Incompressibleflow

Incompressible flow refers to fluid motion in which the fluid density is constant within the flow field and over time. This is a good approximation for liquids and for gases moving at low speeds where density variations are negligible relative to the flow. Under incompressibility, changes in volume of fluid elements are ignored.

The continuity equation for a fluid with constant density ρ reduces to ∇·v = 0, where v is the velocity field. This divergence-free condition means that the volume of any material element is preserved as it moves. The momentum balance for an incompressible, Newtonian fluid is given by ρ Dv/Dt = -∇p + μ ∇^2 v + f, with the constraint ∇·v = 0. Here p is pressure, μ is dynamic viscosity, and f represents body forces. Incompressibility also implies that pressure acts, in part, as a Lagrange multiplier to enforce the divergence-free velocity field.

Special cases include potential flow, where the flow is irrotational (∇×v = 0) and can be described by a velocity potential φ with v = ∇φ; φ then satisfies Laplace’s equation ∇^2 φ = 0. In many practical problems, especially in computational fluid dynamics, the incompressible Navier–Stokes equations are solved using projection methods that first advance the velocity and then solve a pressure Poisson equation to enforce ∇·v = 0.

Applications of incompressible flow theory cover many engineering and natural systems, such as water transport in pipes, cardiovascular blood flow, and low-speed aerodynamics. The incompressible assumption has limits; it becomes inaccurate when density variations are significant, such as in high-speed gas flows (high Mach numbers) or strong compressible effects.