# Exact Solutions of the Oberbeck–Boussinesq Equations for the Description of Shear Thermal Diffusion of Newtonian Fluid Flows

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## Abstract

**:**

## 1. Introduction

## 2. Problem Statement

**k**is the unit vector of the Oz axis (applicate); $\nu $ is kinematic (molecular) viscosity; $\mathsf{\chi}$ is the thermal diffusivity, $d$ is the diffusion coefficient; $\mathrm{\alpha}$ is the thermal diffusion coefficient (the Soret parameter); $\mathrm{\delta}$ is the Dufour parameter; $\nabla =\left(\frac{\partial}{\partial x},\frac{\partial}{\partial y},\frac{\partial}{\partial z}\right)$ is the Hamilton operator; $\left(\mathbf{V}\cdot \nabla \right)={V}_{x}\frac{\partial}{\partial x}+{V}_{y}\frac{\partial}{\partial y}+{V}_{z}\frac{\partial}{\partial z}$ is the convective derivative; and $\Delta =\frac{{\partial}^{2}}{\partial {x}^{2}}+\frac{{\partial}^{2}}{\partial {y}^{2}}+\frac{{\partial}^{2}}{\partial {z}^{2}}$ is the Laplace operator.

## 3. Construction of the Exact Solution

**Theorem 1.**

**Proof.**

## 4. Boundary Value Problem

_{x}. The corresponding term u

_{2}y, according to solution (30), is a linear dependence on the vertical coordinate Z, which leads to a “straightening” of the velocity profile U with distance from the control section y = 0 (Figure 5, Figure 6, Figure 7, Figure 8, Figure 9 and Figure 10).

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Ershkov, S.; Burmasheva, N.; Leshchenko, D.D.; Prosviryakov, E.Y.
Exact Solutions of the Oberbeck–Boussinesq Equations for the Description of Shear Thermal Diffusion of Newtonian Fluid Flows. *Symmetry* **2023**, *15*, 1730.
https://doi.org/10.3390/sym15091730

**AMA Style**

Ershkov S, Burmasheva N, Leshchenko DD, Prosviryakov EY.
Exact Solutions of the Oberbeck–Boussinesq Equations for the Description of Shear Thermal Diffusion of Newtonian Fluid Flows. *Symmetry*. 2023; 15(9):1730.
https://doi.org/10.3390/sym15091730

**Chicago/Turabian Style**

Ershkov, Sergey, Natalya Burmasheva, Dmytro D. Leshchenko, and Evgeniy Yu. Prosviryakov.
2023. "Exact Solutions of the Oberbeck–Boussinesq Equations for the Description of Shear Thermal Diffusion of Newtonian Fluid Flows" *Symmetry* 15, no. 9: 1730.
https://doi.org/10.3390/sym15091730