From our point of view it is only important that it 33 CONSERVATION OF MASSTHE

From our point of view it is only important that it

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. From our point of view it is only important that it
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3.3. CONSERVATION OF MASS—THE CONTINUITY EQUATION 59 have fixed mass ; it does not matter whether it is the same mass at all times—only that the amount is the same. It is convenient for our purposes to relate the mass of the system to the density of the fluid comprising it via m = integraldisplay R ( t ) ρ dV . (3.14) We emphasize that R ( t ) and ρ may both change with time, but they must do so in a way that leaves m unchanged if we are to have conservation of mass. An example of this might be a balloon filled with hot air surrounded by cooler air. As heat is transferred from the balloon to its surroundings, the temperature of the air inside the balloon will decrease, and the density will increase (equation of state for a perfect gas). At the same time the size of the balloon will shrink, corresponding to a change in R ( t ). But the mass of air inside the balloon remains constant—at least if there are no leaks. We can express this mathematically as dm dt = d dt integraldisplay R ( t ) ρ dV = 0 . (3.15) That is, conservation of mass simply means that the time rate of change of mass of a system must be zero. Application of General Transport Theorem To proceed further toward our goal of obtaining a differential equation expressing mass conser- vation, we apply the general transport theorem to obtain integraldisplay R ( t ) ∂ρ ∂t dV + integraldisplay S ( t ) ρ W · n dA = 0 . (3.16) In fluid systems it is often useful to take the velocity field W to be that of the flowing fluid, which corresponds to locally viewing R ( t ) as an arbitrary fluid element. When this is done Eq. (3.16) becomes integraldisplay R ( t ) ∂ρ ∂t dV + integraldisplay S ( t ) ρ U · n dA = 0 . (3.17) Use of Gauss’s Theorem At this point we recognize that our form of conservation of mass contains physical fluid proper- ties that are useful for engineering analyses, namely the flow velocity U and the fluid density ρ . But the form of Eq. (3.17) is still somewhat complicated. In particular, it contains separate integrals over the fluid element’s volume and over its surface. Our next step is to convert the surface integral to a volume integral by means of Gauss’s theorem: recall that the form of this theorem will give, in the present case, integraldisplay S ( t ) ρ U · n dA = integraldisplay R ( t ) ∇ · ρ U dV , and substitution of this into Eq. (3.17) leads to integraldisplay R ( t ) ∂ρ ∂t + ∇ · ρ U dV = 0 . (3.18)
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60 CHAPTER 3. THE EQUATIONS OF FLUID MOTION The Differential Continuity Equation We now recall that the region R ( t ) was arbitrary ( i.e. , it can be made arbitrarily small— within the confines of the continuum hypothesis), and this implies that the integrand must be zero everywhere within R ( t ). If this were not so ( e.g. , the integral is zero because there are positive and negative contributions that cancel), we could subdivide R ( t ) into smaller regions over which the integral was either positive or negative, and hence violating the fact that it is actually zero. Thus, we conclude that ∂ρ ∂t + ∇ · ρ U = 0 . (3.19) This is the differential form of the continuity equation, the expression for mass conservation in a flowing system.
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