Incompressible

Incompressible describes a material whose density remains unchanged under applied pressure. In practice, most liquids are treated as incompressible because their volumes vary only slightly with pressure over ordinary conditions. Incompressibility is an idealization used to simplify the equations of motion in fluid mechanics and materials science.

In hydrodynamics, incompressible flow means the density is constant and does not change with time or space. This leads to the continuity equation reducing to the condition that the velocity field is divergence-free: ∇·v = 0. The momentum equations then involve pressure as a Lagrange multiplier enforcing this constraint, while density is fixed to a reference value ρ0.

The concept is an approximation. A strictly incompressible material would have infinite bulk modulus, which is not physical; most substances are compressible to some extent. Liquids such as water and oil are “nearly incompressible” in many engineering contexts, while gases are highly compressible.

Quantifying incompressibility involves the bulk modulus B = -V dP/dV. A large B indicates small volume change under pressure. In gas dynamics, the speed of sound c = sqrt(∂P/∂ρ) is finite, and density variations cannot be ignored at typical flow speeds, making the incompressible assumption unsuitable. In solids, an approximate incompressibility corresponds to a Poisson's ratio near 0.5, reflecting limited volumetric change under loading.

Applications include hydraulic systems, pipe flows, and many oceanic and atmospheric flows where density variations are negligible. In numerical modelling, treating a flow as incompressible reduces the governing equations and eliminates the need to solve a dynamic equation for density.