An Exact Solution of Stikker's Nonlinear Heat Equation

An Exact Solution of Stikker's Nonlinear Heat Equation Author(s): Allan R. Willms Source: SIAM Journal on Applied Mathematics, Vol. 55, No. 4 (Aug., 1995), pp. 1059-1073 Published by: Society for Industrial and Applied Mathematics Stable URL: http://www.jstor.org/stable/2102638 . Accessed: 26/04/2013 09:42 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp

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SIAM J. APPL. MATH. Vol. 55, No. 4, pp. 1059-1073,

AN EXACT

(?) 1995 Society for Industrial and Applied Mathematics

August 1995

010

SOLUTION

OF STIKKER'S EQUATION*

NONLINEAR

HEAT

ALLAN R. WILLMSt Abstract. Exact solutions are derived fora nonlinearheat equation where the conductivityis a linear fractionalfunctionof (i) the temperaturegradientor (ii) the product of the radial distance and the radial component of the temperaturegradient forproblems expressed in cylindricalcoordinates. It is shown that equations of this form satisfy the same maximum principle as the linear heat equation, and a uniqueness theorem foran associated boundary value problem is given. The exact solutions are additivelyseparable, isolatingthe nonlinearcomponentfromthe remainingindependent variables. The asymptotic behaviour of these solutions is studied, and a boundary value problem that is satisfiedby these solutions is presented. Key words. nonlinear heat conduction,exact solution, diffusion AMS subject classifications.

35C05, 80A20, 35K05

1. Introduction. Considerableworkhas been done in the analysis of certain typesof nonlinearheat conduction(diffusion)equationswherethe conductivity(diffusivity)is a functionof the temperature(concentration)[2], [6]. The porous media equation, a nonlineardiffusion equation of this type,is an example [1], [9]. A review of some nonlinearequations that admit exact solutionshas been completedrecently by Rogersand Ames [10]. Some analysishas also been done forequationswherethe conductivityis a functionof the temperaturegradientor its magnitude[3]. In thispaper we shall developexact solutionsforthe following twononlinearheat conductionequationsin 1R3: (1) (2)

(C

+ bu_,

+

)s

1 +brurur r (ac + drur

+U yy

+ 1 r

r

where (x, y,z) and (r,0, z) are Cartesian and cylindricalcoordinates,respectively; all variables are nondimensional;and the subscriptsdenote partial differentiation; nonzeroconstantsa, b,c, and d satisfycertainconditionsgivenbelow. The constructed solutionto (2) will be radiallysymmetric.An equation of type (2) was derivedby Stikkerforthe problemof heat conductionin steel coils duringthe batch annealing process [11]. The remainderof this paper is organizedas follows.In ? 2 we shall give Stikker's physicalderivationof the nonlinearradial heat conductivity,considersome of the physicalconstraintson the constants,and simplify(1) and (2). Some propertiesof solutionsto these equations are givenin ? 3, exact solutionsforboth equations will be derivedin ? 4, and ? 5 will discussthe typesof boundaryvalue problemsthat these solutionswill satisfy. 2. Derivation and simplification. Since Stikker'sworkis not generallyavailable, we shall summarizehis derivationofthe heat conductionequationin ? 2.1 below. * Received by the editors May 19, 1993; accepted forpublication (in revised form)June 24, 1994. This research was supported by a Natural Sciences and Engineering Research Council of Canada Postgraduate fellowshipat the Universityof Waterloo, Waterloo, Ontario, Canada. t Center forApplied Mathematics, Cornell University,Ithaca, New York 14853. 1059


A. R. WILLMS

1060

2.1. Stikker's derivation. Stikkerwas concernedwith the heat conduction equation relevantto the tight-coilbatch annealingprocess. Batch annealing is a stage in the productionof steel whose purposeis to restorethe steel's ductilityafter cold rolling.In this process,steel coils are heated to a temperatureof about 7000 C and then,aftersome time,allowedto cool. To simplify the mathematics,the steel coil is viewedgeometrically as a collection of concentricringsratherthan a spiral, thus achievingaxial symmetry.This type of approximationhas also been employedin the numericalworkby Harvey [5] and Jaluria [7], [8]. Forte [4] gives some discussionof how the conductivityin the radial directionforStikker'smodel may be modifiedto account forits failureto reflectthe directconnectionbetweenthe coil windings. Since the gaps betweenthe windingsof the coil have only a veryslightinfluence on the specificheat ofthe coils,forall practicalpurposes,the specificheat ofthe coils to increase may be takenas that of the steel,c, whichhas been foundexperimentally in the axial direction, between200 and 7000 C by a factorof1.7. The heat conductivity or z-direction,is the heat conductivityof the steel, A), whichdecreasesbetween200 and 7000 C by a factorof 1.5. The heat conductivity ofthe coils in the radial direction is, however,highlydependenton the widthof the gaps betweenthe windings.The heat conductionequation then reads

(3)

OT OT Fr AeqrOT1 pc at = 10ar ] + Oz aAZ[sOT1]

of wherepc is the specificheat ofthe steelper unitvolume,A, is the heat conductivity ofthe coil in the radial direction. the steel,and Aeq is the equivalentheat conductivity Figure 1 illustratesthe situationin the coils. The resistanceforthe heat flow betweenthe planes r1 and r2 is givenby d1

I

As

d2 A'

:
Ag, the denominatorof the expressionon the rightside of (7) reaches zero beforethe numeratordoes, hence Aeq-* +oo as r2AAT/Ar-- -(d2o/d1o+ Ag/As)+.To avoidthissituationStikker imposesthe physicalconstraintthat the gap betweenthe successivewindings,d2, is bounded frombelow by a minimumdistance,dmin,determinedby the roughnessof the steel (- 10' cm). Consequently,(7) holds for d2= d20+ rAAT > dmin or + rA A

(8)

do

Ar

-do

dmin

whileforsmallervalues Aeq is givenby 9e=As

1 + (dmin/dio) 1 + (As/Ag)(dmin/dio)

=

Aeq-

Note thenthat the domainofvalidityof (7) is exclusiveofthe singularity that occurs at AT +rA Ar do

d2_

=

Ag 0, i.e., df(0) d7

(12)

a 0, by (14). For Stikker'sequation, (11), the parameter3 is givenby (As/Ag - 1)/(1 + (As/Ag)(d2O/dio))the firsttermof (15) may be Owing to the particularformof the nonlinearity, writtenas [(1+

1

)uxUx-

(1+Ux)

+(1Ux)2)

2Ux

hence (15) becomes Ut=

(17)

1+ A-(1

(1 +u)2) 2+ux (

= Au-

d

1 + UX

U + Uyy+ Uzz A x

Similarly,(16) is equivalentto Ut(

11(3\ + (I + u)2

(18)\( -

=AU

+

U)

f

dA

r (i+

(rur)r +

2uOO + uzz

rurjr

Note from(17) and (18) that the effectof the nonlinearityis the introductionof a "perturbation"termthat vanishesas 3 -* 0. The translationintoour newvariablesofcondition(13) withregardto the domain of validityof -yis givenby (19)

-1 < ru,v

- 1 < ux,

forequations (17) and (18), respectively. 3. Properties of solutions. Solutionsto (17) and (18) satisfythe same maximumprincipleas the standardlinearheat equation. Let Q be an open set in R3,and definethe sets D, DT, and CT (the parabolic boundaryof DT) by

D = Q x(, O)),

DT=Q

x (0,T), T

> O,

CT = ODT \{(x,T) I x E Q}.


EXACT SOLUTION

OF STIKKER'S

1065

HEAT EQUATION

THEOREM3.1. If u E C(D) nC 2(D) and satisfies(17) or (18), then u(x, t) < maxu,

V (x,t) E DT, T > .

CT

Proof. Note that if a maximumoccurs at a pointp on the interiorof DT or on {(x,T) I x E Q}, then we have ur(p) = ux(p) = 0, and (17) and (18) evaluated at p are givenby Ut = (1 + 3)u,,

+ uSy + Uzz

and 1

Ut = -(1 + 3) (rur)r +

1

2 Uoo + Uzz

respectively.The resultthenfollowsin the same manneras it does forthe linearheat equation. 0 A uniquenessresultmay also be given. THEOREM 3.2. For a boundedregionD in 1R3,arndconstant3> 0, thereexists at most one solutionu E C(D) nc2(D) to each of theproblems

{

u satisfies(17) or (18) u(x,O) =f(x) u(x, t) = g(x, t) > or y< -1 Y >-1,

in D x R+, in D, on OD x R+, in D x R+,

wherey = ux or rur corresponding to u satisfying(17) or (18). Proof. Set

a = 1, a = r,

(c,nl, ) = (X,Y,z) (C r7, ) = (r,0,z)

ifu satisfies(17) or ifu satisfies(18).

Suppose thereare two solutionsul, u2. Set v = v satisfies |Vt = A\v+ aE(1

1)1+

-

u2 and -yi=

i = 1,2; then

in D x R+,

2)

in D, on OD x R+.

(I ?) = ? V(c,71, t) = O

(I tV(C,71,

Considerthe function (20)

I(t)=

2Jv2dV. D

Clearly,I(t) > 0 and I(0) = 0. Differentiating (20) yields (21)

I'(t)

vvt dV

JvAv dV +

v

((a?

?

Using Green's firstidentity,

J

(fAg + Vf Vg) dV-Jf 49

D

ds,

aD


)

dV.

1066

A. R. WILLMS

and notingthat dV = cedfdr1d(, (21) becomes I'(t)

v

? dV +

=-|IVVI2 D

ds

AD

+0[A , v(( +71)(1+72,)e~ d]d

(22)

dV +Iv

=- -IVVI2

D

n ds

AD

+IJ[(1

+ _/1)(1+ 72)

Since v

|6(4j

(1 +-Yl)(l +-2)

(n(

0 on oD, and thereforezero on (j(7r, () and 6(q, 0), (22) collapses to

-

I'(t)

(1V2

?(1

+ 71)

+ 72) )

Finally,by the theoremassumptions,since 3> 0 and both 1 +-yi and 1 + Y2 have the same sign,it followsthat I'(t) < 0, implyingthat I(t) 0_ and thereforev =_0, hence 1l= U2? Remark.The Dirichletboundaryconditionin Theorem3.2 may be replacedby a Neumanncondition thatn = ?f at all points(6j(71, provided (), 7I, ( 2 (r, 0), m(), that is, ifthe boundaryofD is composedofsurfaces( = constantand surfacestangent to the (-direction. 4. Exact solutions. In this section we shall constructanalyticalsolutionsof (17) and radiallysymmetricanalyticalsolutionsof (18) by assumingseparabilityof the x /r dependencefromthe remainingindependentvariables. 4.1. Solutions of (17). Suppose u(x, y,z, t) = X(x)V(y, z, t); thensubstitution into equation (17) yields xvt

-V%

vt-V

V

x = VX(

_

X//

Xt

+ X (Vyy + VZZ)

+ (1 +V')

_

A

A

(1 +VX/)2y

Note that the V,X dependencedoes not completelyseparate,but one possible solution is (23)

X" = O = Vt-Vyy-Vzz,

implying X(x) = Ax + B. Solutionsof (23) are, however,not veryinteresting, since they are also. solutionsof the linearheat equation, 3 = 0.


EXACT SOLUTION

OF STIKKER'S

HEAT EQUATION

1067

Solutionsto (17) that are not also solutionsto the linearheat equation may be foundby assumingan additiveseparabilityofthe formu(x, y,z, t) = X(x) +V(y, z, t). In this case, substitutioninto (17) yields + vzz(? VY +?)

vt=XI

whichmay be writtenas

+I X'- 1 X)

vt-vyy-Vzz = d

(24)

achievingcompleteseparationof the x dependence. Settingboth sides of (24) equal to a constant,4A, and integrating the X equation once gives Vt-Vyy-Vzz = 4A,

(25)

1 + X'I

(26)

= ,-4(Ax

+ A),

whereA is a constant. Equation (25) is an inhomogeneous linearheat equationin twospatial dimensions, forwhichthereare standardmethodsof solving,givena set of boundaryand initial conditions. nonlinearordinarydifferential We nowturnour attentionto solvingthe first-order equation (ODE) (26). Multiplyingby 1 + X' gives (1 + X')2

-

4(Ax+ A)(1 + X')-

= 0;

hence, (27)

1+Xi

4(Ax?A)2?+3.

=2Ax+2A?

Integrationyields (28)

X?(x) = Ax2?+(2A-1)x

4(Ax?A)2?+3dx

+ B,

whereB is a constant. Note from(27) that since 3 > 0, 1 + X' is eitheralways positiveor always negativeso that thereare no singularitiesof (17). Furthermore, regardingthe domain of validityof our equation, given by (19), we see that X+ is the valid solutionwhile X_ is a nonphysicalsolution. Note also that if A = 0, then assume equations (25) and (27) collapseto the linearequation (23); we shall therefore that A 740. Completingthe integralin (28) gives X?(x)

= Ax2 + (2A-1)x

+B

[(Ax + A) 4(AxA)2

(29)

1

?n+ In2(Ax + A) +

2

Two classes of exact solutionsto (17) are therefore givenby

(30)

UE? (X, y, z, t) = X?(x)

+ V(y, z, t),

whereV(y, z, t) is any solutionto (25).


A. R. WILLMS

1068

4.2. Solutions of (18). We shall restrictourselvesto radiallysymmetricsolutionsof (18); hence,the uoo termvanishes.As in the previoussection,a multiplicative solutionof the formu(r,z, t) = R(r)V(z, t) does not completelyseparate,yielding

RVt= V

) (1

+

? + (1

VrR)2)

?RVzz,

Vt-%zz-R (R" +r ) 1+(1 + VrR/)2) V Rk\rJk\(1?VrR')2J whichadmitssolutionsto

R"

(31)

zz,

r

implying R(r) = Alnr +B. Again,these solutionsare also solutionsto the linearproblem. Assumingan additiveseparabilityofthe formu(r, z, t) = R(r) + V(z, t) gives,on substitutioninto (18) and manipulationidenticalto that in the previoussection,the equation (32)

-

Vt-Vzz=

1+ rR'--

_

R)

the R equation once yields Settingboth sides of (32) equal to 4A and integrating Vt-Vvzz=4A,

(33)

1 + rR' -

(34)

= 2(Ar2 + A). l?+rR,2ArA

linearheat equation. Multiplying(34) by 1+ rR' As before,(33) is an inhomogeneous gives (1 +?rR')2 -2(Ar2 + A)(1 +?rR') -'

=0;

hence, 1 + rR = Ar2+

(35)

A

(r2A)2?+

and therefore R' =Ar? +

(36)

A -i 1 ?(Ar2+A)2 +/. Ir Ir

Integrationyields (37)

R?(r)

=

(A-1)lnr+

r2J

(Ar2 ? A)2?+

3dr + B.

Again, (35) showsthat since a > 0, 1+rR' is eitheralwayspositiveor alwaysnegative; hence, there are no singularitiesof (18). From (19) we see that R+ representsthe


EXACT SOLUTION

OF STIKKER'S

1069

HEAT EQUATION

valid solutionwhile R_ is a nonphysicalsolution. Also, as before,A = 0 yieldsonly solutionsof the linearproblem(31) so that we assume A 780. Using the transformation y = Ar2,the integralin (37) becomes 1=]-

X]

(r2?+A)2+?Adr=

y2?+2Ay?+A2?+

dy.

The above integralcan be foundin mostpublishedtables of indefinite integrals,from which,aftertransforming back to r, we get R?(r)=

(38)

2 -

nr +B(r2 21

r2+A 2 2

Ar2(A +

A2?3In

+

+Aln[A+Ar2

?A)2?+

+ AAr?+/(A2?+3)((Ar2?A)2?!3))

V(Ar2 +AA)2 +]}

Two classes of exact radiallysymmetricsolutionsto (18) are therefore givenby uE?(r, z, t) - R?(r) + V(z, t),

(39)

whereV(z, t) is any solutionto (33). 4.3. Asymptotic behaviour of X? and R?. It is immediatelyclear that X? givenby (29) is well behaved as x -* 0. By expressing(29) in the form

(A AA2 X+ (z) = + (xn

_ A

- - AX ?+ -x2 Ax+A+

B ?+

FA (Ax+A) A) (Ax+A)2

+

\ [?Ax(A)A2?+?

as x withsome analysisit can be shownthat the solutionsX? divergeto infinity ?00.

From (27) we may also establishthe asymptoticbehaviourof 1 + X. lim(1 +X)

= 2A?

lim (1 +X

= if{0

X-4+00

lim(1 + X' X--+0

=

lim (1 + X) lim (

X

_

X-+00( ?x)=

AK>O,

ifA 0,

0

ifA > 0,

-00

if A > 0,

-ooif A 0, thenwe need b = 2 to controlthese terms;i.e., we require (44)

-(A-1)

= ?A.


EXACT SOLUTION

OF STIKKER'S

1071

HEAT EQUATION

For R+ then we need A = 1/2, while forR_, equation (44) cannot be satisfied.We that it is impossibleto controlboth the algebraicand the logarithmic see, therefore, termssimultaneously;the formerare bounded forR+ when A < 0 and forR_ when A > 0, whilethe latterare bounded onlyforR+ withA > 0 and A = 1/2. Hence, R? can be bounded as r -* 0 but are unboundedas r -* 00. The asymptoticbehaviourof 1 + rR' is easily determinedfrom(35). lim(1?+rR')

Ai?,3,

-A?

> = 10o ifAKO, ifA >