On the propagation of transient acoustic waves in isothermal bubbly liquids

Physics Letters A 350 (2006) 56–62 www.elsevier.com/locate/pla

On the propagation of transient acoustic waves in isothermal bubbly liquids P.M. Jordan ∗ , C. Feuillade Naval Research Laboratory, Stennis Space Center, MS 39529, USA Received 28 July 2005; received in revised form 3 October 2005; accepted 4 October 2005 Available online 10 October 2005 Communicated by R. Wu

Abstract The dynamic propagation of acoustic waves in a half-space filled with a viscous, bubbly liquid is studied under van Wijngaarden’s linear theory. The exact solution to this problem, which corresponds to the compressible Stokes’ 1st problem for the van Wijngaarden–Eringen equation, is obtained and analyzed using integral transform methods. Specifically, the following results are obtained: (i) van Wijngaarden’s theory is found to be ill-suited to describe air bubbles in water; (ii) At start-up, the behavior of the bubbly liquid is similar to that of a class of non-Newtonian fluids under shear; (iii) Bounds on the pressure field are established; (iv) For large time, the solution exhibits Taylor shock-like (i.e., nonlinear) behavior.  2005 Elsevier B.V. All rights reserved. PACS: 02.30.Uu; 43.20.+g; 47.55.Kf Keywords: Linear acoustics; bubbly liquids; Stokes’ 1st problem; integral transform methods

1. Introduction Based in part on the earlier work of Lord Rayleigh [1] and Foldy [2], van Wijngaarden [3] showed in 1972 that, in the case of one spatial dimension, the propagation of linear acoustic waves in isothermal bubbly liquids, wherein the bubbles are of uniform radius, is described by the PDE 4 ∂ 2u ∂ 2u ∂ 3u 2 ∂ u + r − + ν = 0, e 0 ∂x 2 ∂t 2 ∂x 2 ∂t ∂x 2 ∂t 2 where v = (u(x, t), 0, 0) is the velocity vector and

ce2

p0 , ρl β0 (1 − β0 ) R20 r02 = . 3β0 (1 − β0 ) ce2 =

νe =

(1.1)

4νl , 3β0 (1 − β0 ) (1.2)

Here, R0 (> 0) is the (constant) equilibrium bubble radius; ce and νe , respectively, denote the effective values of the (isothermal) sound speed and kinematic viscosity; νl = µl /ρl denotes the kinematic viscosity of the liquid phase, where the constants * Corresponding author. Tel.: +1 228 688 4338; fax: +1 228 688 5049.

E-mail address: [email protected] (P.M. Jordan). 0375-9601/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.10.004

ρl and µl (
0), respectively, denote the density and dynamic viscosity of the surrounding liquid; the constant p0 (> 0) is the equilibrium pressure in the liquid/gas mixture; and the constant β0 , where β0 ∈ (0, 1) is the bubble volume fraction, is neither very close to zero nor to unity. In 1985, Caflish et al. [4] extended van Wijngaarden’s theory to include heat conduction and surface tension effects. Subsequently, Eringen [5] rederived the multi-dimensional version of Eq. (1.1) based on a microcontinuum theory, and considered the case of plane waves in an unbounded, three-dimensional domain. In 1994, Saccomandi [6] investigated acoustic acceleration waves under the nonlinear version of Eringen’s [5] theory. More recently, Jordan and Feuillade [7] obtained the exact solution to Eq. (1.1), which they termed the van Wijngaarden– Eringen (VWE) equation, in the context of the compressible version of Stokes’ 2nd problem. For a comprehensive listing (up to 1992) of works on acoustic propagation in bubbly liquids, we note the review paper by Miksis and Ting [8]. Other recent, in-depth, works in this area include that of Llewellin et al. [9], in which a constitutive model describing the viscoelastic (i.e., non-Newtonian) rheology of bubbly liquids/suspensions is developed, and the papers by Brenner et al. [10] and Karpov et al. [11].

P.M. Jordan, C. Feuillade / Physics Letters A 350 (2006) 56–62

It is of interest to note that Hayes and Saccomandi [12] showed that Eq. (1.1) also governs the propagation of damped, transverse plane waves in a particular class of viscoelastic solids. Additionally, it should be noted that the special case of Eq. (1.1) for which µl = 0 and β0  1 was presented by van Wijngaarden [13] in 1968 as the PDE governing acoustic waves in inviscid bubbly liquids. Whitham [14] noted that the inviscid version of the VWE arises in the study of plasma waves, longitudinal waves in elastic bars (see also [15,16]), and in the linear theory of water waves under the Boussinesq approximation for long waves. For applications of the R0 → 0 limiting case, known as Stokes’ equation, see [17] and the references therein. To the best of our knowledge, only time-harmonic solutions of Eq. (1.1) have thus far been obtained. Hence, our aim here is to examine van Wijngaarden’s theory in the context of a dynamic, yet still analytically tractable, flow setting. Specifically, we solve and analyze the compressible version of Stokes’ 1st problem [18] involving the (viscous) VWE. We also derive a number of asymptotic results, including recovery of the (known) inviscid solution. To this end, the present Letter is arranged as follows. In Section 2, the exact solution to the above-mentioned initial-boundary value problem (IBVP) is obtained using integral transform methods. In Section 3, analytical results are presented including large- and small-t expressions. In Section 4 numerical results are presented and in Section 5 conclusions are stated. Lastly, in Section 6, the major results are discussed. 2. Mathematical formulation and solution 2.1. Problem formulation We begin this study by taking the positive z-axis of a Cartesian coordinate system in the upward direction and assuming that an isothermal, homogeneous, viscous bubbly liquid fills the half-space x > 0. Initially, the mixture is in its equilibrium state. At time t = 0+ , the pressure at the boundary x = 0 suddenly assumes, and is maintained at, the constant value pmax (= 0); i.e., the boundary condition (BC) for the pressure at x = 0 is pmax H (t), where H (·) denotes the Heaviside unit step function. We seek to determine the motion of the bubbly liquid at all points in the half-space for all t > 0. To this end, we are lead to consider the following IBVP involving the VWE equation expressed in terms of the acoustic pressure p = ℘ − p0 : ∂ 2p ce2 2 ∂x

∂ 3p ∂ 4p ∂ 2p − 2 + νe 2 + r02 2 2 = 0 (x, t > 0), (2.1) ∂t ∂x ∂t ∂x ∂t

p(0, t) = pmax H (t),

p(x, 0) = ∂p(x, 0)/∂t = 0 (x > 0).

t  = t (ce2 /νe ), we recast our IBVP in dimensionless form as 4 ∂ 2p ∂ 2p ∂ 3p 2 ∂ p + R − + = 0 (x, t > 0), ∂x 2 ∂t 2 ∂x 2 ∂t ∂x 2 ∂t 2

p(0, t) = H (t),

(2.2)

Here, ℘ is the thermodynamic pressure, we now require µl > 0, and we note that ∇ × v is identically zero. Employing the nondimensional variables p  = p/pmax , x  = x(ce /νe ), and

(2.3)

p(∞, t) = 0 (t > 0),

p(x, 0) = ∂p(x, 0)/∂t = 0 (x > 0),

(2.4)

where all primes have been omitted for convenience and the dimensionless bubble radius is given by 
√ ce2 R20 R0 3p0 ρl 2 2 = R ≡ 2 (2.5) . 4µl 3νe β0 (1 − β0 ) 2.2. Exact solution using integral transform methods We will now solve the above IBVP using a dual integral transform approach (see, e.g., Duffy [19]). Hence, applying first the spatial sine transform, which reduces Eq. (2.3) to an ODE, and then using the temporal Laplace transform to solve this ODE, we obtain the dual transform domain solution √  1 ¯pˆ = ξ 2/π 1 + ξ 2 R 2 s(s − s1 )(s − s2 )  1 R2s + , + (2.6) (s − s1 )(s − s2 ) (s − s1 )(s − s2 ) where ξ and s are the sine and Laplace transform parameters, respectively,  −ξ 2 ± ξ ξ 2 (1 − 4R 2 ) − 4 s1,2 = (2.7) , 2(1 + ξ 2 R 2 ) and a hat (respectively, bar) superposed over a quantity denotes the image of that quantity in the sine (respectively, Laplace) transform domain. Obtaining first the Laplace inverse of Eq. (2.6) using a table √ of inverses (see, e.g., [19,20]), multiplying the result by 2/π sin[ξ x], and then integrating with respect to ξ from zero to infinity, we find the exact xt-domain solution to be p(x, t) 

2 = H (t) 1 − π

 ξ ∗ e

−at


 a sin[ξ x] dξ cos[bt] + sin[bt] b ξ

0

∞ +




e−at cosh[bt] +

ξ∗

 ξ ∗ − 0

∞

p(∞, t) = 0 (t > 0),

57

+ ξ∗

ξ e−at sin[bt] sin[ξ x] dξ b 1 + ξ 2R2

ξ e−at sinh[bt] sin[ξ x] dξ b 1 + ξ 2R2  ξ ∗

−R

 a sin[ξ x] dξ sinh[bt] b ξ

2

ξe 0

−at




 a sin[ξ x] dξ cos[bt] − sin[bt] b 1 + ξ 2R2

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P.M. Jordan, C. Feuillade / Physics Letters A 350 (2006) 56–62

∞ +

ξe ξ∗

−at




sin[ξ x] dξ a cosh[bt] − sinh[bt] b 1 + ξ 2R2

(R < Rc ),

(2.8)

where a = a(ξ ) and b = b(ξ ) are given by ξ2 , 2(1 + ξ 2 R 2 )  ξ |ξ 2 (Rc2 − R 2 ) − 1| b(ξ ) = , (1 + ξ 2 R 2 )

a(ξ ) =

(2.9)

ξ ∗ = (Rc2 − R 2 )−1/2 is the integration breakpoint, and Rc = 1/2 is the (same) critical value of R given in [7]. The reader will note that we have given only the solution for R < Rc in Eq. (2.10). The reason for this will become clear in Section 4.1. However, while it is not presented here, we would like to point out that the solution for R
Rc can be obtained in the same manner. 2.3. Solution in Laplace transform domain Finally, we note for future use that by applying only the temporal Laplace transform to Eq. (2.3) and the BCs given in Eq. (2.4), we get, after employing the ICs and solving the resulting subsidiary equation, the exact transform domain solution   −sx 1 (x > 0), p(x, ¯ s) = exp √ (2.10) s R (s − σ1 )(s − σ2 ) where σ1,2 ∈ R are given by  −1 ± 2 Rc2 − R 2 σ1,2 = . 2R 2

(2.11)

3. Analytical results 3.1. Temporal limits and small- and large-time asymptotic results Using the properties of the Laplace transform (see, e.g., [20]), it can be shown from Eq. (2.10) that lim p(x, t) = exp[−x/R]

t→0+

and

lim p(x, t) = 1.

t→∞

(3.1)

From Eq. (3.1)1 , and the fact that limt→0− p(x, t) = 0, we see that at time t = 0+ (i.e., at start-up) p suffers a (nonpropagating) jump discontinuity of amplitude lim p(x, t) − lim p(x, t) = exp[−x/R].

t→0+

t→0−

(3.2)

Having found that p exhibits a jump across t = 0, it is of interest now to examine the behavior of p just after start-up. Hence, we expand the argument of the exponential in Eq. (2.10) in powers of 1/s and then neglect terms O[s −2 ]. On inverting, we obtain, for t (< R 2 ) sufficiently small, the approximation 

p(x, t) ≈ H (t) exp[−x/R]I0 2xt/R 3 , (3.3) where I0 [ · ] denotes the modified Bessel function of the first kind of order zero. From Eq. (3.3) it is evident that just after

start-up, p is an increasing function of time, as can also be inferred from Eqs. (3.1). If, instead, we expand Eq. (2.10) for small s, i.e., in powers of s, we find that   1 −sx −1 s + sx + · · · . p(x, ¯ s) ≈ e (3.4) 2 On inversion, we get the large-t asymptotic expression   1 ˙ p(x, t) ∼ H (t − x) 1 + x δ(t − x) + · · · (t → ∞), (3.5) 2 ˙ ) denotes the where δ(·) denotes the Dirac delta function, δ(ζ derivative of δ with respect to ζ , and x is assumed fixed. From Eq. (3.5) we see that p exhibits a wave-like behavior, and is essentially independent of R, as t becomes large. 3.2. Large bubble radius asymptotics If we allow µl → 0, then R becomes very large and it can be easily shown that    −4  1 1 i 1− σ1,2 ∼ − ± +O R (R → ∞). (3.6) 2R 2 R 8R 2 If we now neglect terms of O[R −2 ], then σ1,2 ∼ ±i/R and Eq. (2.10) becomes   1 −sx (R → ∞), p(x, ¯ s) ∼ exp √ (3.7) s R s 2 + R −2 which we note is the exact transform domain solution to the present IBVP involving the inviscid VWE. (From Eq. (3.7) it is apparent that the start-up jump given in Eq. (3.2) also occurs when viscosity is absent.) Inverting Eq. (3.7) using the complex inversion formula, we find that [16]  

 1   2 ξ(x/R) dξ p(x, t) ∼ H (t) 1 − cos ξ(t/R) sin  π 1 − ξ2 ξ 0

(R → ∞).

(3.8)

3.3. Pressure pulse and impulse response solutions If the BC p(0, t) = H (t) is replaced with p(0, t) = H (t) − H (t − t0 ), where t0 > 0, then the solution to the modified problem is exactly given by P (x, t; t0 ) ≡ p(x, t) − p(x, t − t0 ),

(3.9)

which corresponds to a pressure pulse of duration (or “width”) t0 dimensionless time units. In addition, we observe that limt0 →0 {p(x, t) − p(x, t − t0 )}/t0 gives the exact solution corresponding to the impulse response BC p(0, t) = δ(t). 4. Numerical results 4.1. Physically allowable values of the bubble radius In Table 1, Rc is the value of R0 corresponding to Rc (= 1/2); L(β0 ) = νe /ce is the characteristic length used in the

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59

Fig. 1. p vs. x for R = RKn and t = 0.1, 1.0, 2.5, 10.0. Table 1 Rc , L(β0 ), and R∗0 , for various liquids at p0 = 1 atm Liquid

Water (fresh)

Glycerin

Golden syrup Honey

µl (Pa s) ρl (kg/m3 ) Rc L(0.5) R∗0

0.001 (20 ◦ C) 998 (20 ◦ C) 0.115 µm 0.265 µm 2.3 nm

1.2 (20 ◦ C) 1260 (20 ◦ C) 0.123 mm 0.283 mm 2.45 µm

75.3 (20 ◦ C) 1441 (20 ◦ C) 7.2 mm 16.6 mm 0.144 mm

249 (23 ± 1 ◦ C) 1400 (23 ± 1 ◦ C) 2.41 cm 55.8 mm 0.483 mm

(<0.01) corresponds to a value of R0 that satisfies R0 < R∗0 in the case of golden syrup. The pressure p is plotted, at four values of time t, as a function of the penetration distance x into the bubbly liquid half-space. Clearly, p = 1 when x = 0. Note however, that while p exhibits the exponential decay indicated by Eq. (3.1)1 for small values of t, we see that as t is increased a shoulder develops and the profile assumes the shape of a Taylor shock.1

nondimensional scaling of x (see Section 2.1); Kn(R∗0 ) = 0.01, where the Knudsen number of the flow is defined here as  R0 3p0 r0 = = R, Kn(R0 ) ≡ (4.1) L(β0 ) 4νl ρl

4.3. Half-peak point

and we note that the continuum assumption demands Kn < 0.01 (see, e.g., Zucrow and Hoffman [21]). Also, 1 atm = 1.013 × 105 Pa, the values for both fresh water and glycerin were obtained from p. 462 of [22], for golden syrup from [9], and for honey from [23]. From an experimental standpoint, this restriction on Kn severely restricts the range of physical applicability of the VWE in the case of bubbly liquids. This is clearly evident from Table 1. In the case of both fresh water and glycerin, the values of R∗0 needed to satisfy the continuum assumption are far too small to be either covered by the theory or of practical importance. For golden syrup and honey, the values of R∗0 needed are experimentally achievable but, in both cases, the bubble radii are more than an order of magnitude smaller than would be required to reach the critical radius Rc (= 1/2), where several interesting phenomena might be expected to occur [7].

In Fig. 2, the behavior of the Taylor shock profile is examined by plotting p as a function of x, for three values of t, and R = RKn , along with the corresponding tangent lines at the halfpeak point p = 0.5. In each case, as x increases, we see that the pressure drops rapidly from very nearly unity to essentially zero over a narrow transition region centered on the half-peak point p = 0.5, which is located at x ≈ t. This is shown more explicitly in Fig. 3. Here, the x-coordinate of the half-peak point x50% = x50% (t), which is defined as the solution of the equation p(x50% , t) = 0.5 for given t and R, is plotted as a function of t for R = RKn . While a minor departure from linear behavior is observed for small values of t, due to the time needed for the Taylor shock to develop (see Fig. 1), the linear steady-state behavior is rapidly established, and we see from the slope that the waveform propagates into the half-space with a speed that tends to unity as t is increased; i.e., x˙50% → 1 as t → ∞.

4.2. Temporal evolution of the pressure profile The sequence shown in Fig. 1 depicts the temporal evolution of the pressure profile for fixed R = RKn , where RKn = 0.00983

1 By which we mean, broadly speaking, any strictly monotonic, bounded function that approaches constant, but unequal, limits at ∓∞; also known as a “diffusive soliton”; see, e.g., [24], and also [25] and the references therein.

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Fig. 2. p vs. x for R = RKn . Bold: t = 30.0, thin-solid: t = 20.0, broken: t = 10.0.

Fig. 3. x50% vs. t . Bold: R = RKn , broken: the line x50% = t .



4.4. Taylor shock width and ad hoc approximation

+ Re

Now Fig. 2 also shows that the slope of the tangent line appears to decrease as t increases, implying that the width of the transition region increases with time. Using concepts from classical shock theory, the Taylor shock width (or thickness), which we denote by , may be defined as [26] 

−1  ∂p(x, t)  . (t) ≡ − ∂x x=x50%

(4.2)

In additional numerical work that was carried out, but not presented here, we inverted ∂ p/∂x ¯ numerically, where p¯ is given by Eq. (2.10), using Tzou’s [27] Riemann sum inversion algorithm  e4.7 1 ¯ f (x, t) ≈ f (x, 4.7/t) t 2

N 



(−1) f¯(x, (4.7 + imπ)/t) m

,

(4.3)

m=1

where N (1) is an integer, Re[ · ] denotes the real part of a complex quantity, and x, t > 0. As suggested by Fig. 2, (t) was found to be, in fact, an increasing function of t. Hence, based on Fig. 2, the behaviors of x50% and , and guided by classical shock theory, it is not difficult to show using numerical methods that for t sufficiently large, p admits the ad hoc approximation p(x, t) ≈

  1 1 − tanh 2(x − x50% )/ . 2

(4.4)

From Fig. 4 it is clear that Eq. (4.4) is a more useful approximation than the (analytically derived) asymptotic expression given in Eq. (3.5). In fact, the former appears to become more accurate as t is increased.

P.M. Jordan, C. Feuillade / Physics Letters A 350 (2006) 56–62

61

4.5. Rectangular pressure pulse case Fig. 5 depicts the time variation of the pressure pulse field P at x = 0.5 for R = RKn and for three different values of t0 . Initially, i.e., when the pulse is first applied, P increases and begins to approach its steady-state value of unity. However, after time t = t0 , i.e., after the pulse is discontinued, P continues to increase for a short time before it begins to fall back to zero. It is interesting to note that the duration of this time lag after t = t0 is reached, when P continues to increase, decreases as t0 is increased. 5. Conclusions 1. Van Wijngaarden’s [3] linear theory of acoustic propagation in viscous, isothermal bubbly liquids is, due to the violation of the continuum assumption, ill-suited for describing air bubbles in water. However, it is applicable to gas bubbles in liquids of relatively high viscosity (see Section 4.1). 2. At start-up, p suffers a jump of magnitude exp[−x/R] (see Section 3.1). 3. Based on Eqs. (3.1), we have found that p is an increasing function of time that is bounded in the following way: exp[−x/R] < p(x, t) < 1

Fig. 4. p vs. x for t = 10.0, 20.0, 30.0 and R = RKn . Solid: Eq. (2.8), broken: Eq. (4.4). (a): x50% ≈ 10.374 and  ≈ 7.889, (b): x50% ≈ 20.375 and  ≈ 11.184, (c): x50% ≈ 30.376 and  ≈ 13.707.

(x, t > 0).

(5.1)

4. While p initially exhibits a decaying exponential character, our numerical studies indicate that as t increases, p begins to behave in a wave-like manner. Specifically, a shoulder forms and the profile assumes the shape of a Taylor shock (see Figs. 1 and 2). 5. The Taylor shock width  increases with time (see Section 4.4 and Fig. 2). Moreover, x˙50% , the propagation speed of the Taylor shock’s half-peak point, rapidly tends to unity as t → ∞ (see Figs. 2 and 3); i.e., x˙50% → ce , as t → ∞, in terms of the dimensional variables. Additionally, during the Taylor shock phase p is approximately in the form of “tanh[ · ]” (see Eq. (4.4)), and this approximation appears to become a more accurate one as t → ∞ (see Fig. 4).

Fig. 5. P vs. t for x = 0.5 and R = RKn . Bold: t0 = 0.05, thin-solid: t0 = 0.1, broken: t0 = 0.15.

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6. For t  R, the pressure p is, essentially, independent of t (see Eq. (3.3)). 7. As the viscosity µl becomes very small (⇒ R → ∞), the known solution for the inviscid case of IBVP (2.3), (2.4) is recovered (see Eqs. (3.7)–(3.9)). 8. In the pressure pulse case, P continues to increase for a brief period after t = t0 is reached, indicating inertia, after which it slowly decays back to zero. However, the interval of time over which P is still increasing decreases as t0 is increased (see Section 3.3 and Fig. 5).

that a jump of this type occurs. In fact, the theoretical findings presented here are consistent with recent experimental evidence which indicates that bubbly liquids admit a viscoelastic rheology (see [9] and the references therein).

Acknowledgements

6. Discussion of major results

The authors are grateful to Prof. R.Q. Wu and the anonymous referee for their helpful suggestions, as well as to Prof. M.J. Buckingham for his instructive comments. This research was supported by ONR/NRL funding (PE 061153N).

6.1. Large-time pressure profile: further remarks

References

The transition of p over time from a decaying exponentiallike profile to one that is wave-like, i.e., in the form of a Taylor shock, is an important finding of the present work. However, it is instructive to note that similar behavior has been reported, with regards to the velocity field, in the case of a viscous, heat conducting relaxing gas (see [28] and the references therein). Another interesting aspect of the present study is the following: the Taylor shock phase of the profile’s evolution is, qualitatively, very similar to traveling wave solutions of the KdV–Burgers’ equation [29], which is the PDE governing nonlinear acoustic waves in viscous bubbly liquids [3]. 6.2. Start-up jump: further remarks The jump suffered by p at start-up is due to the idealized Heaviside BC coupled with the fact that the VWE contains a 4th order mixed derivative term of a specific form. If either of these features are eliminated, then so is the jump. For if we replace the BC at x = 0 (see Eq. (2.4)) with the ramp-type BC   p(0, t) = t1−1 tH (t) − (t − t1 )H (t − t1 ) , (6.1) where t1 is a positive constant, or if instead we set R ≡ 0, then p is continuous at start-up, i.e., the jump ceases to exist. What is more, such a jump has been found to occur in other areas of continuum mechanics, and in all cases the governing PDEs contain a mixed derivative term [30–32]. Moreover, it should be noted that a similar paradox, i.e., an apparently infinite signal speed in the medium, but where such a start-up jump does not occur, arises in the corresponding IBVP involving the well-known diffusion equation, where the dependent variable is continuous across t = 0. However, for every t > 0, no matter how small, the solution is everywhere non-zero (see, e.g., [33,34]). In spite of the problem’s idealized nature, physical insight can, nevertheless, be gained. From Eq. (3.2), we see that limt→0+ p is essentially zero for x  R. Consequently, only a very thin slab, of thickness ∼ O[R], experiences an exponentially decreasing (over x) pressurization at start-up. Also, it should be noted that certain non-Newtonian fluids have been predicted to exhibit a start-up jump under Heaviside shearing (see [32] and the references therein). Hence, given that the bubbly liquid considered here consists of a viscous, incompressible fluid containing compressible gas bubbles, it is not unexpected

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