J. Fluid Mech. (2012), . 697, pp. doi:10.1017/jfm.2012.36 ...

J. Fluid Mech. (2012), vol. 697, pp. 60–91. c© Cambridge University Press 2012 60doi:10.1017/jfm.2012.36

Effects of heat conduction in a wall onthermoacoustic-wave propagation

N. Sugimoto† and H. Hyodo

Department of Mechanical Science, Graduate School of Engineering Science,University of Osaka, Toyonaka, Osaka 560-8531, Japan

(Received 13 July 2011; revised 20 November 2011; accepted 14 January 2012;first published online 6 March 2012)

This paper examines the effects of heat conduction in a wall on thermoacoustic-wave propagation in a gas, as a continuation of the previous paper (Sugimoto, J.Fluid Mech., 2010, vol. 658, pp. 89–116), enclosed in two-dimensional channelsby a stack of plates or in a periodic array of circular tubes, both being subjectto a temperature gradient axially and extending infinitely. Within the narrow-tubeapproximation employed previously, the linearized system of fluid-dynamical equationsfor the ideal gas coupled with the equation for heat conduction in the solid wallare reduced to single thermoacoustic-wave equations in the respective cases. In thisprocess, temperatures of the gas and the solid wall are sought to the first orderof asymptotic expansions in a small parameter determined by the square root ofthe product of the ratio of heat capacity of gas per volume to that of the solid,and the ratio of thermal conductivity of the gas to that of the solid. The effectsof heat conduction introduce into the equation two hereditary terms due to triplecoupling among viscous diffusion, thermal diffusion of the gas and that of the solid,and due to double coupling between thermal diffusions of the gas and solid. Whilethe thermoacoutic-wave equations are valid always for any form of disturbancesgenerally, approximate equations are derived from them for a short-time behaviourand a long-time behaviour. For the short-time behaviour, the effects of heat conductionare negligible, while for the long-time behaviour, they will affect the propagation asa wall becomes thinner. It is unveiled that when the geometry of the channels or thetubes, and the combination of the gas and the solid satisfy special conditions, theasymptotic expansions exhibit non-uniformity, i.e. a resonance occurs, and then thethermoacoustic-wave equations break down. Discussion is given on modifications inthe resonant case by taking full account of the effects of heat conduction, and also onthe effects on the acoustic fields.

Key words: acoustics, channel flow, instability

1. IntroductionThis paper examines, as a continuation of the previous paper (Sugimoto 2010,

referred to as I hereafter), the effects of heat conduction in a solid wallon thermoacoustic-wave propagation in gas-filled channels or tubes subject to atemperature gradient axially. Neglecting the effects, in I, the linearized system of

† Email address for correspondence: [email protected]

mailto:[email protected]

Effects of heat conduction in a wall on thermoacoustic-wave propagation 61

fluid-dynamical equations for the gas under a temperature gradient was reduced toa single thermoacoustic-wave equation for excess pressure by using the narrow-tubeapproximation in the sense that a typical axial length is much longer than a spanlength. This is a one-dimensional wave equation for propagation in a gas non-uniformin temperature and subject to shear stress and heat flux on the wall surface includedin the form of hereditary integrals. This equation is always valid for any form ofdisturbances generally, as long as the linearization does not break down.

Effects of heat conduction in a wall used in the title of this paper imply that thethermal conductivity of the wall is regarded as being large but finite in comparisonwith that of the gas, and so is the heat capacity of the wall per volume. When they areregarded as infinitely large, no variations in the wall temperature occur irrespective oftemperature variations in gas. This assumption may be relevant usually in the contextof classical acoustics. In recent thermoacoustic devices, however, so-called stacks areexploited (e.g. Swift 2002), in which ceramics or polymers are used for wall materials,or a wall thickness is comparable with a span length of the channels or the tubes. Insuch a situation, it is unclear as to whether or not the above assumptions are satisfiedfully. The purpose of this paper is to examine the effects on thermoacoustic-wavepropagation in gas enclosed in two-dimensional channels by an infinite stack of platesor in a periodic array of circular tubes, and to clarify how they affect the propagation.

There are few studies on the effects of heat conduction in a wall. Henry (1931)checked them as one of the various effects affecting sound speed given by Kirchhoff’s(1868) theory (see Rayleigh 1945) in the search for causes of discrepancy inmeasurements of the ratio of specific heats. The effects appear through the squareroot of the product of the ratio of the heat capacities of the gas to the solid pervolume, and the ratio of the thermal conductivities of the gas to the solid. Becausethis ratio, denoted by ε, is usually very small (of the order of 10−4) for air and metalsat room temperature in atmosphere, the effects were considered to be negligible. Inthe context of thermoacoustics, Rott (1973) examined the effects in cryogenic regimebecause some solids (e.g. copper) exhibit a significant decrease in heat capacity atlow temperature whereas the thermal conductivity increases very dramatically (AIPHandbook 1982). As a result, it happens that this ratio becomes rather greater thanunity. While such a special situation exists, there are cases in which the ratio is smalleven at cryogenic temperature.

Rott included the effects of heat conduction by restricting substantially to a casewhere the ratio is small, because instability would be suppressed if the ratio werelarge. Later Swift (1988) also included these effects in developing the linear theoryfor two-dimensional channels bounded by plates taking account of the finite thicknessof them. The level of approximation of Swift’s theory is the same as that of Rott’stheory and the present narrow-tube approximation. Using Swift’s theory, Gopinath, Tait& Garrett (1998) calculated the second-order thermoacoustic streaming in a resonantchannel and examined the temperature distribution in the stack. Karpov & Prosperetti(2002) took account of the effects to seek the emergence of nonlinear oscillations in aprime mover. Marx & Blanc-Benon (2004, 2005) examined thermoacoustic streamingin refrigerators to check the validity of the linear theory for a low amplitude. ButMarx & Blanc-Benon (2005) reported that the disagreement was not necessary due tononlinear effects.

It is seen in Swift’s (1988) theory in a frequency domain that the effects ofheat conduction introduce a factor {1+ ε tanh[H (−iω/κe)

1/2]/ tanh[d (−iω/κs)1/2]}−1

,where H and d denote, respectively, half the width of the channel and of the solid

62 N. Sugimoto and H. Hyodo

wall, and κe and κs denote, respectively, the thermal diffusivities of the gas and of thesolid, ω being a typical angular frequency and i an imaginary unit. For a small valueof ε, it appears to be appropriate to expand the factor asymptotically in terms of εand to take account of its first-order effects. In fact, this will be done in this paper.But because the factor has poles in a complex plane of the frequency, non-uniformityarises in the expansion, depending on values of d/H and κe/κs. This implies that asort of resonance occurs. Because ω then takes a pure imaginary value, this suggeststhe diffusion, of course. The diffusion in such a case is different from that in thenon-resonant case in the sense that their effects do not remain within the order ε.

In what follows, models for the problem are presented in § 2 with data regardingthe thermal properties of typical gases and solids. Two models are considered, one forpropagation in gas enclosed in two-dimensional channels, and the other in a periodicarray of circular tubes. In § 3, the system of linearized equations for gas supplementedby the equation of heat conduction in a solid wall in the narrow-tube approximationis summarized. By using the method of Fourier transform, in § 4, all field variablesare expressed in terms of the excess pressure and thermoacoustic-wave equations arederived in both cases. In § 5, relaxation functions involved in the heat flux on the wallsurface are explicitly evaluated in non-resonant cases, and the thermoacoustic-waveequations are approximated for a short-time and a long-time behaviour. In § 6, resultsare summarized and discussion is given on the modifications in the resonant case andthe effects of the heat condition on the acoustic fields.

2. Models of the problem

2.1. Geometry of the models

To examine the effects of heat conduction in a wall on thermoacoustic-wavepropagation, two models are considered. One is for propagation in two-dimensionalchannels between parallel plates of thickness 2d stacked in the direction normal tothe plates periodically with distance 2H apart. Taking the y-axis along this direction,the x-axis is taken parallel to the plates along the direction of wave propagation. Theorigin of the axes is taken at a midpoint spanwise in one ‘cell’ of the channels in thestack of plates. Figure 1 shows the geometrical configuration of the channels where novariations are assumed in the direction normal to the sheet of paper.

The other model is for propagation in a periodic array of circular tubes. Twoarrangements are conceivable, one being a square array in a plane normal to the axisof the tubes, and the other a staggered array, as shown in figures 2(a) and 2(b),respectively. In the square array, circular tubes of radius R and of thickness d areembedded in a solid matrix. The centres of the tubes are located at four corners ofsquare of side length 2(R + d). In the staggered array, the centres are located at threecorners of triangle of side length 2(R + d). If the material of tubes is the same as thatof the matrix, then the term ‘bores’ may be more suitable.

In both models, the channels and the tubes extend infinitely not only in thex-direction but also periodically in a plane normal to the x-axis so that no outerboundaries are considered. Solid walls are assumed to be rigid with smooth surface.A phenomenon occurring in one cell of the channels or the tubes is assumed to beidentical to those in the other cells. Because of the spatial periodicity in arrangements,only a unit cell with the origin of the coordinate axes taken is considered.


y

x

HTw

Te

Ts

Plate

Plate

Plate

Gas

Gas

3H 2d

H 2d

HO

FIGURE 1. Illustration of two-dimensional channels of width 2H separated by an infinitestack of solid plates of thickness 2d subjected to non-uniform temperature distribution in thex-direction where Te, Ts and Tw denote, respectively, the temperatures of gas, solid plate andwall surfaces in a quiescent state, and the origin of the coordinate axes x and y is chosen inone cell of the channels.

z

yy

rd

o

(a) (b) z

R R rd

o

FIGURE 2. Arrangements of the circular tubes in the axial cross-section where (a) and (b)show, respectively, the square and staggered arrays of the tubes of radius R and of thickness d,which are embedded in a solid matrix (shaded area), and the origin of the radial coordinate ris chosen in one cell bounded by the square in (a) and the hexagon in (b).

2.2. Steady temperature fieldsThe solid wall is subjected to a temperature gradient in the x-direction. In a quiescentstate of the gas where no gravity is assumed, steady temperature fields in the gas andthe solid are sought. Letting the temperature at the wall surface between the gas andthe solid be Tw, axial variation of this is assumed to be gentle enough over a distancecomparable with a span length to satisfy the following condition:

H2

Tw

∣∣∣∣∂2Tw

∂x2

∣∣∣∣� H

Tw

∣∣∣∣∂Tw

∂x

∣∣∣∣� 1, (2.1)

where Tw is a function of x only for the channels. For the tubes, H is replaced by Rbut Tw will depend not only on x but on a circumferential coordinate of the tube. This


will be discussed later. No boundary conditions in the axial direction are consideredsince the channels and tubes are assumed to be long enough.

The temperature field in the solid, denoted by Ts, is first sought. Because the heatflux −ks∇Ts must be divergence-free, where ks is a thermal conductivity of the solidand is assumed to be a constant independent of the temperature, Ts satisfies theLaplace equation 1Ts = 0. As long as the assumption (2.1) is valid, Ts is obtained forthe case of the channels as

Ts = Tw + 12

d2Tw

dx2

[d2 − (y− H − d)2

]+ · · · , (2.2)

for H < y< H + 2d, where Tw(x) denotes a temperature on the wall surfaces at y= Hand y = H + 2d (also at y = −H), which is an unknown function of x, and thesymmetry with respect to y= H + d has been used.

For the gas, the temperature field also satisfies ∇ · (k∇T) = 0, k being the thermalconductivity of the gas. This conductivity and a shear viscosity µ are assumed to bedependent on the temperature in the form of a power law given by

k

k0=(

T

T0

)βand

µ

µ0=(

T

T0

)β, (2.3)

where β is a positive constant between 0.5 and 0.6 for air, and the subscript 0 is usedto imply a value of a quantity or a variable attached in a quiescent, reference state.With (2.3), T1+β satisfies the Laplace equation 1T1+β = 0.

In the same way as that leading to (2.2), T1+β may be obtained as

T1+β = T1+βw + 1

2d2

dx2(T1+β

w )(H2 − y2

)+ · · · , (2.4)

where T is equal to Tw at the wall surfaces y=±H. Since the second term is assumedto be small, (2.4) is expanded into

T = Tw + T−βw

2(1+ β)d2

dx2(T1+β

w )(H2 − y2)+ · · · . (2.5)

For this distribution, the continuity of heat fluxes through the wall surfaces is required,i.e. −k∂T/∂y=−ks∂Ts/∂y. Thus, it follows that

k0HT0

1+ βd2

dx2

(Tw

T0

)1+β=−ksd

d2Tw

dx2. (2.6)

Equation (2.6) is readily integrated to yield an equation which determines thetemperature distribution Tw on the wall as

11+ β

(Tw

T0

)1+β+ ks

k0

d

H

Tw

T0= c1 + c2x, (2.7)

where c1 and c2 are arbitrary constants to be determined by boundary conditions atboth ends of the channels, although not specified in the present context.

If ksd/k0H� 1, (Tw/T0)1+β is negligible so that Tw/T0 may be approximately given

by a linear function of x as (k0H/ksd)(c1 + c2x). If ksd/k0H is comparable with unity,then (2.7) should be solved for Tw/T0, which is given by a nonlinear function of x.Note that this distribution is monotonic in x.

For the case of tubes or bores, the temperature fields are not so easily obtained. Thetemperature field of the solid must satisfy, by symmetry, no heat flux −ks∂Ts/∂n = 0


Density Specific heat Thermalconductivity

Thermaldiffusivity

ρ (kg m−3) cp (J (kg K)−1) k (W (m K)−1) k/ρcp (m2 s−1)

Air 1.176 1007 0.02638 2.228× 10−5

Argon 1.623 521.5 0.01784 2.107× 10−5

Helium 0.1625 5193 0.1560 1.849× 10−4

Nitrogen 1.138 1041 0.02597 2.192× 10−5

Ceramics (SiC) 3200 660 170 8.0× 10−5

Copper (pure) 8933 385 401 1.17× 10−4

Polyimide (Kapton) 1420 1090 0.12 7.75× 10−8

Steel (AISI 304) 7900 477 14.9 3.95× 10−6

TABLE 1. Thermal constants of some gases and solids at 1 atm and 300 K.

along each side of the square or hexagonal cell, n denoting the normal coordinatealong the periphery of the cell. While the temperature field is uniform over a cross-section to the lowest approximation, higher-order corrections similar to (2.2) and (2.4)are no longer functions of the radial coordinate r only, but periodic functions in acircumferential coordinate. They are difficult to obtain.

If the boundary conditions are replaced by ∂Ts/∂r = 0 at r = R + d, then theaxisymmetric temperature field is readily obtained as

Ts = Tw + 14

d2Tw

dx2

[R2 − r2 + 2 (R+ d)2 log

( r

R

)]+ · · · , (2.8)

where Tw(x) denotes the temperature along the wall surface at r = R. This is asituation in which the circular tubes having thickness d and thermal conductivity ks

are embedded in a non-heat-conducting matrix (shaded area in figure 2). Such a modelwill be treated in the following for the sake of simplicity. This will be closer to asituation in the staggered array than that in the square array because the fraction of theshaded area to the total area of the unit cell is smaller in the former case.

For the gas, the temperature field is obtained as

T = Tw + T−βw

4(1+ β)d2

dx2(Tw

1+β)(R2 − r2

)+ · · · . (2.9)

The continuity of heat fluxes at r = R leads to

11+ β

(Tw

T0

)1+β+ ks

k0

[(R+ d)2−R2]R2

Tw

T0= c1 + c2x, (2.10)

c1 and c2 being arbitrary constants. Although the coefficient in the second term isdifferent from (2.7), the analysis will be made in parallel.

2.3. Thermal properties of materialsHere we refer briefly to material constants related to thermal properties of gasesand solids used in thermoacoustic heat engines. For four gases (air, argon, heliumand nitrogen) and four solids (ceramics, copper, polyimide and steel), table 1shows numerical values of the density ρ, specific heat at constant pressure cp,thermal conductivity k and thermal diffusivity κ (= k/ρcp) at atmospheric pressure0.1013 MPa and temperature 300 K. Among the gases, helium is exceptional in allaspects. Among the solids, copper and polyimide are two extremes and the thermal


diffusivity differs by a value of the order 10−4. It is noted that the thermal conductivityof the ceramics is comparable in order with copper.

The values for the gases are taken from the database of the National Instituteof Standards and Technology (NIST, US Department of Commerce), Thermophysicalproperties of pure fluids – NIST12 Version 5.2 (see also Turns 2006). The values ofthe pure copper and the stainless steel (AISI 304) are taken from Incropera & DeWitt(1990). Because values of the ceramics depend on additives, mean values of the datagiven by Zhou et al. (2004) are listed up to two significant figures. The values of thepolyimide are taken from the Technical Data Sheet of DuPont Kapton HN.

For air at a higher pressure, e.g. at 1 MPa but at 300 K, ρ = 11.64 kg m−3, cp =1021 J (kg K)−1, k = 0.02668 W (m K)−1 so that κ = 2.245 × 10−6 m2 s−1, while forhelium, ρ = 1.597 kg m−3, cp = 5192 J (kg K)−1, k = 0.1567 W (m K)−1 and κ =1.889 × 10−5 m2 s−1. It is found from these that ρ becomes heavier by a factor ofalmost 10 but cp and k do not change little so that κ decreases inversely proportionallyto the pressure. Conversely, κ increases as the pressure is decreased.

3. Summary of the equations in the narrow-tube approximation3.1. Narrow-tube approximation and dimensionless parameters

At first, we recapitulate the narrow-tube approximation introduced in I. There arethree length scales in the problem. One is a typical span length, H (or R), another atypical axial length L in temperature gradient or a typical axial wavelength of pressuredisturbances a0/ω, a0 and ω being an adiabatic sound speed and a typical angularfrequency, and the other a typical thickness of the viscous diffusion layer

√ν/ω or

thermal layer√κ/ω, respectively, ν being a kinematic viscosity. Because the Prandtl

number Pr(=ν/κ) is of the order of unity for gases, the diffusion layer is representedby the viscous one.

In I, the three dimensionless parameters are introduced as follows:

H

L≡ λ� 1,

ωL

a≡ 1χ

6 O(1) and

√ν/ω

H≡ δ = O(1), (3.1)

where χ takes a value larger than unity but it takes unity if the temperature gradient isabsent. It should be emphasized here that the term ‘narrow tube’ in this paper means acase with λ� 1 not δ� 1. The latter will be called a case of a thick diffusion layer.

When the heat conduction in the wall is taken into account, it will be revealed thatthe introduction of the following parameters combined is useful rather than simpleratios:

ks/ρscs

ke/ρecp= κs

κe≡ K,

√ρecpke

ρscsks=√K

ke

ks≡ ε and

√κe

κs

d

H= 1√

K

d

H≡ Ge, (3.2)

where the subscripts e and s designate the value of quantities for the gas in quiescentstate and that for the solid, respectively, and these parameters vary along the channelsor tubes, since the quantities with e are dependent on temperature. While the necessityfor the introduction of K is obvious, ε measures the square root of the product of tworatios, one being the ratio of the heat capacity of the gas per volume ρecp to that ofthe solid ρscs, and the other the ratio of the thermal conductivity of the gas ke to thatof the solid ks. Because both ratios are very small, admittedly, ε is a small parameter.The last parameter Ge measures the geometry of the two-dimensional channels. Forthe case of the circular tubes, Ge will be defined later by (5.25).


K Ceramics Copper Polyimide Steel

Air 3.61 5.234 3.48× 10−3 0.178Argon 3.82 5.532 3.68× 10−3 0.188Helium 0.44 0.631 4.19× 10−4 0.0214Nitrogen 3.67 5.319 3.54× 10−3 0.180

TABLE 2. Numerical values of K(=κs/κe) for combinations of the gas in the left columnand the solid in the top row at 1 atm and 300 K.

ε × 104 Ceramics Copper Polyimide Steel

Air 2.95 1.505 130 7.46Argon 2.05 1.046 90.2 5.19Helium 6.06 3.090 266 15.3Nitrogen 2.93 1.494 129 7.40

TABLE 3. Numerical values of ε(=√ρecpke/ρscsks) times 104 for combinations of the gasin the left column and the solid in the top row at 1 atm and 300 K.

Tables 2 and 3 show numerical values of K and ε, respectively, calculated for 16combinations of the gases and the solids given in table 1. It is seen that K takes valuesof order unity except for the case of polyimide and for the combination of the heliumwith steel. It is remarked that K for the ceramics is close to that for the copper. On theother hand, ε takes a small value of order 10−4 except for the case of polyimide. Evenfor this case, ε takes a value of order 10−2. This fact suggests us to use ε as a smallparameter and to expand solutions in terms of it.

3.2. Basic equations and boundary conditions

In a quiescent state, let the pressure in the gas p take a uniform value p0 throughout,and let the temperature of the gas be equal to that of the solid wall, as long asd2Tw/dx2 is neglected. According to our notation used so far, the temperature of thegas in the quiescent state is denoted by Te rather than Tw so that T = Te and Ts = Te.The linearized equations for the gas in the narrow-tube approximation are given in I.In presenting the equations again here, the integer j is prepared to distinguish betweenthe cases of channels and of tubes by j = 0 and j = 1, respectively. Although thenotation of y and H is often used even for the case of j= 1, they should be understoodto be replaced by r(>0) and R, respectively. All field variables are written without thesubscript j because no confusion would occur.

The equations for the gas are given by the following equations of continuity, motionand energy together with the equation of state for an ideal gas as

∂ρ ′

∂t+ ∂

∂x(ρeu

′)+ 1yj

∂

∂y(yjρev

′)= 0, (3.3)

ρe∂u′

∂t=−∂p′

∂x+ µe

yj

∂

∂y

(yj ∂u′

∂y

), (3.4)

0=−∂p′

∂y, (3.5)


ρecp

(∂T ′

∂t+ u′

dTe

dx

)= ∂p′

∂t+ ke

yj

∂

∂y

(yj ∂T ′

∂y

), (3.6)

p′

p0= ρ

′

ρe+ T ′

Te, (3.7)

in |x|<∞ and |y|< H where ρ ′, p′, T ′, u′ and v′ denote, respectively, disturbances indensity, pressure, temperature, axial velocity in the x-direction and spanwise velocityin the y-direction from those in the quiescent state, t being the time −∞ < t, and thesubscript e for the quiescent state implying functions of x determined by Te.

These equations are now supplemented by the equation for the heat conduction inthe solid walls. Because a span length of the wall is assumed to be comparable withthat of the channels or tubes and much shorter than the typical axial length L, thenarrow-tube approximation is also employed. Then the equation is approximated as

ρscs∂T ′s∂t= ks

yj

∂

∂y

(yj ∂T ′s∂y

), (3.8)

in |x| <∞ and H < y < H + d where T ′s denotes disturbance in temperature of thesolid wall from Te, and ρs, cs and ks are assumed to be constant.

Boundary conditions on the wall surface(s) require non-slip of the gas as

u′ = v′ = 0 at y= H and y= (−1)j+1 H, (3.9)

and the continuity of the temperatures and of the heat fluxes as

T ′ = T ′s and −ke∂T ′

∂y=−ks

∂T ′s∂y

at y= H and y= (−1)j+1 H. (3.10)

In addition, no heat flux is required as

−ks∂T ′s∂y= 0 at y= H + d. (3.11)

Noting that p′ is uniform in y from (3.5), and averaging (3.3)–(3.7) over the cross-section, the system of equations is reduced to a prototype of the thermoacoustic-waveequation for p′ with a dipole s and a monopole q as

∂2p′

∂t2− ∂

∂x

(a2

e

∂p′

∂x

)= 22j−1

H

[− ∂∂x

(a2

es)+ a2

e

cpTe

∂q

∂t

], (3.12)

where ae denotes a local adiabatic sound speed defined by√γ p0/ρe or

√(γ − 1)cpTe,

γ being the ratio of specific heats, and s and q denote, respectively, shear stress actingon the gas at the wall surface(s) and heat flux flowing into the gas from the wall as

s= µe∂u′

∂y

∣∣∣∣y=+H

+ (j− 1)µe∂u′

∂y

∣∣∣∣y=−H

, (3.13)

and

q= ke∂T ′

∂y

∣∣∣∣y=+H

+ (j− 1)ke∂T ′

∂y

∣∣∣∣y=−H

. (3.14)

Integrating (3.8) over the thickness, on the other hand, it follows that

∂

∂t

(∫ H+d

HρscsT

′s yjdy

)=−2j−1Hjq, (3.15)


where (3.11) and (3.14) have been used. Equation (3.12) for p′ is coupled with thetemperature variation in the solid wall through q in (3.15). If the heat capacity of thesolid per volume ρscs is very large, then T ′s does not change against the heat flux q.If the thermal conductivity ks is very large in comparison with ke, then the secondcondition of (3.10) suggests no temperature gradient normal to the wall surface.

4. Derivation of the thermoacoustic-wave equationsTo derive the closed form of the thermoacoustic-wave equation, we proceed to

express s and q in terms of p′. Following in the same way as demonstrated in I, themethod of Fourier transform is employed. It is defined by

F {u′} = 1√2π

∫ ∞−∞

u′(x, y, t)eiωtdt ≡ u′(x, y, ω), (4.1)

with its inverse transform given by

F−1{u′} = 1√2π

∫ ∞−∞

u′(x, y, ω)e−iωtdω = u′(x, y, t). (4.2)

Making use of the result that p′ is uniform over a cross-section, a first step is toexpress an axial velocity u′ in terms of p′ by solving (3.4) together with (3.9). Thenext step is to express temperature T ′ and T ′s by solving (3.6) with u′ obtained andsimultaneously (3.8) so as to satisfy the boundary conditions (3.10) and (3.11). Withboth velocity and temperature fields available, a final step is to express s and q interms of p′ and to substitute them into (3.12). Then a thermoacoustic-wave equationfor p′ is derived. This equation may alternatively be derived without use of (3.12) bysolving v′ on substituting u′ and ρ ′ (from (3.7)) into (3.3) and applying the boundarycondition (3.9) for v′.

4.1. Case of the two-dimensional channelsAt the outset, the axial velocity u′ is obtained from (3.4). Even when the effects ofheat conduction in the walls are taken into account, it is unchanged and given by

u′ =− 1ρeσ−1 ∂ p′

∂xf , (4.3)

with σ =−iω where f is defined as

f (x, y)= 1− cosh(y/Hδe)

cosh(1/δe), (4.4)

and δe is defined by

δe(x)= 1H

(νe

σ

)1/2, (4.5)

with νe(x) = µe/ρe. Here the dependence of f on σ has been suppressed and σ−1/2 isdefined to take a positive real part for a positive value of ω.

On the other hand, the temperature of the gas T ′ must be sought by solving (3.6)simultaneously with (3.8) for that of the solid wall so that the boundary conditions(3.10) may be fulfilled. Making use of the smallness of ε, an asymptotic expansionwith respect to it is made and terms proportional to ε2 or higher than it are neglected.


It then follows that

T ′ = 1ρecp

p′fP + 1ρe

dTe

dxσ−2 ∂ p′

∂x

(− Pr

1− Prf + 1

1− PrfP

)+ εT ′ε, (4.6)

with Pr = νe/κe and

fP(x, y)= 1− cosh(y√

Pr/Hδe)

cosh(√

Pr/δe), (4.7)

where εT ′ε is a modification due to the thermal coupling with the solid wall, and thesubscript P implies the Prandtl number and thermal origin. In passing, no subscriptimplies viscous origin. The lowest terms in (4.6) correspond to the solution withoutthe coupling and they vanish at y= H where f = fP = 0. The modification is given by

εT ′ε =ε

ρecpp′fKP + ε

ρe

dTe

dxσ−2 ∂ p′

∂x

(−√

Pr

1− PrfK + 1

1− PrfKP

)+ O(ε2), (4.8)

where fK and fKP are defined, respectively, by[fK(x, y)fKP(x, y)

]=[

C

CP

]cosh(y

√Pr/Hδe)

cosh(√

Pr/δe), (4.9)

with

C = tanh(1/δe)

tanh(1/δs), (4.10)

CP = tanh(√

Pr/δe)

tanh(1/δs), (4.11)

and

δs = 1d

(κs

σ

)1/2. (4.12)

Here C denotes the coupling between the viscous diffusion and the thermal diffusionin the solid wall, while CP denotes that between the thermal diffusions of the gas andthe solid, and δs is independent of x because κs is assumed to be a constant.

In consistent with (4.6) and (4.8), T ′s is obtained as

T ′s =ε

ρecpp′fSP + ε

ρe

dTe

dxσ−2 ∂ p′

∂x

(−√

Pr

1− PrfS + 1

1− PrfSP

)+ O(ε2), (4.13)

where fS and fSP are defined, respectively, by[fS(x, y)fSP(x, y)

]=[

C

CP

]cosh[(y− H − d)/dδs]

cosh(1/δs), (4.14)

in H < y< H + 2d, and εT ′ε matches with T ′s at y= H where fK = fS and fKP = fSP.With u′ and T ′ available, the shear stress on the wall surfaces is obtained as

s= 2√νeσ

−1/2 ∂ p′

∂xg(x,H), (4.15)


while the heat flux through the wall surfaces is obtained as

σ q=−2cpTe√νe

a2e

{γ − 1√

Prσ 3/2p′gP(x,H)− a2

e

Te

dTe

dxσ−1/2 ∂ p′

∂x

×[

11− Pr

g(x,H)− 1

(1− Pr)√

PrgP(x,H)

]}+ εσ qw, (4.16)

with

σ qw =−CPσ q+ O(ε), (4.17)

where g and gP are defined, respectively, as

g(x, y)= sinh(y/Hδe)

cosh(1/δe), (4.18)

gP(x, y)= sinh(y√Pr/Hδe)

cosh(√Pr/δe)

. (4.19)

It is found from these relations that the effects of heat conduction in the wall on thegas appear in the temperature field and therefore the heat flux, but not in the velocityfield and the shear stress. If the temperature gradient dTe/dx is absent, the heat flux isdetermined only by the thermal diffusivity κe and the pressure p′ itself. This is seenin (4.16) by noting

√νe/Pr = √κe and

√Pr/δe = (σ/κe)

1/2 H. But when the gradientis present, the viscous diffusivity νe comes into play in the heat flux through g(x,H)together with the pressure gradient ∂ p′/∂x. This is because the temperature field isaffected by convection due to the second term on the left-hand side of (3.6). Thisconvection yields one of the very features of thermoacoustic phenomena.

By the lowest temperature distribution in T ′, i.e. without the thermal coupling, thereflows the lowest heat flux q in (4.16). This gives rise to temperature variation in thewalls through (3.15). Because the heat capacity of solid per volume or its thermalconductivity is large, T ′s is much smaller than T ′ by the order of ε. This feeds backto T ′ε in the gas and the heat flux εqw flows through the wall surface. This heatflux gives rise to temperature variation in the wall of the order of ε2 and, in turn,to temperature variation in gas and heat flux of ε2, and this cycle continues. Theexpansion is truncated at the order of ε, and no higher-order coupling between the gasand the solid is taken into account. Remark that if CP in (4.17) diverges, then a strongcoupling is expected to take place.

Since s and q are now available, the next step is to make their inverse transforms.To do this, use is made of the following formula ((2.39) in I):

1√2π

F−1{σ−1/2 tanh(

√Pr/δe)

}=Φ

( νet

PrH2

)h(t), (4.20)

with h(t) being a unit step function, where Φ is defined by

Φ( νet

PrH2

)= 2√νe√

PrH

∞∑n=1

exp

[−(2n− 1)2π2

4νet

PrH2

]h(t). (4.21)

Because this sum diverges as t→ 0, (4.21) is alternatively written as

Φ = 1√πt

G( νet

PrH2

)= 1√πt

[1+ 2

∞∑n=1

(−1)n exp(−n2PrH2

νet

)]. (4.22)


To obtain qw, transforms of σ−1/2CPg(x,H) and σ−1/2CPgP(x,H) must be evaluated.They are given by the following two inverse transforms, which are set formally as

1√2π

F−1

{σ−1/2 tanh(

√Pr/δe)

tanh(1/δs)tanh(1/δe)

}=MT(t)h(t), (4.23)

1√2π

F−1

{σ−1/2 tanh(

√Pr/δe)

tanh(1/δs)tanh(√Pr/δe)

}=MD(t)h(t), (4.24)

where explicit expressions of MT and MD are given in § 5. Here MT represents thetriple coupling among the viscous diffusion, the thermal diffusion of the gas and thatof the solid, while MD represents the double coupling between the thermal diffusionsof the gas and of the solid.

Using these formulas, the inverse transform of s is expressed in the form of aconvolution integral given by

s= 2√νeM

(∂p′

∂x

), (4.25)

where M (φ) designates a functional of a function φ(x, t), which is defined as aspecial case of a following functional MP(φ) by setting Pr to be equal to unityformally:

MP [φ(x, t)]≡ 1√π

∫ t

−∞

G[νe(t − τ)/PrH2]√t − τ φ(x, τ ) dτ. (4.26)

Using the transforms (4.23) and (4.24), new functionals MT(φ) and MD(φ) aredefined, respectively, as

MI [φ(x, t)]≡∫ t

−∞MI(t − τ)φ(x, τ ) dτ, (4.27)

where the subscript I takes T or D.With these definitions of the functionals, the inverse transform of σ q is expressed as

∂q

∂t=−2cpTe

√νe

a2e

{γ − 1√

PrMP

(∂2p′

∂t2

)− a2

e

Te

dTe

dx

[1

1− PrM

(∂p′

∂x

)− 1

(1− Pr)√

PrMP

(∂p′

∂x

)]}+ ε ∂qw

∂t, (4.28)

where qw is set in the form of

∂qw

∂t= 2cpTeH

a2e

W + O(ε), (4.29)

with

W =√νe

H

{γ − 1√

PrMD

(∂2p′

∂t2

)− a2

e

Te

dTe

dx

[1

1− PrMT

(∂p′

∂x

)− 1

(1− Pr)√

PrMD

(∂p′

∂x

)]}. (4.30)

While the subscripts T and D attached to M imply the origin from the heat flux,respectively, due to the triple and double couplings among the three diffusions, P


implies the origin from the heat flux without the coupling and M without thesubscript implies the origin from the shear stress. Each term in εW has the counterparton the left-hand side. The reason for this will be revealed later.

Substituting s and q into (3.12), the thermoacoustic-wave equation for the case ofthe two-dimensional channels is obtained up to the first-order terms in ε as

∂2p′

∂t2− ∂

∂x

(a2

e

∂p′

∂x

)+ ∂

∂x

[a2

e

√νe

HM

(∂p′

∂x

)]+√νe

H

{γ − 1√

PrMP

(∂2p′

∂t2

)− a2

e

Te

dTe

dx

[1

1− PrM

(∂p′

∂x

)− 1

(1− Pr)√

PrMP

(∂p′

∂x

)]}= εW. (4.31)

4.2. Case of the circular tubesAnalyses in this case can be executed in parallel to the previous case but are a littlecomplicated. The transformed solutions u′ and T ′ are given by (4.3), (4.6) and (4.8),respectively, with f , fP, fK and fKP replaced by the following functions:

f (x, r)= 1− I0(r/Rδe)

I0(1/δe), (4.32)

fP(x, r)= 1− I0(r√

Pr/Rδe)

I0(√

Pr/δe)= 1− I0(r/Rδκ)

I0(1/δκ), (4.33)[

fK(x, r)fKP(x, r)

]=[

C

CP

]I0(r/Rδκ)

I0(1/δκ), (4.34)

with [C

CP

]=[

I1(1/δe)/I0(1/δe)

I1(1/δκ)/I0(1/δκ)

]Z(R, 1/δs), (4.35)

where Ii and Ki below (i = 0, 1) denote the modified Bessel functions of the ith order,respectively, and Z(r, z) is a solution of (3.8) transformed with (3.11) and is given by

Z(r, z)= I1(ζ )K0(rz/R)+ K1(ζ )I0(rz/R)

I1(ζ )K1(z)− K1(ζ )I1(z), (4.36)

with z = 1/δs, ζ = (1 + η)z and η = d/R. Here δe is defined by (4.5) with H replacedby R, while δκ is newly introduced to avoid use of the factor

√Pr in

√Pr/δe, and δs

is defined differently from (4.12) as

δe = 1R

(νe

σ

)1/2, δκ = 1

R

(κe

σ

)1/2 = δe√Pr

and δs = 1R

(κs

σ

)1/2. (4.37)

For T ′s, fS1 and fS2 are replaced, respectively, by[fS(x, r)fSP(x, r)

]=[

C

CP

]Z(r, 1/δs)

Z(R, 1/δs). (4.38)

The shear stress and heat flux are given by half of (4.15) and (4.16) with g and gP

replaced, respectively, by

g(x, r)= I1(r/Rδe)

I0(1/δe), (4.39)


gP(x, r)= I1(r√Pr/Rδe)

I0(√Pr/δe)

. (4.40)

To make inverse transforms of them, the following relation ((4.13) in I) is used:

1√2π

F−1

{σ−1/2 I1(

√Pr/δe)

I0(√Pr/δe)

}=Θ

( νet

PrR2

)h(t), (4.41)

where Θ is defined as

Θ( νet

PrR2

)= 2√νe√

PrR

∞∑n=1

exp(−j2

n

νet

PrR2

), (4.42)

and jn (n = 1, 2, 3, . . .) denote roots of J0(jn) = 0 (0 < j1 ≈ 2.40 < j2 ≈ 5.52 < j3 ≈8.65 . . .). As this sum diverges as t→ 0, its asymptotic behaviour is already availableas

Θ( νet

PrR2

)= 1√πt−√νe

2√PrR− νe

4PrR2

√t

π+ · · · . (4.43)

Using (4.41), the functional NP is introduced as

NP[φ(x, t)] ≡∫ t

−∞Θ

[νe(t − τ)PrR2

]φ(x, τ ) dτ, (4.44)

and the functional N (φ) denotes NP(φ) with Pr = 1 set equal to unity formally.After the case of the channels, new functionals are defined by

NI[φ(x, t)] ≡∫ t

−∞NI(t − τ)φ(x, τ ) dτ, (4.45)

where I stands for T and D, and NT and ND are defined in terms of the followinginverse transforms, respectively, as

1√2π

F−1

{σ−1/2 I1(1/δe)

I0(1/δe)

I1(1/δκ)I0(1/δκ)

Z(R, 1/δs)

}= NT(t)h(t), (4.46)

1√2π

F−1

{σ−1/2 I2

1(1/δκ)I2

0(1/δκ)Z(R, 1/δs)

}= ND(t)h(t). (4.47)

Explicit expressions of NT and ND will be given later.Using these definitions, the shear stress and the heat flux on the wall surface are

expressed by (4.25), (4.28) and (4.29) with the functionals designated by M replacedby their corresponding ones by N . But remark that the factor 2 in (4.25) and in frontof cpTe

√νe in (4.28) and (4.29) should be removed. Substituting these into (3.12) for

j = 1, the thermoacoustic-wave equation for the case of the circular tubes is obtainedup to the first-order terms in ε as

∂2p′

∂t2− ∂

∂x

(a2

e

∂p′

∂x

)+ ∂

∂x

[2a2

e

√νe

RN

(∂p′

∂x

)]+ 2√νe

R

{γ − 1√

PrNP

(∂2p′

∂t2

)− a2

e

Te

dTe

dx

[1

1− PrN

(∂p′

∂x

)− 1

(1− Pr)√

PrNP

(∂p′

∂x

)]}= εW, (4.48)


with

W = 2√νe

R

{γ − 1√

PrND

(∂2p′

∂t2

)− a2

e

Te

dTe

dx

[1

1− PrNT

(∂p′

∂x

)− 1

(1− Pr)√

PrND

(∂p′

∂x

)]}. (4.49)

5. Evaluation of the relaxation functions and approximation of thethermoacoustic-wave equations

The thermoacoustic-wave equations have been derived formally by introducingthe functionals MI and NI where the subscript I takes T or D. Since explicitexpressions of the functions involved are left unspecified, they are evaluated in§ 5.1. The thermoacoustic-wave equations are valid at any time and for any formof disturbances temporarily as well as spatially, as long as they may be regardedas being infinitesimally small. While they are general, it will be illuminating toderive from them approximate equations for short- and long-time behaviours. Thisapproximation is made according as the Deborah number De defined by H2/νet (orR2/νet) is much larger or smaller than unity. Such an approximation corresponds to thecase in which the span length is much larger or smaller than the thickness of diffusionlayer, respectively.

5.1. Evaluation of the relaxation functionsThe kernel functions MI in MI and NI in NI where I takes T or D, are calledrelaxation functions in this paper. Just as in the case of the inverse transforms of (4.20)and (4.41), they are reduced to the inverse Laplace transforms along the imaginaryaxis in the complex σ -plane, which are easily evaluated by Cauchy’s theorem (see(4.17) in I). Looking for simple poles of the integrands, it is found that they arelocated on the negative axis so that the integrals vanish for t < 0. In all cases, nobranch point exists at σ = 0.

5.1.1. Case of the two-dimensional channelsFor MT(t), there are simple poles at σ = −(n− 1

2)2π

2νe/H2, σ = −(n− 1

2)2

π2κe/H2 and σ = −(nπ)2 κs/d2, (n = 1, 2, 3, . . .), which originate from the zeros ofcosh(1/δe) = 0, cosh(

√Pr/δe) = 0 and sinh(1/δs) = 0, respectively. When they are

different from each other, then the theorem yields

MT(t)= 2√νe

H

∞∑n=1

tan[(n− 1/2)π√νe/κe]

tan[(n− 1/2)π√νe/κeGe] exp

[−(2n− 1)2π2

4νet

H2

]

+ 2√κe

H

∞∑n=1

tan[(n− 1/2)π√κe/νe]

tan[(n− 1/2)πGe] exp

[−(2n− 1)2π2

4κet

H2

]

− 2√κs

d

∞∑n=1

tan( nπ

Ge

)tan(

nπ√νe/κeGe

)exp

[−(nπ)2 κst

d2

]. (5.1)

Here all coefficients of the exponential functions are assumed to be finite, which is anon-resonant case. It is found that there appears a new relaxation time d2/κs due tothe heat conduction in the solid wall in addition to the viscous and thermal relaxation


times of the gas H2/νe and H2/κs, respectively. Note that because d2/κs is written asGe2PrH2/νe, this time becomes longer in proportion to Ge2 compared with H2/νe asthe wall becomes thicker and the diffusion in the solid wall lasts longer.

For MD, the poles of second order exist at σ = −(n− 1/2)2π2κe/H2 so that the

theorem yields

MD(t)= 2√κe

H

∞∑n=1

(2n− 1)πtan[(n− 1/2)πGe]

κet

H2exp

[−(2n− 1)2π2

4κet

H2

]

+ 2√κe

H

∞∑n=1

Ge

sin2[(n− 1/2)πGe] exp

[−(2n− 1)2π2

4κet

H2

]

− 2√κs

d

∞∑n=1

tan2( nπ

Ge

)exp

[−(nπ)2 κst

d2

]. (5.2)

The poles of second order contribute to algebraic growth κet/H2, although multipliedby the decaying exponential function, and give rise to a slower behaviour. For amoderate or large value of t(> 0), the exponential functions decay rapidly as nincreases so that the sums converge quickly. For a small value of t, however, theconvergence becomes so slow that they tend to diverge as t→ 0. Further it may occurthat the sums do not converge if one of the coefficients of the exponential functionshappens to diverge. This is called a resonant case and will be examined in detail in thenext section.

To examine their asymptotic behaviours as t→ 0, their Fourier transforms are useful.Expanding them in δ−1

e , δ−1k and δ−1

s by using tanh X = 1 + 2∑∞

n=1 (−1)n exp(−2nX)for |X| � 1, and transforming inversely each exponential function, it then follows that

MT(t)= 1√πt

[1− 2 exp

(−H2

νet

)− 2 exp

(−H2

κet

)+ 2 exp

(− d2

κst

)+ · · ·

], (5.3)

and

MD(t)= 1√πt

[1− 4 exp

(−H2

κet

)+ 2 exp

(− d2

κst

)+ · · ·

], (5.4)

as t→ 0. Thus, it is found that MT and MD tend to diverge as (πt)−1/2 as t→ 0 andthis behaviour is the same as that in (4.22).

For air enclosed by the ceramic plates with d/H = 1, Pr = νe/κe = 0.72,K = κs/κe = 3.61 and Ge = d/H

√K = 0.526 (see table 2), figure 3(a,b) show the

logarithmic plots of MT(t) and MD(t) against t over the range 10−3 6 t 6 10,respectively, where the factor νe/H2 in the abscissa νet/H2 and ordinate (H/

√νe)MT

is omitted in figure 3(a), while κe/H2 is omitted in figure 3(b), for simplicity. Thefunctions MT and MD consist of three respective sums. The first sum in (5.2) changessign so that the absolute value is shown with the label |∑1| attached to the curve. Thearrow indicates the vanishing point. The last sums (excluding the minus sign in frontof the summation symbol) are negative in MT and positive in MD. The broken curvesrepresent the asymptotic expressions (5.3) and (5.4) but they are invisible for t < 0.5because they almost coincide with MT and MD.

The figures show which sum contributes to each function most. For a small valueof t, the first sums in (5.1) and (5.2) are much smaller than the others, while the


t

(a)

0

MT (t)

MD (t)MT (t)

0 0

(b)

10–3 10–2 10–1 100 101

t10–3 10–2 10–1 100 101

10–7

10–5

10–3

10–1

101

103

10–6

10–4

10–2

100

102

104

1

3

3

2

FIGURE 3. Logarithmic plots of the functions MT(t) and MD(t) against t over the interval10−3 6 t 6 10 in (a) and (b), respectively, where the abscissa and ordinate in (a) measureνet/H2 and (H/

√νe)MT(t), while those in (b) measure κet/H2 and (H/

√κe)MD(t), but the

both factors νe/H2 and κe/H2 are omitted for simplicity. Here the subscripts 1, 2 and 3 of thesummation symbol labelled to the curves designate, respectively, the first, second and thirdsums in (5.1) and (5.2), the symbol | · · · | or the minus sign designating the absolute value orthe value with sign reversed, and the broken lines represent the asymptotic expressions (5.3)and (5.4).

second and third sums cancel with each other to yield the asymptotic expressions1/√πt + · · ·. For a large value of t, on the other hand, both functions decay rapidly

and the first sums survive over the others.

5.1.2. Case of the circular tubesFor the case of the circular tubes as well, the functions NT and ND can be

evaluated by the same method. For NT , the simple poles occur at σ = −j2nνe/R2 and

σ = −j2nκe/R2 (n = 1, 2, 3, . . .) from the zeros of I0[(σ/νe)

1/2 R] and I0[(σ/κe)1/2 R],

respectively. Further because Z(R, 1/δs) has poles, their contributions should beincluded. Setting 1/δs = z, it is found that the logarithmic singularity at z = 0 inKi (i= 0, 1) cancels out to disappear and a simple pole appears at z= 0. As z→ 0, Zbehaves as

Z(R, z)= A

(1z+ Bz+ · · ·

), (5.5)

with

A= 2

[(1+ η)2−1] and B= 18− 3

8(1+ η)2+(1+ η)

4 log(1+ η)2 [(1+ η)2−1]2

, (5.6)

while as |z| →∞, Z behaves as

Z(R, z)= 1− 12z+ 3

8z2+ · · · , (5.7)

if Re{z} > 0. There exist an infinite number of simple poles at z = zn = ±ikn

(n = 1, 2, 3, . . .) where zn satisfies I1(z)/I1(ζ ) = K1(z)/K1(ζ ) with ζ = (1 + η)z. Fromthe asymptotic expressions of the ratios of the two modified Bessel functions for|z| � 1 (e.g. Abramowitz & Stegun 1972), kn are approximated to be given by(nπ/η){1+ 3η2/[8n2π

2(1+ η)] + · · ·}.


Taking account of the residues, thus, NT is evaluated as

NT(t)= 2√νe

R

∞∑n=1

X(ijn

√νe/κe)Z(R, ijn

√νe/κs) exp

(−j2

n

νet

R2

)+ 2√κe

R

∞∑n=1

X(i jn

√κe/νe)Z(R, ijn

√κe/κs) exp

(−j2

n

κet

R2

)+ 2√κs

R

∞∑n=1

X(i kn

√κs/νe)X(i kn

√κs/κe)Yn exp

(−k2

n

κst

R2

), (5.8)

where X denotes I1(z)/I0(z) and Yn denotes limz→zn(z − zn)Z(R, z), which is evaluatedas

Yn = I1(ζn)K0(zn)+ K1(ζn)I0(zn)

(1+ η)[I0(ζn)K1(zn)+ K0(ζn)I1(z)] − K1(ζn)I0(zn)− I1(ζn)K0(zn), (5.9)

with ζn = (1 + η)zn, zn being ikn (n = 1, 2, 3, . . .). For evaluation of ND, on the otherhand, there appear poles of second order at σ =−j2

nκe/R2 from zeros of I20[(σ/κe)

1/2 R],which give rise to a slower decay. Taking these into account, ND is expressed as

ND(t)= 2√κe

R

∞∑n=1

ijn

(1+ 2j2

n

κet

R2

)Z(R, ijn

√κe/κs) exp

(−j2

n

κet

R2

)+ 2√κe

R

∞∑n=1

√κe

κs

ddz

Z(R, z)

∣∣∣∣∣z=ijn

√κe/κs

exp(−j2

n

κet

R2

)

+ 2√κs

R

∞∑n=0

X2(i kn

√κs/κe) Yn exp

(−k2

n

κst

R2

). (5.10)

Because the sums on the right-hand sides of (5.8) and (5.10) diverge as t→ 0,their asymptotic behaviours are examined by expanding the transformed expressions interms of δ−1

e , δ−1κ and δ−1

s . Then it follows that

σ−1/2 I1(1/δe)

I0(1/δe)

I1(1/δκ)I0(1/δκ)

Z(R, 1/δs)= σ−1/2 − υ11

2Rσ−1 − υ12

8R2σ−3/2 + · · · , (5.11)

withυ11 =√νe +√κe +√κs, (5.12)

υ12 = νe + κe − 3κs − 2(√νeκe +√κeκs +√κsνe), (5.13)

and

σ−1/2 I21(1/δκ)

I20(1/δκ)

Z(R, 1/δs)= σ−1/2 − υ21

2Rσ−1 + υ22

8R2σ−3/2 + · · · , (5.14)

with υ21 = 2√κe + √κs and υ22 = 4

√κeκs + 3κs. The inverse transforms yield the

asymptotic expressions of NT and ND, respectively, as

NT = 1√πt− υ11

2R− υ12

4R2

√t

π+ · · · , (5.15)

ND = 1√πt− υ21

2R+ υ22

4R2

√t

π+ · · · , (5.16)


T(a)

3

0

0

(b)

0

N

10–6

10–8

10–4

10–2

100

102

104

t10–3 10–2 10–1 100

t10–3 10–2 10–1 100

N

1

3

2

2

ND (t)NT (t)

10–7

10–5

10–3

10–1

101

103

FIGURE 4. Logarithmic plots of the functions NT(t) and ND(t) against t over the interval10−3 6 t 6 5 in (a) and (b), respectively, where the abscissa and ordinate in (a) measureνet/R2 and (R/

√νe)NT(t), while those in (b) measure κet/R2 and (R/

√κe)ND(t), but both

factors νe/R2 and κe/R2 are omitted for simplicity. Here the subscripts 1, 2 and 3 of thesummation symbol labelled to the curves designate, respectively, the first, second and thirdsums in (5.8) and (5.10), the symbol | · · · | or the minus sign designating the absolute value orthe value with sign reversed, and the broken lines indicate the asymptotic expressions (5.15)and (5.16).

for t→ 0. The leading asymptotic expressions are found to be given commonly by1/√πt.

Figure 4(a,b) show the logarithmic plots of NT(t) and ND(t) against t over therange 10−3 6 t 6 5, respectively, where the factor νe/R2 in the abscissa νet/R2 andordinate (R/

√νe)NT is omitted in figure 4(a), while κe/R2 is omitted in figure 4(b),

for simplicity. Here we are concerned with the case with air enclosed in the ceramictube with d/R = η = 1 where the values of Pr and K are the same as employed infigure 3.

As NT and ND consist of three respective sums, each sum is designated by thesummation symbol consecutively from the first. As the sum changes sign, the absolutevalue or the value with sign reversed is shown with the labels attached to the curves.The broken curves represent the asymptotic expressions (5.15) and (5.16). Unlike thecase of MT and MD, three terms are necessary to find agreements with the numericalvalues.

5.2. Approximation of the thermoacoustic-wave equations

5.2.1. Approximate equations for a short-time behaviourFor a short time stipulated by H2/νet � 1, the asymptotic expressions of the

relaxation functions are truncated at the leading terms. But if a value of Ge isvery small, as (5.3) and (5.4) suggest, this condition should be replaced by d2/κst� 1,i.e. Ge2PrH2/νet� 1. Then all functionals for j = 0 are reduced to the derivatives ofminus half-order as (Gel’fand & Shilov 1964)

M (φ)≈MI(φ)≈ 1√π

∫ t

−∞

φ(x, τ )√t − τ dτ ≡ ∂

−1/2φ

∂t−1/2, (5.17)

where I takes P, T and D, and so is the case with j= 1 by replacing M by N .


Using (5.17), the thermoacoustic-wave equations are approximated into

∂2p′

∂t2− ∂

∂x

(a2

e

∂p′

∂x

)+ 2ja2

e

√νe

H

[C∂−1/2

∂t−1/2

(∂2p′

∂x2

)+ (C + CT)

Te

dTe

dx

∂−1/2

∂t−1/2

(∂p′

∂x

)]= 0, (5.18)

with

C = 1+ (1− ε)(γ − 1)√Pr

and CT = 12+ β

2+ 1− ε√

Pr + Pr. (5.19)

Here since the contributions from the functionals remain small, the leadingbalance occurs between the first two terms so that ∂2p′/∂t2 involved in MP in(4.31) and NP in (4.48) has been replaced by (∂/∂x)(a2

e∂p′/∂x) and the relation(a2

e

√νe)−1(d/dx)(a2

e

√νe) = (3/2 + β/2)T−1

e dTe/dx has been used. It is revealed thatthe effect of the heat conduction in the wall gives rise to only small correctionsof order ε in the coefficients C and CT and therefore the short-time behaviour isaffected little, as expected. In passing, as was found by Henry and remarked by Rott,it is interesting to find that because C is engaged in damping in the absence of thetemperature gradient, the effects of heat conduction decreases the values of C (andalso CT) to reduce the damping, although very slightly.

5.2.2. Approximate equations for a long-time behaviourOur next concern is approximation for a long time H2/νet� 1. This corresponds

to the case with |δe| � 1, and also to the case in which the span length is muchsmaller than the thickness of the diffusion layer. But if a value of Ge is very large,this condition should be replaced by Ge2PrH2/νet � 1. To derive the approximateequations, it is convenient to work with (3.12) transformed.

Treating both cases for j = 0 and j = 1 simultaneously, and replacing H with R forj= 1, s in (4.15) is expanded in terms of 1/δe up to 1/δ4

e as

s= H

22j−1

(1− 1

3+ 5j

H2σ

νe+ 2

15+ 81j

H4σ 2

ν2e

+ · · ·)∂ p′

∂x, (5.20)

while σ q in (4.16) is expanded in a similar fashion as

σ q= cpTeH

22j−1a2e

{−(γ − 1)

(1− Pr

3+ 5j

H2σ

νe+ · · ·

)σ 2p′

− a2e

Te

dTe

dx

[1

3+ 5j− 2(1+ Pr)

15+ 81j

H2σ

νe+ · · ·

]H2σ

νe

∂ p′

∂x

}+ εσ qw. (5.21)

Since qw =−CPq in (4.17), CP is expanded similarly so that qw is obtained as

εσ qw = εcpTeH

23j−1a2eGe

{(γ − 1)

[1−

(2Pr

3+ 5j− Qj

)H2σ

νe+ · · ·

]σ 2p′ + a2

e

Te

dTe

dx

×[

13+ 5j

−(

Pr

(3+ 5j)2+ 2(1+ Pr)

15+ 81j− Qj

3+ 5j

)H2σ

νe+ · · ·

]H2σ

νe

∂ p′

∂x

},

(5.22)


where Qj are defined as

Q0 = νed2

3κsH2= 1

3PrGe2, (5.23)

Q1 = νe

κs

{18− 3

8(1+ η)2+(1+ η)

4 log(1+ η)2[(1+ η)2−1]

}, (5.24)

and Ge for the case of the circular tubes is defined as

Ge=√κe

κs

[(1+ η)2−1]2

=√κe

κs

[d

R+ 1

2

(d

R

)2]. (5.25)

For η� 1, Q1 ≈ Q0 = Pr(κe/κs)η2/3.

Before deriving the approximate equations, note that, in (3.12), the first term on theleft-hand side stems from the adiabatic density change in the equation of continuity,while the second term stems from the pressure gradient in the equation of motion.When the span length is very narrow, the pressure gradient almost balances with theshear stress on the wall surfaces. Substituting s in (5.20) into (3.12), in fact, it is seenthat the first term in (5.20) cancels with the term due to the pressure gradient.

On the other hand, the form of heat flux (5.21) (more generally from (4.16))depends not only on the pressure but also on the product of the temperature gradientdTe/dx and the pressure gradient ∂p′/∂x. The magnitude of the latter is of order(χ/|δe|)2 in comparison with the former and is comparable if χ becomes of order|δe|. Given a pressure gradient, the heat flux changes its sense of flow according to alocal value of the temperature gradient. This is another feature of the thermoacousticphenomena.

For a very small span length, the temperature of the gas is almost equal to the localtemperature of the wall so that the temperature variation in the gas is negligible andthe thermal process is regarded as being isothermal locally. Yet the heat flux flowsthrough the wall surface. Noting the sign of the heat flux (taken positive into the gas),and cpTe = a2

e/(γ − 1), it flows into the solid when the pressure tends to increasetemporarily because q is given by −21−2jH∂p′/∂t to the lowest relation of (5.21).

Substituting (5.21) into (3.12), the first term in (5.21), i.e. the term proportionalto −(γ − 1)σ 2p′, and the first term on the left-hand side of (3.12) yield γ σ 2p′.The factor γ implies the isothermal sound speed a2

e/γ jointly with the second termdue to the pressure gradient if the shear stress were absent. However, the shearstress does exist to cancel with the pressure gradient. Thus, the lowest relationof (3.12) is reduced to γ σ 2p′ = 0, i.e. γ ∂2p′/∂t2 = 0, not to the wave equationγ ∂2p′/∂t2 − (∂/∂x)(a2

e∂p′/∂x)= 0. This means that the isothermal sound speed ae/√γ

has no meaning physically.Keeping these relations in mind, we proceed to seek higher-order terms in (3.12).

Substituting (5.20) and (5.21) with (5.22) into (3.12) transformed, and dividing itby γ σ , we arrive at an equation in the same form as (3.17) in I but with an additionalterm on the right-hand side, denoted by εwj, due to the effects of heat conduction inthe wall. To treat both cases with j= 0 and j= 1 simultaneously, α is now replaced byαj given by

αj = a2eH2

(3+ 5j)γ νe, (5.26)

while the coefficient 2/5 in (3.17) in I is replaced by (6+ 10j)/(15+ 81j).


Transforming these inversely, the approximate equations for the long-time behaviourare obtained as

∂p′

∂t− ∂

∂x

(αj∂p′

∂x

)+ αj

Te

dTe

dx

∂p′

∂x− (γ − 1)Pr

αj

a2e

∂2p′

∂t2

+ 6+ 10j

15+ 81j

[∂

∂x

(αjH2

νe

∂2p′

∂t∂x

)− (1+ Pr)

αjH2

νeTe

dTe

dx

∂2p′

∂t∂x

]= εwj, (5.27)

where εwj are given by

εwj = ε

2jGe

{γ − 1γ

∂p′

∂t+ αj

Te

dTe

dx

∂p′

∂x− (γ − 1)Pr[2− (3+ 5j)Qj]αj

a2e

∂2p′

∂t2

−[

Pr

3+ 5j+ 6+ 10j

15+ 81j(1+ Pr)− Qj

]αjH2

νeTe

dTe

dx

∂2p′

∂t∂x

}. (5.28)

Because each term in εwj has a counterpart on the left-hand side of (5.27), andno new terms in form appear, the effects of heat conduction in the wall will notgive rise to qualitative changes. But quantitative differences will occur through thecoefficients. Their effects appear through the parameter Ge. As Ge becomes smaller tobe comparable with ε, they give rise to an appreciable difference from those withouttheir effects. Noting that

ε

2jGe=

ρecp

ρscs

H

dfor j= 0,

ρecp

ρscs

R2

[(R+ d)2−d2] for j= 1,(5.29)

by (3.2) and (5.25), ε/2jGe implies the heat capacity of the wall per unit axial length.Thus, the effects of heat conduction in the wall become pronounced as the thickness ofwall becomes thinner.

6. Results and discussion6.1. Summary of the results

In this section, the effects of heat conduction in the wall are summarized anddiscussed. Since they are qualitatively common to both cases of channels and tubes,the discussion is focused mainly on the former case. As the effects of heat conductionare often neglected, they appear through the small parameter ε defined by (3.2). Forthe gases and solids employed usually in thermoacoustic devices, ε takes a small value,of the order of 10−4–10−2, depending on the combination of the gas and solid, asshown in table 3.

Making use of the smallness of ε, the temperatures of the gas and solid wall areobtained in the form of the asymptotic expansion of ε up to its first order. Thetemperature variation in the solid is found to be of order ε in comparison with that inthe gas. Consistent with this variation, the temperature of the gas is also subjected tochanges of the order of ε, and so is the heat flux through the wall surfaces. This heatflux introduces new additional terms to the thermoacoustic-wave equation.

The thermoacoustic-wave equation is the one-dimensional wave equation for gas thatis non-uniform in temperature, to which the contributions from the shear stress atthe wall surfaces and the heat flux through them are added. The shear stress and theheat flux introduce hereditary effects represented by the functionals M and MP. The


relaxation functions Φ(νet/H2) and Φ(νet/PrH2) involved in them are expressed interms of the sum of exponential functions and they decay monotonically as t increaseswith typical relaxation times H2/νe and H2/κe, respectively.

It has been revealed that the effects of heat conduction in the wall introduce thetwo new terms represented by MT and MD resulting from the heat flux due to thetriple coupling among the viscous diffusion, the thermal diffusions of gas and solid,and the double coupling between the thermal diffusions of gas and solid, respectively.The relaxation functions MT and MD involved decay monotonically as t increases, andeach function consists of three sums. A new relaxation time d2/κs or R2/κs comes inthrough the third term in the cases of the channels and of the tubes, respectively.

Because the geometry parameter Ge is chosen arbitrarily, it occurs that dependingon the values of Pr , K and Ge, one of the coefficients of the exponential functions inthe respective sums may diverge and then the expansion exhibits non-uniformity. Thissuggests that the effects of heat conduction no longer remain of the order of ε. Such adivergence may be regarded as a sort of ‘resonance’. This case is discussed below in§ 6.2.

For the short-time behaviour, however, the effects of heat conduction remain to besmall, of the order of ε, irrespective of the resonance. This is because all relaxationfunctions behave as 1/

√πt for a short time. Therefore, the approximate equations

are subject to slight change through the coefficients C and CT . For the long-timebehaviour, on the other hand, the effects are expected to be significant. This becomespronounced as the value of Ge becomes smaller than unity, i.e. a wall is much thinnerthan a span length or a thermal diffusivity of the solid is much greater than a thermaldiffusivity of the gas. In view of the numerical values in tables 2 and 3, such a caseis plausible in reality if copper or ceramics are used and the ratio d/H is chosen to beextremely small.

6.2. Resonant case and unusual diffusion

6.2.1. Relations in the case without expansion in terms of εWhen the resonance takes place, the expansion in terms of ε becomes invalid and

the temperature distributions should be sought without expansion. Then T ′ in (4.6) ismodified by replacing fP in (4.7) with

fP = 1− 1A

cosh(y√Pr/Hδe)

cosh(√Pr/δe)

, (6.1)

where

A = 1+ ε tanh(√Pr/δe)

tanh(1/δs)= 1+ εCP, (6.2)

while εT ′ε in (4.8) is modified by replacing fK in (4.9) with

fK = 1A

tanh(1/δe)

tanh(1/δs)

cosh(y√Pr/Hδe)

cosh(√Pr/δe)

, (6.3)

and setting fKP to be zero. For T ′s, on the other hand, fS and fSP in (4.14) arereplaced by A −1fS and A −1fSP, respectively. Since the replacements above give exactexpressions to T ′ and T ′s, the symbol O(ε2) in (4.8) and (4.13) is unnecessary.


For the case of the tubes, fP is replaced by

fP = 1− 1A

I0(r/Rδκ)

I0(1/δκ), (6.4)

with

A = 1+ ε I1(1/δκ)I0(1/δκ)

Z(R, 1/δs)= 1+ εCP, (6.5)

and fK is replaced by

fK = 1A

[I1(1/δe)

I0(1/δe)Z(R, 1/δs)

]I0(r/Rδκ)

I0(1/δκ), (6.6)

with setting fKP = 0, while fS and fSP in (4.38) are replaced by A −1fS and A −1fSP,respectively.

By expanding 1/A as 1 − εCP + O(ε2), it is verified that T ′ and T ′s in § 4 arerecovered. Thus, each term in W on the right-hand side of (4.31) has a counterpart onthe left-hand side. The replacements by 1/A modify the heat flux in (4.16) throughgP, which affect the thermoacoustic-wave equations. But no changes occur in f andtherefore in the shear stress and g. Noting that gP results from differentiation of fP

with respect to y, gP is multiplied by A −1. In addition, (4.17) is replaced by

σ qw =−2cpTe√νe

a2e

[1

1− Pr

a2e

Te

dTe

dxσ−1/2 ∂ p′

∂x

CP

Ag(x,H)

], (6.7)

without the symbol O(ε). For the case of the circular tube, the factor 2 should bedeleted.

6.2.2. Modifications of relaxation functionsTo derive the thermoacoustic-wave equations, the inverse transform of the heat flux

(4.16) is necessary, and the inverse transform (4.20) is modified by 1/A . Thus, thefollowing transform must be evaluated:

1√2π

F−1

{σ−1/2

1+ ε tanh(√Pr/δe)/ tanh(1/δs)

tanh(√Pr/δe)

}. (6.8)

It is readily seen that the poles of tanh(√Pr/δe) and tanh(1/δs) are no longer poles of

the integrand. Instead new poles appear from zeros of the denominator by competitionbetween tanh(

√Pr/δe) and tanh(1/δs).

For the denominator to vanish: tanh(1/δs) + ε tanh(√Pr/δe) = 0; there are two

cases, one being a case in which tanh(√Pr/δe) takes a large value of order ε−1,

or the other a case in which tanh(1/δs) takes a small value of order ε. In theformer case, simple poles are located near those of tanh(

√Pr/δe) and are given

by −{(2n− 1)2π2/4 − ε(2n − 1)π/ tan[(n − 1/2)πGe] + O(ε2)}κe/H2. The poles are

shifted by the order of ε. Noting that the factor 1/ tan[(n − 1/2)πGe] results from1/ tanh(1/δs) and assuming this does not vanish for any positive integer of n, these


poles yield the inverse transform as

2√κe

H

∞∑n=1

{1+ ε Ge

sin2[(n− 1/2)πGe] + O(ε2)

}−1

× exp

{−(2n− 1)2π2

4κet

H2− ε (2n− 1)π

tan[(n− 1/2)πGe]κet

H2+ O(ε2)

}. (6.9)

In the latter case, simple poles are located near the zeros of tanh(1/δs). They arefound to be given by σ =− (nπ)2[1− 2ε tan(nπ/Ge)/nπ+ O(ε2)]κs/d2 and shifted bythe order of ε. Noting that tan(nπ/Ge) results from tanh(

√Pr/δe) and assuming this

does not diverge for any n, these poles yield

ε2√κs

d

∞∑n=1

tan2( nπ

Ge

)exp

[−n2π

2 κst

d2+ 2εnπ tan

( nπ

Ge

) κst

d2+ O(ε2)

]. (6.10)

Thus, the inverse transform of (6.8) is given by the sum of (6.9) and (6.10). If theexponential function in (6.9) is expanded in terms of ε, the leading term of orderε0 corresponds to (4.21), which yields the functional MP. The first-order terms inε correspond to the first and second sums of MD(t) in (5.2) with sign reversed,respectively, while (6.10) corresponds to the third sum with the sign reversed.

For the heat flux q, there is another contribution from qw given in (6.7), althoughsmall, of the order of ε. To evaluate this, the following inverse transform must beexecuted:

1√2π

F−1

{σ−1/2

1+ ε tanh(√Pr/δe)/ tanh(1/δs)

tanh(√Pr/δe) tanh(1/δe)

tanh(1/δs)

}. (6.11)

This is reduced to (4.23) to the lowest order in ε. The transform (6.11) can beevaluated in a similar fashion to (6.8). In this case, however, there occur poles oftanh(1/δe) in addition to those in (6.8), and there are contributions from three types ofpoles. It is verified that these correspond to the three sums in (5.1) to the lowest orderin ε.

It is found from (6.9) and (6.10) that the coefficients of ε diverge whentan[(n − 1/2)πGe] vanishes or tan(nπ/Ge) diverges for a special value Ge. Theseare the resonant cases seen in (5.2). For Ge = 2, for example, they correspondto cases where the zeros of cosh(

√Pr/δe) coincide with those of sinh(1/δs) and

κe/H2 = 4κs/d2. Then the simple poles of the integrand in (6.8) are located atσ =−[(n− 1/2)2π2±(2n− 1)π

√ε/2+ · · ·]κe/H2, and the integral is evaluated as

2√κe

H

∞∑n=1

exp

[−(2n− 1)2π2

4κet

H2

]cosh

[(2n− 1)π

√ε

2κet

H2

]+ O(ε). (6.12)

The relaxation becomes slower by the order of√ε in the hyperbolic function than the

order of ε in (6.9) and (6.10) in the non-resonant case. At resonance, the diffusion isunusual in the sense that the effects of heat conduction no longer remain of the orderof ε. This difference becomes significant as the value of ε becomes larger.

Profiles of the relaxation function are displayed in non-resonant cases. Taking thevalue of ε to be 0.01, figure 5 depicts logarithmic plots of the function Φ(t) − εMD(t)


t

2.5

3.5

4.5

10–110–2 100 102101

104

10–6

10–8

10–10

10–12

10–14

10–16

10–20

10–18

10–4

10–2

100

102

10–3

FIGURE 5. Logarithmic plots of the relaxation function Φ(t) − εMD(t) in the non-resonantcases with ε = 0.01 and Ge = 1.5, 2.5, 3.5 and 4.5 against t over the interval 10−3 6 t 6 102

where the abscissa and ordinate measure κet/H2 and (H/√κe)(Φ−εMD), respectively, but the

factor κe/H2 is omitted.

for four values of Ge(=1.5, 2.5, 3.5 and 4.5), where the factor κe/H2 in the abscissaκet/H2 and ordinate (H/

√κe)(Φ − εMD) is omitted. For t less than 5, all behaviours

are almost the same as 1/√πt. For t greater than 5, it is seen that as the value of

Ge becomes larger, slower relaxations appear. The curve of Φ(t) without the effects ofheat conduction almost coincides with the curve for Ge= 1.5.

The relaxation function in the resonant case for Ge = 2 is compared with those innon-resonance cases in the vicinity of it. Figure 6 displays the logarithmic plots of thefunction determined by (6.12) and Φ(t)− εMD(t) for the values of Ge= 1.95 and 2.05against t over the interval 0.1 6 t 6 100 where ε is chosen to be 0.2 and the factorκe/H2 is omitted from the abscissa and ordinate.

For reference, Φ(t) is drawn in the broken curve. The open and solid dots represent,respectively, the inverse Laplace transforms of (4.20) and (6.8) for Ge= 2 and ε = 0.2evaluated numerically by the double exponential formulas (Ooura & Mori 1991). Theopen dots almost lie on the broken curve for Φ(t) but the dots in both cases scatterfor the ordinate below 10−15, which is the limit of accuracy in double precision.Because the formulas provide very accurate results just as seen from the agreementsbetween the open dots and the broken curve, it is conjectured that differences betweenthe curve for Ge = 2 and the dots would stem from higher-order terms in (6.12).When (6.8) is evaluated numerically for Ge = 1.95 and Ge = 2.05, they are close tothe solid dots away from the curves with Ge= 1.95 and 2.05.

6.2.3. Resonance conditions and modified thermoacoustic-wave equationsResonance conditions are found from the coefficients of the exponential functions in

MT and MD. They are given for√

PrGe and Ge which coincide with 2m/(2n − 1) forany positive integers m and n, unless

√Pr takes (2m − 1)/(2n − 1). It is noted that

because the value of Ge varies with x, the resonance conditions are met at somewherealong the channel. For the resonance conditions in the case of the tubes, they areidentified by the relaxation functions NT and ND. The resonance occurs when jn

√Pr


2

2.05

1.95

10–6

10–8

10–10

10–12

10–14

10–16

10–20

10–18

10–4

10–2

100

102

t10–1 100 102101

FIGURE 6. Differences of the relaxation functions in the vicinity of the resonance atGe = 2 for ε = 0.2. Logarithmic plots of the relaxation function determined by (6.12)in the resonant case and Φ(t) − εMD(t) in the non-resonant cases for Ge = 1.95 and2.05 against t over the interval 0.1 6 t 6 100 where the abscissa and ordinate measureκet/H2 and (H/

√κe)(Φ − εMD), respectively, but the factor κe/H2 is omitted. The

open and solid dots represent, respectively, the inverse Laplace transforms of (4.20)and (6.8) evaluated numerically by the double exponential formulas where the formeralmost lie on the curve Φ(t) in the broken line but both dots scatter for the ordinatebelow 10−15.

or jn/√Pr hit one of the roots of J0 or when kn

√κs/νe or kn

√κs/κe hit one of those

roots. Further jn√νe/κs or jn

√κe/κs hit the poles of Z.

To derive the thermoacoustic-wave equation valid both in non-resonant and resonantcases, the asymptotic expansion in ε should be abandoned. Then the functionalsMP − εMD and MT in (4.31) and (4.30) are replaced, respectively, by MP and MT ,whose relaxation functions MP and MT are to be determined by the inverse transformsof (6.8) and (6.11), respectively. Then the thermoacoustic-wave equation takes thefollowing form:

∂2p′

∂t2− ∂

∂x

(a2

e

∂p′

∂x

)+ ∂

∂x

[a2

e

√νe

HM

(∂p′

∂x

)]+√νe

H

{γ − 1√

PrMP

(∂2p′

∂t2

)− a2

e

Te

dTe

dx

[1

1− PrM

(∂p′

∂x

)− 1

(1− Pr)√

PrMP

(∂p′

∂x

)

− ε

1− PrMT

(∂p′

∂x

)]}= 0. (6.13)

Although explicit expressions of MP and MT are not given, they are determined by thezeros of the function tanh(1/δs)+ε tanh(

√Pr/δe) and the poles of tanh(1/δe). Equation

(6.13) is the full thermoacoustic-wave equation taking account of the effects of heatconduction without making an expansion in terms of ε, and therefore it is valid for anyvalue of Ge. For the case of the circular tubes, similar replacements are necessary in(4.48) and (4.49). The relaxation functions in ˜NP and ˜NT are simply replaced by theinverse transforms of (4.46) and (4.47) with the factor 1/A multiplied.


Equation (6.13) has been derived by evaluating the heat flux fully. Using this heatflux, the mean temperature of the solid wall T ′s is obtained from (3.15) as

∂2T ′s∂t2= ε

Ge

Te

ρea2e

√νe

H

{γ − 1√

PrMP

(∂2p′

∂t2

)− a2

e

Te

dTe

dx

[1

1− PrM

(∂p′

∂x

)− 1

(1− Pr)√Pr

MP

(∂p′

∂x

)− ε

1− PrMT

(∂p′

∂x

)]}. (6.14)

The terms in the curl brackets may be replaced by the first three terms in (6.13) withsign reversed. The relation (6.14) holds for the case of the circular tubes by replacingM with N , and H with 2R. Then the mean temperature is defined by integrating2πrT ′s over the thickness and dividing it by π[(1+ η)2−1]R2, whereby the integral ofrZ over the thickness is equal to δ−1

s .

6.3. Effects of heat conduction on acoustic fieldsFinally we discuss how the effects of heat conduction in the wall affect the acousticfield. They are discussed in two cases where the thickness of the diffusion layer is thinand thick, which corresponds to the short- and long-time behaviours, respectively.

6.3.1. Case of a thin diffusion layerIn this case with |δe| � 1, the diffusion layer is treated as a boundary layer. In the

outside of this layer where |y| < H and |δe| � (H − |y|)/H, f in (4.4) and fP in (4.7)may be set to be unity, while fK and fKP may be dropped because errors involved areexponentially small. Then u′ and T ′ satisfy relations for lossless wave propagation in agas non-uniform in temperature, i.e. (3.4) and (3.6) with the y-dependence dropped.

A spanwise velocity v′ is available by multiplying (3.3) with yj to integrate it withrespect to y from y = 0. The result is given by (2.27) in I (with constant = 0) but yis now replaced by y/2j and an additional term ερe

√νe/Pr σ 1/2 (T ′ε|y=H/Te)gP appear

on the right hand due to the effect of heat conduction. In the outside of the boundarylayer, g and gP are exponentially small so that v′ is found to vary linearly in y. At theedge of the boundary layer where |y| ≈ H but |δe| � (H − |y|)/H, v′ is approximatedto be

∂v′

∂t≈− H

2jρea2e

[∂2p′

∂t2− ∂

∂x

(a2

e

∂p′

∂x

)]sgn(y). (6.15)

Using (5.18), v′ is alternatively given by (5.17) in I but C and CT are now modifiedto be (5.19). Similarly the shear stress s and the heat flux q are given by (5.18) and(5.19) in I, respectively, but with new C and CT and the factor 2 replaced by 21−j.

In the boundary layer, v′ decays rapidly toward the wall surface to vanish thereand so do u′ and T ′. In the solid wall, T ′s depends on δs, which is related to δe byδs = δe/

√PrGe. If a value of Ge is moderate, then T ′s decays exponentially from the

wall surface toward the interior of wall so that temperature variations on the wallsurface do not penetrate into the solid beyond a distance of order |δs|d just like in theboundary layer of the gas.

Setting tanh(1/δe)≈ tanh(√Pr/δe)≈ tanh(1/δs)≈ 1 for a moderate value of Ge, the

temperature variation on the wall surface T ′ at y= H is given by εT ′ε. This, denoted byT ′w, is given as

∂2T ′w∂t2= εTe

ρea2e

[(γ − 1)

∂2p′

∂t2+ 1

1+√Pra2

e

Te

dTe

dx

∂p′

∂x

]. (6.16)


The first term in the square brackets shows that T ′w varies with the pressure in phase,while the second term lags due to time-integration. When the temperature gradient issteep, the second term is larger than the first by χ 2.

For T ′s in (6.14), on the other hand, MT and MD are reduced to the minus half-orderderivative for a short time. Thus, T ′s is given by the minus half-order derivative of T ′w.The magnitude of T ′s/Te is found to be of the order of ε|δe|χ 2/

√PrGe relative to p′/p0.

As the value of Ge becomes smaller, the temperature variation becomes pronounced.

6.3.2. Case of a thick diffusion layerFor |δe| ∼ χ � 1, u′ is simply the velocity profile of the Poiseuille flow to the lowest

approximation. A higher-order term is not parabolic and given by Uj(y) as

u′ =− 12j+1µe

∂p′

∂x(H2 − y2)+ 1

8(3+ 5j)ρeν2e

∂2p′

∂t ∂xUj(y), (6.17)

where Uj(y)= (H2 − y2)[(5− 2j)H2 − y2], while v′ is given as

v′ = γ

2ρea2eH2

[∂p′

∂t− ε

2jGe

(γ − 1γ

∂p′

∂t+ αj

Te

dTe

dx

∂p′

∂x

)](H2 − y2)y. (6.18)

There appear peaks in |v′| at |y| = H/√

3, detached from the wall surface, whereas, forthe case of the boundary layer, the peaks appear at its edge.

The temperature T ′ vanishes to the lowest order. A higher-order approximation isgiven by

T ′ = 12j+1ke

∂p′

∂t(H2 − y2)+ Pr

8(3+ 5j)ρeν2e

dTe

dx

∂p′

∂xUj(y)+ εT ′ε, (6.19)

with εT ′ε and T ′s given by

ε∂T ′ε∂t≈ ∂T ′s∂t≈ ε

2jρecpGe

(∂p′

∂t+ γ

γ − 1αj

Te

dTe

dx

∂p′

∂x

). (6.20)

Thus, the temperatures of the solid wall is uniform over the thickness to the lowestorder. The leading term in (6.19) relative to Te is of the order of |δe |−2 p′/p0.Comparing it with εT ′ε, εT

′ε becomes larger if ε/PrGe� |δe |−2.

Finally the shear stress and heat flux are given, to the higher-order approximation,by

s= H

22j−1

[∂p′

∂x− H2

(3+ 5j)νe

∂2p′

∂t∂x

], (6.21)

and

q=−(

1− ε

2jGe

) H

22j−1

(∂p′

∂t+ γ

γ − 1αj

Te

dTe

dx

∂p′

∂x

). (6.22)

While the shear stress is subject to no changes due to the effects of heat conduction,the heat flux is modified by the factor ε/Ge. Thus, as this value becomes larger, theeffects of heat conduction cannot be ignored.

7. ConclusionsIn this paper we have examined the effects of heat conduction in the wall on

acoustic-wave propagation in gas enclosed in the two-dimensional channels and


circular tubes subject to an axial temperature gradient. On the basis of the narrow-tube approximation, the linearized basic equations in both cases have been reduced tosingle thermoacoustic-wave equations for the excess pressure. While the equations arevalid generally at any time and for any form of temporal and spatial variations, theapproximate equations have been derived from the respective wave equations for short-and long-time behaviours specified according to whether the Deborah number is muchlarger or smaller than unity.

The present analysis has clarified quantitatively under what conditions the effectsof heat conduction are negligible or become pronounced to be primary. The effectsappear through the parameter ε defined by the square root of the product of tworatios, one being the ratio of the heat capacities per volume and the other the ratio ofthe thermal conductivities. This is not the only parameter for the effects as they aredetermined by the ratio of thermal diffusivities and especially the geometry parameterGe defined by the relative thickness of the wall.

Because the heat capacity per volume and thermal conductivity of the solid wall areusually greater than those of gas, the value of ε is very small so that the effects ofheat condition have been taken into account to the first order of ε. But it has beenunveiled that when the geometry parameter takes special values, the expansion exhibitsnon-uniformity, i.e. resonance, and then the effects become enhanced up to the orderof√ε. In comparison with the diffusion of ε in non-resonant case, the effects give

rise to unusual diffusion in a transient behaviour and modify the thermoacoustic-waveequations.

In addition to such a resonance, the effects become pronounced in a transientbehaviour as the value of Ge becomes larger, whereas in a long-time behaviour, theybecome pronounced as the value of Ge becomes smaller and ε/Ge becomes larger.When the value of Ge is small, the heat capacity of the solid wall per unit axial lengthis not large enough compared with that of the gas. Thus, the condition for a long-timebehaviour, Ge2PrH2/νet� 1, is quickly satisfied after a short time. If the value of Geis large, it takes a longer time for a long-time behaviour to be realized but then theeffects are smaller relatively. For the short-time behaviour, in contrast, the effects giverise to only small effects of order ε and therefore they may always be negligible.

In conclusion, the effects of heat conduction in the wall may be neglected as long asthe value of ε is small enough. But as it becomes larger, although smaller than unity, itmay happen that the effects appear enhanced, depending through Ge on the geometryof the channels and the tubes, and the combination between the gas and the solid.The quantitative results about the effects provide useful information in estimating theirinfluence on thermoacoustic-wave propagation and also as a guideline in designingthermoacoustic devices.

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