Math 2.0

Fluid mechanics Definition

Compressible flow

Builds on Fluid dynamicsIncompressible flowMach numberSpeed of soundEuler equations

Definition

Let a fluid flow in a region Ω⊆ℝ3\Omega \subseteq \R^3 be described by a density ρ(x,t)>0\rho(x, t) > 0, a velocity 𝐮(x,t)\mathbf{u}(x, t) and a pressure p(x,t)p(x, t). The flow is compressible when the density is not constant along particle paths, that is, DρDt=∂tρ+𝐮⋅∇ρ≠0\frac{D\rho}{Dt} = \partial_t \rho + \mathbf{u} \cdot \nabla\rho \neq 0 somewhere. By conservation of mass, ∂tρ+∇⋅(ρ𝐮)=0\partial_t \rho + \nabla \cdot (\rho\mathbf{u}) = 0, this is the same as ∇⋅𝐮≠0\nabla \cdot \mathbf{u} \neq 0 somewhere. In compressible flow theory the density is an unknown. An equation of state links it to the pressure and the temperature TT. An example is the ideal gas law p=ρRTp = \rho R T, with RR the specific gas constant.

Remarks

Example: in steady isentropic flow of air, with ratio of specific heats γ=1.4\gamma = 1.4, the density at Mach number MM is ρ=ρ0(1+0.2M2)−2.5\rho = \rho_0 (1 + 0.2 M^2)^{-2.5}, where ρ0\rho_0 is the density at rest. At M=0.3M = 0.3 this is about 0.956 ρ00.956\,\rho_0, a change of about 44 percent, so the flow is compressible, though only mildly. Non-example: incompressible flow, where ∇⋅𝐮=0\nabla \cdot \mathbf{u} = 0, for instance the flow of water in a pipe at ordinary pressures. Inviscid compressible flow is governed by the compressible Euler equations, which form a hyperbolic system of conservation laws. The size of the density changes is set by the Mach number M=∣𝐮∣/cM = \abs{\mathbf{u}} / c, with cc the speed of sound: for M→0M \to 0 the relative density changes are of order M2M^2.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Euler equations, concept, no evidence levelEuler equationsconceptFluid dynamics, concept, no evidence levelFluid dynamicsconceptIncompressible flow, concept, no evidence levelIncompressible flowconceptMach number, concept, no evidence levelMach numberconceptSpeed of sound, concept, no evidence levelSpeed of soundconceptCompressible flow, definition, no evidence levelCompressible flowdefinition
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  1. Euler equations concept
  2. Fluid dynamics concept
  3. Incompressible flow concept
  4. Mach number concept
  5. Speed of sound concept
  6. Compressible flow definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Compressible_flow
About this page
TagP9220P0MB4
URIm2://P9220P0MB4
Kinddefinition
WritingW0 stub
MSC76N15, 76N06
DefinesCompressible flow
UsesFluid dynamics, Incompressible flow, Mach number, Speed of sound, Euler equations
Changed3 November 2026

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