When the low velocity is zero, the Navier–Stokes momentum equation for incompressible flow \(\boxed{\rho \frac{D\mathbf{u}}{Dt} = -\nabla p + \rho \mathbf{g} + \mu \nabla^2 \mathbf{u} \quad \text{(incompressible)}}\) reduces to a balance between hydrostatic pressure \( p_s \) and the steady body force acting on the hydrostatic density \( \rho_s \) \[ 0 = -\nabla p_s + \rho_s \mathbf{g} \] Subtracting this hydrostatic balance from the Navier–Stokes momentum equation for incompressible flow, define pressure difference: \( p' = p - p_s \) density difference: \( \rho' = \rho - \rho_s \) \[\Rightarrow \rho \frac{D \mathbf{u}}{Dt} = -\nabla p' + \rho' \mathbf{g} + \mu \nabla^2 \mathbf{u} \] If the fluid density is constant (\( \rho' = 0 \)), the gravitational body force term vanishes, \[ \rho \frac{D \mathbf{u}}{Dt} = -\nabla p' + \mu \nabla^2 \mathbf{u} \] The Boussinesq Approximation The Boussinesq set involves

• Incompressibility: \( \nabla \cdot \mathbf{u} = 0 \)
• Momentum equation: The original momentum equation \(\rho \frac{D\mathbf{u}}{Dt} = -\nabla p + \rho \mathbf{g} + \mu \nabla^2 \mathbf{u} \quad \text{(incompressible)}\) Subtract hydrostatic pressure balance, and introduce the perturbations \( p' = p - p_s \), \( \rho' = \rho - \rho_s \) Then the perturbation momentum equation becomes \[ \rho \frac{D\mathbf{u}}{Dt} = -\nabla p' + \rho' \mathbf{g} + \mu \nabla^2 \mathbf{u} \]
• Small variations: \( \alpha \delta T \ll 1 \), \( \rho' \ll \rho_0 \)