Momentum, on the other hand, presents us with a me ompliated ase sine we have to deal with a vetial quantity. The problem is simplified if we treat eah of the omponents of the vet independently. s you reall from the brief presentation in Chapter 3, a omponent of momentum an be thought of as flowing through matter muh like entropy harge do. The flow of eah of the omponents of momentum results in a flow field like those shown in Chapter 3 and below in Fig. 11.7. Here are the equations f a single omponent of momentum. You an build the omplete result f all three omponents by ombining the parts. If we hoose the x-omponent, the speifi value of x-momentum p x is the x-omponent of the veloity. Therefe, we have px x px In etion 11.1.2, we transfmed a surfae integral into an integral over the volume bounded by the surfae. We treated the simple example of purely one-dimensional mi = ρv v+ (11.36) Figure 11.7: Flow pattern of one omponent of momentum resulting from tension in a flat strip having a noth. The omponent of momentum whose flow is depited here is the one identified with the diretion of tension. ee also Fig. 3.8. + x In this ase it might be instrutive to present all three omponents of the urrent density: pxx = ρvv x x + pxx pxy = ρvv x y + pxy pxz = ρvv x z + pxz (11.37) These quantities have a simple graphial representation; pxx, f example, represents the urrent density of x-momentum flowing in x-diretion, while pxy is the urrent density of x-momentum flowing in y-diretion (see Fig. 11.7) ine there are three omponents of urrent density vets belonging to the three omponents of momentum, a total of nine omponents 7 fm the momentum urrent density tens. 11.2.5 Transfmation of a urfae Integral (Divergene Theem) 486 THE DYNMIC OF HET
11.2 DENITIE ND CURRENT DENITIE gration of lousts. ine the number of lousts is a salar quantity, its urrent density is a vet desribing the three possible diretions of flow of this fluidlike quantity. If the urrent density vet has only one omponent, then lousts move in only one diretion. In this ase, the lousts flux is I = n d = d Lx L x The seond fm on the right has been introdued to shten the notation. This integral an be transfmed into a volume integral ading to I = d= We used this relation to derive the loal fm of the equation of balane of lousts above in etion 11.1.2 (see Equ.(11.6)). In this fm, the transfmation is the simplest example of what is alled the divergene theem Gauss s theem. Let me briefly write down this relation without giving a proof. 8 If we define a urrent density vet Q on the losed surfae of a body, the surfae integral an be transfmed into an integral over the volume enlosed by the surfae: def x L Lx Lx Lx d nd = d Q Q (11.38) 7. This quantity annot be represented as a vet anyme; rather, it is a tens whih may be written in matrix fm J p ρvv x x + ρvv + ρvv + = ρvv y x + ρvv + ρvv + ρvv z x + ρvv + ρvv + pxx x y pxy x z pxz pyx y y pyy y z pyz pzx z y pzy z z pzz The negative ondutive part of this quantity is ommonly alled the stress tens t t t T = t t t t t t xx xy xz yx yy yz zx zy zz pxx pxy pxz = pyx pyy pyz pzx pzy pzz while the omplete quantity would be alled the momentum urrent tens. The surfae integral of a row of the tens (f one of the omponents of the odinate systems) is alled the omponent of the surfae fe F x = T x n d (T x is the first row of the stress tens), while the surfae integral f the stress tens is the surfae fe vet F= T n d PRT III 487
where Q is alled the divergene of Q. In retangular Cartesian odinates = + + x y z Q Q x Q y Q z (11.39) The divergene of a vet written in omponent fm is often abbreviated as follows: x x + y y + z z x Q Q Q Qi i (11.40) In this notation it is assumed that a summation is arried out over all indies whih appear twie in the same term; x i, i = 1,2,3 stands f the three omponents (x,y,z) of the odinate system. In this fm, the divergene looks like the expression used in single-dimensional ases. In fat, the simplest examples usually suggest the proper fm of me ompliated ases. 11.3 THE BLNCE OF M Control volume Let us start with the first of the three fluidlike quantities f whih we have to obtain laws of balane, namely the amount of substane. The balane of amount of substane is a neessary prerequisite f fmulating theies appliable to fluid otherwise defmable media. If we wish to quantify onvetive urrents assoiated with proesses in open systems, we have to be able to write down the urrents of amount of substane. F pratial reasons, however, engineers ommonly use mass as a substitute f amount of substane, and as long as there are no hemial reations taking plae inside the material, there is no problem in doing so. Therefe, we will use a fmulation based on mass. In the previous setion we introdued the onepts and tools needed to fmulate the ontinuum fms of the laws of balane of fluidlike quantities. tarting with the integrated fm of the balane of mass ṁ= I m (11.41) + x Figure 11.8: imple one-dimensional flow with respet to an open and stationary ontrol volume. Imagine a fluid flowing in the x-diretion only. we an easily show how to obtain the appropriate loal equation appliable to the ontinuous ase. Let us apply this law to a stationary ontrol volume of simple shape (Fig. 11.8) and assume the flow field to be one-dimensional. In this equation, m is the mass inside the ontrol volume, while I m is the net urrent of mass aross the surfae of the ontrol volume. We shall replae the mass by the volume integral of the mass density, and the flux by the surfae integral of the flux density, as in Equ.(11.26). With Equ.(11.34) this leads to d dt ρd + ρv d = 0 (11.42) If we use the divergene theem f the surfae integral and apply the time derivative 8. F a derivation of the divergene theem see Marsden and Weinstein (1985), ol. III, p. 927. 488 THE DYNMIC OF HET
11.4 THE BLNCE OF ENTROPY to the integrand of the first integral, we obtain ρ t d ρ t ρv d 0 x + = ρv d = 0 x + ine the integral must be zero f arbitrary volumes, the last expression an only be satisfied if the terms in brakets are equal to zero: ρ + ( ρv )= 0 t x (11.43) You an easily apply the transfmations to the me general three-dimensional ase ρ + ( ρv i)= 0 t x i ρ + ( ρ )= t v 0 (11.44) (11.45) This looks very similar to the simpler expression. In ontrast to Equ.(11.42) whih is the integral fm of the law of balane of mass, Equ.(11.43) and its ounterpart in Equations (11.44) (11.45) represent the loal differential fm of this law. The balane of mass often is alled the equation of ontinuity. 11.4 THE BLNCE OF ENTROPY Entropy is a salar quantity ust like mass, so the derivation of the loal fm of the law of balane should lead to a result similar to what we have ust seen. Consider as we did in Fig. 11.8, the flow of a fluid in the x-diretion only. s far as entropy is onerned, we will inlude ondutive and onvetive transpts in the derivation, and prodution of entropy in irreversible proesses. oures of entropy from radiation, however, will be exluded here. The integral fm of the equation of balane of entropy f the ontrol volume in Fig. 11.8 then looks like = I + I +Π, onv, ond (11.46) If we introdue densities and urrent densities as in etion 11.2, the law beomes d dt = + + ρsd sρv d d (11.47) s is the speifi entropy of the fluid, () and represent the ondutive entropy urrent density and the density of the entropy prodution rate, respetively. Remember that we are dealing with a purely one-dimensional ase. If we now apply the transfmation of the surfae integral, we obtain PRT III 489
ρ s ρ t x s + v + d = 0 The expression in brakets must be zero, whih yields the loal fm of the law of balane: ρ ( s)+ ρ t x s v + ( ) = (11.48) The general three-dimensional ase an be written in a fm that looks ust like the one derived f purely one-dimensional transpts. pplying the divergene theem to the generalized fm of Equ.(11.47) yields = ρ ρ ( s)+ s v+ t (11.49) I p,onv ( i ) = i ρ ( s)+ sρv i + t x (11.50) Extending this result to inlude the effets of soures from radiation is pretty simple. How this is done will be demonstrated below f the ase of momentum (remember that gravity leads to soures of momentum). v Σ p + z I p,ond Figure 11.9: Flow lines depiting the onvetive and ondutive transpts of momentum are shown together with a soure of momentum due to the interation of the fluid with the gravitational field. The fluid is flowing downward leading to the onvetive downward flow of momentum together with the fluid (dashed lines). ine the material is under ompression, momentum flows ondutively in the positive diretion (downward; solid lines). With the positive diretion as hosen, the gravitational field supplies momentum to the fluid. 11.5 THE BLNCE OF MOMENTUM Basially, the law of balane of momentum is derived analogously to what you have seen so far. While the fundamental ideas do not hange, the urrent ase an be rather omplex if we try to deal with it in the most general fm. It is therefe all the me imptant to disuss the simplest possible nontrivial ase. Ftunately, purely one-dimensional flow of momentum is meaningful in physial terms, so let us deal with this ase in some detail. One-dimensional onvetive transpt of momentum is a simple onept: if a fluid flows in one diretion only, it arries only one single omponent of momentum. The ase of one-dimensional ondutive transpt is ust as well known. Let the diretion of fluid flow define the spatial omponent we are talking about. Having the same omponent of momentum flowing through the fluid simply means that the material is under ompression tension in the same diretion. fritionless fluid flowing through a straight pipe demonstrates what we mean: the ondutive momentum urrent density of the omponent parallel to the pipe s axis is the pressure of the fluid. In addition to ondutive and onvetive modes of transpt, we will onsider soures of momentum due to the interation of the fluid with a field. If you imagine the fluid flowing through a vertial pipe (Fig. 11.9), the ation of the gravitational field leads to the flow of momentum of the same (vertial) omponent diretly into out of the body. If we ollet the different terms, the integral equation of balane of momentum f the z-diretion looks like 490 THE DYNMIC OF HET