We discuss the full Lagrangian approach for calculating the velocity and density fields in a medium without self-stresses, which is usually used to describe the dispersed phase in dilute disperse systems. The method consists in using the Lagrangian form of the momentum and continuity equations for the dispersed phase and deriving additional equations for the components of the Jacobian of transformation from the Eulerian to Lagrangian variables. The method allows to calculate the dispersed-phase density along individual particle trajectories from the solution of systems of ordinary differential equations. This makes it possible to study the behavior of dispersed particles in an essentially multidimensional and unsteady flows with discontinuities, multiple intersections of particle trajectories, and local particle accumulation zones. Examples are given of applying the method to problems in two-phase gas dynamics, in which «folds» in the dispersed-phase medium, caustics, localized regions of accumulation of the particles, and integrable singularities in the particle density arise. We also discuss the possibility of applying a similar Lagrangian approach for modeling the evolution of other mechanical objects (deformable material volumes and surfaces in moving media, collisionless particle systems described by kinetic equations, differential characteristics of passive-scalar fields, density gradients in stratified flows, etc.).
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