# Self-similar analysis of a viscous heated Oberbeck-Boussinesq flow system

## TL;DR

This paper extends the Oberbeck-Boussinesq system by including a viscous heat source, deriving self-similar solutions that could impact microfluidics, nanofluidics, and climate modeling.

## Contribution

It generalizes the OB system with a viscous heat source and derives self-similar solutions, advancing understanding of coupled heat and fluid dynamics.

## Key findings

- Derived analytic self-similar solutions for the extended system.
- Provided explanations for convection patterns using self-similarity.
- Potential applications in microfluidics, nanofluidics, and climate studies.

## Abstract

The simplest model to couple the heat conduction and Navier-Stokes equations together is the Oberbeck-Boussinesq(OB)system which were investigated by E.N. Lorenz and opened the paradigm of chaos. In our former studies - Chaos, Solitons and Fractals 78, 249 (2015), ibid, 103, 336 (2017) - we derived analytic solutions for the velocity, pressure and temperature fields. Additionally, we gave a possible explanation of the Rayleigh-B\`enard convection cells with the help of the self-similar Ansatz. Now we generalize the OB hydrodynamical system, including a viscous source term in the heat conduction equation. Our results may attract the interest of various fields like micro or nanofluidics or climate studies.