Fluid-particle dynamics for passive tracers advected by a thermally fluctuating viscoelastic medium

Many biological fluids, like mucus and cytoplasm, have prominent viscoelastic properties. As a consequence, immersed particles exhibit subdiffusive behavior, which is to say, the variance of the particle displacement grows sublinearly with time. In this work, we propose a viscoelastic generalization of the Landau-Lifschitz Navier-Stokes fluid model and investigate the properties of particles that are passively advected by such a medium. We exploit certain exact formulations that arise from the Gaussian nature of the fluid model and introduce analysis of memory in the fluid statistics, marking an important step toward capturing fluctuating hydrodynamics among subdiffusive particles. The proposed method is spectral, meshless and is based on the numerical evaluation of the covariance matrix associated with individual fluid modes. With this method, we probe a central hypothesis of passive microrheology, a field premised on the idea that the statistics of particle trajectories can reveal fundamental information about their surrounding fluid environment.