Finite Control Volume – Conservation of Linear Momentum (continued)

There are differing versions of this integral form for conservation of linear momentum depending on the conditions of the flow and of the fluid.

If the flow is steady (no change with time), then ∂/∂t ∫ ρ V dV is zero.

cv

Steady flow exists when the flow is established and doesn’t change with time.

For steady flow then gives the following relation (in integral form)

∫ V ρ V . n dS = Σ F or ∫ V ρ V . n dS  +   ∫ V ρ V . n dS = Σ F

cs cs in cs out

1. Steady, Incompressible Flow ρ = constant (integral form)

∫ V ( V . n ) dS   +      ∫ V ( V . n ) dS = (1/ρ) Σ F

cs in cs out

Note: This integral form holds for a “nonuniform” velocity profile.

2. Steady, compressible, Uniform Flow (uniform velocity profile across cs)

ρ_out A_out V²_out - ρ_in A_in V²_in = Σ Fx

3. Steady, Incompressible, Uniform Flow (uniform velocity profile across cs)

A_out V²_out - A_in V²_in = (1/ρ) Σ Fx

Click here for examples with non-deforming, non-moving control volumes.

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