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On the stability of thermal conduction
Volume 87A, number 9
PHYSICS LETTERS
8 February 1982
ON THE STABILITY OF THERMAL CONDUCTION S. SIMONS Department ofApplied Mathematicr, Queen Mary College, London, El 4NS, England Received 23 September 1981
It is shown that previous treatments of the stability of thermal conduction will break down when very rapid changes in the physical parameters can occur. A new approach is developed which remedies this.
Recent papers by Lebon et al. [1] and Sieniutycz [2] have been concerned with the question of stability during thermal conduction. That is, if we apply a spatially varying but temporally constant temperature over the boundary of a body, there wifi exist a stationary distribution for the temperature T(r) which satisfies Laplace’s equation with the given boundary conditions. However, if the initial temperature distribution within the body is not this stationary one, will T(r, t) tend towards the stationary solution as t 00? If the answer is in the affirmative we consider the stationary state to be stable, and it is known that such stability will exist if the time-dependent situation is described by the usual time-dependent conduction equation. The above authors, however, have realised that the validity of this latter equation is limited by the requirement that all changes in T should occur slowly, so that thermal equilibrium is effectively established at each point throughout the process. In order to tackle the problem when this is not so, they have therefore used the modification in the conduction equation = K dT q gra ~
The first purpose of the present communication is to point out that if physical parameters are changing sufficiently rapidly, then even eq. (2) is an inadequate description of the situation. The reasons for this are given in detail by Simons [3] where it is shown that the form of the required corrections to eq. (1) to cope with rapid changes depends on the variation with energy of the relaxation time of the particles (for example, electrons or phonons), which conduct the heat. If this variation is too great then the form of the correction term in eq. (2) will always be incorrect, and it appears that in the majority of cases for phonons this will in fact be the case. If, however, the variation is sufficiently weak, and in particular if the relaxation time is a constant, then the correction to eq. (1) may be expressed as a power series in r(a/at) of which the first term (apart from a numerical factor) is the correction term given in eq. (2). It follows therefore that the validity of eq. (2) is at best restricted to changes which are ~ot too rapid, and thus the approach of Lebon et al. and Sieniutycz will be inadequate for the general case where arbitrarily rapid changes in T may occur. In view of the above comments we proceed now to
where q is the heat flux and K the thermal conductivity, suggested by various authors to cope with rapid changes in the physical parameters. This modification takes the form
formulate a new proof of the stability of thermal conduction which will apply to all situations where the variation of temperature throughout the specimenis small comparedto the mean temperature. We suppose
q + r(dq/dt)
the specimen to be in the shape of a parallelepiped with a temporally constant temperature T(R) given at all points R on the surface, and in the steady state there will be a stationary distribution T(r) inside the specimen. We suppose, however, that the temperature insideis in fact an arbitrary distribution T*(r, t) and we define
-~
—
‘
=
—K
ad T
(2)
where r is the thermal flux relaxation time, and on the basis of this equation they have tackled the stability question. -
0 03’i-9163/82/0000—0000/$ 02.75 © 1982 North-Holland
453
Volume 87A, number 9
O(r, t) = T*(r, t)
PHYSICS LETTERS
The boundary conditions will then ensure that for all t
Here, v is the particle velocity, L is the linearized collision operator which describes interactions between the particles and, for the case of electrons,M is the
O’R 1’~ ~ 0
linearonoperator field the particles. describing We obtain the effect a wave of asolution magnetic of eq.
—
—
T(r)
(3)
8 February 1982
.
(4~
.
‘~ ‘
It is clear that to prove stability of thermal conduction we require to show that for alir
(9) by letting 4(k, r, t) = Re[~z(k) ei~~)t)J
tim O(r, t)
which yields
0
(5)
.
i+i(ug)~z+L~1+M~1=0.
00
In view of eq. (4) we can perform a three-dimensional Fourier analysis of U (r, 0), which takes the form O(r,0)=~Agewr. (6) g Here the summation is over all vectors g of the lattice which is reciprocal to the lattice whose unit cell is the specimen, and Ag are suitable constants. For t> 0 each of the terms in the summation (6) will give rise to a thermal wave in the medium so that for general t
o (r,
,
t) = ~ Ag e1(Z.r1~,t)
(10)
(11)
We now define the scalar product of two functions a(k) and 13(k) by (a,13)=f(_aF/8E)a*(k)13(k)dk,
(12)
and proceed to take the scalar product with ~iof eq. (11). This gives + g (u~= i(~i,Lj.~)+i(p, Mp). (13) Now,
(7) g where w(g) is the angular frequency corresponding to wave number g. To investigate the g dependence of w it is necessary to consider the time-dependent Boltzmann equation which describes the distribution of the particles (usually electrons or phonons) which transport energy in the medium. If f(k, r, t) is the distribution function for particles of wave number k, we let
It is known that L is a hermitian positive definite operator, while Mis a skew hermitian operator. Thus is real and positive and (j.t, Mgi) is purely imaginary. It follows therefore from eq. (13) that
f(k, r,
References
t)
= F(E,
T)
—
T’(aF/aE)q(k, r,
t),
(8)
where F(E, T) is the equilibrium distribution (Fermi— Dirac or Bose—Einstein) for particles of energy E, and the relevant Boltzmann equation then takes the form
(aØ/at) + u• grad q5 + Lçb + Mc~
454
0.
(9)
Im[co(g)] = (ii, Lp)/ 0. (14) Referring to eq. (7), it is seen that eq. (14) implies the result (5). The stability of thermal conduction follows immediately.
[1] G. Lebon and J. Casas-Vazquez, Phys. Lett. 55A (1976)
L21 S. Sieniutycz, Phys. Lett. 78A (1980) 433. [3J S. Simons, Phys. Lett. 66A (1978) 453.
PHYSICS LETTERS
8 February 1982
ON THE STABILITY OF THERMAL CONDUCTION S. SIMONS Department ofApplied Mathematicr, Queen Mary College, London, El 4NS, England Received 23 September 1981
It is shown that previous treatments of the stability of thermal conduction will break down when very rapid changes in the physical parameters can occur. A new approach is developed which remedies this.
Recent papers by Lebon et al. [1] and Sieniutycz [2] have been concerned with the question of stability during thermal conduction. That is, if we apply a spatially varying but temporally constant temperature over the boundary of a body, there wifi exist a stationary distribution for the temperature T(r) which satisfies Laplace’s equation with the given boundary conditions. However, if the initial temperature distribution within the body is not this stationary one, will T(r, t) tend towards the stationary solution as t 00? If the answer is in the affirmative we consider the stationary state to be stable, and it is known that such stability will exist if the time-dependent situation is described by the usual time-dependent conduction equation. The above authors, however, have realised that the validity of this latter equation is limited by the requirement that all changes in T should occur slowly, so that thermal equilibrium is effectively established at each point throughout the process. In order to tackle the problem when this is not so, they have therefore used the modification in the conduction equation = K dT q gra ~
The first purpose of the present communication is to point out that if physical parameters are changing sufficiently rapidly, then even eq. (2) is an inadequate description of the situation. The reasons for this are given in detail by Simons [3] where it is shown that the form of the required corrections to eq. (1) to cope with rapid changes depends on the variation with energy of the relaxation time of the particles (for example, electrons or phonons), which conduct the heat. If this variation is too great then the form of the correction term in eq. (2) will always be incorrect, and it appears that in the majority of cases for phonons this will in fact be the case. If, however, the variation is sufficiently weak, and in particular if the relaxation time is a constant, then the correction to eq. (1) may be expressed as a power series in r(a/at) of which the first term (apart from a numerical factor) is the correction term given in eq. (2). It follows therefore that the validity of eq. (2) is at best restricted to changes which are ~ot too rapid, and thus the approach of Lebon et al. and Sieniutycz will be inadequate for the general case where arbitrarily rapid changes in T may occur. In view of the above comments we proceed now to
where q is the heat flux and K the thermal conductivity, suggested by various authors to cope with rapid changes in the physical parameters. This modification takes the form
formulate a new proof of the stability of thermal conduction which will apply to all situations where the variation of temperature throughout the specimenis small comparedto the mean temperature. We suppose
q + r(dq/dt)
the specimen to be in the shape of a parallelepiped with a temporally constant temperature T(R) given at all points R on the surface, and in the steady state there will be a stationary distribution T(r) inside the specimen. We suppose, however, that the temperature insideis in fact an arbitrary distribution T*(r, t) and we define
-~
—
‘
=
—K
ad T
(2)
where r is the thermal flux relaxation time, and on the basis of this equation they have tackled the stability question. -
0 03’i-9163/82/0000—0000/$ 02.75 © 1982 North-Holland
453
Volume 87A, number 9
O(r, t) = T*(r, t)
PHYSICS LETTERS
The boundary conditions will then ensure that for all t
Here, v is the particle velocity, L is the linearized collision operator which describes interactions between the particles and, for the case of electrons,M is the
O’R 1’~ ~ 0
linearonoperator field the particles. describing We obtain the effect a wave of asolution magnetic of eq.
—
—
T(r)
(3)
8 February 1982
.
(4~
.
‘~ ‘
It is clear that to prove stability of thermal conduction we require to show that for alir
(9) by letting 4(k, r, t) = Re[~z(k) ei~~)t)J
tim O(r, t)
which yields
0
(5)
.
i+i(ug)~z+L~1+M~1=0.
00
In view of eq. (4) we can perform a three-dimensional Fourier analysis of U (r, 0), which takes the form O(r,0)=~Agewr. (6) g Here the summation is over all vectors g of the lattice which is reciprocal to the lattice whose unit cell is the specimen, and Ag are suitable constants. For t> 0 each of the terms in the summation (6) will give rise to a thermal wave in the medium so that for general t
o (r,
,
t) = ~ Ag e1(Z.r1~,t)
(10)
(11)
We now define the scalar product of two functions a(k) and 13(k) by (a,13)=f(_aF/8E)a*(k)13(k)dk,
(12)
and proceed to take the scalar product with ~iof eq. (11). This gives + g (u~= i(~i,Lj.~)+i(p, Mp). (13) Now,
(7) g where w(g) is the angular frequency corresponding to wave number g. To investigate the g dependence of w it is necessary to consider the time-dependent Boltzmann equation which describes the distribution of the particles (usually electrons or phonons) which transport energy in the medium. If f(k, r, t) is the distribution function for particles of wave number k, we let
It is known that L is a hermitian positive definite operator, while Mis a skew hermitian operator. Thus
f(k, r,
References
t)
= F(E,
T)
—
T’(aF/aE)q(k, r,
t),
(8)
where F(E, T) is the equilibrium distribution (Fermi— Dirac or Bose—Einstein) for particles of energy E, and the relevant Boltzmann equation then takes the form
(aØ/at) + u• grad q5 + Lçb + Mc~
454
0.
(9)
Im[co(g)] = (ii, Lp)/
[1] G. Lebon and J. Casas-Vazquez, Phys. Lett. 55A (1976)
L21 S. Sieniutycz, Phys. Lett. 78A (1980) 433. [3J S. Simons, Phys. Lett. 66A (1978) 453.