The finite velocity of propagation of thermal perturbations in media with constant thermal conductivity

Propagationof thermal pe~ur~atio~s

141

6.

BAKLANOVSKAYA, V. F., Study of the mesh method for solving the Fist boundary value problem for equations of the non-stationary seepage type, in: Numerical methods for solving differential and integral equations and quadratureformulae (Chislennye metody resheniya differents. i integr. urnii i kvadraturnye f-ly), Nauka, 228-243, Moscow, 1964.

7.

COURANT, R., FRIEDRICHS, K., and LEVI, G., On difference equations of mathematical physics (Russian translation), Usp. matem. Nuuk, No. 8,125-160,194l.

THE FINITE VELOCITY OF PROPAGATION OF THERMAL PERTURBATIONS IN MEDIA WITH CONSTANT THERMAL CONDUCTIVITY L. K. MARTINSON

Moscow (Received 27 December 1974)

TIIE SOL~IO~S of the parabolic heat conduction equations are considered for a medium with constant thermal conductivity with a volume heat absorption which is a non-linear function of temperature. The solutions found have the form of temperature waves whose fronts propagate with a finite velocity relative to the zero unperturbed background. The conditions for the existence of such solutions are analyzed. 1. As shown in [l-4] , the quasilinear parabolic equation

describing, in particular, the heat conduction process in media with a thermal conductivity which is a power function of temperature, has for n> 1 generalized solutions in the form of temperature waves whose fronts propagate with finite velocity relative to the zero unperturbed background. The existence of such solutions is due to the finite velocity of propagation of heat perturbations in such media [4,5 J . For n< 1 the heat perturbations propagate with infmite velocity [6] and the front solutions of Eq. (1.1) do not exist. The physical inte~retation of the effect of the fmite velocity of propa~tion of the heat perturbations for n) 1 is usually associated with the fact that on the temperature wavefront (where u = 0 and Eq. (1.1) is degenerate) the thermal conductivity vanishes and this leads to retardation of the heat propagation process. In [7] the possibility was indicated that temperature waves with a fmite velocity of propagation of the front may also exist in media with constant thermal conductivity (n = 1) when volume heat absorption occurs in the medium whose power is a non-linear function of temperature. This means that an equation of the form

~t=w--f(4

(1.2)

subject to certain conditions imposed on Au), may also have generalized solutions of wave type with a ftite velocity of propagation of the temperature wavefront. 2. We consider a series of exact non-stationary self-similar solutions of Eq. (1.2) for some actual non-linear relations of a special form for flu). These solutions correspond to temperature 9% v.?chisLMat. mat. Fin, l&5,1233-1241,

1976.

142

L. K. Martinson

waves whose fronts propagate with constant fmite velocity relative to the zero unperturbed background. Let

f=f*(U)=f*Ua(aa-‘-I),

f*=const~0,

*/6cKl.

(2.1)

Relation (2.1) corresponds to volume heat sources of dipole type. For u> 1 heat dissipation occurs in the medium, and for UC 1 heat absorption occurs, where fl0) = 0. Equation (1.2) for the case (2.1) has a solution in the half-plane R+‘= { (t, z) : OG<+a, XEij’) :

u (t, 2) =

u#J (t-

x/c) i’(1-a),

1

0,

XC&

Q-2)

xiwt,

If we now denote by m the greatest integer satisfying the condition n2< (l-a) -*, then (2.2) implies that ufS?‘(R+Z) and UEC” (R+Z\IT), where f’ is the set of points in R+2 satisfying the condition x = ct. The line x = ct is a line of weak discontinuity of the function U, and, since m>Z, in the case considered, the physical conditions of continuity of temperature and heat flow are satisfied for the solution (2.2) at any point. The front of the temperature wave (2.2) propagates in the medium with the constant finite velocity c. We note that close to the temperature wavefront, where OtuCl, and f(u) >O, volume absorption of heat occurs. It may therefore be supposed that it is precisely the absorption of heat close to the temperature wavefront which ensures the existence of a front propaga~ng with ftite velocity, although the thermal conductivity in this case does not vanish on the temperature wavefront. Indeed, we consider the case where there exists in the medium only volume absorption of heat whose power depends on temperature so that

f=fi+

1,

r>O,

0,

60.

e(r)={

(u>=fr(u)0 (l--u>,

Here fl%CO (R,‘) , where R+‘= (u: 0.(UC-+},

and also, fit(u)

(2.3)

20 for all r.GR+‘.

Equation (1.2) has a solution, also for heat sinks of the form (2.3), which corresponds to a temperature wave whose front propagates with fmite velocity, independent of time, relative to the zero unperturbed background:

I+(exp[e”(g-%*)

u (4 xl -(%I =

where

E=t--x/c, c2=af,c+y

(E&o) *‘(-, i

&,=[fo(l-~)

0,

1-l.

I-l}a-‘,

%>%o, O<%G%o, $60,

(2.4)

143

Propagationof thermal perturbations

( E=tO) the solution (2.4) has in the general (f;=O) ands=ct-ego On the two lines s=ct case a weak discontinuity along derivatives of higher than the second order. It is obvious that solutions of plane temperature wave form with constant velocity of propagation of the front may also be found for other functions j(u) in (1.2). In particular, it is interesting to consider the case where j(u) is a step function, possessing a discontinuity of the first kind at two points such as u = 0 and u = u:

fo= const f(u)={

o

> 0,

o
7

u&L..

In this case for OCU-CU. heat absorption takes place in the medium with constant power of the volume heat sinks, and Eq. (1.2) also has in R+* a generalized solution in the form of a plane temperature wave with the self-similar variable &t--x/c:

b-+f0E0) exp[c’(Ho) 1-foEo,

u (t, 5) =u &) =

where the quantity to>0

!

$ ~exp(cZg)-w-foE,

Pb7 o-Qeo,

(2.5)

EGO,

0,

is defined as the solution of the transcendental equation (U*+f0E0)

c2=fo[exp

(2.6)

(cl&J -I].

It follows from (2.6) that to-+ao as ~.*m andEo-+O as u&+0. The first passage to the limit in (2.5) leads to the solution previously obtained in [8] : u(t,x)=

/ ${exp[c2(t-$)]

-l)-j0(r-f)

7

scct~ x2ct.

0,

The second passage to the limit gives a non-trivial solution, if fO-+~~ as u.+O, so that the quantity Q-f&, describing the quantity of heat absorbed close to the front, remains finite. The limiting solution corresponding to this delta-shaped nature of the dependence of the power of the volume heat sinks on temperature, namely

u (t, x) =

lp{exp[cz(~L$)J-~}I

0,

X
(2.7)

x>ct,

describes a temperature wave for which the heat absorption takes place in an infinitely narrow layer close to the front. In this case the heat flux has a discontinuity at the temperature wavefront (the line x = ct), equal in amount to the quantity of heat absorbed at the temperature wavefront. Despite the special form of the dependence of the power of the volume heat sinks on the temperature, the solutions (2.2), (2.4) and (2.5) can be given a deftite physical meaning. Firstly, these solutions indicate the possibility of the propagation, in a medium with constant thermal conductivity, of temperature waves with finite velocity of displacement of the wavefront, if from some region close to the wavefront, where the temperature is less than some definite value, there

144

L. K. Martinson

occurs a radiation of energy to infinity equivalent to the presence of heat sinks close to the front. Because of the effect of screening of the radiation by non-transparent high-temperature regions the radiation process in the propagation of heat waves is qualitatively of precisely this nature [9]. Moreover, the solutions (2.4) and (2.5) may be regarded as solutions of the problem of the propagation of a phase transition front with heat absorption (Stefan’s problem), if the heat absorption occurs close to the phase transition front in a layer of finite width [lo] . The limiting solution (2.7) then corresponds to the case where the heat absorption occurs in an infinitely narrow layer close to the front of the phase traction. In the examples considered for some specific forms of non-linearity offlu) exact selfsimilar solutions of Eq. (1.2) were found in the form of plane temperature waves propagating relative to the zero unperturbed background with constant ftite velocity of displacement of the front. In the case of an arbitrary function AU) it is possible to formulate some necessary conditions on the function f, for the satisfaction of which Fq. (1.2) can have a front solution of the form u (E, 2) = u(i), where E=t--r/c, c=const, such that u>O for all PER’, and

u(E)=

mi),

1

P-0,

w9

EGO.

o t

In the general case it is also necessary to require that the solution sought u&’ (R*), This requirement ensures the satisfaction of the physical conditions of continuity of temperature and heat flow at all points, including the point of the temperature wavefront t = 0. But if, near to zero, flu) has a singularity of &function type, then it is sufficient to require that ued.T (R’) flCi (R’\@) , where $3 is the zero element of the set R 1 (see the limiting solution of (2.7)). Since the function u(t) must satisfy the ordinary differential equation (the primes denote differentiation with respect to 4) a”-c2u’-c2f

(22)=o,

(2.9)

then the existence of a front solution of the form (2.8) for (2.9) first of all requires that the conditionfi0) = 0 be satisfied. Therefore, u=O is a solution of Bq. (29). On the other hand, the solution u=O’must be a singular solution-of Bq. (2.9). Then the integral curve corre~onding to a temperature wave of the form (2.8) may be composed of a singular solution u = 0 for tO, joined at the point of the temperature wavefront 5 = 0. Therefore the conditions necessary for the existence of front solutions of the type indicated for (l-2), are the conditions for the violation of the uniqueness of the solution of the Cauchy problem for Eq. (2.9) at the point $ = 0 where tl = 0. The conditions for the existence and uniqueness of the solution of the Cauchy problem for (2.9) at the point t = 0 are violated if flu) has a discontinuity at the point u = 0 or if at this point f does not satisfy a L.ipschitz condition (in the weaker formulation: if f&O) is not bounded) [ 111. The examples of exact selfsimilar solutions considered above correspond to precisely this type of singularities off. Assuming that (2.9) has a frontal solution of the form (2.8), it may be noticed that if@) has a discontinuity of the first hind at the point u = 0, then the second derivative u”(t) has a discontinuity on the tern~ra~~ wavefront. Denoting by curly brackets the difference in the values of the quantities at the jump, we have

{u”}=c2{f}.

(2.10)

Propagationof thermal perturbations

145

Since for a solution of the form (2.8) we have {u”} >O, then (2.10) implies that {j}>O, that is, absorption of heat must occur close to the front of the temperature wave, and close to the point C;= 0 the asymptotic expression for fi (k) has the form

E(E)3

$

{f}E”.

A similar investigation of (2.9) for a functionfof the form f(u) =fOuy, where fO= const>0, O-Cr< 1, leads to the following asymptotic expression for iZ(g) :

foC2 w--7)2 i’(i-~)g2,(,_,)~ E(E)-

[

(2.11)

I

2(1+7)

3. We consider in a more general formulation the question of the fmite velocity of propagation of thermal perturbations in a medium with constant thermal conductivity in the presence of volume absorption of heat, depending on the temperature. We reduce the investigation of the one-dimensional non-stationary process of the propagation of heat perturbations relative to a zero unperturbed background for this case, to the solution of the first boundary value problem for u(x, r) in the half-strip G= { (t, Z) : O
&=&c-f(U),

u(t, O)=q$),

We confine ourselves to the case where f(u) >O for UN,

u(t, +=J)=o.

(3.1)

cp(t) >O for DO, and @J(O) = 0.

The velocity of propagation of the heat perturbation from the wall x = 0, the source of perturbations, will be finite if for all TV [ 0, T] we can fmd an x=x, (t) C+OO such that u=O for Z>Z, (t) . In other words, the fmite velocity of propagation of the heat perturbations corresponds to a finite solution of (3.1) for all t=[O, T]. We will solve the boundary value problem (3.1) by Rothe’s method [12], based on the discretization of the time variable. Introducing the mesh t=th=kr, k = 0, 1, . . . , N, with sufficiently small step T and replacing the operator Nat by its finite-difference analog, for the determination of the approximate value Q(X) of the function u(t, x) at the points t = fk we obtain the following semidiscrete approximation of the boundary value problem (3.1) as a system of boundary value problems for the ordinary differential equations:

et

u,-Uk-I

uk (0) =(Pkr

I=

(Uk)xc-f (Uk), U,,(+OO) -0,

k=l,2

,...,

N, (3.2)

where uu = 0, and ‘pk=‘p (k7). It can be shown [ 13,141, that the error of this method is of the order of O(T) and that as r-+0 the convergence r&+-u holds. The velocity of propagation of the front of the heat perturbations in (3.1) is finite, if at each step in k the system (3.2) has a frontal finite solution, that is, if for any k we can find an &<+oO, such that Uk = 0 for all C?%zk, and in accordance with the requirements of continuity of the temperature and heat flow,ukECi (R+‘) . If such a point x exists, then it is the frontal point separating the region where at the instant t = tk the heat perturbation has reached, from the unperturbed region a+&.

L. K. Martinson

146

To clarify the conditions for which the velocity of propagation of heat perturbations in processes described by the boundary value problem (3.1) will be finite, it is essential to analyze (3.2) for the first step (k = l), since it can be shown that the conditions for the existence of a frontal solution of (3.2) at succeeding steps (k> 1) do not differ from the corresponding conditions at the first step. On the other hand, if a frontal point zlC-l-~, is not found for ul(x), then the velocity of propagation of the heat perturbations for (3.1) will be infmite. For the first step from (3.2) we have

(UJd- z-‘u,-f(u,)

us (+w)

111(O) =(pit

4,

-0.

(3.3)

Problem (3.3) can have a finite solution only if Us-0 is a singular solution of (3.3). Then the frontal solution (in the general case a generalized solution) of problem (3.3) will be a particular solution of Eq. (3.3), whose integral curve passes through the points ($1~0) and (0,x1), joined to the singular solution u,=O for zX%. This implies that the condition AO) = 0 is one of the necessary conditions for the existence of a finite frontal solution of (39, which can be represented in the form “1

s

lp-” (q) dq = xi-x,

X--l,

(3 *4)

0

ui--0,

x2x1,

where

Therefore, the boundary value problem (3.3) has a finite solution (3.4), if

(3.5) Because of the conditions imposed on f and @J,the integrand in the quadrature (3.5) has a singular@ only at the point r) = 0. Condition (3.5) will be satisfied if this singularity is integrable. In particular, we note that if f-0 condition (3.5) cannot hold, which corresponds to an infinite velocity of propagation of heat perturbations for this case [lo] . The function u r(x), defined by the implicit expression (3.4), is continuous together with its first derivative for all. zeJ?+‘. It can be shown that also at the succeeding steps for k = 2,3, . . . ,N on satisfaction of condition (3.5) the boundary value problems (3.2) will have frontal solutions of the form

u,(x)=

h(x)

1 o

9

9

X-=X&,

xaq.

It is easy to see that condition (3.5) is equivalent to the condition j [ jf(V)&]-% 0 0

d?l <+w.

(3.6)

&O&?QgQtiOll

Of thtV?lQ~

147

pertU&UtionS

The value of the upper limit of integration with respect to n in (3.6) is immaterial and without loss of generality may be put equal to unity. Therefore, (3.6) for j(O) = 0 is a condition of finite velocity of propagation of heat perturbations in problem (3.1). Thereby it imposes constraints on the nature of the behaviour off close to u = 0. In particular, for f(u) =I&‘, where f0=const>0, and TX, the velocity of propagation of heat perturbations will be finite, if 7< 1. Indeed, in this case relation (3.6) assumes the form

(3.7) Condition (3.7) is satisfied if

7
Also, condition (3.6) is satisfied by an flu) which has a di~ont~ui~ of the first kind at the point u = 0. The solution of (3.4) is then a solution of (3.3) in a generalized sense, since the second derivative has a discontinuity at the point x = xl. These results are in accordance with the results of section 2 of the present paper. Therefore, a finite velocity of propagation of heat perturbations in non-stationary processes described by quasilinear parabolic equations, may be connected not only with the vanishing of the thermal iconducti~ty on the temperature wave front, that is, with the ~~1~~ of the equation at the points where u = 0, but also with the presence in the medhrm of non-&rear heat absorption which is a defmite function of the temperature close to the front of the temperature wave. In the problems considered (with the exception of the l~it~g problems with a singularity off of S-function type) the finite frontal solutions r.z~C’~(G)have physical significance. In other words, in these problems the temperature waves cannot have steep fronts, as in probIems for equations of the form (1 A). Therefore finding gently sloping fronts of temperature waves by numerical finite-difference methods with complete discretization of the time and space variables involves certain difficulties, while Rothe’s semidiscrete semianafytic method enables the motion of such a gently sloping front to be traced. We aiso note that the conditionsff0) = 0 and (3.6) are by f 151 sufficient conditions for the spatial localization of the heat ~~urbations in the Cauchy problem for Eq. (1.2), if the initial distribution is a finite function. This testifies to the fact that the effect of spatial localization of heat perturbations is associated with the finite velocity of propagation. 4. The question of the ftite velocity of propa~tion of the front of the temperatum wave for processes described by Eq. (I ‘2) can also be investigated by considering the velocity of motion of the isotherms on the propagation of the heat perturbation [3] . Indeed, let the relation x = x0(r) correspond to the law of motion of the isotherm u = UI-J= const. Then, differentiating the equation u (t, z. (t) ) =ao, we obtain for the velocity of motion of the isotherm

(4.1)

148

L. K. Martinson

If the front of the temperature wave z=t (t) propagates relative to an unperturbed background, then its velocity of motion may be regarded as the velocity of motion dcldt of the isotherm u = 0. Since on the front of the temperature wave u=dulc?z=O, the expansion of u(t, x) in the perturbed region close to the frontal point z=t (t) has the form nctA=(W~

ak(5-z)k,

a>l.

(4.2)

h-0

It follows from Eqs. (4.1) and (4.2) that the velocity of motion of the temperature wavefront propagating relative to a zero unperturbed background can be finite only if f+O. Indeed, for +O from (4.1) taking into account (4.2) we have the known result dgldt+co. But if f+O and close to u = 0 the asymptote offhas the form f(u) =:fOuT,where O
This is found to be in agreement (see, for example, (2.11)) with the results obtained above. Translatedby J. Berry REFERENCES 1.

ZEL’DOVICH, Ya. B. and KOMPANEETS, A. A., On the theory of the propagation of heat for a thermal conductivity dependent on temperature. In: Collection dedicated to the 70.th birthday ofA. F. Ioffe (Sb. posvyashchennom 70-letiyu A. F. Ioffe), 61-71, Izd-vo Akad. Nauk SSSR, Moscow, 1950.

2.

BARENBLATT, G. I., On some non-stationary Mekh., 16, 1,67-78, 1952.

3.

SAMARSKII, A. A. and SOBOL, I. M., Examples of the numerical calcuktion of temperature waves. Zh. vychisl.Mat. mat. Fiz., 3,4,702-719, 1963.

4.

OLEINIK, 0. A., KALASHNIKOV, A. S. and CHOU YU-LIN., The Cauchy problem and boundary value problems for an equation of unsteady filtration type. Izv. Akad. Nauk SSSR. Ser. Matem., 22,5, 667-704,1958.

5.

BARENBLATT, G. I. and VISHIK, M. I., On the fmite velocity of propagation in problems of the non-stationary filtration of a liquid and gas Prikl Mat. Mekh., 20,3,411-417, 1956.

6.

KALASHNIKOV, A. S., On equations of the unsteady filtration type with an infinite velocity of propagation of perturbations. Vestn MGU. Matem, mekhan., No. 6,45-49,1972.

7.

MARTINSON, L. K. and PAVLOV, K. B., The problem of the threedimensional localization of thermal perturbations in the theory of non-linear heat conduction. Zh. vj%hisl Mat. mat. Fiz., 12.4, 1048-1053, 1972.

8.

PAVLOV, K. B., The spatial localization of transition layers in problems of the non-linear theory of thermal conductivity. PrikL mekhan. tekhn. fiz,, No. 4,179-181,1973.

9.

ZEL’DOVICH, Ya. B. and RAIZER, Yu. P., The physics of shock wavesand high-temperature hydrodynamic phenomena (Fizika udamykh voln i vysokotemperatumykh gklrodinamicheskikh yavlenii), “Nauka”, Moscow, 1966.

motions of a liquid and gas in a porous medium. Rikl Mat.

10. TIKHONOV, A. N. and SAMARSKII, A. A., The equationsof mathematicalphysics (Uravneniya matematicheskoi fiziki), “Nauka”, Moscow, 1966.

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Gal&kin’smethod in problems of diffraction 11. MATVEEV, N. M., Methods of integration of ordinary differential equations (Metody integrirovaniya obyknovennykh differentsial’nykh uravnenii), “Vysshaya shkola”, Moscow, 1967. 12. ROTHE, E., Wiirmeleitungsgleichungen 1931.

mit nichtkonstanten

Koeffizienten.

Math. Ann., 104, 340-342,

13. LADYZHENSKAYA, 0. A., SOLONNIKOV, V. A. and URAL’TSEVA, N. N., Linear and non-linear equations of the parabolic type (Lineinye i nehneinye uravneniya parabolicheskogo tipa), “Nauka”, Moscow, 1967. 14. LIONS, J.-L. Some methods of solvingnon-linear boundary value problems (Nekototye metody resheniya nelineinykh kraevykh zadach), “Mir”, Moscow, 1972. 15. KALASHNMOV, A. S., The propagation of disturbances in problems of non-linear heat conduction with absorption. Z/r. v?chisL Mat. mat. Fis., 14,4,891-905, 1974.

GALERKIN’S METHOD IN PROBLEMS OF DIFFRACTION AT SMOOTH WAVE-LIKE SURFACES* 0. B. POPOV Armavir (Received

A COMPLETE justification

1 November

1974)

of a direct method, similar to Gal&kin’s method, for the solution of problems of diffraction by a wide class of reflecting gratings. is given

At the present time there is great interest in the problems of diffraction by various types of periodic structures, in particular ribbon and reflecting gratings, in the long waveband commensurate with the period of the grating. In the case of inhomogeneous media direct numerical methods are the most efficient for the construction of approximate solutions. For a fairly extensive class of such problems it is possible to reduce the boundary value problem of diffraction in an unbounded region to an interior boundary value problem. This is achieved by substituting the radiation conditions in a special form of partial conditions [l] . In particular, this reduction is possible for problems of diffraction by smooth wave-like surfaces of arbitrary profile, when the region of inhomogeneity along the lattice is a layer of finite width.

We consider the diffraction of a scalar field by a wave-like surface (u) whose profile in the plane (x, y) is defmed by the equations s,=Aa(u),

yo=Au,

a(v)>O,

o(u+2n)=a(L7),

--oc
A SO, [A ] =L, (Tand v are dimensionless. In the region of diffraction (to the left of (0)) we assume that the medium is inhomogeneous. The field Jl(x, y) is described by the equation

[Ax,,+k%, Y)I$=0 with periodic

coefficient

k (5,

y) =k (z,

y+2nA),

constant in the region z
k (2, y) =const=ko. On

the surface (a) let the impedance boundary conditions

*Zh. vychisL Mat. mat. Fiz., 16,5, 1242-1251,

1976.

(1)