Modeling and Inversion of History Forces in Particle-Laden Flows

The dynamics of particles immersed in an incompressible flow are essential to a wide range of natural and industrial processes, such as combustion, aerosol transport, sedimentation, and pollutant dispersion. Accurate prediction of particle trajectories in such flows requires capturing nonlinear drag, which reflects viscous resistance, and the unsteady history force, which accounts for the memory effect and added-mass-like behavior associated with the particle's acceleration through the fluid. Incorporating the non-local history force transforms the governing equations into a system of delay differential equations (DDEs), whose direct numerical evaluation is computationally expensive due to the long-memory convolution with the classical Basset kernel. We address this challenge by applying the well-known linear chain trick to transform the delay differential equations (DDEs) into ordinary differential equations (ODEs), and approximate the history force as a superposition of memory states. Previous research indicates that the history force may involve a memory kernel decaying as $t^{-2}$ at large time, in contrast to the classical $t^{-\frac 1 2}$. Our recursive formulation enables the direct inference of this kernel from particle trajectory data. Preliminary results include numerical comparisons between our recursive-memory approach and the classical Basset kernel (evaluated using the trapezoidal rule), as well as synthetic tests of an inverse problem framework to recover the underlying history force kernel.