Lie symmetries and optimal classifications with certain modal approaches for the three-dimensional gas-dynamical equations

This paper is devoted to analyzing the solution framework of the gas-dynamic equations for a three-dimensional unbounded homentropic sheared flow using the Lie group approach. An extensive symmetry analysis of the system of governing PDEs is performed to decrease the number of independent variables. The classification of inequivalent subalgebras into an optimal set called the optimal set of subalgebras, is essential. We have constructed the one-dimensional, two-dimensional, and three-dimensional optimal set of subalgebras for the model PDEs. The three-dimensional optimal set of subalgebras is very useful as it directly transforms the system of governing PDEs into a system of ODEs. Consequently, we obtain closed-form exact solutions of the governing model. Alternatively, the two-dimensional optimal subalgebras yield some solution ansatz, which describes various physical modes such as Kelvin mode and certain other modes and their typical combinations. The three-dimensional normal mode approach is justified using a combined ansatz in the limiting case. Moreover, we acquire the conserved quantities corresponding to the governing model by performing the conservation laws multiplier technique.