Group classification of self-similar, linear velocity fluid flow.

Self-similar motions describe flow in which the spatial profiles of velocity, pressure and density vary with time while remaining geometrically similar to themselves. Viewed in the appropriately scaled coordinate frame, the spatial distributions of these variables are invariant with respect to time. Under the ansatz of a linear velocity flow, a flow in which the velocity is linearly proportional to the spatial parameter multiplied by R’(t)/R(t), the Euler equations admit self-similar solutions for certain R. Such solutions appear across the literature. For example, density profiles associated with a linear velocity are used to model uniform heating compression of ICF targets. This work unifies these solutions under a common framework and classifies them according to their group properties. To achieve this, we apply the symmetry analysis methods of Lie within the context of equivalence transformations proposed by Ovsiannikov. We then construct and solve a system of constraining equations to find the admissible functions R leading to solutions that are invariant under the determined group. As well as group characterization, the applied analysis easily identifies the vector fields pertaining to each solution, the characteristics of which correspond to the propagation of the flow.