## Reynolds Transport Theorem

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# Reynolds Transport Theorem

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 (G

_{0}+ δG, t + δt) - N (G

_{0}t] / δt

Δt → 0

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

Adding and subtracting N (G

_{0}, t + t) in the term under the limit and rearranging them as following two terms.

Lim [N (G

_{0}+ δG, t + δt) - N (G

_{0}.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 (G

_{0}+ δG, + δt) - N (G

_{0}, t + t) = N

_{m}= N

_{1}

Where N

_{1}and N

_{2}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

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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