Reynolds Transport Theorem

Consider a system ABCD bounded by control surface ‘cs’ at a time t.

Let ‘cv’ denotes its control volume.

The system occupies new position due to fluid motion at time t + ( ) as shown by dotted surface. Let G define the location of volume in the flow field. An extensive property N (dependent upon mass) is a function of intensive properties n (independent of mass) such that

N = ∫ n ρ dV = N ( G. t )

The rate of change of N system is given by

(dN / dt)  = (DN / Dt) =  Lim   [N (G₀ + δG, t + δt) - N (G₀ t] / δt
                                        Δt → 0

Where G0 and G0 + ( ) locate the volumes t and t + ( ) respectively.

Adding and subtracting N (G₀ , t + t)   in the term under the limit and rearranging them as following two terms.

Lim   [N (G₀ + δG, t + δt) - N (G₀ .t] / δt = (∂N / ∂t) _G0 = ∂ / ∂t  ∫ n ρ dV
Δt → 0

Which means the local rate of change of N in the control volume and

N [N (G₀ + δG, + δt) - N (G₀ , t + t) = N_m = N₁

Where  N₁ and N₂ refers the value of N for regions 1 and II respectively, the region II being common to both positions.

Nm  = ∫ n ρ ( dA u δt)  =  ∫   n ρ dt (UdA)
                                         CDA

N1  =   ∫  n ρ dt (UdA)
          ABCD

Subtracting I from III

Nm - N1  =  ∫    n ρ dt (UdA)
                   ABCD

Which reduces the second limit term to  ∫  n ( ρUdA )

We get,  DN / Dt = ∫  n (ρUdA) + ∂ / ∂t  ∫ n ρ dv
                               cs                              cs

This is the Reynolds transport equation. it shown that the rate of change N (extensive property) for a system equals the sum f the efflux of N across the control surface and the rate of change of N within the control volume.

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