Conservation Law. Chapter Goal. 6.2 Theory


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1 Chpter 6 Conservtion Lw 6.1 Gol Our long term gol is to unerstn how mthemticl moels re erive. Here, we will stuy how certin quntity chnges with time in given region (sptil omin). We then first erive the very importnt conservtion lw which is use in mny moels. Lter on, we will use the conservtion lw to erive more specific moels. The first one will be the het eqution. 6.2 Theory Mny PDE moels involve the stuy of how certin quntity chnges with time n spce. This chnge follows bsic lw clle the conservtion lw. Simply put, this lw sys tht the rte t which quntity chnges in given omin must equl the rte t which the quntity flows cross the bounry of tht omin plus the rte t which the quntity is crete or estroye, insie the omin. For exmple, consier the stuy of the popultion of certin niml species within fixe geogrphic re (our omin). The popultion in this geogrphic re will be etermine by how mny nimls re born, how mny ie, how mny migrte in n out. The conservtion lw pplie to this exmple sys tht the rte of chnge of the niml popultion is equl to the rte t which nimls migrte in the region minus the rte t which they migrte out plus the birth rte minus the eth rte. Similr sttements cn be me bout mny other quntities such s het energy, the mss of chemicl, the number of utomobiles on freewy,... Now, we trnsform such sttement into equtions, tht is we quntify it. For this, let u (x, t) enote the ensity of certin quntity (mss, energy, nimls,...). Recll tht ensity is mesure in mount of quntity per unit volume or per unit length. So tht if we know the ensity of the quntity n the volume of the region where it is contine, then we lso know the mount of the quntity. Let us ssume for now tht ny vrition in the ensity be restricte 29
2 30 CHAPTER 6. CONSERVATION LAW Figure 6.1: Tube with crosssectionl re A to one sptil imension we will cll x. Tht is, we ssume oneimensionl omin, ech cross section being lbele by the sptil vrible x. Figure 6.1 illustrtes this ie in the cse of tube s our omin. It s crosssectionl re is clle A. We ssume tht the lterl sies re insulte so tht the quntity being stuie only vries in the xirection n in time. For ech vlue of x, u (x, t) oes not vry within the cross section t x. Remrk 31 Let us mke some remrks n introuce further nottion: 1. The omin escribe here hs constnt crosssectionl re A. In more complex omin, the crosssectionl re might epen on x (see problems). 2. The mount of the quntity t time t in smll section of with x will be u (x, t) Ax for ech x. It follows tht the mount of the quntity in n rbitrry section x b will be u (x, t) Ax. 3. Let φ = φ (x, t) enote the flux of the quntity t x, t time t. It mesures the mount of the quntity crossing the section t x, t time t. Its units re mount of quntity per unit re, per unit time. So, the ctul mount of the quntity crossing the section t x, t time t is given by Aφ (x, t). By convention, flux is positive if the flow is to the right, n negtive if the flow is to the left. 4. Let f (x, t) be the given rte t which the quntity is crete or estroye within the section t x, t time t. It is mesure in mount of quntity per unit volume, per unit time. f is clle source if it is positive n sink if it is negtive. So, the mount of the quntity being crete in smll section of with x for ech x is f (x, t) Ax per unit time. It follows tht the mount of the quntity being crete in n rbitrry section x b will be f (x, t) Ax. We re now rey to formulte the conservtion lw in smll section of the tube, of re A for ech x such tht x b. The conservtion lw sys tht
3 6.2. THEORY 31 the rte of chnge of the mount of the quntity in tht section must be equl to the rte t which the quntity flows in t x = minus the rte t which it flows out t b plus the rte t which it is crete within the section x b. Using the remrks n nottion bove, the lw becomes t u (x, t) Ax = Aφ (, t) Aφ (b, t) + f (x, t) Ax (6.1) Let us first notice tht since A is constnt, it cn be tken out of the integrls n cncele from the formul to obtin t u (x, t) x = φ (, t) φ (b, t) + f (x, t) x (6.2) This eqution is the funmentl conservtion lw. It simply inictes blnce between how much goes in, how much goes out n how much is chnge. Eqution 6.1 is n integrl eqution. We cn reformulte it s PDE if we mke further ssumptions. We begin by remining the reer of theorem known s Leibniz rule, lso known s "ifferentiting uner the integrl". Theorem 32 (Leibniz Rule) If (t), b (t), n F (x, t) re continuously ifferentible then t (t) (t) F (y, t) y = (t) (t) F t (y, t) y+f (b (t), t) b (t) F ( (t), t) (t) (6.3) This is often known s "tking the erivtive insie the integrl". If we ssume tht u hs continuous prtil erivtives, then using Leibniz rule, the left sie of eqution 6.2 becomes t u (x, t) x = u t (x, t) x If we lso ssume tht φ hs continuous prtil erivtives, then using the funmentl theorem of clculus, we cn write φ (, t) φ (b, t) = Therefore, eqution 6.2 cn be rewritten φ x (x, t) x [u t (x, t) + φ x (x, t) f (x, t)] x = 0 It is possible for n integrl to be 0 without the integrn being equl to 0. However, in our cse, the intervl of integrtion ws rbitrry. In other wors,
4 32 CHAPTER 6. CONSERVATION LAW we re sying tht no mtter wht n b re, this integrl must be 0. Since the integrn is continuous, it follows tht it must be 0 in other wors or u t (x, t) + φ x (x, t) f (x, t) = 0 u t (x, t) + φ x (x, t) = f (x, t) (6.4) It is importnt to remember tht this eqution ws obtine uner the ssumption tht u n φ re continuously ifferentible. Eqution 6.4 will be clle the funmentl conservtion lw. Summry 33 We stuy how certin quntity chnges with time in given region. We mke the following ssumptions: 1. u (x, t) enotes the ensity of the quntity being stuie. u is ssume to be continuously iff erentible. 2. φ (x, t) is the flux of the quntity t time t t x. It mesures the mount of the quntity crossing cross section of our region t x. φ is ssume to be continuously iff erentible. 3. f (x, t) is the rte t which the quntity is crete or estroye within our region. f is ssume to be continuous. 4. We ssume the quntity being stuie only vries in the x irection. 5. Then, the eqution escribing how our quntity chnges with time in the given region is u t (x, t) + φ x (x, t) = f (x, t) Remrk 34 Let us mke few remrks before looking specific exmples. 1. Eqution 6.4 is often written s for simplicity. u t + φ x = f 2. The functions φ n f re functions of x n t. Tht epenence my be through the function u. For exmple, we my hve f = f (u). Similrly, we my hve φ = φ (u). These epenencies my le to nonliner moel. We will see one in the exmples. 3. Eqution 6.4 involves two unknown functions: u n φ, usully the source f is ssume to be given. This mens tht nother eqution relting u n φ is neee. Such n eqution usully rises from physicl ssumptions on the meium.
5 6.3. ADVECTION EQUATION Eqution 6.4 is in its most generl form. As we look t specific moels, it will tke on ifferent forms. For exmple, when we tlke bout flow in this section, we i not specify how the quntity trvele. In the next sections we will consier vrious possibilities incluing vection n iff usion. Also, the source term cn tke on ifferent forms. We now look t specific exmples. Here, we will simply write own the eqution corresponing to the moel. We will iscuss how to solve them little lter. 6.3 Avection Eqution Avection refers to trnsport of certin substnce in flui (wter, ny liqui, ir,...). An exmple of vection is trnsport of pollutnt in river. The flow of the river crries the pollutnt. A moel where the flux is proportionl to the ensity is clle n vection moel. It is esy to unerstn why. Thinking of the exmple of the river crrying pollutnt, the mount of pollutnt which crosses the bounry of given region in the river clerly epens on the ensity of the pollutnt. In this cse, we hve for some constnt c: φ = cu The constnt c is the spee of the flui. In our exmple bove, it will be how fst the river flows. Eqution 6.4 becomes: We look t specific exmples. u t + cu x = f (x, t) Avection When f (x, t) = 0 (No Source) In the bsence of sources, Eqution 6.4 becomes: u t + cu x = 0 (6.5) Eqution 6.5 is clle the vection eqution. Note tht c must hve velocity units (length per time). As note bove, c is the spee t which the flui is flowing Exmple: Avection n Decy Recll from elementry ifferentil equtions tht ecy is moele by the lw u t = λu
6 34 CHAPTER 6. CONSERVATION LAW where λ is the ecy rte. For exmple, substnce vecting through tube t velocity c, n ecying t rte of ecy λ woul be moele by the vectionecy eqution u t + cu x = λu (6.6) Here, λu correspons to the source term (the function f in eqution 6.4) Generl Avection eqution In its most generl form, the vection eqution is where: n c re constnts. u t + cu x + u = f (x, t) (6.7) cu x is the term which correspons to the flux. Recll, in the vection moel, the flux φ is φ = cu, c being the spee of the flow of the flui in the vection moel. u n f (x, t) correspon to the source. u sys tht the rte of chnge of the quntity within the omin is proportionl to the quntity. Exmples inclue rioctive ecy, popultion growth. f (x, t) sys tht the rte of chnge of the quntity within the omin is some given function. An exmple of this woul be pollutnts flowing in river, some pollutnts being umpe within the omin. f (x, t) woul give the rte t which the pollutnt is being umpe. Exmple 35 If the eqution is u t +cu x +u = f (x, t), then it mens tht there is vection (cu x ), ecy or growth (u) s well s quntity being e t rte given by f (x, y). Exmple 36 If the eqution is u t + cu x + u = 0, then it mens tht there is vection n growth or ecy. Exmple 37 If the eqution is u t + cu x = 0, then then it mens tht there is only vection. 6.4 Problems 1. How oes the conservtion lw in eqution 6.4 chnge if the tube hs vrible crosssectionl re A = A (x) inste of constnt one?
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