## Reynolds Transport Equation

Consider N as properties of a substance whose measure depends on the amount of the substance presented in a system. An arbitrary flow field of v = v(x, y, z, t)is observed from some reference xyz. We consider a system of fluid with finite mass at time t and t + At (figure 2.3). The streamlines correspond to those at time t. The control volume is considered with system at time t and is fixed.

The distribution of N per unit mass will be given as n, such that N = HI npdV with dV representing an element of volume with mass density of p. We are trying to establish a relation between the rate of change of N in system with variation of this property inside the control volume.

The system at time t and t + at can be divided into three regions. Region II is common to the system at both time t and t + At. Rate of change of N with respect to time for the system can be written as:

dt ) system Dt

UHiiinpdV + !UnnpdV)t+M -(///,npdV + ;;;nnpdV)t llm -:-

At^0 At

The above equation can be arranged as following:

dn ((fffii npdV) t+At (IUii npdV) \ , Dt a™ I At i + ( . )

In the first term on the right hand side of equation. 2.1, we can say that, as At goes to zero the region II becomes that of control volume ,cv,. Then:

The second term in equation 2.1 shows the amount of the property N that has crossed part of the control surface, cs, shown as ARB. Therefore, the second term is the average of efflux of N across ARB during time interval of At. As At goes to zero this becomes exact rate of efflux of N through the cs. Similarly, the third term approximates the amount of N that has passed into the cv during At through remaining of the cs (influx). Thus, the lase two integrals give the net rate of efflux of N from cv at time t as:

-^ t+At_ iim (Uh nP-)i = Net efflux rate of N from cs

In conclusion, the rate of change of N for system at time t are sum of first, rate of change of N inside cv having the shape of the system at time t (equation 2.2)and second, the net rate of efflux of N through the cs at time t (equation 2.3).

In equation 2.3, the term dV can be written as:

which is the volume of fluid that has crossed the cs in time dt. dA is the normal outward vector on control surface. Multiplying (equation 2.4) by p and dividing by dt then gives the instantaneous rate of mass flow of fluid pv.dA leaving the control volume through the indicated area dA. For the fluid entering the control volume the expression dV = v.dAdt is negative while for the fluid going out of the control volume the expression is positive. Therefore equation 2.3 can be written as follows:

Efflux rate through cs / / npv.dA

J J ARB

Influx rate through cs / / npv.d A

J J ALB

Net efflux rate through cs / / npv.dA+ npv.dA

### ARB ALB

In the limit as At ^ 0, the approximation becomes exact, so we can express the right side of the above equation as JJ n(pv.dA), where the integral is the closed integral over the entire control surface. Now the equation 2.1 can be written as:

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