- Email: [email protected]
Heat generation by impulse ultrasound
Ultrasonics 43 (2004) 95–100 www.elsevier.com/locate/ultras
Heat generation by impulse ultrasound Anna Perelomova Faculty of Applied Physics and Mathematics, Department of Theoretical Physics, Technical University of Gdansk, ul. G. Narutowicza 11/12, 80-952 Gdansk, Poland Accepted 12 April 2004 Available online 28 April 2004
Abstract An original method allowing to get a system of nonlinear evolution equations for the interacting modes applies to a problem of the heat generation by non-periodic ultrasound, including impulse one. The basic idea and final equations for the thermoviscous plane flow are presented. The limit of periodic source is traced. The numerical calculations were based on the pulse solution of the Burgers equation as an ultrasound source. Some illustrations on temporal behavior of the medium expansion caused by the pulse ultrasound are presented. Ó 2004 Elsevier B.V. All rights reserved. Keywords: Acoustical heating; Impulse ultrasound; Nonlinear flow
1. Introduction Acoustic heating is a phenomenon appearing due to the sound absorption. It is important in many applications of ultrasound, also in medical therapy, where an increase of the background temperature should be evaluated. An elevation of temperature is caused by a heat producing per unit volume and followed by a decrease of density. The change of density caused by absorption was not taken into account in many sources [1,4] since the ambient state was traditionally considered as a purely incompressible liquid. Recently, the distortion of density was proved to be important when studying the acoustic heating for the majority of fluids [5]. The secondary processes in the sound field such as acoustic heating change the background of acoustic wave propagation and therefore influence on the primary wave itself. The novel phenomena as focusing of an acoustic beam are observed due to the strong ultrasound absorption [12]. Usually, the theory applies to the periodic acoustic wave caused by transducer. The averaging over one period (or over the integer number of periods) allows to separate non-acoustic and acoustic parts of the overall perturbation. The interval of averaging must be small in
E-mail address: [email protected] (A. Perelomova). 0041-624X/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.ultras.2004.04.002
comparison to the characteristic time of the non-acoustic slowly varying field. In an absorbing fluid, the basic relation to be averaged is the total energy conservation ~~ law: oE=ot þ r J ¼ 0. Here, E ¼ qe þ qð~ v ~ vÞ=2 is the total energy volume density, ~ J ¼ p~ v þ E~ v is energy flux density vector, e, q,~ v, p are internal energy per mass unit, mass density, velocity, and pressure, correspondingly. All variables are thought as a sum of the acoustic (subscript a) and non-acoustic parts. Averaging over one period for the periodic sound wave (marked by square brackets) yields in the result for the instantaneous rate per unit volume q at which heat is produced in a medium ~ ~ by ultrasound: hqi ¼ r J a with an acoustic source in the right-hand side, ~ J a ¼ hpa~ va i [3]. This way to evaluate heat generation is suitable for the periodic acoustic waves. We may say that the very procedure of temporal averaging serves as a projecting to ‘slow’ and ‘quick’ parts of the overall flow and fits in many real applications concerning periodic ultrasound but has essential shortage. The temporal averaging fails when pulses or other non-periodic acoustic waves are the source of heating. In medical therapy, single short pulses are of great importance; actually almost all experimental data deal with wave packets. Temporal averaging says nothing about temporal behavior of both acoustic and heat modes. We present here a way to evaluate heating caused by any acoustic field including non-periodic one. The basic idea is to separate modes on the level of the initial
96
A. Perelomova / Ultrasonics 43 (2004) 95–100
system of conservation laws. Modes as all possible flows following from the linear conservation laws in fluid are thought as eigenvectors of the evolution operator relating to the basic system of hydrodynamic equations. Matrix projectors allow to separate every mode from the overall perturbation at any moment and to get dynamic equations for the interacting modes. The idea has been developed by the author in some applications to flows over inhomogeneous and stratified media [7,8]. The steps to get approximate solution are also pointed out there. The procedure does not need temporal averaging and allows to consider non-isentropic initial conditions. Therefore, the heat mode is not certainly secondary mode caused by sound. We do not concern here the three-dimensional flow where rotational flow of liquid appears though an idea is fruitful also in the case of multi-dimensional movement [11,15]. Note that rotational modes are slowly varying as well as heat one so the temporal averaging could not separate these ‘slow’ modes in the consistent manner. To avoid involving of the heat mode, the initial point for problems relating to streaming are dynamic equations for flow over incompressible liquid. The inconsistency of this approach was underlined in many sources [6,14]. We may add that the heating following ultrasound changes both temperature and density of the medium so the new background of the acoustic and rotational perturbations appear that could not be taken into account in the incompressible model of fluid.
2. Theory The basic system of conservation equations in the differential form looks: o~ v l~ ~ v þ rp ~ lD~ þ qð~ vrÞ~ v r div~ v ¼ 0; ot 3 oe ~ v vDT Q ¼ 0; q þ pr~ ot oq ~ þ rðq~ vÞ ¼ 0; ot
q
ð1Þ
where, among already mentioned variables, T means temperature, v is the coefficient of heat conductivity, Q is so-called dissipation function determined by formula [2]: Q¼
3 X
sik vik ;
i;k¼1
with vik being tensor of deformations given by vik ¼ 0:5ðovi =oxk þ ovk =oxi Þ, and sik is as follows: sik ¼ 2lvik ~ v=3. The fluids obeying the (i 6¼ k), sii ¼ 2lvii 2lr~ p equation of state for an ideal gas e ¼ qðc 1Þ will be considered for simplicity (c ¼ Cp =Cv is the ratio of specific heats at constant pressure and constant volume, respectively). There is no any difficulties in evaluating
any other fluid by expansion of internal energy and temperature in series, see [9]. For the plane flow depending on one spatial co-ordinate x over uniform background (p0 ; q0 ) the system (1) goes to the equivalent system in non-dimensional variables v , p , q , x , t (v ¼ v=c, p ¼ ðp p0 Þ=c2 q0 , q ¼ ðq q0 Þ=q0 , x ¼ x= k, t ¼ tc=k, asterisks for dimensionless variables will be later omitted): o ~ w þ Lw ¼ w: ð2Þ ot Here, k means a characteristic scale of disturbance, c is adiabatic sound velocity, rffiffiffiffiffiffiffiffi p0 c¼ c ; q0 0
d1 o2 =ox2 @ o=ox L¼ o=ox
o=ox cd2 2 c 1 o =ox2 0
1 0 d2 2 o =ox2 A c 1 0
is the linear matrix operator with d1 ¼
4l ; and 3q0 ck
v d2 ¼ q0 ck
1 1 ; Cv Cp
0 1 v w ¼ @pA q
is column of perturbations. The right-hand nonlinear vector 0 1 o2 v v oxo v þ q oxo p d1 q ox 2 2 C ~ ¼B w @ v oxo p cp oxo v þ d1 ðc 1Þ ov A ox o o v ox q q ox v includes only quadratic nonlinear terms that are of the major importance in nonlinear acoustics. The possible linear motions of fluid follow from the linearized version of system (2): o w þ Lw ¼ 0: ð3Þ ot There are two acoustic modes relating to rightwards and the leftwards progressive (acoustic) waves as well as the heat (called also entropy) mode: 1 0 1 0 v1 ðx; tÞ 1 ðb=2Þo=ox C B C B w1 ¼ @ p1 ðx; tÞ A ¼ @ 1 d2 o=ox Aq1 ðx; tÞ; q1 ðx; tÞ 1 1 1 ðb=2Þo=ox B C w2 ¼ @ 1 þ d2 o=ox Aq2 ðx; tÞ; 0
0 B w3 ¼ @
1 d2 o=ox c 1
0 1
1 C Aq3 ðx; tÞ;
ð4Þ
A. Perelomova / Ultrasonics 43 (2004) 95–100
where b ¼ d1 þ d2 . An overall flow now is thought as a T sum of all modes: wðx; tÞ ¼ ð vðx; tÞ pðx; tÞ qðx; tÞ Þ ¼ w1 þ w2 þ w3 . Next, matrix projectors are determined due to equations: Pn wðx; tÞ ¼ wn ðx; tÞ (n ¼ 1, 2, 3) that separate modes from the overall perturbations in every moment: 0
d2
b4 o=ox
1
d2 d2 o=ox o=ox 12 2ðc 1Þ 2ðc 1Þ C C b cd2 d2 1 1 C; o=ox o=ox 2 P1;2 2ðc 1Þ 2ðc 1Þ 2 4 C A b d2 d2 d2 1 1 2 þ 2 o=ox 4 2ðc 1Þ o=ox 2ðc 1Þ o=ox 2 0 1 d2 d2 0 o=ox c 1 o=ox c 1 B C ð5Þ P3 ¼ @ A: 0 0 0 1 1 d2 o=ox 1
B2 B ¼B B @
2
The operators possess general properties of orthogonal projectors: P1 þ P2 þ P3 ¼ eI ; P1 P2 ¼ P1 P3 ¼ . . . ¼ ~0, where eI and ~ 0 are unit and zero matrices. The projectors P1 , P2 , P3 do commute both with L and oto , so one can act by any projector at the basic system immediately, going to the linear evolution equation for every mode: o w þ Lwn ¼ 0 ðn ¼ 1; 2; 3Þ: ð6Þ ot n We fix relations (4) going to the nonlinear flow. The nonlinear evolution equations for interacting modes follow from Eq. (2) when corresponding projector acts on both sides of this equation: o ~ w þ Lwn ¼ Pn w: ð7Þ ot n ~ depends on a sum of The right-hand nonlinear vector w all modes. If the rightwards progressive acoustic wave is dominant, one can keep only this mode input in the nonlinear part. For the first mode evolution, Eq. (7) yields in the well-known Burgers evolution equation (cross nonlinear–viscous terms are omitted): oq1 oq1 c þ 1 o b o2 q1 q1 þ þ q ¼ 0: 2 ox 2 ox2 1 ot ox
ð8Þ
3. General formulae and the limit of periodic acoustic source To get an evolution equation for the heat mode generation, one acts by the projector P3 on the system (7) assuming that the nonlinear right-hand vector consists of the rightwards progressive acoustic mode inputs only. For the density perturbation of the heat mode, an equation looks: oq3 d2 o2 q3 op1 ov1 oq ov1 þ cp1 v1 1 q1 ¼ v1 ot c 1 ox2 ox ox ox ox
2 ov1 ; ð9Þ ðc 1Þd1 ox
97
or, taking into account relations inside the acoustic mode given by Eq. (4), in terms of the dominant mode pressure: oq3 d2 o2 q3 ot c 1 ox2
2 ! op1 b o2 p1 op1 ¼ ðc 1Þ p1 b : p1 ox 2 ox2 ox
ð10Þ
Only quadratic nonlinear terms are left in the evolution equation (10). To compare Eq. (10) with the known results for the periodic wave, let us take the approximate periodic solution of the Burgers equation (8) (suitable for the small Reynolds numbers, see [13]) since the acoustic pressure p1 ðx; tÞ has to satisfy this equation: p1 ðx; tÞ ¼
4P0 expð bx=2Þ sinðx tÞ: Reðc þ 1Þ
ð11Þ
The non-dimensional Reynolds number is Re ¼ P0 =b, P0 is an amplitude on a transducer. The formula (11) is correct beyond some vicinity of transducer where the nonlinear distortions are strong, and the amplitude of 4P0 solution (11) p10 ¼ Reðcþ1Þ is independent on P0 . Temporal average of both sides of (10) gives: 2 oq3 d2 o2 q3 op1 ¼ ðc 1Þ ot c 1 ox2 ox b 2 expð bxÞ: ð12Þ ¼ ðc 1Þp10 2 In all calculations quadratic viscous terms as well as cubic nonlinear ones were left out of account. To calculate Eq. (10), note that the periodic perturbation (11) satisfies the relations below: *
2 + op1 b o2 p 1 b op1 p1 2 ¼ : ð13Þ p1 ¼ 2 2 ox ox ox The formula (12) goes to the known result for the periodic ultrasound given in introduction. It is known that heating is an isobaric process as proved by the eigenvector w3 ðx; tÞ defined by Eq. (4). The rate of heat and temperature distortions (both dimensional) per unit volume in our variables looks: c oT c3 q0 oq3 q ¼ q0 C p ¼ : ð14Þ k ot kðc 1Þ ot From the other hand, a gradient of dimensional acoustic energy flux relating to the rightwards plane wave is 1 o q c3 o q c3 o 2 hJ1 i ¼ 0 hp1 v1 i ¼ 0 p ; k ox k ox k ox 1
ð15Þ
where the last calculation ignores terms of order b2 . Since q ¼ 1k oxo hJ1 i, the formula (12) follows from D Eqs. E d2 o2 q3 (14) and (15), it involves an additional term c 1 ox2 which is not usually involved in Eq. (12) when periodic
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A. Perelomova / Ultrasonics 43 (2004) 95–100
acoustic sources considered. This term is responsible for the expansion due to effects of the thermal conductivity. 4. Heating caused by pulses. Illustrations As we already proved, the evolution equation (10) goes to the known case of periodic acoustic source but is suitable for any ultrasound source including non-peri1 odic one. At the first view, the term ðc 1Þp1 op ox in the right-hand side of Eq. (10) is expected to be dominant in comparison to two others because the last two terms are multiplied by the small thermoviscous coefficients. But 1 for periodic motions, the first term ðc 1Þp1 op ox provides only one half of the known result as follows from the discussion before since the temporal averages of all three terms are of the same order. To simplify calculations, let a medium of ultrasound propagation does not possess thermal conductivity: d2 ¼ 0. Relating to the boundary regime problem, we rewrite an evolution equation (10) going to the convenient variables (slowly varying n ¼ bx and the retarded time s ¼ t x) in the leading order as follows:
2 ! oq3 op1 op1 b o2 p1 op1 ¼ ðc 1Þ bp1 p1 b : p1 os on os 2 os2 os ð16Þ with the acoustic pressure being a solution of the Burgers equation which in the new variables looks
op1 o c þ 1 p12 b o p1 ¼ 0: ð17Þ þ b 2 2 2 os on os For the small Reynolds numbers Eq. (17) gives o 1 o2 p1 p1 ; on 2 os2
ð18Þ
that agrees with the first relation from Eq. (13) for the averaged values. In fact, as it follows from Eq. (17), a 2 value bp1 opon1 b2 p1 oosp21 in the right-hand side of Eq. p13
and therefore is beyond the preci(16) is of order of sion of calculations and should be ignored. 1 To interpret properly the term ðc 1Þp1 op os in the right-hand side of Eq. (16), some notes about general features of linear projecting should be done. Linear projecting can not in principle support nonlinear relations of perturbations providing the isentropic (quasi-isentropic due to attenuation) wave. The relation between dimensionless perturbations of pressure and density in isentropic waveform in are as follows: 2
p ¼ q þ ðc 1Þq =2 þ ; v ¼ q þ ðc 3Þq2 =4 þ
ð19Þ
Linear projecting keeps exact relations specific for modes but results in the fact that every mode becomes a sum of parts moving with approximate velocities of
other modes due to nonlinear interactions. Indeed, assuming that the rightwards progressive mode is dominative, the quadratic parts of two other modes that follow the acoustic one, are: q2;fol ¼ ðc þ 1Þq21 =8;
q3;fol ¼ ðc 1Þq21 =2:
Calculations are based on the dynamic equation (7). Keeping all rightwards parts, one goes to a vector of perturbations as follows (relations (4) for non-viscous flow are used): 0 1 v w ¼ @pA q 0 1 0 1 0 1 1 1 0 ¼ @ 1 Aq1 þ @ 1 Aq2;fol þ @ 0 Aq3;fol 1 1 1 0 1 0 1 c 1 1 1 ¼ @ 1 Aq1 þ @ c þ 1 Aq21 : ð20Þ 8 5 3c 1 It may be easily checked that relations of p, q, v given by Eq. (20) are isentropic with the accuracy of second-order terms and satisfy relations (19). All higher-order corrections may be written on in the same way. 1 After all the said, it is clear, that term ðc 1Þp1 op os in the right-hand side of Eq. (16) does not participate in the heated trace after acoustic pulse passing, it is actually a part of rightwards motion keeping a wave quasi-isentropic. In the case of a pulse, integration of this term over time that includes a pulse gives zero. For an unlimited in time acoustic source, this term does contribute to a local heating. In the frames of pulses, Eq. (16) yields in the next dynamic equation true for times after pulse passing: 2 Z t x
op1 q3;trace ðx; tÞ ¼ ðc 1Þb ds: ð21Þ os x The subtle structure of expansion at any time includes both terms and looks as follows:
2 ! Z t x op1 op1 p1 q3 ðx; tÞ ¼ ðc 1Þ ds: ð22Þ þb os os x To give an example of a mono-polar source, let us take a solution of the Burgers equation in the form of selfsimilar solution of Eq. (17), see [13]: 2b p1 ðn; sÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðn þ n0 Þ
ðsþs0 Þ2 exp 2ðnþn 0Þ
; p ffiffiffiffiffi ffi pffiffiffi sþs0 p ffiffiffiffiffiffiffiffiffiffiffi 2 bC cþ1 2p erf 2
ð23Þ
2ðnþn0 Þ
with n0 , s0 , C being constants. Note that in contrast to periodic sound (11), the solution (23) does not relate to
A. Perelomova / Ultrasonics 43 (2004) 95–100 ∆ T/T0
(a)
p/( ρ0c2)
99
0.0014 0.02
0.0012 0.5
1
1.5
2
tc/λ
0.001 0.0008
-0.02
0.0006 -0.04
0.0004 -0.06
0.0002
-0.08
1 p/( ρ0c2)
2
3
4
5
6
x
Fig. 2. Stationary disturbance of the relative temperature DT =T0 versus dimensionless distance from a transducer: bold line corresponds to b ¼ 0:1, thin one corresponds to b ¼ 0:05.
(b)
0.015 0.01
5. Discussion
0.005 4
5
6
7
tc/λ
-0.005 -0.01 -0.015
Fig. 1. (a) Dimensionless pressure of acoustic pulse p=ðq0 c2 Þ via dimensionless time tc=k at distance x=k ¼ 1 from a transducer (bold line). A thin line marks change in density of background corresponding to acoustic heating q3;trace =q0 multiplied by 300 (value given by Eq. (21)), a dashed line marks q3 =q0 multiplied by 300 (value given by Eq. (22)). In these series, b ¼ 0:01. (b) The same as at (a) at distance x=k ¼ 5 from a transducer.
large or small Reynolds numbers. The absolute value of C is responsible for the symmetry of the impulse: jCj 1 gives a curve close to the Gauss one, and a sign of C determines qaffiffiffiffipolarity of the pulse. Let us consider p e replacing C by the new constant large C ¼ cþ1 C 2 2b 0 e Oðb Þ. In calculations of Eqs. (21) and (22) with C ultrasound source given by Eq. (23), the next values of e ¼ 2, n0 ¼ s0 ¼ 0. parameters are taken: c ¼ 1:4, C Calculations based on formulae (21) and (22) are presented by Fig. 1(a) and (b). There are curves of pressure of acoustic pulse and corresponding decrease in the density of the background. For convenience, p and t in the axes labels mark dimension perturbation of pressure of acoustic wave and time. Fig. 1 demonstrates the approximate equality of expansion given by formulae (21) and (22) at times after pulse passing. The relative stationary increase of the temperature of the background follows from Eq. (21): DT ðxÞ ¼ ðc 1Þb lim t!1 T0
Z
t x
x
op1 os
2 ds:
ð24Þ
Calculations based on formula (24) are shown at the Fig. 2 for two different values of dimensionless attenuation b. More b, more increase in temperature of the background expected.
Illustrations show that starting far after pulse passing, the trace followed the ultrasound pulse approaches a certain limit: a negative value relating to the rarefaction of the medium (and to the increase in temperature, meaning that the process occurs under a constant pressure). In contrast to the periodic ultrasound, the rarefied trace goes to a limit and does not grow with time. For a single acoustic pulse, the large values of heating of the background are hardly expected. Nevertheless, calculations of the heating given rise by single pulse or series of pulses, may be useful in medical therapy and other applications of acoustics of pulses. The calculations given in the article are just an example of evaluating of this subtle phenomenon in the ideal gas. Any other medium of sound propagation may be easily considered by involving the correspondent thermodynamic equations of state (see [10]). Account of thermoconductivity would lead to (1) somewhat changed thermoviscous coefficients in the right-hand side of Eq. (10) and, the more important, (2) the natural process of slow heat transfer depending on the temperature gradient caused by account of d 2 o 2 q3 the term c 1 in the linear left-hand side of the ox2 dynamic equation (10). While unlimited in time signals treated, a term 1 ðc 1Þp1 op os is important, it does participate in the local heating. It should be stressed that general formula (10) goes to Eq. (16) which was used in calculations when acoustic wave may be thought as a progressive wave slightly varying with distance from a transducer: p1 ¼ p1 ðt x; bxÞ. Note that there are other non-periodic waves like shock one that could not be in principle treated in this way. Then formula (10) should be used. The advantages of the method are obvious if nonperiodic acoustic source is considered: the final equations are obtained without a procedure of temporal averaging, so the detail temporal behavior of all modes
100
A. Perelomova / Ultrasonics 43 (2004) 95–100
may be investigated. The heating is one of the possible results of modes interaction in the plane flow, when the acoustic mode is dominant. Some other applications of linear projecting in the multi-dimensional flow were recently published by the author, where acoustic streaming from the non-periodic acoustic beams considered [11].
References [1] N.S. Bakhvalov, Y.M. Zhilejkin, E.A. Zabolotskaya, Nonlinear Theory of Sound Beams, Am. Inst. Phys., New York, 1987. [2] D.I. Blokhintsev, Acoustics of Non-uniform Moving Medium, Nauka, Moscow, 1981 (in Russian). [3] I. Gyarmati, Nonequilibrium Thermodynamics, Field Theory and Variation Principles, Springer-Verlag, Berlin, 1970. [4] A.A. Karabutov, O.V. Rudenko, O.A. Sapozhnikov, Thermal self-focusing of weak shock waves, Soviet Physics Acoustics 34 (1989) 371–374. [5] S. Makarov, M. Ochmann, Nonlinear and thermoviscous phenomena in acoustics, Part I, Acta Acustica 82 (1996) 579–606. [6] L. Menguy, J. Gilbert, Nonlinear acoustic streaming accompanying a plane stationary wave an a guide, Acta Acustica 86 (2000) 245–259.
[7] A.A. Perelomova, Nonlinear dynamics of vertically propagating acoustic waves in a stratified atmosphere, Acta Acustica 84 (1998) 1002–1006. [8] A.A. Perelomova, Projectors in nonlinear evolution problem: acoustic solitons of bubbly liquid, Applied Mathematical Letters 13 (2000) 93–98. [9] A.A. Perelomova, Nonlinear dynamics of directed acoustic waves in stratified liquids and gases, Optics of Atmosphere and Ocean 13 (2) (2000) 133–138. [10] A. Perelomova, Interaction of modes in nonlinear acoustics: theory and applications to pulse dynamics, Acta Acustica united with Acustica 89 (2003) 86–94. [11] A. Perelomova, Acoustic radiation force and streaming caused by non-periodic acoustic source, Acta Acustica united with Acustica 89 (2003) 754–763. [12] O.V. Rudenko, M.M. Sagatov, O.A. Sapozhnikov, Thermal selffocusing of saw-tooth waves, Soviet Physics JETP 71 (1990) 449– 451. [13] O.V. Rudenko, S.I. Soluyan, Theoretical Foundations of Nonlinear Acoustics, Consultants Bureau, New York, 1977. [14] J.N. Tjotta, S. Tjotta, Nonlinear equations of acoustics. in: Proceedings of the 12th International Symposium on Nonlinear Acoustics, 1990, pp. 80–97. [15] I.S. Vereschagina, A.A. Perelomova, Investigation of influence of space modes interaction on nonlinear dynamics of sound beams, Acoustical Physics 48 (2) (2002) 147–154.
Heat generation by impulse ultrasound Anna Perelomova Faculty of Applied Physics and Mathematics, Department of Theoretical Physics, Technical University of Gdansk, ul. G. Narutowicza 11/12, 80-952 Gdansk, Poland Accepted 12 April 2004 Available online 28 April 2004
Abstract An original method allowing to get a system of nonlinear evolution equations for the interacting modes applies to a problem of the heat generation by non-periodic ultrasound, including impulse one. The basic idea and final equations for the thermoviscous plane flow are presented. The limit of periodic source is traced. The numerical calculations were based on the pulse solution of the Burgers equation as an ultrasound source. Some illustrations on temporal behavior of the medium expansion caused by the pulse ultrasound are presented. Ó 2004 Elsevier B.V. All rights reserved. Keywords: Acoustical heating; Impulse ultrasound; Nonlinear flow
1. Introduction Acoustic heating is a phenomenon appearing due to the sound absorption. It is important in many applications of ultrasound, also in medical therapy, where an increase of the background temperature should be evaluated. An elevation of temperature is caused by a heat producing per unit volume and followed by a decrease of density. The change of density caused by absorption was not taken into account in many sources [1,4] since the ambient state was traditionally considered as a purely incompressible liquid. Recently, the distortion of density was proved to be important when studying the acoustic heating for the majority of fluids [5]. The secondary processes in the sound field such as acoustic heating change the background of acoustic wave propagation and therefore influence on the primary wave itself. The novel phenomena as focusing of an acoustic beam are observed due to the strong ultrasound absorption [12]. Usually, the theory applies to the periodic acoustic wave caused by transducer. The averaging over one period (or over the integer number of periods) allows to separate non-acoustic and acoustic parts of the overall perturbation. The interval of averaging must be small in
E-mail address: [email protected] (A. Perelomova). 0041-624X/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.ultras.2004.04.002
comparison to the characteristic time of the non-acoustic slowly varying field. In an absorbing fluid, the basic relation to be averaged is the total energy conservation ~~ law: oE=ot þ r J ¼ 0. Here, E ¼ qe þ qð~ v ~ vÞ=2 is the total energy volume density, ~ J ¼ p~ v þ E~ v is energy flux density vector, e, q,~ v, p are internal energy per mass unit, mass density, velocity, and pressure, correspondingly. All variables are thought as a sum of the acoustic (subscript a) and non-acoustic parts. Averaging over one period for the periodic sound wave (marked by square brackets) yields in the result for the instantaneous rate per unit volume q at which heat is produced in a medium ~ ~ by ultrasound: hqi ¼ r J a with an acoustic source in the right-hand side, ~ J a ¼ hpa~ va i [3]. This way to evaluate heat generation is suitable for the periodic acoustic waves. We may say that the very procedure of temporal averaging serves as a projecting to ‘slow’ and ‘quick’ parts of the overall flow and fits in many real applications concerning periodic ultrasound but has essential shortage. The temporal averaging fails when pulses or other non-periodic acoustic waves are the source of heating. In medical therapy, single short pulses are of great importance; actually almost all experimental data deal with wave packets. Temporal averaging says nothing about temporal behavior of both acoustic and heat modes. We present here a way to evaluate heating caused by any acoustic field including non-periodic one. The basic idea is to separate modes on the level of the initial
96
A. Perelomova / Ultrasonics 43 (2004) 95–100
system of conservation laws. Modes as all possible flows following from the linear conservation laws in fluid are thought as eigenvectors of the evolution operator relating to the basic system of hydrodynamic equations. Matrix projectors allow to separate every mode from the overall perturbation at any moment and to get dynamic equations for the interacting modes. The idea has been developed by the author in some applications to flows over inhomogeneous and stratified media [7,8]. The steps to get approximate solution are also pointed out there. The procedure does not need temporal averaging and allows to consider non-isentropic initial conditions. Therefore, the heat mode is not certainly secondary mode caused by sound. We do not concern here the three-dimensional flow where rotational flow of liquid appears though an idea is fruitful also in the case of multi-dimensional movement [11,15]. Note that rotational modes are slowly varying as well as heat one so the temporal averaging could not separate these ‘slow’ modes in the consistent manner. To avoid involving of the heat mode, the initial point for problems relating to streaming are dynamic equations for flow over incompressible liquid. The inconsistency of this approach was underlined in many sources [6,14]. We may add that the heating following ultrasound changes both temperature and density of the medium so the new background of the acoustic and rotational perturbations appear that could not be taken into account in the incompressible model of fluid.
2. Theory The basic system of conservation equations in the differential form looks: o~ v l~ ~ v þ rp ~ lD~ þ qð~ vrÞ~ v r div~ v ¼ 0; ot 3 oe ~ v vDT Q ¼ 0; q þ pr~ ot oq ~ þ rðq~ vÞ ¼ 0; ot
q
ð1Þ
where, among already mentioned variables, T means temperature, v is the coefficient of heat conductivity, Q is so-called dissipation function determined by formula [2]: Q¼
3 X
sik vik ;
i;k¼1
with vik being tensor of deformations given by vik ¼ 0:5ðovi =oxk þ ovk =oxi Þ, and sik is as follows: sik ¼ 2lvik ~ v=3. The fluids obeying the (i 6¼ k), sii ¼ 2lvii 2lr~ p equation of state for an ideal gas e ¼ qðc 1Þ will be considered for simplicity (c ¼ Cp =Cv is the ratio of specific heats at constant pressure and constant volume, respectively). There is no any difficulties in evaluating
any other fluid by expansion of internal energy and temperature in series, see [9]. For the plane flow depending on one spatial co-ordinate x over uniform background (p0 ; q0 ) the system (1) goes to the equivalent system in non-dimensional variables v , p , q , x , t (v ¼ v=c, p ¼ ðp p0 Þ=c2 q0 , q ¼ ðq q0 Þ=q0 , x ¼ x= k, t ¼ tc=k, asterisks for dimensionless variables will be later omitted): o ~ w þ Lw ¼ w: ð2Þ ot Here, k means a characteristic scale of disturbance, c is adiabatic sound velocity, rffiffiffiffiffiffiffiffi p0 c¼ c ; q0 0
d1 o2 =ox2 @ o=ox L¼ o=ox
o=ox cd2 2 c 1 o =ox2 0
1 0 d2 2 o =ox2 A c 1 0
is the linear matrix operator with d1 ¼
4l ; and 3q0 ck
v d2 ¼ q0 ck
1 1 ; Cv Cp
0 1 v w ¼ @pA q
is column of perturbations. The right-hand nonlinear vector 0 1 o2 v v oxo v þ q oxo p d1 q ox 2 2 C ~ ¼B w @ v oxo p cp oxo v þ d1 ðc 1Þ ov A ox o o v ox q q ox v includes only quadratic nonlinear terms that are of the major importance in nonlinear acoustics. The possible linear motions of fluid follow from the linearized version of system (2): o w þ Lw ¼ 0: ð3Þ ot There are two acoustic modes relating to rightwards and the leftwards progressive (acoustic) waves as well as the heat (called also entropy) mode: 1 0 1 0 v1 ðx; tÞ 1 ðb=2Þo=ox C B C B w1 ¼ @ p1 ðx; tÞ A ¼ @ 1 d2 o=ox Aq1 ðx; tÞ; q1 ðx; tÞ 1 1 1 ðb=2Þo=ox B C w2 ¼ @ 1 þ d2 o=ox Aq2 ðx; tÞ; 0
0 B w3 ¼ @
1 d2 o=ox c 1
0 1
1 C Aq3 ðx; tÞ;
ð4Þ
A. Perelomova / Ultrasonics 43 (2004) 95–100
where b ¼ d1 þ d2 . An overall flow now is thought as a T sum of all modes: wðx; tÞ ¼ ð vðx; tÞ pðx; tÞ qðx; tÞ Þ ¼ w1 þ w2 þ w3 . Next, matrix projectors are determined due to equations: Pn wðx; tÞ ¼ wn ðx; tÞ (n ¼ 1, 2, 3) that separate modes from the overall perturbations in every moment: 0
d2
b4 o=ox
1
d2 d2 o=ox o=ox 12 2ðc 1Þ 2ðc 1Þ C C b cd2 d2 1 1 C; o=ox o=ox 2 P1;2 2ðc 1Þ 2ðc 1Þ 2 4 C A b d2 d2 d2 1 1 2 þ 2 o=ox 4 2ðc 1Þ o=ox 2ðc 1Þ o=ox 2 0 1 d2 d2 0 o=ox c 1 o=ox c 1 B C ð5Þ P3 ¼ @ A: 0 0 0 1 1 d2 o=ox 1
B2 B ¼B B @
2
The operators possess general properties of orthogonal projectors: P1 þ P2 þ P3 ¼ eI ; P1 P2 ¼ P1 P3 ¼ . . . ¼ ~0, where eI and ~ 0 are unit and zero matrices. The projectors P1 , P2 , P3 do commute both with L and oto , so one can act by any projector at the basic system immediately, going to the linear evolution equation for every mode: o w þ Lwn ¼ 0 ðn ¼ 1; 2; 3Þ: ð6Þ ot n We fix relations (4) going to the nonlinear flow. The nonlinear evolution equations for interacting modes follow from Eq. (2) when corresponding projector acts on both sides of this equation: o ~ w þ Lwn ¼ Pn w: ð7Þ ot n ~ depends on a sum of The right-hand nonlinear vector w all modes. If the rightwards progressive acoustic wave is dominant, one can keep only this mode input in the nonlinear part. For the first mode evolution, Eq. (7) yields in the well-known Burgers evolution equation (cross nonlinear–viscous terms are omitted): oq1 oq1 c þ 1 o b o2 q1 q1 þ þ q ¼ 0: 2 ox 2 ox2 1 ot ox
ð8Þ
3. General formulae and the limit of periodic acoustic source To get an evolution equation for the heat mode generation, one acts by the projector P3 on the system (7) assuming that the nonlinear right-hand vector consists of the rightwards progressive acoustic mode inputs only. For the density perturbation of the heat mode, an equation looks: oq3 d2 o2 q3 op1 ov1 oq ov1 þ cp1 v1 1 q1 ¼ v1 ot c 1 ox2 ox ox ox ox
2 ov1 ; ð9Þ ðc 1Þd1 ox
97
or, taking into account relations inside the acoustic mode given by Eq. (4), in terms of the dominant mode pressure: oq3 d2 o2 q3 ot c 1 ox2
2 ! op1 b o2 p1 op1 ¼ ðc 1Þ p1 b : p1 ox 2 ox2 ox
ð10Þ
Only quadratic nonlinear terms are left in the evolution equation (10). To compare Eq. (10) with the known results for the periodic wave, let us take the approximate periodic solution of the Burgers equation (8) (suitable for the small Reynolds numbers, see [13]) since the acoustic pressure p1 ðx; tÞ has to satisfy this equation: p1 ðx; tÞ ¼
4P0 expð bx=2Þ sinðx tÞ: Reðc þ 1Þ
ð11Þ
The non-dimensional Reynolds number is Re ¼ P0 =b, P0 is an amplitude on a transducer. The formula (11) is correct beyond some vicinity of transducer where the nonlinear distortions are strong, and the amplitude of 4P0 solution (11) p10 ¼ Reðcþ1Þ is independent on P0 . Temporal average of both sides of (10) gives: 2 oq3 d2 o2 q3 op1 ¼ ðc 1Þ ot c 1 ox2 ox b 2 expð bxÞ: ð12Þ ¼ ðc 1Þp10 2 In all calculations quadratic viscous terms as well as cubic nonlinear ones were left out of account. To calculate Eq. (10), note that the periodic perturbation (11) satisfies the relations below: *
2 + op1 b o2 p 1 b op1 p1 2 ¼ : ð13Þ p1 ¼ 2 2 ox ox ox The formula (12) goes to the known result for the periodic ultrasound given in introduction. It is known that heating is an isobaric process as proved by the eigenvector w3 ðx; tÞ defined by Eq. (4). The rate of heat and temperature distortions (both dimensional) per unit volume in our variables looks: c oT c3 q0 oq3 q ¼ q0 C p ¼ : ð14Þ k ot kðc 1Þ ot From the other hand, a gradient of dimensional acoustic energy flux relating to the rightwards plane wave is 1 o q c3 o q c3 o 2 hJ1 i ¼ 0 hp1 v1 i ¼ 0 p ; k ox k ox k ox 1
ð15Þ
where the last calculation ignores terms of order b2 . Since q ¼ 1k oxo hJ1 i, the formula (12) follows from D Eqs. E d2 o2 q3 (14) and (15), it involves an additional term c 1 ox2 which is not usually involved in Eq. (12) when periodic
98
A. Perelomova / Ultrasonics 43 (2004) 95–100
acoustic sources considered. This term is responsible for the expansion due to effects of the thermal conductivity. 4. Heating caused by pulses. Illustrations As we already proved, the evolution equation (10) goes to the known case of periodic acoustic source but is suitable for any ultrasound source including non-peri1 odic one. At the first view, the term ðc 1Þp1 op ox in the right-hand side of Eq. (10) is expected to be dominant in comparison to two others because the last two terms are multiplied by the small thermoviscous coefficients. But 1 for periodic motions, the first term ðc 1Þp1 op ox provides only one half of the known result as follows from the discussion before since the temporal averages of all three terms are of the same order. To simplify calculations, let a medium of ultrasound propagation does not possess thermal conductivity: d2 ¼ 0. Relating to the boundary regime problem, we rewrite an evolution equation (10) going to the convenient variables (slowly varying n ¼ bx and the retarded time s ¼ t x) in the leading order as follows:
2 ! oq3 op1 op1 b o2 p1 op1 ¼ ðc 1Þ bp1 p1 b : p1 os on os 2 os2 os ð16Þ with the acoustic pressure being a solution of the Burgers equation which in the new variables looks
op1 o c þ 1 p12 b o p1 ¼ 0: ð17Þ þ b 2 2 2 os on os For the small Reynolds numbers Eq. (17) gives o 1 o2 p1 p1 ; on 2 os2
ð18Þ
that agrees with the first relation from Eq. (13) for the averaged values. In fact, as it follows from Eq. (17), a 2 value bp1 opon1 b2 p1 oosp21 in the right-hand side of Eq. p13
and therefore is beyond the preci(16) is of order of sion of calculations and should be ignored. 1 To interpret properly the term ðc 1Þp1 op os in the right-hand side of Eq. (16), some notes about general features of linear projecting should be done. Linear projecting can not in principle support nonlinear relations of perturbations providing the isentropic (quasi-isentropic due to attenuation) wave. The relation between dimensionless perturbations of pressure and density in isentropic waveform in are as follows: 2
p ¼ q þ ðc 1Þq =2 þ ; v ¼ q þ ðc 3Þq2 =4 þ
ð19Þ
Linear projecting keeps exact relations specific for modes but results in the fact that every mode becomes a sum of parts moving with approximate velocities of
other modes due to nonlinear interactions. Indeed, assuming that the rightwards progressive mode is dominative, the quadratic parts of two other modes that follow the acoustic one, are: q2;fol ¼ ðc þ 1Þq21 =8;
q3;fol ¼ ðc 1Þq21 =2:
Calculations are based on the dynamic equation (7). Keeping all rightwards parts, one goes to a vector of perturbations as follows (relations (4) for non-viscous flow are used): 0 1 v w ¼ @pA q 0 1 0 1 0 1 1 1 0 ¼ @ 1 Aq1 þ @ 1 Aq2;fol þ @ 0 Aq3;fol 1 1 1 0 1 0 1 c 1 1 1 ¼ @ 1 Aq1 þ @ c þ 1 Aq21 : ð20Þ 8 5 3c 1 It may be easily checked that relations of p, q, v given by Eq. (20) are isentropic with the accuracy of second-order terms and satisfy relations (19). All higher-order corrections may be written on in the same way. 1 After all the said, it is clear, that term ðc 1Þp1 op os in the right-hand side of Eq. (16) does not participate in the heated trace after acoustic pulse passing, it is actually a part of rightwards motion keeping a wave quasi-isentropic. In the case of a pulse, integration of this term over time that includes a pulse gives zero. For an unlimited in time acoustic source, this term does contribute to a local heating. In the frames of pulses, Eq. (16) yields in the next dynamic equation true for times after pulse passing: 2 Z t x
op1 q3;trace ðx; tÞ ¼ ðc 1Þb ds: ð21Þ os x The subtle structure of expansion at any time includes both terms and looks as follows:
2 ! Z t x op1 op1 p1 q3 ðx; tÞ ¼ ðc 1Þ ds: ð22Þ þb os os x To give an example of a mono-polar source, let us take a solution of the Burgers equation in the form of selfsimilar solution of Eq. (17), see [13]: 2b p1 ðn; sÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðn þ n0 Þ
ðsþs0 Þ2 exp 2ðnþn 0Þ
; p ffiffiffiffiffi ffi pffiffiffi sþs0 p ffiffiffiffiffiffiffiffiffiffiffi 2 bC cþ1 2p erf 2
ð23Þ
2ðnþn0 Þ
with n0 , s0 , C being constants. Note that in contrast to periodic sound (11), the solution (23) does not relate to
A. Perelomova / Ultrasonics 43 (2004) 95–100 ∆ T/T0
(a)
p/( ρ0c2)
99
0.0014 0.02
0.0012 0.5
1
1.5
2
tc/λ
0.001 0.0008
-0.02
0.0006 -0.04
0.0004 -0.06
0.0002
-0.08
1 p/( ρ0c2)
2
3
4
5
6
x
Fig. 2. Stationary disturbance of the relative temperature DT =T0 versus dimensionless distance from a transducer: bold line corresponds to b ¼ 0:1, thin one corresponds to b ¼ 0:05.
(b)
0.015 0.01
5. Discussion
0.005 4
5
6
7
tc/λ
-0.005 -0.01 -0.015
Fig. 1. (a) Dimensionless pressure of acoustic pulse p=ðq0 c2 Þ via dimensionless time tc=k at distance x=k ¼ 1 from a transducer (bold line). A thin line marks change in density of background corresponding to acoustic heating q3;trace =q0 multiplied by 300 (value given by Eq. (21)), a dashed line marks q3 =q0 multiplied by 300 (value given by Eq. (22)). In these series, b ¼ 0:01. (b) The same as at (a) at distance x=k ¼ 5 from a transducer.
large or small Reynolds numbers. The absolute value of C is responsible for the symmetry of the impulse: jCj 1 gives a curve close to the Gauss one, and a sign of C determines qaffiffiffiffipolarity of the pulse. Let us consider p e replacing C by the new constant large C ¼ cþ1 C 2 2b 0 e Oðb Þ. In calculations of Eqs. (21) and (22) with C ultrasound source given by Eq. (23), the next values of e ¼ 2, n0 ¼ s0 ¼ 0. parameters are taken: c ¼ 1:4, C Calculations based on formulae (21) and (22) are presented by Fig. 1(a) and (b). There are curves of pressure of acoustic pulse and corresponding decrease in the density of the background. For convenience, p and t in the axes labels mark dimension perturbation of pressure of acoustic wave and time. Fig. 1 demonstrates the approximate equality of expansion given by formulae (21) and (22) at times after pulse passing. The relative stationary increase of the temperature of the background follows from Eq. (21): DT ðxÞ ¼ ðc 1Þb lim t!1 T0
Z
t x
x
op1 os
2 ds:
ð24Þ
Calculations based on formula (24) are shown at the Fig. 2 for two different values of dimensionless attenuation b. More b, more increase in temperature of the background expected.
Illustrations show that starting far after pulse passing, the trace followed the ultrasound pulse approaches a certain limit: a negative value relating to the rarefaction of the medium (and to the increase in temperature, meaning that the process occurs under a constant pressure). In contrast to the periodic ultrasound, the rarefied trace goes to a limit and does not grow with time. For a single acoustic pulse, the large values of heating of the background are hardly expected. Nevertheless, calculations of the heating given rise by single pulse or series of pulses, may be useful in medical therapy and other applications of acoustics of pulses. The calculations given in the article are just an example of evaluating of this subtle phenomenon in the ideal gas. Any other medium of sound propagation may be easily considered by involving the correspondent thermodynamic equations of state (see [10]). Account of thermoconductivity would lead to (1) somewhat changed thermoviscous coefficients in the right-hand side of Eq. (10) and, the more important, (2) the natural process of slow heat transfer depending on the temperature gradient caused by account of d 2 o 2 q3 the term c 1 in the linear left-hand side of the ox2 dynamic equation (10). While unlimited in time signals treated, a term 1 ðc 1Þp1 op os is important, it does participate in the local heating. It should be stressed that general formula (10) goes to Eq. (16) which was used in calculations when acoustic wave may be thought as a progressive wave slightly varying with distance from a transducer: p1 ¼ p1 ðt x; bxÞ. Note that there are other non-periodic waves like shock one that could not be in principle treated in this way. Then formula (10) should be used. The advantages of the method are obvious if nonperiodic acoustic source is considered: the final equations are obtained without a procedure of temporal averaging, so the detail temporal behavior of all modes
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A. Perelomova / Ultrasonics 43 (2004) 95–100
may be investigated. The heating is one of the possible results of modes interaction in the plane flow, when the acoustic mode is dominant. Some other applications of linear projecting in the multi-dimensional flow were recently published by the author, where acoustic streaming from the non-periodic acoustic beams considered [11].
References [1] N.S. Bakhvalov, Y.M. Zhilejkin, E.A. Zabolotskaya, Nonlinear Theory of Sound Beams, Am. Inst. Phys., New York, 1987. [2] D.I. Blokhintsev, Acoustics of Non-uniform Moving Medium, Nauka, Moscow, 1981 (in Russian). [3] I. Gyarmati, Nonequilibrium Thermodynamics, Field Theory and Variation Principles, Springer-Verlag, Berlin, 1970. [4] A.A. Karabutov, O.V. Rudenko, O.A. Sapozhnikov, Thermal self-focusing of weak shock waves, Soviet Physics Acoustics 34 (1989) 371–374. [5] S. Makarov, M. Ochmann, Nonlinear and thermoviscous phenomena in acoustics, Part I, Acta Acustica 82 (1996) 579–606. [6] L. Menguy, J. Gilbert, Nonlinear acoustic streaming accompanying a plane stationary wave an a guide, Acta Acustica 86 (2000) 245–259.
[7] A.A. Perelomova, Nonlinear dynamics of vertically propagating acoustic waves in a stratified atmosphere, Acta Acustica 84 (1998) 1002–1006. [8] A.A. Perelomova, Projectors in nonlinear evolution problem: acoustic solitons of bubbly liquid, Applied Mathematical Letters 13 (2000) 93–98. [9] A.A. Perelomova, Nonlinear dynamics of directed acoustic waves in stratified liquids and gases, Optics of Atmosphere and Ocean 13 (2) (2000) 133–138. [10] A. Perelomova, Interaction of modes in nonlinear acoustics: theory and applications to pulse dynamics, Acta Acustica united with Acustica 89 (2003) 86–94. [11] A. Perelomova, Acoustic radiation force and streaming caused by non-periodic acoustic source, Acta Acustica united with Acustica 89 (2003) 754–763. [12] O.V. Rudenko, M.M. Sagatov, O.A. Sapozhnikov, Thermal selffocusing of saw-tooth waves, Soviet Physics JETP 71 (1990) 449– 451. [13] O.V. Rudenko, S.I. Soluyan, Theoretical Foundations of Nonlinear Acoustics, Consultants Bureau, New York, 1977. [14] J.N. Tjotta, S. Tjotta, Nonlinear equations of acoustics. in: Proceedings of the 12th International Symposium on Nonlinear Acoustics, 1990, pp. 80–97. [15] I.S. Vereschagina, A.A. Perelomova, Investigation of influence of space modes interaction on nonlinear dynamics of sound beams, Acoustical Physics 48 (2) (2002) 147–154.