# On the Use and Misuse of the Oberbeck–Boussinesq Approximation

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## Abstract

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## 1. Introduction

## 2. A Minimalistic Survey of the Oberbeck–Boussinesq Model

#### 2.1. The Buoyant Flow Governing Equations

**u**is the velocity, P is the pressure, T is the temperature, t is time, $\mu $ is the dynamic viscosity,

**g**is the gravitational acceleration, and $\alpha $ is the thermal diffusivity. There are two densities in Equations (1)–(3). One is the reference density, ${\rho}_{0}$, i.e., the fluid density evaluated at the reference temperature, ${T}_{0}$. On the other hand, $\rho $ denotes the fluid density evaluated at the local temperature T through the linear equation of state,

#### 2.2. The Buoyant Seepage Flow in a Porous Medium

**U**denotes the seepage, or Darcy’s, velocity in the porous medium. The seepage velocity results from a representative-elementary-volume average of the pore-scale fluid velocity,

**u**. Furthermore, the local momentum balance equation may be modelled through Darcy’s law, namely,

#### 2.3. Hydrostatic Pressure, Buoyancy Force

## 3. The Rectangular Cavity with Side Heating

**u**. Then, we may wonder how we could evaluate the mass flow rate, $\dot{m}$, across S. There is a correct way and an incorrect way. The correct way is by applying the principle that the fluid density is to be considered constant and equal to ${\rho}_{0}$, so that Equation (12) yields

## 4. Mixed Convection Duct Flow

## 5. A Vertical Porous Slab Separating Two Fluid Reservoirs

**U**, across the porous material. By recalling that the fluid density is to be intended as uniform with the value ${\rho}_{0}$, Equation (23) implies that the mass flow rates across ${S}_{1}$ and ${S}_{2}$ are equal as required by mass conservation,

## 6. The Froude Number and the Rayleigh Number

**F**in order to obtain a dimensionless formulation of the local momentum balance equation. The dimensionless

**F**denoted with an asterisk is given by

**∇**P to be equal to $-{\rho}_{0}g{\hat{\mathbf{e}}}_{y}$ whenever the Oberbeck–Boussinesq approximation is used. More precisely, one must have

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Batchelor’s Profile

## References

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**Figure 1.**Sketch of the rectangular cavity with side heating discussed in Section 3.

**Figure 2.**Sketch of the mixed convection duct flow discussed in Section 4.

**Figure 3.**Sketch of the porous slab separating two fluid reservoirs discussed in Section 5.

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**MDPI and ACS Style**

Barletta, A.; Celli, M.; Rees, D.A.S.
On the Use and Misuse of the Oberbeck–Boussinesq Approximation. *Physics* **2023**, *5*, 298-309.
https://doi.org/10.3390/physics5010022

**AMA Style**

Barletta A, Celli M, Rees DAS.
On the Use and Misuse of the Oberbeck–Boussinesq Approximation. *Physics*. 2023; 5(1):298-309.
https://doi.org/10.3390/physics5010022

**Chicago/Turabian Style**

Barletta, Antonio, Michele Celli, and D. Andrew S. Rees.
2023. "On the Use and Misuse of the Oberbeck–Boussinesq Approximation" *Physics* 5, no. 1: 298-309.
https://doi.org/10.3390/physics5010022