Stretching and breaking of particles in compressible random flows

A key feature of turbulent suspensions that involve floating particles on the surface or inertial particles in the bulk is the compressibility of the effective particle-phase velocity field. Little, however, is known about the effects of small-scale flow compressibility on the stretching and breaking of particles. Here, we gain insight into the nature of these effects by studying the deformation of tiny particles in model fluctuating flows. We consider a generic particle with extensional dynamics that is governed by a vector model, which accounts for elasticity, internal viscosity and non-affine deformation. Applying the dynamical systems approach of Balkovsky et al. ( Phys. Rev. Lett ., vol. 84, 2000, p. 4765), we first obtain general results for the stationary statistics of particle extension in compressible chaotic flows. We then specialise to a time-decorrelated Gaussian random flow and derive an exact solution for the Batchelor regime of the compressible Kraichnan model. We also perform numerical simulations for a time-correlated renewing flow. While straining is suppressed on the average in compressible flows, our results show that large deviations of the strain rate strongly stretch particles and give rise to a power-law distribution of extensions. Extreme straining events are particularly important for stiff particles and, in the examples considered here, give rise to a counterintuitive effect: stiff particles stretch more and break faster in flows of increasing compressibility. Highly elastic particles, whose deformation is dictated by the mean straining, stretch less and break slower. As a consequence, the shrink–stretch transition that occurs on varying a particle’s relaxation time becomes increasingly shallow with compressibility, ultimately vanishing in the strongly compressible regime (wherein line elements contract on the average). Though based on specific random flows, our work shows how compressibility can affect the extensional dynamics of particles by altering the fluctuations of the strain rate, including its large deviations.