Analytic self-similar solutions of the Oberbeck-Boussinesq equations

TL;DR

This paper derives new two-dimensional analytic self-similar solutions for the Oberbeck-Boussinesq equations, revealing detailed pressure, temperature, and velocity fields with damped oscillations, advancing understanding of fluid dynamics and heat transfer.

Contribution

It introduces a novel two-dimensional self-similar Ansatz for the Oberbeck-Boussinesq equations, providing explicit analytic solutions expressed via error functions.

Findings

Pressure, temperature, and velocity fields are explicitly derived.

Temperature exhibits strongly damped oscillations.

Solutions are expressed analytically using error functions.

Abstract

In this article we will present pure two-dimensional analytic solutions for the coupled non-compressible Newtoniain Navier-Stokes --- with Boussinesq approximation --- and the heat conduction equation. The system was investigated from E.N. Lorenz half a century ago with Fourier series and pioneered the way to the paradigm of chaos. We present a novel analysis of the same system where the key idea is the two-dimensional generalization of the well-known self-similar Ansatz of Barenblatt which will be interpreted in a geometrical way. The results, the pressure, temperature and velocity fields are all analytic and can be expressed with the help of the error functions. The temperature field has a strongly damped oscillating behavior which is an interesting feature.