l,2 or 3, such that the joint law of the motion of two particles allows the particles to meet and coalesce in finite time. The coalescent setJtis a random subset of Rdconsisting of the initial positions of particles which have coalesced by timetwith the particle which started at 0. We show that the expected volume ofJtgrows at a rate proportional to whend=1, and at rates close to proportional tot/logt(resp.t) whend= 2 (resp.d=3). We give an example of a coalescing stochastic flow whend= 3. These results are analogous to growth rates of expected population size of a surviving type in the "invasion process" described by Clifford and Sudbury