Hydrodynamics of an ideal incompressible fluid with a linear velocity field

In this study, a three-dimensional gas-dynamic model of an ideal incompressible fluid is proposed, where the solution is sought in the form of a linear velocity field with inhomogeneous deformation. The problem is formulated in both Eulerian and Lagrangian variables. Exact solutions are obtained for a special linearity matrix, generalizing previously known solutions. The equations of world lines for these solutions are derived, the trajectories of fluid particle motion are constructed, and the evolution of the initial spherical particle volume is investigated. The equations of constant pressure surfaces are presented and their time dynamics is analyzed. Special attention is paid to the analysis of particle motion in an ideal incompressible fluid and to obtaining new, more general solutions.