# Fluid dynamics solutions obtained from the Riemann invariant approach

## TL;DR

This paper develops a method using Riemann invariants and the generalized method of characteristics to find explicit solutions for fluid dynamics equations in (3+1) dimensions, accounting for gravitational and Coriolis forces.

## Contribution

It introduces a systematic approach to derive rank-2 solutions of hydrodynamics equations using Riemann invariants and the generalized method of characteristics.

## Key findings

- Derived conditions for existence of Riemann invariant solutions.
- Generated classes of wave solutions in fluid dynamics.
- Provided a detailed theory of simple wave solutions.

## Abstract

The generalized method of characteristics is used to obtain rank-2 solutions of the classical equations of hydrodynamics in (3+1) dimensions describing the motion of a fluid medium in the presence of gravitational and Coriolis forces. We determine the necessary and sufficient conditions which guarantee the existence of solutions expressed in terms of Riemann invariants for an inhomogeneous quasilinear system of partial differential equations. The paper contains a detailed exposition of the theory of simple wave solutions and a presentation of the main tool used to study the Cauchy problem. A systematic use is made of the generalized method of characteristics in order to generate several classes of wave solutions written in terms of Riemann invariants.