Derivation of Reynolds Transport Theorem

To formulate the relation between the equations applied to a control mass system and those applied to a control volume, a general flow situation is considered in Fig. 10.3 where the velocity of a fluid is given relative to coordinate axes ox, oy, oz. At any time t, a control mass system consisting of a certain mass of fluid is considered to have the dotted-line boundaries as indicated. A control volume (stationary relative to the coordinate axes) is considered that exactly coincides with the control mass system at time t (Fig. 10.3a). At time t+δt, the control mass system has moved somewhat, since each particle constituting the control mass system moves with the velocity associated with its location.

Fig 10.3 Relationship between Control Mass system and control volume concepts in the analysis of a flow field

Consider, N to be the total amount of some property (mass, momentum, energy) within the control mass system at time t, and let η be the amount of this property per unit mass throughout the fluid. The time rate of increase in N for the control mass system is now formulated in terms of the change in N for the control volume. Let the volume of the control mass system and that of the control volume be ₁at time t with both of them coinciding with each other (Fig. 10.3a). At time t + δt, the volume of the control mass system changes and comprises volumes _III and _IV(Fig. 10.3b). Volumes _II and _IV are the intercepted regions between the control mass system and control volume at time t+δt. The increase in property N of the control mass system in time δt is given by

where,d represents an element of volume. After adding and subtracting

                                (10.9)

The left hand side of Eq.(10.9) is the average time rate of increase in N within the control mass system during the time δt.
In the limit as δt approaches zero, it becomes dN/dt (the rate of change of N within the control mass system at time t ).

In the first term of the right hand side of the above equation the first two integrals are the amount of N in the control volume at time t+δt, while the third integral is the amount N in the control volume at time t. In the limit, as δt approaches zero, this term represents the time rate of increase of the property N within the control volume and can be written as

The next term, which is the time rate of flow of N out of the control volume may be written, in the limit as

In which is the velocity vector and is an elemental area vector on the control surface. The sign of vector  is positive if its direction is outward normal (Fig. 10.3c). Similarly, the last term of the Eq.(10.9) is the rate of flow of N into the control volume is, in the limit δt → 0

The minus sign is needed as  is negative for inflow. The last two terms of Eq.(10.9) may be combined into a single one is an integral over the entire surface of the control volume and is written as . This term indicates the net rate of outflow N from the control lume. Hence, Eq.(10.9) can be written as

(10.10)

The Eq.(10.10) is known as Reynolds Transport Theorem

Important Note: In the derivation of Reynolds transport theorem (Eq. 10.10), the velocity field was described relative to a reference frame xyz (Fig. 10.3) in which the control volume was kept fixed, and no restriction was placed on the motion of the reference frame xyz. This makes it clear that the fluid velocity in Eq.(10.10) is measured relative to the control volume. To emphasize this point, the Eq. (10.10) can be written as

(10.11)

where the fluid velocity , is defined relative to the control volume as

(10.12)

and are now the velocities of fluid and the control volume respectively as observed in a fixed frame of reference. The velocity of the control volume may be constant or any arbitrary function of time.

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