Lesson 03 Heat Equation with Different BCs

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1 PDE & Complex Variables P3- esso 3 Heat Equatio with Differet BCs ( ) Physical meaig (SJF ) et u(x, represet the temperature of a thi rod govered by the (coductio) heat equatio: u t =α u xx (3.) where α is the thermal diffusivity (derived by coservatio of eergy, Appedix 3A). Eq. (3.) meas that the time-rate of chage i temperature (u t ) is proportioal to the cocavity (u xx ) of the temperature distributio. By eq. (.), we have: u t [ u( x, u ( x, ] (3.) Therefore, eq. (3.) models two facts: () heat always flows from high- to low-temperature regios; () the flow rate is proportioal to the temperature gradiet. E.g. if the temperature at x is higher tha its surroudig average, u>u, u xx <, u t <, temperature is decreasig at a rate proportioal to [ ( x, u ( x, ] u. I steady state, u t =, u xx =, i.e. heat flow teds to eutralize the curvature of the temperature distributio, leadig to a liear temperature spatial profile. <Commet> I wave equatio [eq. (.)], u xx < does ot guaratee u t < (but u tt <). The solutios to wave equatio ad heat equatio behave very differetly (vibratio vs. diffusio).

2 PDE & Complex Variables P3- Heat Equatio with Various BCs (EK.5, SJF 3) Type (Dirichle BC: temperature is specified o the boudary, i.e. u=g(. Cosider a thi rod of legth with two eds fixed at zero temperature. PDE: u t =α u xx Two homogeeous BCs: {u(,=, u(,=} Oe IC: u(x,)=φ(, for oly first-order partial derivative with respect to t is ivolved. Solvig heat equatio by separatio of variables: ) Separatio of variables: et u(x,=x( T( X + k X =, T& + α k T = (esso ) ) Solvig the ormal modes by homogeeous BCs: () To avoid trivial solutio u(x,=, homogeeous BCs of u(x, BCs of X(: {u(,=, u(,=} {X()=, X()=} () The spatial ODE: X + k X =, X(=Acos(k+Bsi(k; π By BCs, (i) X()= A=; (ii) X()= k = k =, =,, X (=si(k ; The temporal ODE: T & + λ T =, λ =αk = -th ormal mode is u (x,= X ( T (: πα ; T (= exp( λ ) u (x, = A si( k exp( λ t ; (3.3) <Commet> (a) Eq s (.) ad (3.3) have the same spatial shape (due to same spatial ODE ad BCs) but differet forms of temporal evolutio (oscillatio vs. decayig), for the orders of temporal ODEs are two ad oe, respectively. (b) Higher order modes [u (x, with larger ] will decay faster, for the decay costat

3 PDE & Complex Variables P3-3 λ. Temperature profile will be smoother as time elapses (low-pass filterig). 3) Determiig the exact solutio by IC: u(x,= u ( x, = A si( k exp( λ = = (3.4) = Substitute the IC ito eq. (3.4): u(x,)= A si( k =φ(. By Fourier sie series, A = φ ( si( k dx (3.5) x, for < x < / E.g. et iitial temperature distributio is triagular: u(x,)=φ(=. By x, for / < x < ( / 4 ( ) ), if is odd eq. (3.5), A = π. As show below, exact solutio (lef will, if is eve coverge to the fudametal mode (righ as time elapses. [T= λ represets decay time costat of fudametal mode: u (x,t)=e - u (x,).] Type (Neuma) BC: By (experimetal) Fourier s law of coolig, heat flows i the directio where temperature u( r v, decreases most rapidly, ad the flow rate is proportioal to the rate of temperature chage i that directio. Heat flux q v ( r v, (W/m )= κ (W/mK) u( r v, t ) (Appedix 3A).

4 PDE & Complex Variables P3-4 O the boudary, heat flux is i parallel with the outward ormal directio v : q v =q v = κ ( u). Takig ier product with v u, q= κ, i.e. u q( = κ =g(. Cosider a thi rod of legth with two isulated eds. PDE: u t =α u xx Two homogeeous BCs: {u x (,=, u x (,=} Oe IC: u(x,)=φ(. ) Separatio of variables: X + k X =, T& + α k T = ; ) Solvig the ormal modes by homogeeous BCs: Spatial ODE: X + k X =, X=Acos(k+Bsi(k, X ( = k[asi(k Bcos(k]; Trasformatio of BCs: {u x (,=u x (,=} { X () = X () =}; By BCs: (i) () X = B=, (ii) X () π = k = k =, =,,, X (=cos(k, differet BCs produce differet modes! Temporal ODE: T & + λt =, λ =αk = -th ormal mode is u (x,= X ( T (: 3) Determiig the exact solutio by IC: πα ; T (= exp( λ ) u (x, = A cos( k exp( λ t ; (3.6) u(x,= u ( x, = A cos( k exp( λ = = (3.7) = Substitute the IC ito eq. (3.7): u(x,)= A cos( k =φ(. By Fourier sie series,

5 PDE & Complex Variables P3-5 A = φ ( dx, A = φ ( cos( k dx (3.8) x, for < x < / E.g. et iitial temperature distributio is triagular: u(x,)=φ(=. By x, for / < x < eq s (3.7 8), exact solutio will coverge to A (average temperature of iitial distributio) as time elapses. This makes sese, for o eergy escapes from the rod, ad heat flow teds to remove ay temperature curvature. u ( ) Type3 (mixed) BC: temperature of the surroudig medium is specified, i.e. +γ[u+g(]=, where γ = κ h, h is heat-exchage coefficiet, κ is thermal coductivity. ) By the Newto s law of coolig: heat flux is i the directio from high- to low-

6 PDE & Complex Variables P3-6 temperature T, ad is proportioal to its differece T across the boudary. Outward heat flux at x=: h[u(, g (], outward heat flux at x=: h[u(, g (]. u ) By the Fourier s law of coolig (experimetal): q= κ. u(, u(, u(, Outward heat flux at x=: κ =, outward heat flux at x=: κ. ( x x 3) Equatig &, we have: u u x x (, = (, = γ [ u(, g( ] h, where γ = γ [ u(, g ( ] κ (3.9) Cosider a thi rod of legth with oe ed fixed at zero temperature, the other ed immersed i a liquid of zero temperature (SJF 7). PDE: u t =α u xx Two homogeeous BCs: {u(,=, u x (,+γ u(,=} Oe IC: u(x,)=φ(. ) Separatio of variables: u(x,=x( T( X + k X =, T& + α k T = ) Solvig the ormal modes by homogeeous BCs: Spatial ODE: X + k X =, X(=Acos(k+Bsi(k; Trasformatio of BCs: {u(,=, u x (,+γ u(,=} {X()=, X'()+γ X()=} By BCs: (i) A=, (ii) k=k, where {k } are irregularly-spaced discrete roots of a oliear equatio: k ta(k)= (3.) γ X (=si(k [differet from that of eq. (3.3)].

7 PDE & Complex Variables P3-7 Temporal ODE: T + λ T = -th ormal mode: 3) Determiig the exact solutio by IC: &, λ =αk ; T (= exp( λ ) t ; u (x, = A si( k exp( λ (3.) u(x,= u ( x, = A si( k exp( λ = = (3.) = Substitute the IC ito eq. (3.): u(x,)= A si( k =φ(; By orthogoality of X ( i [,] (for the spatial ODE is a Sturm-iouville problem): si( k si( k dx = δ m =, if m = si x, sic(, otherwise x m, si(k ) A = [ ] ( [ ] δ m = δ msic( k) 4k, where φ si( k dx (3.3) sic(k ) x, for < x < / E.g. et iitial temperature distributio is triagular: u(x,)=φ(=. By x, for / < x < eq s (3. 3), exact solutio behaves differetly with those i type- ad type- BCs. Ca you justify why u(, eve the right ed is immersed i a liquid of zero-temperature ad u(,)=?

8 PDE & Complex Variables P3-8 Heat Equatio without Boudary (EK.6) Problem: heat diffusio alog a thi rod of ifiite legth. PDE: u t = α u xx No BC Oe IC: u(x,)=φ( Solvig heat equatios by separatio of variables & Fourier itegrals: ) Separatio of variables: u(x,=x(t( X + k X =, T& + α k T = ) No BC eigevalue k is NOT quatized, but cotiuous: Spatial ODE: X + k X = X k (=A(k)cos(k+B(k)si(k [A(k), B(k) are justified by substitutio]; temporal ODE: T + α k T = -th ormal mode: & T k (= exp( α k u k (x, = [A(k) cos(k+b(k) si(k] exp( α k 3) Determiig the exact solutio by IC:, (3.4) u(x,= u k ( x, t; k) dk = [ A( k) cos( k + B( k) si( k ] e α k t dk (3.5) Substitute the IC ito eq. (3.5): u(x,)= [ A k) cos( k + B( k) si( k ] ( dk =φ(; by

9 PDE & Complex Variables P3-9 Fourier itegrals (EK.7), A(k) = π φ( ξ ) cos( k ξ ) dξ, B(k) = φ( ξ ) si( k ξ ) dξ (3.6) π ( ) Gree s fuctio Itegral solutio of the form of eq. (3.5) ca be simplified as (EK.6): u(x, = α π φ( ξ )exp t 4α ( x ξ ) t dξ = φ( G(x, (3.7) x where G(x,= exp is called the Gree s fuctio of the system, which ca α π t 4α t be iterpreted as the temperature respose to a iitial temperature impulse: φ(=δ(. Sice liear homogeeous PDE + liear homogeeous BCs (like this particular problem) describe liear ad time ivariat (TI) systems, a iput of c δ(x x )+c δ(x x ) results i a output of c G(x x, +c G(x x,. We ca decompose the iitial distributio φ( as a cotiuum of impulses: φ(= φ( ξ ) δ ( x ξ ) dξ. Each impulse φ(ξ) δ(x ξ) [located at x=ξ ad with magitude φ(ξ)] has a output φ(ξ) G(x ξ,, ad their superpositio gives the overall respose u(x,. You may fid that aother equivalet formula is useful i some practical evaluatios: z u(x,= ( π ) φ( x + αz t ) e dz (3.8)

10 PDE & Complex Variables P3- E.g. Fid u(x, if iitial temperature distributio φ(= U, for < x <, otherwise ( As: By eq. (3.8): u(x,= x (+ π ) / α t / α t e z dz temperature profile u( gets smoother as t icreases. = x x erf erf α t α t, Error fuctio is defied as: erf( π x e t dt.

11 PDE & Complex Variables P3- Appedix 3A Derivatio of coductive heat equatio (by D. W. Trim) To describe coductive heat flow i a medium, we defie a 3-D heat flux vector q v ( r v, (W/m ). Its directio ad magitude represet the directio of heat flow ad the amout of heat per uit time crossig uit area ormal to the directio of q v. By the (experimetal) Fourier s law of coolig, heat flows i the directio where temperature u( r v, decreases most rapidly, ad the flow rate is proportioal to the rate of temperature chage i that directio: q v ( r v, = κ u( r v, t ) (3A.) where κ (W/mK)> is the thermal coductivity. Heat is accumulated i a volume V eclosed by surface S because of: (i) flux across S, (ii) iteral heat source g( r v, (W/m 3 ): S v v q ( ds ) + V g dv (W) The heat accumulatio causes temperature icrease to maitai eergy coservatio: V u v s ρ dv = ( v ) t q ds S + V g dv where s (J/kgK) ad ρ (Kg/m 3 ) are specific heat ad desity of the medium. By eq. (3A.) v v ad divergece theorem, q ( ds ) = ( κ u) dv, S V V [ ρ su g ( κ u) ] dv = (3A.) t If the itegrad is cotiuous throughout a arbitrarily small volume V, eq. (3A.) becomes: u t = α g u + κ (3A.3) where α (m /sec) is the thermal diffusivity.

Chapter 10 Partial Differential Equations and Fourier Series

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