M445: Heat equation with sources

Transcription

1 M5: Heat equation with sources David Gurarie I. On Fourier and Newton s cooling laws The Newton s law claims the temperature rate to be proportional to the di erence: d dt T = (T T ) () The Fourier law postulates the heat- u to be proportional to the temperature gradient: d dt Q = Z rt NdS; () Q = ct with coe cients c=heat capacity; =heat conductivity. Two descriptions deal with di erent time scales: fast for the Fourier and slow for the Newton. A physical model could be a uid undergoing turbulent miing as it cools down, e.g. buoyancy-driven convection in a pool with a freezing surface. Call temperatures T (air) and T b (initial water). We consider a circulation pattern that randomly replaces surface parcels at a constant rate (so called renewal model ). The uid patches that come to the surface could have the ambient water temperature T>T, and thus capable of releasing heat. Or they ve already participated in the heat-echange on the previous time-step, which brought their temperature down to T ¼ T, hence rendered them incapable of further cooling. Let Á (t) denotes a fraction of the surface eposed (and capable) of the heat echange. A constant replacement rate makes Á (t) an eponential function, Á (t) = e t=,where is the slow time-scale. We take the standard erf -solution of the half-space problem and its heat- u µ z T (z; t) = (T T b )erf + T b p t Q (;t) = k (@ z T )j z= = k p ¼ t (T T b ) Here denotes the heat-di usivity, and k -heat conductivity.

2 Averaging out the fast time scales we get T (z) = Q () = Z Z Á (t) T (z; t) dt =(T T b ) e z=p Á (t) Q (;t) dt = p k (T T )= c p½ b (T T b ) l µ () where l µ is the length scale of the average temperature pro le, c p -speci c heat at constant pressure and ½ -density. The second line of () is essentially the Newton s law. Mathematically, the erf-solution (e.g. for a cooling bar [ a; a]) T (; t) =erf µ + a p t erf µ a p ¼ p a + O t = t t has a polynomial fall-o in t (the same would hold for the space-average temperature over [ a; a]). However, averaging over the slow (Newton) time scale, i.e. taking time-convolution of T with e t=,weget Z t µ e (t s)= p ds ¼ e t= c + c p + ::: s t i.e. Newton s eponential fall-o. II. Heat equation with delta-sources We write a typical heat-di usion problem using symbolic operator notation u t + L [u] = F () uj t= = f Here L could be an ordinary di erential + q on [;l] with suitable boundary conditions at f; lg, or more general elliptic pde L = r pr + q, on region D ½ R n with boundary, and boundary condition B [u] =(a + b@ n ) uj. The formal (ODE-type) solution of () is given by the operator-eponential u = e tl [f]+ analogous to the matri-eponential. Z t e (t s)l [F (s)] ds (5)

3 Such operator-eponential represents a fundamental solution of problem (). One could show that operator e tl acting on functions ff ()g isgivenbyanintegralkernel G (;»; t), calledgreen s function of the problem, Z G [f] = G (;»; :::) f (») d» (6) D We are interested in the delta-source F = h (t) ± ( ). If G (;»; t)denotes the Green s function of L B, then solution (5) u (; t) = Z t G (; ;t s) h (s) ds (7) We evaluate (7) for the standard Gaussian G = (¼ t) n= e = t i.e. L = on R n and treat cases. Z t u (; t) = (¼ ) n= e = s s n= h (t s) ds (8) A. Constant source h =Const Here (8) yields after the change s! z = s u = jj n Z = t µ z n= e z dz = jj n n ; t epressed in terms of incomplete Euler gamma function (º;p)= Z p e z z º dz of order º = n, depending on space-dimension. Dimensions n =; ; can be epanded for small as ;p = p p p ¼ + p p p= + (;p) = µ +ln p 5 p5= + O p 7= + p p + 8 p + O p ;p = p ¼ p p + p= 5 p5= + p7= + O p

4 with Euler constant = : 577. We plot all gammas Incomplete gamma-functions (º;p) for º = ;; and write the corresponding solutions u epanded in small p = t dim q t p ¼ jj + =p ³ t 96 : ln t + t ³ p¼ p jj t + jj ( t) = u + ::: 5= t = + ::: 6 t jj + ::: 6( t) 5= Notice that in D solution has an asymptotic limit as t! r p t ¼ u (; t) ' jj whereas D-one converges to a potential-type equilibrium p ¼ u (; t)! jj The eact solutions for n =; ; u = jj µ ; t u = µ; t u = jj µ ; t

5 are plotted below as radial pro les u (r; t) in D-space-time view t t D D 5.5 t D Temperature pro le for D, D and D steady point sources 5

6 We also show their time snapshots D D D Time slices of temperature pro les in, and D A typical pattern shows accumulation of heat near the source and its spread outward. The rate of accumulation depends on space dimension and steepens with theincreaseofn. B. Time periodic source: h =cos!t Here solution u = Z t cos! (t s) e = s (¼ s) n= ds The integral has no closed form epression in known (elementary or special) functions. But its large-time asymptotics could be reduced to Fourier transforms of function: f (t) = e = t (¼ t) n=. Namely, Z u ¼ cos!t cos!s f (s) dsc A sin!t sin!s f (s) dsc A {z } {z } ^f c (!) Z ^f s (!) 6

7 The complete (half-line Fourier) transform of f in epressed through the modi ed Bessel (Kelvin) function K n= ^f (!) = Z e i!t =t e n =e i¼( t n= 8 ) µp n=! ³ K n= p i! jj jj In special cases, e.g. D K = is an elementary function, so integral (9) is simpli ed to ^f = +i ³ p e p!jj cos p! jj + i sin p! jj! We have thus shown that the asymptotic pattern consists of eponentially attenuated propagating heat-waves ³p u ¼ e p!jj cos! jj!t (9).5 - t Modulated heat-wave and its time snapshots Let us remark that the relation between the wave number k ¼ p! and frequency! is consistent with the heat-di usion dispersion law: i! = k. III. Equilibria for heat-di usion problems We use operator formalism (5) for a typical heat-di usion problem () to write its formal solution in terms of operator eponentials, analogous to the matri-eponential. All functions u; f; F could be epanded in terms of eigenfunctions fã k g of operator L, (rather eigenmodes of the boundary value problem L; B) In particular, Green s function is epanded as G (;»; t) = X k L [à k ] = kã k () Bà k j = e t k à k () ¹ à k (») kã k k 7

8 and solution (??) becomes u (; t) = X Z t ^fk e t k + e (t s) k ^Fk (s)i dsa à k () : () k n o n o Here ^fk ; ^Fk (t) denote generalized Fourier coe cients of f and F ^f k = hf ()j à ki kã k k in the sense of L (square-mean) inner product. A simple equilibrium solution v of problem () with a stationary (time-independent) source F is given by L [v] =F ) v = L [F ] () By analogy with eponential e tl operator L could be represented by an integral kernel (Green s function) K (;») = X k k à k () ¹ à k (») kã k k epanded in eigenmodes of L. Hence v () = X k ^F k à k () k The latter is easily shown to be a limit of solution (??) ast!, provided all eigenvalues f kg of L are positive. Indeed, convolution integral (??) becomes u = I e tl L A. Periodic equilibria [F ]+e tl [f]! L [F ]=v, ast! More interesting case arises for a periodic source F (; t). One asks the same two questionsasabove. whether periodic solutions v eist for (). whether they are stable, in the sense that any u (; t)! v as t! Both are easily answered using the above operator (ODE)-formalism. We rst consider a single frequency case F = F () e i!t in the comple form, ½ ut + L [u] =Fe i!t () uj = f 8

9 Formal solution of IVP () u = ei!t e Lt i! + L [F ]+e Lt [f] () µ µ = e i!t [F ]+e f Lt F i! + L i! + L is decomposed into the periodic component v () e i!t, where equilibrium v satis es (i! + L) v = F ) v =(i! + L) F (5) and negative eponential e Lt [:::]. As above operators (i! + L) ; e Lt are given by (comple-valued) Green s functions K (;»; i!); G (;»; t), or else could be epanded in eigenmodes v () = X ^F k à i! + k () (6) k k From comple form (6) one could easily get the real periodic solution ½ ut + L [u] =F cos!t uj = f (7) by taking the real and imaginary parts of () µ e i!t L Re = i! + L L +! cos!t +! sin!t L +! This yields the IVP-solution (7) written as µ L u = cos!t L +! +sin!t! L +! F + e Lt µ f in the operator-form, or an equivalent series epansion u = X k + e kt ½ k cos!t +! sin!t µ ^f k k +! ^Fk L L +! F ^Fk (8) k +! ¾ k à k () The latter clearly demonstrates that u (; t) converges to a periodic equilibrium v (;t) = Re (v () e i!t ),providedalleigenvalues k are positive, so eponential terms drop in ()-(8). 9

10 B. Multiple frequency case Here F is represented by a time-fourier series F = X m e i!mt F m () In the periodic case all frequencies are multiples of a single (lowest) one! m = m! and the period of F is T = ¼!. More generally, f! mg are arbitrary real numbers, the so called frequency spectrum of F. We seek partial (periodic) solution v in the form v = X m e i!mt v m () (9) with undetermined Fourier coe cients fv m g. The substitution in () determines each one of them via (5) v m =(i! m + L) F m So v is epanded in the time Fourier series (9) with the same period (or quasiperiods) as F. An interesting eample of multiple frequencies arises for a periodically moving point-source F = ± ( a cos!t) () We consider it on a symmetric interval [ l; l] with amplitude of oscillation a < l. P Generalized time-periodic function () has a frequency Fourier epansion F = m eim!t F m () with coe cients a F m () = cos m cos a ¼ p = T m a ¼ p a whose numerators are made of the classical Tchebyshev polynomials of the rst kind. We plot a few of them Function F is called quasi-periodic if its spectrum is made of linear combinations of a nite (basic) set f! ; :::;! p g px! m = n k! k with integer coe cients n k. Otherwise, it is called almost periodic. k=

11 As a consequence we get the periodic equilibrium for the moving-source problem, epanded in the double series v (; t) = ¼ X X µ k cos m!t + m! sin m!t m= k Tm a assuming all eigenvalues of L positive. Problems: k +(m!) p = a à k kã k k à k (). Specify epansion () for the Dirichlet and Neumann problem on [ l; l], L :. Compute the rst 5 frequency modes m =; :::; 5.. Plot approimate periodic equilibrium () by truncating both series. Use Mathematica! ()