Bubbles effect on sound dispersion in thin-walled tube with polymeric liquid and elastic central rod

Journal of Sound and Vibration 330 (2011) 3155–3165

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Bubbles effect on sound dispersion in thin-walled tube with polymeric liquid and elastic central rod S.P. Levitsky n, R.M. Bergman, J. Haddad Shamoon College of Engineering, Bialik/Bazel Streets, Beer-Sheva 84100, Israel

a r t i c l e i n f o

abstract

Article history: Received 25 May 2010 Received in revised form 19 January 2011 Accepted 23 January 2011 Handling Editor: L. Huang Available online 25 February 2011

The present study is devoted to the investigation of fine air bubbles effect on sound propagation in thin-walled elastic tube with compressible polymeric liquid and cylindrical elastic rod in the central part of the tube. The problem formulation and solution method follow the previous paper of the authors (S.P. Levitsky, R.M. Bergman, J. Haddad, Sound dispersion in deformable tube with polymeric liquid and elastic central rod, Journal of Sound and Vibration 275 (1–2) (2004) 267–281). In order to account for the bubbles’ influence on sound dispersion and attenuation, dynamic equation of state of the mixture, formulated within homogeneous approximation, is used. It is assumed that the volume gas concentration is small. The resulting dispersion equation for the waveguide with viscoelastic liquid–gas mixture is studied in a long-wave range. Results of simulations illustrate the influence of free gas concentration, bubble radius and rheological properties of the liquid on sound dispersion and attenuation in the system. & 2011 Elsevier Ltd. All rights reserved.

1. Introduction Presence of microbubbles in polymeric liquids is a rule rather than exception [1]. It is explained by different reasons, including chemical reactions, small wettability of solid boundaries, usually large viscosity of such liquids, prohibiting gravitational evacuation of free gas. Because bubbles possess much larger compressibility than liquid, they strongly change its acoustic properties. Characteristic features of sound propagation in liquid–bubble mixtures are low sound speed in a long-wave range and high dispersion at the frequencies close to the resonance frequency of bubbles [2]. Dissipation at gas– liquid dynamic interaction in the wave far exceeds dissipation at wave propagation in a pure liquid in a wide frequency range; it is governed by rheological, heat and acoustic losses at the bubble–liquid interface. Therefore, proper description of sound waves propagation in waveguides with polymeric liquids must account for the free gas effect, which can be essential even at minor volume concentrations of bubbles. The target of the paper is generalization of the model [3] of sound propagation in a gap between thin elastic circular shell and coaxial central rod, filled with vicoelastic polymeric liquid, in order to incorporate in the model the effect of gas microbubbles. The flow geometry is typical for certain extrusion processes in polymer processing technology and rheological measurements [4]; it is widely used in heat exchangers, different hydraulic systems, oil industry, etc. Wave propagation in a waveguide with cylindrical rod centered in a fluid-filled annulus is strongly influenced by structure coupling, which changes essentially the liquid response to acoustic excitation. Therefore, proper description of pressure signals propagation in such a system must account for coupled liquid–shell and liquid–rod interactions [5]. Additional kind of interaction introduce microbubbles; in the present paper description of the liquid–gas mixture dynamics in the wave is based on a quasi-homogeneous model [6].

n

Corresponding author. Tel.: + 972 8 6475734; fax: +972 8 6475754. E-mail address: [email protected] (S.P. Levitsky).

0022-460X/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsv.2011.01.019

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Nomenclature

Greek letters

equilibrium bubble radius dimensionless sound speed in the system sound speed in pure liquid complex sound speed in a mixture of bubbles in liquid cpg specific heat capacity of gas at a constant pressure Es Young module of the tube material FðyÞ non-dimensional spectrum of relaxation times f frequency, Hz Gðtt1 Þ relaxation function G complex dynamic module of viscoelastic liquid h half of the shell width K wavenumber, m  1 Km mixture dynamic bulk module kg gas thermal conductivity L tube length l wave length pf , p0 pressure in the liquid and equilibrium pressure in the waveguide pg0 equilibrium gas pressure Dpc contact pressure equal to normal stress in liquid at the pipe wall R1 radius of the middle surface of the tube R2 radius of the rod r radial coordinate of the cylindrical coordinate system sij deviator of rate-of-strain tensor in liquid T0 equilibrium temperature of the gas phase t time ux, ur longitudinal and transverse displacements of the shell middle surface V mean relative flow velocity in the wave Vg gas volume per unit mixture mass DVg gas specific volume increment ! v , vx , vr liquid velocity vector and its longitudinal and transverse components Dv variation of the axial component of mixture velocity in the wave wr , wx radial and longitudinal displacements in the rod x axial coordinate of the cylindrical coordinate system

a1

a0 C cf cm

g Zp Zs y lr

mr ns r Dr rf rg0 rr rs s srr tij trr f

w O

o

parameter of the spectral distribution of relaxation times adiabatic exponent of gas low-frequency Newtonian viscosity of liquid low-molecular solvent viscosity relaxation time first Lame constant of the rod material second Lame constant of the rod material Poisson module of the tube material density of the bubbly mixture mixture density variation in the wave density of the pure liquid equilibrium gas density density of the rod material density of the tube material liquid–gas surface tension coefficient normal component of the stress tensor in the rod deviator of stress tensor in liquid the normal component of deviatoric stress in liquid volume concentration of bubbles dimensionless sound attenuation in the system frequency, s  1 dimensionless frequency

Subscripts 0 f g m r s

equilibrium state fluid gas mixture rod shell

Superscripts ^ –

complex amplitude dimensionless parameter

Linear acoustics of different waveguides with viscoelastic liquid was a subject of the studies [7–9]. In Ref. [7] stationary sound wave propagation in an elastic cylindrical tube with viscoelastic liquid inside was investigated. In Ref. [8] propagation of a pressure pulse, generated at the initial moment of time in a tube was described within the same model; the problem was solved by Laplace transform method. Sound dispersion and attenuation in cylindrical gap between two elastic shells, filled with polymeric liquid, was studied in Ref. [9]. Both internal and external walls of the waveguide were supposed to be thin-walled shells, characterized by different geometrical and elastic parameters. Propagation of acoustic waves in a thin-walled elastic tube with coaxial internal elastic rod was investigated in Ref. [3]. It was shown, in particular, that the width of the gap between the pipe wall and the rod highly influences both sound speed and its attenuation. The narrowing of the gap enhances the liquid viscoelasticity effect, leading to sound speed growth and attenuation reduction. Another conclusion, following from the study, is that elastic properties of the rod can be disregarded only if its relative radius is small; otherwise, they essentially influence the wave propagation. As distinct to Ref. [3], the current study is based on a more realistic liquid description, allowing for the microbubbles presence. Survey of papers, concerning bubble dynamics in polymeric liquids, can be found in Ref. [10]; the model of sound propagation in a cylindrical tube, filled with bubbly viscoelastic liquid, was reported in Ref. [11]. The approach used below,

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is similar to Ref. [11], but relates to a more complicated waveguide with internal elastic rod. The present study is limited to small concentrations of free gas and low-frequency range. 2. Problem formulation and the solution scheme Since the current study is based on the paper [3], it is necessary to introduce briefly the scheme of the problem solution in the case of polymeric liquid without bubbles. Consider sound wave propagation along thin-walled circular cylindrical tube with radius of the middle surface R1 and the wall width 2h (e1 ¼ h=R1 51). It is supposed that the tube wall is only slightly deformable; R1 5L, where L is the tube length, and R1 can be used instead of internal radius of the tube. The central part of the tube is occupied by coaxial elastic circular rod with radius R2 and the gap between the tube wall and the rod is filled with polymeric solution, described by generalized Maxwell rheological model, accounting for the liquid compressibility. It is supposed that the wave length l ¼ 2p=K bR1 , where K is the wavenumber. Sound propagation in the system in this case can be characterized as a fluid–structure interaction problem, and elastic deformations of the shell and the rod must be coupled with the flow field in the gap. To do this, dynamic equations for each region are formulated and solved separately and their solutions are coupled by appropriate boundary conditions. Axisymmetric dynamics of a thin-walled elastic cylindrical circular shell is described by Kirchhoff–Love equations ! Es @2 ux ns @ur @2 ux þ (1) ¼ rs 2 R1 @x 1n2s @x2 @t   Es h @u u E h3 @4 ur 1 @2 ur ns x þ r þ s 2  Dpc ¼ rs h 2 2 4 2 @x R1 ð1ns ÞR1 3ð1ns Þ @x @t

(2)

Here ux ,ur are displacements in axial and radial directions, respectively; rs ,Es , ns are the density, Young and Poisson module of the tube material, respectively; Dpc is the contact pressure equal to normal stress in liquid at the pipe wall. Kinematic and dynamic boundary conditions for Eqs. (1) and (2) are formulated at the liquid–shell interface for r ¼ R1 h  R1 . They insure structure coupling at this interface and have the form vr ¼

@ur , @t

vx ¼

@ux , @t

Dpc ¼ Dpf trr9R1 , Dpf ¼ pf p0

(3)

where trr9R1 is the normal component of deviatoric stress in liquid at the shell surface; pf ,p0 are the pressure in liquid and equilibrium pressure in the waveguide, respectively; vx ,vr are the longitudinal and transverse components of the liquid velocity, respectively. Dynamic equations for elastic rod are formulated in a long-wave approximation (in this case it can be assumed ð@wr =@rÞ bð@wx =@xÞ, ð@2 wr =@r 2 Þ b ð@2 wr =@x2 Þ, ð@2 wr =@r 2 Þ b ð@2 wx =@x @rÞ), and can be reduced to !   @2 wr 1 @wr 1 @2 wr  2 wr rr lr þ 2mr þ ¼0 (4) 2 r @r r @r @t 2

srr  ðlr þ2mr Þ

@wr wr þ lr @r r

(5)

Here wr ,wx are the radial and longitudinal displacements in the rod, respectively; srr is the normal component of the stress tensor; rr is the density of the rod material; lr , mr is the Lame constants, coupled with the Young and Poisson modules of the rod, respectively, Er , nr by the relations

mr ¼

Er , 2ð1 þ nr Þ

lr ¼

2mr nr 12nr

Solution of Eq. (4) must be also coupled with the hydrodynamic solution, describing fluid flow in the wave. This is achieved by the following boundary conditions at the rod surface r ¼ R2 : vx ¼

@wx , @t

vr ¼

@wr , @t

srr9R2 ¼ trr9R2 Dpf

(6)

Hydrodynamic problem for a polymeric liquid flow in the gap is solved in quasi-one-dimensional approximation [3]. It is supposed that the liquid follows linear generalized Maxwell model   Z t 1 1 @vi @vj ! tij ¼ 2 Gðtt1 Þsij ðt1 Þ dt1 þ2Zs sij , s ¼ e ðrU v ÞI, eij ¼ þ (7) 3 2 @xj @xi 1 Here Gðtt1 Þ is the relaxation function; tij ,sij are the deviators of stress and rate-of-strain tensors, respectively; Zs is the ! low-molecular solvent viscosity; v is the liquid velocity. Influence of microbubbles on the friction losses at fluid motion in the waveguide is neglected; this assumption is valid for sufficiently small volume concentrations of free gas [12]. In this case [6] r0 =rf 0 ¼ 1f  1, where f is the volume concentration of bubbles and r0 , rf 0 is the equilibrium density of bubbly mixture and pure liquid, respectively. Therefore, in the case f51, studied below, the mixture density r0 in linearized momentum balance equation can be identified with rf0. The effect of microbubbles is accounted for through the mixture

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dynamic equation of state, according to quasi-homogeneous approach in wave dynamics of bubbly flows [6]. In view of the above, the mass and momentum balance equations can be written in the form   @r 1@ @vx ðrvr Þ þ þ rf 0 ¼0 (8) r @r @t @x

rf 0

@pf @vx ¼ þ @t @x

Z

t

Gðtt1 Þ 1

! ! @2 vx 1 @vx @2 vx 1 @vx þ þ Z þ dt 1 s r @r r @r @r 2 @r 2

(9)

Usual assumptions of long-wave approximation in liquid dynamics in tube at acoustic excitation are used, which imply vr 5vx ,

@2 vx 1 @vx , 5 r @r @x2

@2 vx @2 vx 5 2 , @x2 @r

u_ x 5V,

@vr @vx 5 @x @r

(10)

Here V is the mean relative flow velocity in the wave. Eqs. (8) and (9) are averaged across the gap cross-section and transient friction at the shell and rod surfaces is found in terms of the average flow velocity from solution of nonstationary hydrodynamic problem for incompressible viscoelastic liquid flow in the gap [3]. Then the full system of equations and boundary conditions for liquid velocity, pressure, density, and displacements in the shell and the rod is brought to dimensionless form with p0, R1 and rs choosed for characteristic parameters, and is solved for harmonic propagating wave with frequency O and wavenumber K. Specifically, momentum and mass balance equations, and state ^ V, ^ r ^ of disturbances of the averaged pressure, flow equation for liquid with bubbles in terms of complex amplitudes P, velocity and density in the wave, respectively, have the form [3] ^ ^ kZDVð1 e22 Þ1 ioV^ ¼ ikkP8   ^ 1R ikV^ ¼ 0, ^ þ2ioð1e22 Þ1 u^ 1 e2 w ior

^ 19x ¼ e2 ^ 1R ¼ w w

^ P^ ¼ k1 c2m r

(11) (12) (13)

k ¼ rs =rf 0 , o ¼ Ot0 , k ¼ KR1 , x ¼ r=R1 , cm ¼ ðt0 =R1 Þcm , t0 ¼ R1 ðrs =p0 Þ1=2 ^ 1 are the complex amplitudes of where cm is the complex sound speed in a bubbly liquid, defined below in Section 3; u^ 1 , w dimensionless radial displacements in the shell and the rod, respectively; dimensionless dynamic viscosity of viscoelastic liquid Z and the rest parameters are defined in Appendix A. When deriving Eq. (12), kinematic conditions for the radial liquid velocity at the shell and rod surfaces were used in the form v^ r9R1 ¼ iou^ 1 ,

^ 1R v^ r9R2 ¼ iow

It is supposed also that the axial component of the liquid velocity at the interfaces vanishes, with account for the obvious relation u_ x 5vx ; small cross effect of liquid viscosity and compressibility and the influence of radial shell and rod displacements on the losses at liquid motion in the waveguide were neglected. Equations for the complex amplitudes u^ 1 , u^ 2 of the radial and axial displacements in the shell, respectively, follows from Eqs. (1) and (2) and have the form ins ku^ 1 þ ðk2 Es 

1

o2 ð1ns 2 ÞÞu^ 2 ¼ 0, Es ¼ Es =p0

  1 1 1þ e21 k4 Es ð1n2s Þo2 u^ 1 ikns u^ 2  ð1n2s Þ=ð2e1 Es Þ p^ c ¼ 0 3

(14) (15)

^ 1 of where p^ c is the complex amplitude of the contact pressure Dpc . And, finally, the equation for the complex amplitude w the radial displacement in the rod follows from Eq. (4) and has the form " # ^1 ^ 1 1 @w @2 w 1 2 ^1 ¼0 þ þ b  2 w (16) 2 x @x @x x

b ¼ o½k1 ðlr þ 2mr Þ1=2 , flr , mr g ¼ p1 k1 ¼ rs =rr 0 flr , mr g, Solution of the Bessel equation (16), bounded at the rod axis, means ^ 1 ¼ A1 J1 ðbxÞ w

(17)

where J1 is the Bessel function of the first kind and first order. The arbitrary constant A1 is found from the dynamic boundary condition (6) for the radial component of stress at the rod surface, which in terms of complex amplitudes has the form

s^ R ¼ p^ c , s^ R ¼ s^ rr9x ¼ e2

(18)

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where the amplitude of the contact pressure at the pipe wall p^ c follows the relation: ^ t^ R p^ c ¼ P

(19)

Here tR is the deviatoric stress component in liquid, averaged across the gap [3] Z R1 1 trr9R1  trr9R2 ¼ tR  2 trr 2pr dr 2 pR1 ð1e2 Þ R2

(20)

The boundary condition (19), in view of relation (20), rheological equation (7), mass balance equation (12) and state equation (13), takes the form !  2iokZ ^ 2ioZ  ^ 1R u^ 1 w (21) p^ c ¼ 1 P ð1e2 Þ 3c2f The expression for the constant A1 follows from Eqs. (17) and (18) with the use of Eq. (5), written in terms of complex amplitudes. Now the problem is closed; its solution after certain routine algebra yields the following dispersion equation for the dimensionless sound speed in the system: az2 bz þd ¼ 0,

z ¼ c2 ,

c ¼ o=k

(22)

Parameters in Eq. (22) are defined by formulas, given in Appendix A. 3. Dynamic equation of state for liquid, accounting for gas microbubbles Dynamic equation of state for liquid with bubbles in the wave is formulated below within homogeneous model approach [6]. It is assumed that small spherical bubbles with equilibrium radius a0 are uniformly distributed in a volume, bubbles interact via pressure field, the mixture is considered as a compressible ideal liquid and the losses can be attributed to bubble–liquid dynamic interactions only; volume concentration of free gas f is small (usually is supposed f r0.01), and influence of bubbles on equilibrium mixture density r0 can be neglected (r0 E rf0). Then linearized equations of motion and continuity for the mixture can be written in the form [6]

rf 0

@ðDvÞ @ðDpÞ ¼ , @t @x

@ðDrÞ @ðDvÞ ¼ rf 0 @t @x

(23)

Here Dv is the velocity and Dr is the mixture density variations in the wave; the latter is coupled with variation of the gas volume per unit mixture mass Vg by the relation

Dr ¼ ð1jÞ2 Drf r20 DVg

(24)

It is well known that bubbles stay for a basic source of dissipation at sound wave propagation in the mixture [2], and input of losses in pure liquid in the total dissipation in the presence of bubbles can be neglected. It means that the state equation for liquid phase in a good approximation can be written in the form

Drf ¼ cf2 Dp, Drf ¼ rf rf 0 , Dp ¼ pp0

(25) na30

Vg0 ¼ ð4=3Þp between the gas Here cf is sound speed in a pure liquid. In view of the obvious relation specific volume increment DVg in the mixture and perturbation of the bubble radius Da, the following equation for the mixture density versus pressure and bubble radius variations in the wave can be obtained from Eqs. (24) and (25)

DVg ¼ 3Vg0 a1 0 Da,

Dr ¼ ð1fÞ2 cf2 Dp3rf 0 fð1fÞa1 0 Da

(26)

Here n is the number of bubbles per unit mass of the mixture. The next step will be to express the bubble radius variation Da in terms of pressure variation in liquid Dp, which is identified with the averaged pressure disturbance in the gap. For harmonic wave we can assume {Dr, Dv, Da, Dp} eiOt and find the necessary relation between Da and Dp from solution of the problem of a single bubble dynamics in acoustic field. For a gas bubble in viscoelastic liquid with rheological equation (7) this solution was found in Ref. [10]. Together with Eqs. (23) and (26) it leads to dynamic equation of state for the mixture (13) with the complex sound speed cm, defined by [10] 1 c2m

nb ¼

DT ¼

¼

1 c2f

þ

3jð1fÞk1

a30 o2 2ImfG g ImfDT g þ þ 2Zs þ , 2o k 2cf o 2 3ðpg0 =p0 Þbg , 2 bg 3ð1 Þðbg cthbg 1Þ

g

g

(27)

2ionb þðwb ða20 =kÞo2 Þ



wb ¼ 4RefG g þRefDT g2s=a0 a



b2g ¼ p0ffiffiffiffi ioPeg , Peg ¼

k

a0 ðp0 =rf 0 Þ1=2 ag

(28)

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ag ¼

kg

rg0 cpg

,

rg0 ¼

pg0 , T0 cpg ð1g1 Þ

s ¼ s=ðp0 R1 Þ, Zs ¼ Zs =ðp0 t0 Þ, cf ¼ cf ðt0 =R1 Þ

Here pg0, rg0, T0 are the equilibrium pressure, density and temperature of the gas phase, respectively; kg, cpg, g are the gas thermal conductivity, specific heat capacity at constant pressure and adiabatic exponent, respectively; s is the liquid–gas surface tension coefficient; a0 ¼ a0 =R1 is the dimensionless equilibrium radius of bubbles; Gnis the complex dynamic module of viscoelastic liquid, related to the non-dimensional spectrum FðyÞ of relaxation times y by the formula, given in Appendix A. Eq. (27) is based on the exact solution of the problem of gas bubble dynamics in viscoelastic liquid, found in Ref. [10]. It is valid in a wide range of frequencies because accounts not only for rheological losses, but also for heat and acoustic dissipation at single bubble interaction with the liquid phase. When deriving Eq. (27), it was assumed that the thermal boundary layer thickness around pulsating bubble is much smaller from its radius and the pressure in bubbles does not depend on the spatial coordinate, that is pg =pg(t) [2]. It is implied also that the equilibrium bubble radius a0 5R1, which allows neglecting by bubbles interaction with the tube wall for a0 50:1 [13]. Parameters nb , wb in Eq. (28) have the meaning of dimensionless effective dissipation coefficient and elastic constant of the bubble, respectively. One can see that within the formulated model, rheology of the liquid influences the above parameters in an additive manner. More exact analyses of bubble dynamics in viscoelastic liquid [10], taking into account cross effects of liquid rheology, heat transfer and liquid compressibility, has shown that their input in the total dissipation and effective elasticity of bubble is minor and can be neglected in the frequency range f5106 Hz, f= O/2p. In the special case Gn = 0(Z ¼ Zs ) Eq. (27) yields the known result for sound speed in a pure viscous liquid with gas bubbles [12]. Both sound speed and attenuation changes in the presence of bubbles can be attributed now to the mixture dynamic bulk module Km, which in the presence of bubbles is frequency-dependent. For monochromatic wave this viscoelastic 2 module is linked to the complex sound speed in the mixture, cm, by equation Km ¼ r0 cm . Using of Km instead of adiabatic bulk module of the pure liquid in the equation of state leads to the change of cf to cm and allows accounting for the bubbles input in the sound dispersion and attenuation in the system. 4. Approximate solution in the limit x-0 The expression for the complex sound speed c, following from Eqs. (22) and (27), is sufficiently cumbersome and does not allow to separate in a simple form the input of different physical factors (shell and rod elasticity, liquid rheology and compressibility, bubbles elasticity and dissipation at liquid–gas and liquid–wall interactions), which are frequency dependent. In order to elucidate some basic features and validate the model, it is useful to consider the low-frequency limit o-0, which may be also considered as a certain benchmark problem. In this case Z-Zp ¼ const, Zp ¼ Zp =ðp0 t0 Þ, where Zp is the low-frequency Newtonian viscosity; and the liquid with microbubbles can be treated as a pure viscous one. Longitudinal deformations and inertia of the pipe wall and the rod can be neglected (n = 0, Q= 1, b = 0), and the dispersion equation (22) combined with Eq. (27) yields the following simple asymptotic solution for the low-frequency sound speed c0 in the system: ! kð1e22 Þe1 1 1 1 1 k 1 ¼ ðioÞ N0 2 þ 2 þ 2 , c2r ¼ 1 , c2b ¼ (29) 2 1 f 2 c0 cr cf cb Es þ e2 e1 ðlr þ mr Þ D0 ¼ ð1þ e22 þð1e22 Þ=ln e2 Þ1

N0 ¼ io þ 8kZp D0 ,

2 2 Here c2 f þc b ¼ c mo , where cmo is the low-frequency sound speed in a homogeneous mixture of bubbles in ideal compressible liquid, following from Eq. (27) at o-0 and f 51, when neglecting all kinds of losses at bubble–liquid interaction. It is taken into account in Eq. (29) that in the low-frequency range it can be assumed g =1 (isothermal gas behavior [14]), and for bubbles with a0 Z10  5 m at normal conditions the effect of surface tension can be neglected [10]. In order to validate relation (29), it was obtained also from the basic system of equations (see Appendix B) at the same assumptions. The inputs of liquid, bubbles, shell and rod elasticity are now separated, and Eq. (29) in special cases yields known relations. First, it predicts qualitatively different results for viscous and inviscid liquids—in the case of viscous liquid c-0 at o-0, which is explained by the transition from inertial to creeping flow regime in the wave in this limit [7,15,16]. For ideal liquid it follows from Eq. (29):

1 c20

¼

1 c2r

þ

1 c2f

þ

1 c2b

(30)

This equation at f = 0 coincides with a similar result of Ref. [3]. Note only that, as it follows from the expression for D0, increase of the e2 value yields enhancement of the liquid viscosity effect, which has clear explanation—the less is the gap the larger are the flow gradients. In the case f = 0, e2 = 0 it follows further from Eq. (30) 1 c20

¼

1 c2r0

þ

1 c2f

,

c2r0 ¼ ke1 Es

(31)

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This relation coincides with the Korteweg formula for the speed cK of water hammer wave in a circular water-filled pipe [17]. Eq. (31) reveals the meaning of the constant cr0 it defines the sound speed in an elastic circular shell filled with incompressible liquid. In the case f =0, e2a0 Eq. (30) defines sound speed in an annular cylindrical channel with elastic shell and elastic central rod, when the gap is filled with compressible liquid 1 c20

¼

1 c2r

þ

1

(32)

c2f

This relation can be considered as a modified Korteweg formula for an annular waveguide. It follows from Eq. (32) that the presence of the rod lowers the sound speed—the more is the value of e2, the less is c0 . Note also that the more is the elastic module of the rod the larger is the sound speed in the system—for non-deformable rod (Er ¼ 1) the relation for cr takes the form c2r1 ¼ kð1e22 Þe1 Es , which means that cr ocr1 . The same conclusion can be made also about the influence of the shell elasticity. In the rigid pipe with a rigid central rod the sound wave propagates with the speed cf . Finally, the case fa0, e2 =0 corresponds to elastic tube with ideal liquid, containing bubbles. The result, following from Eq. (30), is confirmed by numerous experimental data—bubbles provide for a drastic reduction of the low-frequency sound speed in the system [18,19].

5. Numerical results In the general case solution of the dispersion equation (22) was studied numerically and the plots of dimensionless sound speed C = o/Re{k} and attenuation w ¼ Imfkg versus dimensionless frequency o for different volume concentrations of bubbles f and rheological characteristics of liquid were obtained. The values of the system parameters common for all pictures are collected in Table 1; they correspond to aluminum shell and rod, and air bubbles mixture at normal a conditions. The Spriggs law [4] yk = y1/k1 was used to calculate the spectrum of relaxation times. In the case of a discrete spectrum, the relation for the complex dynamic module Gn (see Appendix A) takes the form ðiOÞ1 G ¼

1 Zp Zs X ka1 iOy1 zða1 Þ k ¼ 1 k2a1 þ ðOy1 Þ2

(33)

where z(a1) is the Riemann zeta function of the spectral distribution parameter a1. Results of simulations are presented in Figs. 1–4. The studied non-dimensional frequency range for the chosen parameter values corresponds approximately to 1 Hz ofo80 kHz. The curves in Figs. 1 and 2 were obtained for low viscous polymeric solution (Zp/Zs =10), while the curves in Figs. 3 and 4 correspond to high viscous liquid with Zp/Zs = 103. It follows from Figs. 1 and 3 that microbubbles influence sound dispersion in the waveguide in a different manner at low and high frequencies, respectively. Low-frequency dispersion in an annular gap is connected with transition from creeping to inertial flow regime and is controlled by liquid rheology (let remind that for viscous liquid, as distinct from ideal one, C-0 at o-0). Dispersion on this branch of the dispersion curve demonstrate strong dependency from the volume concentration of free gas (and much less—from the bubbles’ size), and does not change its sign in the presence of bubbles. However, they can lower essentially the sound speed, and this reduction is more prominent for viscoelastic liquid with small Newtonian viscosity in the low-frequency range. For highly viscous elastic liquid a characteristic peak (‘‘overshoot’’) appears on the curve C = C(o) in the region of viscoelastic transition (transition frequency O1 can be estimated from the relation y1 O1  1; note that this estimate is based on the main relaxation time only, while the influence of relaxation spectrum leads to widening of the transition region). Localization of the ‘‘overshoot’’ on the plot C =C(o) depends on the value of y1—a decrease in y1 results in a shift of the ‘‘overshoot’’ towards larger frequencies. Simultaneous increase of the liquid dynamic viscosity leads to a decrease of the sound speed along the curves 200 , 300 in Fig. 3, as compared with the plots 20 , 30 , respectively, but leaves unchanged the dispersion sign, which is positive. Note that this result for sound waves in an elastic tube, filled with a viscous liquid, is confirmed by experimental data [20]. The high frequency dispersion is much more influenced by bubbles. As it is characteristic for bubble–liquid mixtures [2], the sound speed in this region exhibits drastic non-monotonic changes with frequency, which closely follows sound dispersion in a bulk of bubbly liquid with the same properties [10]. As distinct to the low-frequency range, both scale and localization of the high frequency dispersion are sensitive to the size of bubbles. Table 1 Shell (rod), liquid and gas parameter values. Parameter Es,r, N/m

2

e1 rf0, kg/m3 a1

Value

Parameter 10

7  10 0.05 103 2

ns,r e2 Zs, Pa s kg, W/(m K)

Value

Parameter

0.34 0.4 10  3 0.025

3

rs,r, kg/m p0, Pa s, N/m cpg, J/(kg K)

Value 3

2.7  10 105 0.05 1.01  103

Parameter

Value

R1, m T0, K cf, m/s

0.01 293 1490 1.4

g

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Fig. 1. Sound dispersion in the system, Zp = 0.01 Pa s, y1 =0.001 s. For plot 1 f = 0, for 2, 20 —f =10  5, for 3, 30 —f = 10  3. The curves 2, 3 correspond to a0 = 5  10  6 m, the curves 20 , 30 —to a0 = 5  10  4 m.

Fig. 2. Attenuation of sound in the system, Zp = 0.01 Pa s. The plots 1, 2, 20 , 3, 30 correspond to the same parameter values as similar plots in Fig. 1. For plots 200 , 300 y1 = 0.0001 s, the rest parameters are the same as for plots 2, 3.

The presence of bubbles lead to essential increase in attenuation of sound (Figs. 2 and 4), which for viscoelastic liquid with bubbles in a wide frequency range is much less, than for a similar pure viscous one [3]. We can see that the input of losses, resulting from bubbles interaction with liquid in the wave, for the studied waveguide becomes dominant at sufficiently large frequencies and the plot of attenuation versus frequency in tube in this region takes a typical resonant form. Results of simulations clearly show that dissipation in translational liquid flow in the gap is the basic one in the lowfrequency range, where the impact of bubbles on the losses can be neglected. At large frequencies bubbles provide the main input in sound attenuation in the system. The larger are bubbles the less is the transition frequency, separating regions with flow and bubbles controlled dissipation, respectively. The gas concentration acts in the same direction. In order to validate numerical results, typical curves 200 , 300 from Fig. 3 are shown in separate Fig. 5 in comparison with the graphs, calculated for viscous (29) and inviscid (30) asymptotics for the sound speed at the same parameter values. Good matching of exact and approximate solutions for o 51 can be recognized, especially in the case f = 10  3. At a smaller gas content (f =10  5, curves 200 , 2000 ) the effect of liquid rheology on the sound speed becomes dominant in the viscous transition region, which yields more essential discrepancies between exact and approximate solutions, because the later doesn’t account for the frequency dependence of dynamic viscosity [4].

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Fig. 3. Sound dispersion in the system, Zp = 1 Pa s. For plot 1 f =0, for 2, 20 —f = 10  5, for 3, 30 , 300 —f = 10  3. The curves 2, 3 correspond to a0 = 5  10  6 m, the curves 20 , 200 , 30 , 300 —to a0 =5  10  4 m. For plots 1–3, 20 , 30 —y1 = 0.1 s; for plots 200 , 300 y1 = 0.01 s.

Fig. 4. Attenuation of sound in the system, Zp = 1 Pa s. All plots correspond to the same parameter values as similar plots in Fig. 3.

6. Conclusions The paper presents results of modeling of sound dispersion and attenuation in a thin-walled tube with viscoelastic liquid, containing small amount of free gas. The central part of the tube is occupied by coaxial elastic rod. The study was concentrated on numerical analysis of bubbles effect on sound propagation in the system. Results of simulations show that microbubbles even in small amounts essentially lower the sound speed in a low-frequency range, lead to appearance of additional dispersion region at large frequencies and are responsible for increased attenuation of sound. The scale of these effects is sensitive to rheological properties of the liquid. It was shown also that losses, resulting from shear friction in translational liquid flow in the wave provide the dominant part of dissipation in the low-frequency range, while at large frequencies bubbles are responsible for the main input in sound attenuation in the waveguide. The larger are bubbles and gas concentration the less is the transition frequency, separating regions with flow and bubbles controlled dissipation, respectively. Results of the study show that transient processes in annular hydraulic elements with polymeric liquid are changed essentially in the presence of microbubbles, and the model developed contributes to understanding how they can influence technological processes, such as pressure extrusion at metering pump operation etc. Dispersion equation formulated in the paper can be used for simulation of pressure pulses propagation in the system. It opens possibilities for

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Fig. 5. Viscous and inviscid asymptotics for the sound speed at o-0. The curves 200 , 300 are the same as in Fig. 3; the dashed (2000 , 3000 ) and dotted lines correspond to asymptotic solutions for the same system parameters as the graphs 200 , 300 .

optimizing the control and performance of dynamic processes in the presence of free gas traces in liquid, which is especially important in view of the known difficulties with polymeric liquids devolatilization. Theoretical results of the paper, indicating that acoustic properties of the system with annular flow of high viscous polymeric liquid are sensitive even to minor amounts of microbubbles, can find application also in free gas detection by acoustic means in the same manner as for low viscous Newtonian fluids.

Acknowledgment The work was supported by the Shamoon College of Engineering.

Appendix A. Coefficients of the dispersion equation (22)    1n2s Q 1 2J1 ðbe2 ÞioZ o2 Z  a ¼ ioEs e1 Q 1 2 Lð1e2 Þ 1e2 1ns " !#  2e1 Q o2 ZJ1 ðbe2 Þ ð1n2s Þo2 Z 1 J1 ðbe2 Þ 2e2 2ioZ 1n2s Q 1 ioZN N þ 2  b ¼ ioe1 Q þ þ e1 Es QN  2  þ 2 þ 2 2 Lð1e2 Þ L 1ns ð1e2 Þk cm ð1e2 Þ ð1e2 ÞEs cm cm ð1e2 Þ ð1e22 Þk d¼

e1 QN c2m



e1 QNJ1 ðbe2 Þ L

! 2e2 2ioZ ð1ns 2 ÞN þ þ ð1e22 Þk c2m ð1e2 Þ Es

! 1 , þ c2m ð1e2 Þ ð1e22 Þk ioZ

1 L ¼ ðlr þ 2mr Þ½bJ0 ðbe2 Þe1 2 J1 ðbe2 Þlr e2 J1 ðbe2 Þ,

Z ¼ Zs þðioÞ1 G ,

N ¼ io þ

8kZD , 1e22

1

Q ¼ 1Es o2 ð1n2s Þ

e2 ¼ R2 =R1

1 mTðm, e2 Þ D¼ 4 12m1 ð1e22 Þ1 Tðm, e2 Þ Z 1 ðoyÞFðyÞði þ oyÞ G ¼ dy, y ¼ y=t0 , 1 þ ðoyÞ2 0

m ¼ iðio=kZÞ1=2

Tðm, e2 Þ ¼ A½ J1 ðmÞe2 J1 ðme2 Þ þ B½Y1 ðmÞe2 Y1 ðme2 Þ   Y0 ðmÞ Y0 ðmÞJ0 ðme2 Þ 1 J0 ðmÞ A ¼ 1 Y0 ðme2 Þ Y0 ðme2 Þ h i9 8 ðmÞ < = J0 ðme2 Þ 1 YY0 0ðme 1 2Þ 1 B¼ Y0 ðme2 Þ : J0 ðmÞ Y0 ðmÞJ0 ðme2 Þ ; Y0 ðme2 Þ

Here J0 and Y0 are the Bessel functions of first and second kinds of the zero order, respectively, J1(x) and Y1(x) are the Bessel functions of first and second kinds of the first order, respectively.

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Appendix B. Sound speed in the system at x-0 Let simplify the equations for the complex amplitudes of disturbances in the wave at o-0. In this limit longitudinal deformations and inertia of the shell and the rod can be neglected, the radial deformations are quasi-static (the assumptions of the Korteweg model [17]). Then it follows from Eqs. (14) and (15) u^ 1 ¼

1 p^ c 2e1 Es

(B1)

In this frequency region the quasi-static approach can be used for calculation of friction at the shell and rod surfaces in terms of the average flow velocity, which yields instead of Eq. (11) ikkP^ ¼ N0 V^

(B2)

^ 1 ¼ C1 x w

(B3)

Quasi-static solution of Eq. (16) has the form

where the constant C1 is found from the boundary condition (18) C1 ¼ 

1 2ðl þ mÞ

p^ c

(B4)

Neglecting in Eq. (19) by small input of the liquid viscosity in the normal stresses at the shell and rod surfaces, we obtain p^ c ¼ P

(B5)

Let rewrite Eq. (12) in the form 



^ r^ ¼ c1 V2ð1 e22 Þ1 u^ 1 e2 w^ 1R ¼ 0

(B6)

Then it follows from Eqs. (B1), (B3)–(B6), (13): " k 1 V^ ¼ c 2 þ 1e22 cf

e22 1 þ e1 Es l1 þ m1

!# P^

(B7)

Eqs. (B2) and (B7) yield relation (29) for c0 . References [1] R.J. Albalak (Ed.), Polymer Devolatilization, Marcel Dekker, New York, 1966. [2] T.G. Leigton, Acoustic Bubble, Academic Press, San Diego, 1994. [3] S.P. Levitsky, R.M. Bergman, J. Haddad, Sound dispersion in deformable tube with polymeric liquid and elastic central rod, Journal of Sound and Vibration 275 (2004) 267–281. [4] C.D. Han, Rheology in Polymer Processing, Academic Press, New York, 1976. [5] R.D. Blevins, Flow-induced Vibration, Van Nostrand Reinhold Co., New York, 1977. [6] G.K. Batchelor, Compression waves in a suspension of gas bubbles in liquid, Fluid Dynamics Transactions 4 (1968) 65–84. [7] S. Levitsky, R. Bergman, O. Levi, J. Haddad, Pressure waves in elastic tube with polymeric liquid, Applied Mechanics and Engineering 4 (1999) 561–574. [8] S. Levitsky, R. Bergman, J. Haddad, Acoustic waves in thin-walled elastic tube with polymeric solution, Ultrasonics 38 (2000) 857–859. [9] S. Levitsky, R. Bergman, J. Haddad, Low-frequency sound propagation in cylindrical gap, filled by polymeric solution, Acustica—Acta Acustica 93 (2007) 535–541. [10] S.P. Levitsky, Z.P. Shulman, Bubbles in Polymeric Liquids. Dynamics, Heat and Mass Transfer, Technomics Publishing Co., Lancaster, 1995. [11] S. Levitsky, R. Bergman, J. Haddad, Acoustic properties of thin-walled elastic tube, containing polymeric liquid with fine bubbles, in: Proceedings of the 19th International Congress on Acoustics, Madrid, September 2007, 6 pp. [12] V.E. Nakoryakov, B.G. Pokusaev, I.R. Shreiber, Wave Propagation in Gas–liquid Media, CRC Press, New York, 1993. [13] H.M. Oguz, A. Prosperetti, The natural frequency of oscillation of gas bubbles in tubes, Journal of the Acoustical Society of America 103 (1998) 3301–3308. [14] A. Prosperetti, Thermal effects and damping mechanisms in the forced radial oscillations of gas bubbles in liquids, Journal of the Acoustical Society of America 61 (1977) 17–27. [15] S.I. Rubinow, J.B. Keller, Wave propagation in a viscoelastic tube containing a viscous liquid, Journal of Fluid Mechanics 88 (1978) 181–203. [16] M.A. Biot, Theory of propagation of elastic waves in a fluid-saturated porous solid, Journal of the Acoustical Society of America 28 (1956) 168–191. [17] R.E. Goodson, R.G. Leonard, A survey of modeling techniques for fluid line transients, Journal of Basic Engineering, Transactions of the ASME 94 (1972) 474–482. [18] L.V. Wijngaarden, One-dimensional flow of liquids containing small gas bubbles, Annual Review of Fluid Mechanics 4 (1972) 369–396. [19] P.S. Wilson, R.A. Roy, An audible demonstration of the speed of sound in bubbly liquids, American Journal of Physics 76 (2008) 975–981. [20] Y. Iemoto, Y. Watanabe, Measurements of phase velocity of a sound wave propagating in a tube in low frequency region, Japanese Journal of Applied Physics 43 (2004) 401–402.