3 Conservation Laws, Constitutive Relations, and Some Classical PDEs

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1 3 Conservtion Lws, Constitutive Reltions, nd Some Clssicl PDEs As topic between the introduction of PDEs nd strting to consider wys to solve them, this section introduces conservtion of mss nd its differentil form. For specific physicl situtions, constitutive reltions must be introduced. We give here the bsic dvection, diffusion, nd combined dvectiondiffusion cses tht we will be working with throughout these Notes. We hve lso included brief discussion of shock conditions nd boundry conditions, which we will cycle bck to lter in the course. On pge 8 we give summry of wht you should try to get out of this section. 3.1 Bckground on Clssicl Models Involving PDEs: One Spce Dimension Most of the clssicl models come from the use of conservtion principles nd constitutive reltions relevnt to physicl sitution being investigted. Assuming vrious limits hold, differentil form of the principles led to PDEs from which nlysis cn be done on them, nd numericl pproximtions cn be developed. It ll boils down to the power of clculus in the end. We will illustrte some of these in the context of one spce dimension. Consider single quntity (mss, energy, bugs, vehicles, chemicl in some non-moving fluid, or simply think of it s stuff ), nd let ρ = ρ(x, t) be the density of this quntity (mss per unit volume) so the mount of this quntity t loction x t time t is ρ(x, t)adx, where we think of thin tube of uniform cross-section of cross-sectionl re A (see Figure 1). Remrk: We ssume there is lrge enough mount of our stuff round to consider continuum ides, tht is, the quntities defined here exist, re continuous in its vribles, nd ny limits employed re ssumed to exist. Next consider φ = φ(x, t) is the flux of our quntity t loction x t time t. 1

2 Figure 1: Domin imge for the conservtion rgument Tht is, it mesures the mount of quntity crossing unit section t x t time t per unit time. Thus, Aφ(x, t) is the mount of our stuff crossing section of tube t x t time t. Let S = S(x, t)=rte our stuff is being creted t x per unit re t time t. Now consider n rbitrry intervl [, b]. The conservtion of mss lw sttes tht the rte of chnge of totl mount of the quntity in segment [, b] must equl the net rte t which it flows out of the intervl, plus the rte t which it is being creted/destroyed within the segment [, b]. In symbols, since we ssume A is constnt, or or d dt ρ(x, t)adx = Aφ(, t) Aφ(b, t) + b φ (x, t)dx = (x, t)dx + t { t S(x, t)adx (1) S(x, t)dx φ (x, t) + (x, t) S(x, t)}dx = 0 (2) Lemm: Let f be continuous function defined on n intervl [A, B]. If, for every subintervl (, b) [A, B], f(x)dx = 0, then f 0 in [A, B]. Proof is left s n exercise. Given tht the intervl (, b) in (2) is rbitrry, then by the Lemm we hve t (x, t) = φ(x, t) + S(x, t) (3) 2

3 Figure 2: Pure dvection is relly trnslting of initil dt So, t loction x the mount of stuff chnges due to stuff moving round (first term on the right side) plus stuff being either creted or destroyed (the second term on the right side). With S 0, (3) becomes the differentil form of the conservtion principle. Remrk: How is the eqution chnged if we ssume insted tht A = A(x) A 0 > 0, i.e. A is not ssumed constnt? While (1) is the codified conservtion principle (when S 0), (3) is more restrictive form becuse of the continuity condition. However, it is (3) tht is used in prctice. Note tht for the single eqution (3) there re two unknowns, ρ nd φ. This is where constitutive reltions come in the modeling. Now consider few exmples. Exmple: pure dvection or liner trnsport: Let S 0 nd the flux be proportionl to the density: φ = cρ. (Constnt c will hve units of length over time so it is speed of propgtion of the signl.) Then (3) becomes t + c = 0. (4) Eqution (4) is first order, liner PDE, nd represents pure dvection of the initil disturbnce ρ(x, 0) = ρ 0 (x). Hence, solution to the eqution is ρ(x, t) = ρ 0 (x ct), ssuming ρ 0 is differentible function defined on the whole rel line. Such solution is clled trveling wve solution, nd represents right-moving wve if c > 0, nd left-moving wve if c < 0. (See figure 2.) 3

4 Exmple: trffic flow theory: In the most elementry version of trffic flow theory, ρ represents trffic density (the number of vehicles per unit length of highwy), nd φ is the trffic flow rte (φ(x, t) is the number of vehicles pssing given point x t time t). Here highwy mens unidirectionl rodwy of infinite length, nd with no entrnces or exits (think of very long tunnel or bridge). Then n eqution of stte (constitutive reltion) might specify tht flux is function of density only, tht is φ = φ(ρ). A typicl shpe is given by figure 3. Now φ = dφ dρ speed s c(ρ) := dφ/dρ, then (4) becomes t. Define the locl wve + c(ρ) = 0. (5) This is nonliner first-order PDE for the trffic density (nd is nlogous to one-dimensionl gs dynmics). We could consider incorporting entrnces/exits by dding source term S on the right-hnd side of (5). Exmple: Fickin diffusion: Fick s lw sttes tht the flux is proportionl to the grdient of the density. In its simplest form this implies φ = D, where D > 0 is constnt diffusivity coefficient. So the flow, due to the sign convention, will go from plces of high density to plces of low density. Substituting this into (3) yields t D 2 ρ = S(x, t), (6) 2 which is the (nonhomogeneous) diffusion eqution.(see figure 4.) Specil cse: the 1D het eqution If one is concerned with mesuring het flow, then het energy is mesured through temperture u, in some mteril with mteril properties c, the specific het prmeter, nd k, the therml conductivity. Then ρ in (3) is replced by ρcu, here ρ nottionlly mens constnt density of the mteril. Now the pproprite form of Fick s lw, clled Fourier s lw (in one dimension), is φ = k u/. Then, in the cse of no het sources, (3) becomes ρc u t = { k u }. (7) Eqution (7) becomes equivlent to (6) (ssuming k is constnt, nd S 0) if we define D := k/ρc. Then D is clled the therml diffusivity. We will 4

5 Figure 3: This is the fundmentl grph of trffic theory Figure 4: Diffusion spreds the dt out, forgetting the informtion content in the dt in the bsence of source terms, i.e. S 0. generlly solve problem using the form of eqution (6) rther thn (7), but keep these definitions in mind. Remrk: There is probbilistic pproch to deriving the diffusion eqution tht strts with rndom wlk model. This leds into notion of Brownin motion mde fmous by Einstein. Hence, t the microscle, diffusion is bout rndom processes, while t the mcroscle, where verging hs tken plce, diffusion follows conservtion principle. The connection with rndom processes mens there re strong connections between PDEs nd probbility theory (nd hence ppliction res such s mthemticl finnce) tht we will not hve spce to pursue in these Notes. Exmple: dvection-diffusion: This cse ssumes we hve both processes 5

6 Figure 5: The combintion of mechnisms tends to both trnslte nd spred the initil dt working, so φ = φ dv + φ diff = uρ D, where u is mteril speed prmeter, D is diffusivity. Here we might interpret the sitution tht solute is being crried long (dvected) with bulk movement of fluid (solvent), sy with fluid velocity u = u(x, t), nd ρ here mens concentrtion. Now (3) becomes, with S 0, t + (uρ) = D 2 ρ. (8) 2 Exmple: Chemotxis: A lrge number of insects nd nimls rely on cute sense of smell for conveying informtion between members of the species, employing chemicls clled pheromones. At the cellulr level, motility of cells, such s in wound heling, is often controlled by specific chemicl grdients. These cses led to modeling flux due to chemicl ttrctnt (or repellent). In the presence of grdient of ttrctnt = (x, t) (We ll stick to one dimensionl sptil description here, but the mechnism is prticulrly pplicble in R 2 or R 3 ), gives rise to movement of cells, of density u = u(x, t), up the grdient. This suggests flux J chemotxis = bu /, where b = b() is n ffinity sensitivity coefficient tht my depend on the ttrctnt concentrtion. If we consider the totl flux s J = J diffusion + J chemotxis = D u/ + bu /, then (3) becomes, gin setting S 0, nd letting 6

7 Figure 6: Popultion of moebe beginning to ggregte vi chemicl signl, i.e. chemotxis. (Figure from Lin nd Segel s Mthemtics Applied to Deterministic Problems in the Nturl Sciences.) b, D = constnts, u t = D 2 u b 2 (u ). In this cse we would need n eqution for the ttrctnt, for exmple, t = D 2 + k 1u k 2 2. Such systems led to very interesting ptterns of behvior. See cse of this in figure 6. Exmple: 1D diffusion of popultion: In popultion biology nd other disciplines concerned with the growth nd movement of popultions, there is usully growth lw (constitutive reltion) under considertion. If u(t) represents the popultion of some species t time t, couple of dynmic lw exmples would be 1. Mlthusin or exponentil growth: du dt 2. Verhulst or logistic growth: du dt = ru(1 u/u mx) = ru (r=fixed net rte of growth) The second cse recognizes limited resources so tht unbounded growth is not possible (s in the Mlthusin cse). These models re further generlized to llow (rndom) movement of the popultion. The simplest cses re given by u t = D 2 u + ru (9) 2 u t = D 2 u + ru(1 u/u mx) (10) 2 7

8 Eqution (9) will be esily solved by methods introduce in these Notes, prtly to see wht effect the ru term hs on the solution behvior. Eqution (10) is Fisher s eqution, which ws first investigted in the 1920 s with regrd to the propgtion of n undesirble gene within popultion. It is nonliner PDE, nd one member of lrge clss of equtions clled rection-diffusion equtions tht crop up in ll sorts of science nd engineering subdisciplines. However, techniques for nlyzing such equtions will not be developed in these Notes. Summry You should understnd the bsics of getting the differentil form of the conservtion principle. The Lemm will be used gin lter in the course, so you should know its sttement. You should know wht is ment by pure dvection, diffusion, nd dvection-diffusion, nd the chrcter of the solutions s t increses. Exercises (1) In the 1D derivtion of the derivtive form of conservtion lw, (3), wht would the nlogous result if the cross-sectionl re A is smooth function of x, A = A(x)? Wht would the diffusion eqution look like if φ = k/, with k > 0 being constnt, nd S 0? (2) In the pure dvection cse (4), if c = 5 nd initilly ρ(x, 0) = ρ 0 (x) = e x2, wht would be the solution ρ(x, t)? Since ρ 0 (x) is Gussin bump moving out long chrcteristic line, where is the top of the bump t time t = 10? 8