The Boussinesq approximation

The basis of this approximation is that there are flows in which the temperature varies little, and therefore the density varies little, yet in which the buoyancy drives the motion. Thus the variation in density is neglected everywhere except in the buoyancy term. Let $\rho_b$ denotes the density at the bottom where temperature is $T_b$. For small temperature difference between the top and bottom layer we can write

$$\rho = \rho_b [1-\alpha (T-T_b)]$$

where $\alpha=$ coefficient of volume expansion. For liquid $\alpha$ is in the range $10^{-3}$ to $10^{-4}$. For a temperature variation of moderate amount we have

$$\frac{\vert\delta\rho\vert}{\rho_b}=\frac{\vert\rho-\rho_b\vert}{\rho_b}=\alpha\vert T-T_b\vert \ll 1$$

but the buoyancy term $g(\rho-\rho_b)$ is the same order of magnitude as the inertia, acceleration or the viscous stress so is not negligible.

The differential in density in the continuity equation (1) are of the order $\alpha$ and hence neglected to give

$$\frac{\ensuremath{\partial}u_j}{\ensuremath{\partial}x_j}=0$$ (4)

as for an incompressible fluid. Then the stress tensor becomes

$$\tau_{ij}=-p \delta_{ij}+\mu \left(\frac{\ensuremath{\partial}u_i... ...\partial}x_j}+\frac{\ensuremath{\partial}u_j}{\ensuremath{\partial}x_i}\right)$$

Again the momentum equation becomes

$$\rho\left(\frac{\ensuremath{\partial}u_i}{\ensuremath{\partial}t}... ...{i3}-\frac{\ensuremath{\partial}p}{\ensuremath{\partial}x_i} + \mu \Delta u_i,$$

which after using the Boussinesq approximation becomes (after a little manipulation)

$$\frac{\ensuremath{\partial}u_i}{\ensuremath{\partial}t} + u_j \fr... ...\left(\frac{p}{\rho_b}+g z\right)-\alpha g (T_b-T)\delta_{i3} + \nu \Delta u_i,$$ (5)

where $\Delta={\ensuremath{\partial}^2}/{\ensuremath{\partial}x_j^2}$ is the Laplacian operator.

Now we consider the energy equation. The velocity is of the order $[\alpha\vert\delta T\vert g d]^{1/2}$ and thus the ratio of $\Phi$ term to the term due to conduction of heat is of the order of

$$\frac{\mu \alpha g d}{k}$$

and this is in the range from $10^{-7}$ to $10^{-8}$ for gases and liquids and the viscous dissipation is neglected. The term

$$-p\frac{\ensuremath{\partial}u_k}{\ensuremath{\partial}x_k} = \fr... ...\rho}{\ensuremath{\partial}T}\right)_p \frac{DT}{Dt} = -\alpha p \frac{DT}{Dt}$$

Using perfect gas relations $p=\rho R T, R=(c_p-c_v)$ and $\alpha=1/T$ we get

$$-p\frac{\ensuremath{\partial}u_k}{\ensuremath{\partial}x_k}=-\alpha \rho R T \frac{DT}{Dt} = -\rho (c_p-c_v)\frac{DT}{Dt}$$

Thus though ${\ensuremath{\partial}u_k}/{\ensuremath{\partial}x_k}=0$ holds in the equation of continuity, we should not use this relation in the heating due to compression term for gases. For liquids however, this heating is negligible. Thus the final form of the energy equation is

$$\frac{\ensuremath{\partial}T}{\ensuremath{\partial}t} + u_j \frac{\ensuremath{\partial}T}{\ensuremath{\partial}x_j} = \kappa \Delta T,$$ (6)

where $\kappa=k/\rho_b c_p$ for a perfect gas and $k/\rho_b c$ for liquids. Equations (4), (5) and (6) are called Boussinesq equations and describe the motion of a Boussinesq fluid.