ABSTRACT
The classic derivation of the incompressible Euler equations is based on conservation of mass and momentum. Consider an arbitrary but fixed region of space Ω, in the fluid. The mass of the fluid in Ω is https://www.w3.org/1998/Math/MathML"> M = ∫ ∫ ∫ Ω ρ , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315266008/9ef028a4-40db-4499-bfd1-64ec0ed7470c/content/ueqn17_1_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and the total momentum of the fluid in Ω is https://www.w3.org/1998/Math/MathML"> P → = ∫ ∫ ∫ Ω ρ u → . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315266008/9ef028a4-40db-4499-bfd1-64ec0ed7470c/content/ueqn17_2_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> The rate of change of M, as fluid flows in or out of Ω, is given by the integral around the boundary of the speed at which mass is entering or exiting, since mass cannot be created or destroyed inside Ω: https://www.w3.org/1998/Math/MathML"> ∂ M ∂ t = − ∫ ∫ ∂ Ω ρ u → ⋅ n ˆ . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315266008/9ef028a4-40db-4499-bfd1-64ec0ed7470c/content/ueqn17_3_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> Here https://www.w3.org/1998/Math/MathML"> n ˆ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315266008/9ef028a4-40db-4499-bfd1-64ec0ed7470c/content/ineqn17_5_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> is the outward-pointing normal. We can transform this into a volume integral with the divergence theorem: https://www.w3.org/1998/Math/MathML"> ∂ M ∂ t = − ∫ ∫ ∫ Ω ∇ ⋅ ( ρ u → ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315266008/9ef028a4-40db-4499-bfd1-64ec0ed7470c/content/ueqn17_4_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> Expanding M and differentiating with respect to time (recalling that Ω is fixed) gives https://www.w3.org/1998/Math/MathML"> ∫ ∫ ∫ Ω ∂ ρ ∂ t = − ∫ ∫ ∫ Ω ∇ ⋅ ( ρ u → ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315266008/9ef028a4-40db-4499-bfd1-64ec0ed7470c/content/ueqn17_5_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> Since this is true for any region Ω, the integrands must match: https://www.w3.org/1998/Math/MathML"> ∂ ρ ∂ t + ∇ ⋅ ( ρ u → ) = 0. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315266008/9ef028a4-40db-4499-bfd1-64ec0ed7470c/content/ueqn17_6_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> This is called the continuity equation. For an incompressible fluid the material derivative of density Dρ/Dt is zero, i.e., https://www.w3.org/1998/Math/MathML"> ∂ ρ ∂ t + u → ⋅ ∇ ρ = 0. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315266008/9ef028a4-40db-4499-bfd1-64ec0ed7470c/content/ueqn17_7_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> Subtracting this from the continuity equation gives https://www.w3.org/1998/Math/MathML"> ρ ∇ ⋅ u → = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315266008/9ef028a4-40db-4499-bfd1-64ec0ed7470c/content/ineqn17_1_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , or more simply https://www.w3.org/1998/Math/MathML"> ∇ ⋅ u → = 0 , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315266008/9ef028a4-40db-4499-bfd1-64ec0ed7470c/content/ueqn17_8_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> which is termed the incompressibility condition. Note that this is independent of density, even for problems where fluids of different densities mix together.
