The first is that the integral on the left hand side is now being

The first is that the integral on the left hand side

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formula. The first is that the integral on the left-hand side is now being differentiated with respect to the substantial derivative , and the second is that the velocity field in the second integral on the right is now U , which must be the velocity of an arbitrary fluid parcel. These restrictions will play a crucial role in later derivations. Equation (3.10) follows immediately from the general transport theorem using these two re- strictions and the chain rule for differentiation. We demonstrate this for the scalar case, and leave the vector case to ambitious readers. We begin by defining F ( x ( t ) , y ( t ) , z ( t ) , t ) integraldisplay R ( t ) Φ dV (3.11) with R ( t ) taken to be a time-dependent fluid element (and hence the time dependence of the coordinate arguments of F ). Now the general transport theorem applied to the scalar Φ is d dt integraldisplay R ( t ) Φ dV = integraldisplay R ( t ) Φ ∂t dV + integraldisplay S ( t ) Φ W · n dA ,
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58 CHAPTER 3. THE EQUATIONS OF FLUID MOTION but since R ( t ) is a fluid element we can replace W in the above equation with U to obtain d dt integraldisplay R ( t ) Φ dV = integraldisplay R ( t ) Φ ∂t dV + integraldisplay S ( t ) Φ U · n dA . (3.12) We next use the definition of the function F to write d dt integraldisplay R ( t ) Φ dV = dF dt , and now we apply chain-rule differentiation to the right-hand side (just as we did earlier in “deriv- ing” the substantial derivative). We obtain dF dt = ∂F ∂t dt dt + ∂F ∂x dx dt + ∂F ∂y dy dt + ∂F ∂z dz dt = ∂F ∂t + u ∂F ∂x + v ∂F ∂y + w ∂F ∂z = DF Dt D Dt integraldisplay R ( t ) Φ dV , (3.13) as required by Eq. (3.10). Hence, we can replace the ordinary derivative on the left-hand side of Eq. (3.12) with the substantial derivative (but only because R ( t ) is a fluid element traveling with the fluid velocity ( u, v, w ) T ), completing the derivation of Eq. (3.10) for the scalar case. 3.3 Conservation of Mass—the continuity equation In this section we will derive the partial differential equation representing conservation of mass in a fluid flow, the so-called continuity equation . We will then, in a sense, “work backwards” to recover an integral form, often called the “control-volume” form, that can be applied to engineering calculations in an approximate, but very useful, way. We will then consider some specific examples of employing this equation. 3.3.1 Derivation of the continuity equation We begin this section with the general statement of conservation of mass, and arrive at the differ- ential form of the continuity equation via a straightforward analysis involving application of the general transport theorem and Gauss’s theorem, both of which have been discussed in the previous section. Conservation of Mass We start by considering a fixed mass m of fluid contained in an arbitrary region R ( t ). As we have already hinted, we can identify this region with a fluid element, but in some cases we will choose to associate this with a macroscopic domain. In either case, the boundary S ( t ) of R ( t ) can in general move with time. Any such region is often termed a system , especially in thermodynamics contexts, and it might be either open or closed . From our point of view it is only important that it
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