A Riccati-type solution of 3D Euler equations for incompressible flow

TL;DR

This paper derives a Riccati-type analytical solution for non-stationary 3D incompressible Euler equations, emphasizing the invariance of Bernoulli's function and providing conditions for exact solutions in fluid flow modeling.

Contribution

It introduces a novel Riccati-type solution approach for non-stationary 3D Euler equations with Bernoulli invariance, expanding analytical solution methods in fluid mechanics.

Findings

Derived conditions for exact solutions of Euler equations

Established form restrictions based on Bernoulli-function invariance

Provided a method to compute pressure from velocity fields

Abstract

In fluid mechanics, a lot of authors have been reporting analytical solutions of Euler and Navier-Stokes equations. But there is an essential deficiency of non-stationary solutions indeed. In our presentation, we explore the case of non-stationary flows of the Euler equations for incompressible fluids, which should conserve the Bernoulli-function to be invariant for the aforementioned system. We use previously suggested ansatz for solving of the system of Navier-Stokes equations (which is proved to have the analytical way to present its solution in case of conserving the Bernoulli-function to be invariant for such the type of the flows). Conditions for the existence of exact solution of the aforementioned type for the Euler equations are obtained. The restrictions at choosing of the form of the 3D nonstationary solution for the given constant Bernoulli-function B are considered. We should especially note that pressure field should be calculated from the given constant Bernoulli-function B, if all the components of velocity field are obtained.