Incompressibility

In physics, incompressibility is a property of a material or fluid in which its density remains constant when subjected to pressure changes. For a fluid, this idealization leads to a divergence-free velocity field, div u = 0, and a simplified form of the continuity equation. Incompressibility is an approximation rather than an exact law, as all real materials exhibit some volume change under pressure, though liquids often change very little.

In fluid dynamics, the incompressible Navier-Stokes equations describe the motion of an incompressible fluid: ρ(∂u/∂t + u dot grad u) = -grad p + μ grad^2 u + f with div u = 0. For incompressible media, density ρ is constant and the pressure p acts as a Lagrange multiplier enforcing the divergence-free constraint. The relationship between pressure, density, and temperature becomes decoupled through an equation of state; ρ is treated as constant.

In solids and porous media, incompressibility refers to negligible volumetric strain under loading. A material is considered incompressible if the relative volume change ΔV/V is very small; mathematically one may treat det(F) ≈ 1, where F is the deformation gradient, or use a large bulk modulus K with small compressibility. In practice, many polymers and biological tissues are modeled as nearly incompressible.

Limitations and applications: the incompressible approximation breaks down at high speeds, strong shocks, or when density variations drive dynamics such as buoyancy or sound waves. The speed of sound c relates to the bulk modulus by c = sqrt(K/ρ) for small density changes; in computational fluid dynamics, incompressibility simplifies simulations but may require artificial compressibility methods or projection schemes. The Boussinesq approximation treats varying density only in buoyancy terms while using an incompressible velocity field.