The main focus of this thesis is the dynamics of finite-density inertial particles in an arbitrary flow. We derive a closed-form solution for the Lyapunov exponent, which measures the rate of change of volume of a cloud of particles. At an infinite density ratio, it has been shown in M. Esmaily and A. Mani, Physical Review Fluids, 5(8):084303, 2020 that much of the clustering phenomenon in 3D turbulence can be explained using a 1D canonical flow oscillating at a single frequency. We extend this analysis to study the dynamics of various particles, namely light, heavy and neutrally buoyant for a wide range of flows. We know that clustering is negligible at small and large Stokes number and substantial at Stokes number ~ O(1). We analytically explain this non-monotonic behavior for finite-density particles heavier than the fluid and study the effect of density ratio for such particles in turbulence. We also observe a lower bound of -1/2 on the Lyapunov exponent normalized by the particle relaxation time, but there is no such upper bound. We further observe that neutrally buoyant particles disperse in a strongly straining flow. Finally, we explain the effect of the Basset term on our analysis. This thesis also contributes to developing an efficient solver called complex-valued Stokes Solver (SCVS) for cardiovascular flows, which solves the unsteady Stokes equation in the spectral domain instead of solving it in the time domain. SCVS shows an order of magnitude improvement in accuracy and two orders of magnitude improvement in cost compared to a standard stabilized finite element solver called RBVMS. For a more direct one-to-one comparison, we also compare SCVS against an in-house time-based Stokes solver, which uses the same shape functions, where we see three orders of magnitude improvement in the cost and improved accuracy. This improvement in performance stems from the transformation of the governing differential equations into a system of boundary value problems, where only a few modes need to be solved instead of thousands of time steps.