Acoustic waves in thin-walled elastic tube with polymeric solution

Ultrasonics 38 (2000) 857–859 www.elsevier.nl/locate/ultras

Acoustic waves in thin-walled elastic tube with polymeric solution S. Levitsky *, R. Bergman, J. Haddad Negev Academic College of Engineering, 71 Bazel St., PO Box 45, Beer-Sheva 84100, Israel

Abstract Acoustic waves in a pipe with polymeric liquid are investigated within a quasi-one-dimensional approach. Analysis of the dispersion equation has shown that rheological features lead to essential changes in both attenuation and speed of the sound. The results may find application in acoustic rheometry of polymeric liquids and for modeling of fast dynamic processes in polymer production technology. © 2000 Elsevier Science B.V. All rights reserved. Keywords: Acoustic waves; Attenuation; Dispersion; Elastic tube; Polymeric liquid

1. Introduction

equation [2]

Acoustic methods are finding increasing applications in rheometry of polymeric solutions and melts [1]. Existing techniques are based mainly upon measurements of ultrasound dispersion and attenuation in a volume of the testing liquid. The theoretical relations generally used for the data treatment follow from the theory of ultrasound propagation in an unbounded liquid that restricts the dimension of the measuring unit from below – it must be much greater than the sound wavelength. At low and moderate frequencies such a limitation can become critical; therefore this frequency region is more conveniently studied experimentally with small radius liquid-filled tubes. The purpose of the paper is to estimate the ranges of dispersion and attenuation of sound in such a system taking into account the liquid’s viscoelasticity. The problem is solved by the operational method within a quasi-one-dimensional model. The resulting dispersion equation is analyzed for polymeric solutions, describing by the Rouse–Spriggs distribution law.

G(t−t )s (t ) dt +2g s , Dp =c2 Dr , 1 ij 1 1 sv ij f f f −2 Dr =r −r , Dp =p −p , s =e −1 (VV)I. (1) f f f0 f f f0 ij ij 3 Here t and e are deviatoric stress and rate deformation ij ij tensors; V is the velocity; r , p and c are density, f f f pressure and speed of sound in the liquid; G(t−t ) is a 1 relaxation function; g is the solvent viscosity; index 0 sv refers to the equilibrium state. It is supposed that the wavelength l&R. In this case the losses due to the acoustically induced flow in a pipe can be evaluated with neglect of the liquid’s compressibility [3]. The dimensionless linearized equation of non-stationary axisymmetric flow of viscoelastic liquid along axis x of the pipe has the form:

2. Solution of the problem

K 9 =t2 K/R, 0

Consider small radius thin-walled cylindrical tube with internal radius R and the length L (R%L), filled by a viscoelastic compressible liquid with rheological

g: =g /p t , sv sv f0 0

* Corresponding author. E-mail address: [email protected] (S. Levitsky)

t =2 ij

P

t

∂V 9

x =K 9 +k ∂t

P A

+kg: sv

t

A

G 9 (t−t: ) 1

−2 ∂2V 9

1 ∂V 9 x x+ ∂j2 j ∂j

∂2V 9

B

dt: 1

B

1 ∂V 9 x , x+ ∂j2 j ∂j

K=−

1 ∂p

f, V 9 =V t /R, x x 0 r ∂x f0

t=t/t , j=r/R, k=r /r , t =R(r /r )1/2. (2) 0 s f0 0 s f0 In Eq. (2) V is the axial velocity, r the radius, and r x s the density of the pipe material. Laplace transformation (I[F(t)]=∆2exp(−st)F(t) dt) of Eq. (2) leads to Bessel 0

0041-624X/00/$ - see front matter © 2000 Elsevier Science B.V. All rights reserved. PII: S0 0 4 1 -6 2 4 X ( 9 9 ) 0 0 15 5 - 9

858

S. Levitsky et al. / Ultrasonics 38 (2000) 857–859

equation for the image V 9 1x =I(V 9 x ). Solution of the latter with the boundary condition V =0 ( longitudinal 91 x|j=1 displacements in the wall are considered to be small ) permits us to evaluate the wall friction’s transform t: 1 w from rheological Eq. (1). The result can be expressed through the image of the average velocity V 9 1=2∆1 V 9 1 j dj as follows: 0 x mT( m) , t: 1 =−4g: DV 9 1, D=− 1 w 4 1−2m−1T(m) J ( m) , m2=−s/A, T( m)= 1 J (m) 0

g: −g: h: 2 k , sv ∑ t: =t /p , A=kg: , g: =g: + p w w f0 sv h: z(a) k=1 sh: +1 1 k g: =g /p t . (3) p p f0 0 Here s is the Laplace variable, J and J are first kind 0 1 Bessel’s functions of zero and first order, t the shear w stress in the liquid at the wall of the tube, g the p Newtonian viscosity of the solution, h: =h /t the dimenk k 0 sionless relaxation time with the number k. The solution (3) was obtained on the assumption that the shear relaxation spectrum of the liquid is discrete and relaxation times are distributed according to the Rouse– Spriggs law [2] (h =h /ka, z(a) is the Riemann zeta k 1 function of the spectral distribution parameter a). The equations for the propagation of acoustic signals in the tube follow from the linearized momentum and mass balance laws for the liquid, written in a quasi-onedimensional approximation [3] (the term ∂2v /∂x2 in x momentum equation is considered to be small with respect to (1/r)∂v /∂r and ∂2v /∂r2; v %v ). In addition, x x r x the small cross-effect of the liquid’s rheology and compressibility is neglected [4]. As a result in terms of averaged axial velocity V and pressure P the following equations are obtained: r

∂V f0 ∂t

∂r ∂t

+

V=

r=

=−

∂P ∂x

+

2t w, R

2r (V ) ∂V f0 r r=R +r =0, f0 R ∂x

2 R2 2

R2

P P

R

0 R

2 V r dr, P= x R2

r r dr. f

P

R 0

Dp r dr, f (4)

0 To close the problem it is necessary to invoke dynamic equations of the pipe’s wall. Analysis [5] of the tube dynamics in the axisymmetric and long waves case has shown that the bending stresses in the shell are small with respect to membrane stresses. In addition, in such a case the longitudinal deformations and inertia of the

pipe wall can be neglected. As a result, from the momentum balance equation for a thin elastic cylindrical shell, written in the Kirchhoff–Love approximation it follows that u =(R2/2Eh)P, (V ) =u˙ , where u is the disr r r=R r r placement of the shell’s middle surface in the transverse direction, h the half of the shell’s width, and E Young’s module of the tube material. Eqs. (4) were rewritten in the non-dimensional form and solved in the Laplace space taking into account Eqs. (3). For the wave travelling in the positive direction of f=x/R, the solution has the form P9 1=(P /s)exp(−lf), l2=s(s+8kg: D) (c: −2 +c−2 ), 0 f s c: =c t /R, c: =ekE9 , f f 0 s e=h/R, E9 =E/p , P9 1=I(P9 ), P9 =P/p . (5) f0 f0 Relation (5) describes in Laplace space propagation of the step pressure pulse, generated at the initial moment of time at the cross-section f=0 of the tube. Because of the relatively complicated dependence of the transformed phase and amplitude of the pressure in the wave on the transform parameter s, this relation cannot be inverted to obtain exact analytical results [6 ]. Nevertheless, the most important features of the process can be deduced from the dispersion relation for the sound wave s that follows from Eqs. (5) after a change of variables siV, lik: with V=vt , v being the 0 frequency. The result has the form: V/k: =c: [1+8(iV)−1kg: D]−1/2, c: =c: c: (c: 2 +c: 2 )−1/2, K K f s f s (6) where k: is complex dimensionless wave number. For an ideal liquid (g: =0) this equation gives the Korteweg– Joukowski speed c: of a pressure pulse propagating in K an elastic tube [7]. From Eqs. (6) it follows that the non-dimensional speed of sound C=V/Re{k: }0 when V0 – the result that was also reported in Ref. [7] for a pure viscous Newtonian liquid in an elastic tube. Low-frequency dispersion comes about at V~V =8kg: D and is con1 nected with the transition from the creeping to the inertial flow regime in the acoustic wave. Viscoelasticity of the polymeric solution is manifested in this frequency region, if the main relaxation time of the liquid h: is l close to V−1 . 1 3. Simulations and analysis Numerical simulations were performed for thinwalled tube with polymeric liquid, characterized by the following parameters: R=10−2 m, e=0.05, E=7× 1010 N m−2, r =2.7 · 103 kg m−3 (aluminum), p = s f0 105 Pa, r =103 kg m−3, c =1500 m s−1, g =0.1 Pa s. f0 f sv The Newtonian viscosity of the solution was evaluated

S. Levitsky et al. / Ultrasonics 38 (2000) 857–859

from the empirical Martin relation g /g =1+ p sv b exp(k b) with k =0.4 [4]. Here b is the reduced M M polymer concentration ( b=c [g], c is the usual conp p centration and [g] is the characteristic viscosity of the solution [8]). The relaxation times h were found accordk ing to the Rouse theory with a=2 and h: =0.608g: A exp(k b) [2]. The non-dimensional 1 sv M parameter A=[g]Mp /R (M is the molecular mass of f0 G the polymer, R the universal gas constant) was chosen G equal to 500, which corresponds to the usually considered range of variation of this parameter for solutions of high polymers in organic solvents [4]. 3-D plots of the speed of sound C and attenuation x=− Im{k: } as functions of frequency V and reduced polymer concentration b for the described set of parameters is presented in Figs. 1 and 2. The curves 1 and 2 in Fig. 2 were evaluated for h =0, i.e. for a pure viscous k liquid with viscosity equal to Newtonian viscosity of the solution g ; they illustrate influence of the liquid’s viscop elasticity. Curve 1 represents the dependence of attenuation from the polymer concentration b at V=10; curve 2 represents the same as the function of frequency V at

859

b=10. Note here that V=10 for the chosen parameter values corresponds to a dimensional frequency f#1 kHz. As follows from Fig. 1, for low concentration polymer solutions the dispersion is positive, i.e. the speed of sound grows with frequency and tends to the Korteweg– Joukowski speed c =192.3 (the corresponding dimenK sional value is 1170 m s−1). This kind of dependence C=C(V) is quite similar to that for tube filled with a pure viscous liquid [9]. At sufficiently high values of the reduced concentration b the dispersion changes its sign as a result of the overshoot, appearing in the vicinity of V~h: −1 (for the plot in Fig. 1 the relaxation time h: 1 1 is changed approximately from 1 to 10 (the corresponding dimensional range is 0.0016–0.016 s) for 5V and 1 VV it is smaller than for an equivalent Newtonian 1 liquid with g=g . p 4. Summary The dispersion of acoustic waves in a thin-walled elastic tube with polymeric solution can be very higher (with a possible change in sign) than in a similar system with a pure viscous liquid, while the attenuation is smaller. The scale of these features is sensitive to the concentration and molecular characteristics of the polymeric solution. It indicates that elastic tubes of small radius can be used for the characterization of such viscoelastic liquids by sufficiently simple acoustic means.

Fig. 1. Dispersion of sound in a thin-walled aluminum tube with a polymeric solution.

Fig. 2. Attenuation of sound waves in a tube with a viscoelastic polymeric solution. The curves 1 and 2 correspond to a pure viscous liquid with Newtonian viscosity, equal to the viscosity of the polymeric solution g . p

References [1] D.R. Raichel, K. Takabayashi, Discernment of non-Newtonian behavior in liquids by acoustic means, Rheol. Acta 37 (1998) 593–600. [2] A.G. Fredrickson, Principles and Applications of Rheology, Prentice-Hall, New York, 1964. [3] W. Zielke, Frequency-dependent friction in transient pipe flow, J. Basic Eng., Trans. ASME 1 (1968) 109–115. [4] S.P. Levitsky, Z.P. Shulman, Bubbles in Polymeric Liquids. Dynamics Heat and Mass Transfer, Technomics, Lancaster, USA, 1995. [5] R.M. Bergman, Investigation of the free vibrations of noncircular cylindrical shells, J. Appl. Math. Mech. 37 (1973) 1068–1077. [6 ] S.K. Garg, A.H. Nayfeh, A.J. Good, Compressional waves in fluidsaturated elastic porous media, J. Appl. Phys. 45 (1974) 1968–1974. [7] V.E. Nakoryakov, B.G. Pokusaev, I.R. Shreiber, Wave Propagation in Gas–Liquid Media, CRC Press, New York, 1993. [8] G. Strobl, The Physics of Polymers, Springer, Berlin, 1997. [9] S.I. Rubinow, J.B. Keller, Wave propagation in a fluid-filled tube, J. Acoust. Soc. Am. 50 (1971) 198–223.