Reynolds' transport theorem

Let F(r,t) represent the amount of some physical property of a continuous material medium per unit volume. The total amount of this property present in a finite region 𝒱 of the material is obtained through the volume integral.

∫ F (r,t) dV 𝒱

If this property is being transported by the action of the flow of the material with a velocity u(r,t), then Reynolds’ transport theorem states that the rate of change of the total amount of F within the material volume is equal to the volume integral of the instantaneous changes of F occuring within the volume, plus the surface integral of the rate at which F is being transported through the surface 𝒮 (bounding 𝒱) to and from the surrounding region.

d ∫ ∫ ∂F ∫ -- F (r,t) dV = ----dV + F u ⋅ n dS dt 𝒱 𝒱 ∂t 𝒮

Here, n is a unit vector indicating the normal direction of the surface (oriented to point out of the volume).

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