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A note on poroacoustic traveling waves under Forchheimerʼs law
Physics Letters A 377 (2013) 1350–1357
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Physics Letters A www.elsevier.com/locate/pla
A note on poroacoustic traveling waves under Forchheimer’s law P.M. Jordan ∗ Acoustics Div., U.S. Naval Research Laboratory, Stennis Space Ctr., MS 39529, USA
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 13 December 2012 Accepted 29 March 2013 Available online 6 April 2013 Communicated by R. Wu
Acoustic traveling waves in a gas that saturates a rigid porous medium is investigated under the assumption that the drag experienced by the gas is modeled by Forchheimer’s law. Exact traveling wave solutions (TWS)s, as well as approximate and asymptotic expressions, are obtained; decay rates are determined; and acceleration wave results are presented. In addition, special cases are considered, critical values of the wave variable and parameters are derived, and comparisons with predictions based on Darcy’s law are performed. It is shown that, with respect to the Darcy case, most of the metrics that characterize such waveforms exhibit an increase in magnitude under Forchheimer’s law. Published by Elsevier B.V.
Keywords: Nonlinear poroacoustics Forchheimer’s law Traveling wave solutions Acceleration waves
1. Introduction In situations involving high-velocity flow through porous materials, Darcy’s law has been shown to fail [18,21]. In such cases, the consensus in the literature is that Forchheimer’s equation, the most common version of which is [18, §1.5.2]
μχ C f χ 2 |v|v, ∇P = − v− √ K
K
(1)
then becomes the appropriate form of the resistance law. Here, P is an intrinsic pressure; (> 0) is the mass density of the gas; the positive constants μ, K , and χ (< 1) denote, respectively, the shear viscosity, permeability, and porosity; the constant C f ( 0) is the formdrag (or inertia) coefficient; and the intrinsic (i.e., volume-averaged over a volume element consisting of gas only) velocity vector v is related to the filtration velocity vector V via the Dupuit–Forchheimer relationship, namely, V = χ v. Since its introduction in 1901, (1) has been the topic of a great many investigations, both theoretical and experimental; see, e.g., Refs. [6,13,18,21,23] and those therein. After more than a century of effort, however, the question of when to use (1) instead of Darcy’s law, which the former reduces to on setting C f = 0, is one that is still debated. The main difficultly centers on quantifying the transition region between the Darcy and Forchheimer flow regimes, where the former and latter correspond to low and high values, respectively, of the associated Reynolds number; again, see Ref. [18, §1.5.2]. According to Beavers and Sparrow [2], Forchheimer’s law should be used in situations where the following inequality is not satisfied:
(Re K )−1 C f ,
(2)
where Re K , known as the permeability-based Reynolds number, is defined below in (11). It should be noted that this criterion is equivalent to Ruth and Ma’s [24, p. 261] finding that “[Forchheimer’s law should be used] when Fo becomes experimentally significant with respect to 1.” In Ref. [24], Fo := C f Re K defines the Forchheimer number. Our primary aim here is to compare/contrast the Forchheimer and Darcy formulations in the traveling wave context, the results of which we hope sheds light on the transition region issue. Specifically, we consider poroacoustic traveling waves, in one-dimension (1D), for the case in which (1) is the resistance law and the permeating fluid is a perfect gas. As such, the present study extends the findings of Jordan and Fulford [15], who modeled the impact of the porous matrix on the acoustic field using Darcy’s law, to the high Reynolds number regime (i.e., the regime in which the inequality in (2) is not satisfied). To this end, the present Letter is arranged as follows. In Section 2, the governing equations and constitutive relations are stated. In Section 3, a traveling wave analysis is carried out and exact
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P.M. Jordan / Physics Letters A 377 (2013) 1350–1357
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TWSs are presented. In Section 4, a variety of asymptotic/approximate results are derived and special/limiting cases are noted. Then, in Section 5, acceleration wave results are presented and analyzed. Lastly, in Section 6, possible extensions of the present investigation are proposed. Remark 1. It is noteworthy that a resistance law in the form of (1) also arises when modeling compressible flow in deformable pipes; see, e.g., Ref. [10] and those therein. 2. Constitutive assumptions and balance laws Consider a perfect gas1 that both permeates and is flowing through a fixed and rigid, non-thermally conducting, homogeneous and isotropic porous medium, the drag of which on the gas is described by Forchheimer’s law. Assuming the flow can be regarded as homentropic [25], the equation of state for such a gas becomes
℘ = ℘e (/e )γ .
(3)
where ℘ (> 0) is the thermodynamic pressure, γ = c p /c v is a positive constant known as the adiabatic index, and the positive constants e and ℘e denote the equilibrium state values of and ℘ , respectively. In the case of plane waves propagating along the x-axis, to which we henceforth restrict our attention, v = (u (x, t ), 0, 0), while and ℘ are both functions of x and t only. Thus, the velocity field is irrotational and the continuity and momentum equations become
t + u x + u x = 0,
(4)
(ut + uu x ) = −℘ x − au − b|u |u ,
(5)
respectively, where the absence of all external body forces in (5) has been posited. Here, we have set a = μχ K −1 and b
= Cfχ
2
K −1/2 for
convenience; from (3) we find that
℘ = c e2 (/e )γ −1 ,
(6)
√
where c e = γ ℘e /e denotes the sound speed in the undisturbed gas; and by equilibrium state we mean the unperturbed state characterized by u = 0, = e , and ℘ = ℘e . Eliminating ℘ from (5) using (6) and then introducing the following non-dimensional quantities:
u◦ = u/V ,
◦ = /e ,
℘ ◦ = ℘/℘e ,
x◦ = x/ L ,
t ◦ = cet /L ,
(7)
our system (3)–(5) is reduced to
℘ = γ ,
(8)
t + (u )x = 0,
ut +
2
(9)
uu x = −x γ −1 − δ u 1 + Re K C f |u | .
(10)
Here, = V /c e denotes the Mach number, where the positive constant V represents a characteristic speed; δ = χ / Re is the dimensionless Darcy coefficient, where Re = c e ( K / L )/ν is a Reynolds number, the positive constant L represents a characteristic length, and ν = μ/e is the kinematic viscosity; the permeability-based Reynolds number [4] is defined as
Re K := (χ V
√
K )/ν ;
(11)
and all circle superscripts (◦ ) have been suppressed but are to remain understood. Remark 2. The continuum assumption, on which the present poroacoustic model is based, requires Kn 1, where Kn = Knudsen number of the flow. Thus, δ can be re-expressed as
δ = χ 2 / Kn / Re K .
√
K / L is the
(12)
3. Traveling wave analysis 3.1. Ansatzs and associated ODE Further restricting our attention to only those plane waves that are right-running, we set = f (ξ ), u = g (ξ ), and ℘ = p (ξ ), where ξ := x − ct is the wave variable and c, the speed of the traveling waveform, is a positive constant. Now, substituting the first and last ansatzs into (8) gives
p (ξ ) = f γ (ξ ),
(13)
where for future reference we note that
p (ξ ) = γ f γ −1 (ξ ) f (ξ ). 1
By which we mean an ideal gas for which c p > c v > 0, i.e., the specific heats at constant pressure and volume, are constants; see Thompson [25, p. 79].
(14)
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P.M. Jordan / Physics Letters A 377 (2013) 1350–1357
Next, we substitute the first and second ansatzs into both (9) and (10). This yields, after integrating the former with respect to ξ once and simplifying, the following system:
f ( g − c ) = K1 ,
(15)
g f (−c + g ) + f γ −1 f = − δ g 1 + Fo | g | f ,
(16)
where K1 is the constant of integration, a prime denotes d/dξ , and we recall that Fo = Re K C f is the Forchheimer number. Solving for K1 using the equilibrium state conditions f = 1 and g = 0, (15) can be recast as
g = c ( f − 1)/ f
( g < c / ),
(17)
where the restriction now imposed on g ensures f > 0. Finally, after eliminating g between (16) and (17), using the fact that
g = c f / f 2,
(18)
and simplifying, we obtain the associated ordinary differential equation (ODE) for the density field, namely,
c 1 − c −2 f γ +1 f = −δ f (1 − f ) 1 + c λ|1 − f | , where λ := −1 Fo, i.e.,
(19)
λ denotes the -scaled Forchheimer number.
3.2. Phase plane analysis and stability of equilibria An analysis of (19) in the ( f , f )-plane reveals that it admits the equilibrium solutions ¯f = {0, 1}. If f w ∈ (0, 1), where f w := f (0) denotes the value of f at the wavefront x = ct, then f ∈ (0, 1) for every (positive) c = 0, while f is always unbounded for f w > 1. More significantly, however, (19) is found to describe two distinct flow regimes, corresponding to the two cases of c = 1, separated by the degenerate case c = 1. For c > 1, the equilibrium solutions ¯f = {0, 1} are stable and unstable, respectively, and, provided f w ∈ (0, 1), there exists a unique, strictly decreasing kink-type [1] integral curve of (19), such that f ∈ (0, 1) for every ξ ∈ R. For c < 1, it is easily established that ¯f = {0, 1} are both stable and that | f | = ∞ at f = f s , where f s ∈ (0, 1) is given by f s := 2/(γ +1) c . In turns out that, just as in the Darcy case treated by Jordan and Fulford [15], f is dual-valued when c < 1 and f w ∈ (0, 1) hold simultaneously. As these authors went on to show, the physical interpretation of this loss of uniqueness is a propagating shock wave [26, p. 77]. For c = 1, the equilibrium ¯f = 0 is once again stable, with f < 0 for f ∈ (0, 1); however, the stability/instability of ¯f = 1 cannot be determined. To understand how this degeneracy impacts the integral curves, we first expand f γ +1 about ¯f = 1 and then re-express the c = 1 case of (19) as
1 1 (1 − f ) (γ + 1) 1 − γ (1 − f ) + γ (γ − 1)(1 − f )2 + · · · f + δ f 1 + λ|1 − f | = 0 (c = 1), 2
6
(20)
from which it is obvious that f = 1 is a solution; however,
lim
f →1
δ −δ f (1 − f )(1 + λ|1 − f |) =− γ +1 1 − f γ +1
(c = 1).
(21)
Since the right-hand side of (21) is clearly not zero, it follows that taking c = 1 has caused f to suffer a jump discontinuity at ¯f = 1. 3.3. Exact solutions Since our interest lies only with those solutions for f that are both bounded and non-increasing, f w ∈ (0, 1) and c > c s are henceforth posited. Noting that the former assumption implies f ∈ (0, 1), we return to (19), from which the absolute value bars can now of course be omitted, and separate variables. Integrating now subject to f (0) = f w , we find ourselves faced with the task of evaluating
f c fw
(1 − c −2 fγ +1 ) df = −δξ f(1 − f)[1 + c λ(1 − f)]
(0 < f < 1).
(22)
Although its integrand is somewhat more complicated than that of Ref. [15, Eq. (3.11)], which corresponds to the Darcy (i.e., λ = 0) case, this integral can, like the former, be evaluated in closed form with the aid of special functions. Using the symbolic routine Integrate, which is provided in the software package Mathematica (ver. 8.0), it is readily established that the exact, albeit implicit, solution of (19) is given by the following. For c = 1 (c > c s ):
δξ = Z ( f , c , λ) − Z ( f w , c , λ),
f ∈ (0, 1).
(23)
For c = 1:
f (ξ ) = 1,
ξ ∈ (−∞, ξ1 (γ , λ)];
δξ = Z ( f , 1, λ) − Z ( f w , 1, λ),
f ∈ (0, 1).
(24)
P.M. Jordan / Physics Letters A 377 (2013) 1350–1357
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γ = 1.4, δ = 1.0, and f w = 0.5. Blue: λ = 0 (Darcy’s law). Red: λ = 0.5. Gold: λ = λ∗ ≈ 1.461 (see Section 4.3). Green: λ = 2.0. (Color graphics only
Fig. 1. f vs. ξ for c = 1.1, in electronic version.)
Here,
Z ( f , c , λ) :=
2 F 1 (1,
+
γ + 1; γ + 2; f ) f γ +1 λ{2 F 1 [1, γ + 1; γ + 2; c λ f (1 + c λ)−1 ] f γ +1 } − c (γ + 1) (γ + 1)(1 + c λ)
c 1 + cλ
ln
1− f f
− c λ ln
1 + c λ(1 − f ) 1− f
(25)
,
where 2 F 1 denotes the Gauss hypergeometric series;
ξ1 (γ , λ) := −
1
δ
ϑ + ψ(γ + 1) +
2 F 1 (1,
γ + 1; γ + 2; f w ) f wγ +1 γ +1 γ +1
λ{2 F 1 [1, γ + 1; γ + 2; λ(1 + λ)−1 ] − 2 F 1 [1, γ + 1; γ + 2; λ f w (1 + λ)−1 ] f w (γ + 1)(1 + λ) 1 1 − fw 1 + λ(1 − f w ) − λ ln , ln + 1+λ fw 1 − fw
+
}
(26)
where ψ(·) is the digamma function and ϑ ≈ 0.5772 is the Euler–Mascheroni constant; and we observe that f ∈ (0, 1) implies ξ ∈ (ξ1 (γ , λ), ∞) for the case c = 1. To simplify the forthcoming analysis, which will focus primarily on the density2 field and c 1, we take advantage of the fact that (19) is invariant under ξ → ξ + const. and assume henceforth, without loss of generality, f w = 1/2, a consequence of which is that the lower bound on c now becomes c s = 2−(γ +1)/2 . Remark 3. It should be noted that ξ1 (γ , 0) = ξ1 ( f w ), where ξ1 ( f w ) denotes the c = 1 special case of the quantity defined in Ref. [15, Eq. (3.14)]. 3.4. Shock thickness Based on the definition given in Ref. [20, p. 591], the shock thickness, (> 0), associated with our kink-type TWSs for f is given by
:=
4(c 2 − 2−(γ +1) ) c δ(1 + c λ/2)
( f w = 1/2, c 1),
(27)
from which it is clear that is a decreasing function of λ. Thus, for given values of c, γ , etc., Forchheimer’s law predicts a steeper slope at (0, f w ), which is the center point of the transition region (i.e., |ξ | < /2) of the kink’s shock-front, than does Darcy’s law; see Fig. 1, all the curves of which were plotted for the case of a diatomic gas (i.e., γ = 7/5).
2
Of course, corresponding results for the pressure and velocity fields can be derived using (13) and (17), respectively.
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P.M. Jordan / Physics Letters A 377 (2013) 1350–1357
3.5. Special cases of perfect gases In the case of both monatomic (i.e., γ = 5/3) and diatomic (i.e., γ = 7/5) gases under ordinary conditions, the integral in (22) can be evaluated exclusively in terms of elementary functions. Unfortunately, due to the length and complexity of the resulting expressions, neither of these cases can be treated here in detail; however, as shown in Ref. [15, §4.2], the special case λ = 0 (i.e., the Darcy case) does results in analytically tractable solutions for both γ = 5/3 and γ = 7/5. √ It is also noteworthy that the evaluation of (22) for γ = 7/5 yields an expression which contains the famous number ϕ , where ϕ = 12 (1 + 5 ) denotes the Golden ratio; again, see Ref. [15, §4.2]. 4. Analytical results 4.1. Results for small- f Expanding (23) and (24) about f = 0, neglecting terms O ( f 2 ), and then solving for f , the following approximations are readily obtained. For c = 1 (c > c s ):
f (ξ ) ≈
1 + cλ
1 + 2c λ
W 0 exp −
(1 + c λ)( Z ( 12 , c , λ) + δξ )
c
1 + 2c λ
(1 + c λ)1+c λ
,
ξ ξ˜ (c , λ).
(28)
For c = 1:
f (ξ ) ≈
1+λ
W0
1 + 2λ
(1 + 2λ) exp[−(1 + λ)( Z ( 12 , 1, λ) + δξ )] , (1 + λ)1+λ
ξ ξ˜ (1, λ).
(29)
Here, W 0 denotes the principal branch of the Lambert W -function,3
ξ˜ (c , λ) :=
1
c
δ 1 + cλ
−Z
1 2
, c, λ
(30)
,
and it should be noted that, in the case of a perfect gas, ξ˜ (1, λ) > ξ1 (γ , λ) for all λ 0. From these approximations and Ref. [5, Eq. (B.4)], it is easily established that f (ξ ) ∼ A(ξ, λ) as ξ → ∞, where
(1 + c λ) Z ( 12 , c , λ) exp[−δ(1 + c λ)ξ/c ] A(ξ, λ) := exp − c (1 + c λ)c λ
(c > c s ).
(31)
4.2. Asymptotic rate of decay: Forchheimer vs. Darcy In carrying out this phase of our analysis it is useful to introduce the function
(ξ, λ) := A(ξ, λ) − A(ξ, 0) (λ > 0).
(32)
Observing that (ξ , λ) = 0, where ξ = ξ (c , λ) is given by †
ξ † (c , λ) :=
†
†
{2 F 1 [1, γ + 1; γ + 2; c λ(2 + 2c λ)−1 ] − 2 F 1 [1, γ + 1; γ + 2; 12 ]}( 12 )γ +1 1 (λ > 0), + c δ −1 ln 1 + c δ(γ + 1) 1 + cλ
(33)
we recast (32) in the more useful form
(η, λ) = exp −λξ † (η − 1) − 1 A(η, 0) (λ > 0),
(34)
where we have set ξ = ξ † η . By plotting the surface ξ † vs. (c , λ) for different values of γ (not shown), it can be numerically established that ξ † > 0 for all parameter values of interest. It follows, therefore, that if c > c s , then (ξ, λ) < 0 for ξ ∈ (ξ † , ∞). Thus, if c > c s , Forchheimer’s law yields a faster asymptotic rate of decay than Darcy’s, a fact clearly illustrated by the curves of Fig. 1. 4.3. A critical value of λ and small-|ξ | approximations The curves plotted in Fig. 2, which exactly correspond to the f vs. ξ profiles appearing in Fig. 1, illustrate the impact of varying λ on −c f (ξ ) (i.e., t ). Clearly, ξ¯ , where f (ξ¯ ) = 0, and maxξ ∈R [−c f (ξ )] are strictly increasing functions of λ, and thus minλ0 (ξ¯ ) and minλ0 [maxξ ∈R [−c f (ξ )]] both occur under Darcy’s law (i.e., for λ = 0). The gold curve that appears in Fig. 2 was plotted for λ = λ∗ , where ξ¯ = 0 when λ = λ∗ . To learn more about this critical value of the -scaled Forchheimer number, we start by differentiating (19) with respect to ξ , the result of which can be expressed as
3 For a summary of the major properties of this relatively new addition to the family of special functions, see, e.g., Christov and Jordan [5, Appendix B], as well as the references cited therein.
P.M. Jordan / Physics Letters A 377 (2013) 1350–1357
Fig. 2. −c f vs. ξ for c = 1.1, version.)
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γ = 1.4, δ = 1.0, and f w = 0.5. Blue: λ = 0 (Darcy’s law). Red: λ = 0.5. Gold: λ = λ∗ (≈ 1.461). Green: λ = 2.0. (Color graphics only in electronic
[γ (1 + c λ) − (γ − 1)(1 + 2c λ) f + c λ(γ − 2) f 2 ] f γ +1 (c 2 − f γ +1 )2 2 c [1 + c λ − 2(1 + 2c λ) f + 3c λ f 2 ] f (1 − f )[1 + c λ(1 − f )] + . (c 2 − f γ +1 )2 c 2 − f γ +1
2 2
f =c δ
(35)
On setting f = 1/2 and λ = λ∗ , where
λ∗ =
2(γ + 1)
(c > 1),
c 3 2γ +1 − c (γ + 2)
(36)
the sum within the large braces vanishes and (35) becomes f = 0. Thus, with little additional effort, the following can be established:
max −c f (ξ ) = −c f (0) = ξ ∈R
δ c 2 2γ −1 c 2 2γ +1 − (γ + 2)
λ = λ∗ , c > 1 ,
(37)
where we observe that 2γ +1 > γ + 2 for 1 < γ 53 . Evidently, letting λ → λ∗ increases the symmetry of the f vs. ξ curve with respect to the f -axis. In the remainder of this subsection we show how the symmetry introduced by the special case λ = λ∗ allows for relatively simple, but quite accurate, explicit small-|ξ | approximations to be obtained. As the first step in this process, we set f = w + 1/2 and then expand about w = 0; as a result, (19) becomes, on neglecting terms O( w 3 ), a special case of Riccati’s well known equation, namely,
w = f (0) 1 − 4σ w 2 where
| w | 1 , λ = λ∗ ,
(38)
σ (> 0) is given here by
σ := 1 −
γ (γ + 1) 2 |1 − c 2 2 γ + 1 |
(c > 1).
(39)
Separating variables and integrating, it is not difficult to show that
f (ξ ) ≈
1 2
1 + σ −1/2 tanh 2σ 1/2 f (0)ξ
|ξ | 1, λ = λ∗ ,
(40)
from which it follows that
f (ξ ) ≈ f (0) sech2 2σ 1/2 f (0)ξ
|ξ | 1, λ = λ∗ .
(41)
In the vicinity of ξ = 0, (40) and (41) provide very good approximations of the gold (i.e., λ = λ∗ ) curves shown in Figs. 1 and 2, respectively. Remark 4. When λ = λ∗ , Forchheimer’s law yields TWSs for f that are, qualitatively, similar to those of the classic (i.e., parabolic) Burgers’ equation, at least for small-|ξ |.
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5. Acceleration waves: Forchheimer vs. Darcy As a review of the literature will reveal, interest in poroacoustic acceleration waves has been on the rise in recent years; see, e.g., Refs. [7–9,11,15,19,23] and those therein. In this subsection we investigate this special class of singular surface phenomenon in the context of the present traveling wave analysis; in particular, we compare/contrast the two cases of λ 0, i.e., the Forchheimer and Darcy cases, respectively, both of which are capable of admitting acceleration waves. To this end, we introduce the following concepts and formulae from singular surface theory [23]: (i) The plane x = Σ(t ) represents a singular surface (i.e., a wavefront), which propagates along the x-axis with constant speed, across which some function, say, F = F(x, t ), suffers a jump discontinuity. (ii) The amplitude of the jump in F is defined here as
[[F]] := F− − F+ ,
(42)
where F± := limx→Σ ± F are assumed to exist and the ± superscripts correspond to the regions ahead of and behind Σ , respectively. (iii) The jump in the product of two functions, e.g., F and G, across x = Σ(t ) is given by
[[FG]] = F+ [[G]] + G+ [[F]] + [[F]][[G]].
(43)
In the present study, the jumps in f , g , etc., which occur across the plane ξ = ξ1 (γ , λ) when c = 1 (recall Section 3.2), are the acceleration wave amplitudes. Making the identification Σ(t ) = t + ξ1 (γ , λ) and noting that ( f )− = 0, it can be shown that
J f K =
δ
γ +1
JgK =
,
δ
(γ + 1)
,
J pK =
γδ (c = 1), γ +1
(44)
all of which we observe are independent of λ. Here, [[ f ]] follows immediately from (21) and the definition given in (42); [[ g ]] and [[ p ]], on the other hand, were determined using (18) and (14), respectively, in conjunction with (44)1 and (43). While the acceleration wave amplitudes are independent of λ, the location of their common wavefront, i.e., x = t + ξ1 (γ , λ), is not. To understand the nature of the dependence of ξ1 on λ, as well as on γ , we first observe that
min ξ1 (γ , λ) = ξ1
1.454 , ,0 ≈ − 3 δ
5 (γ , λ) ∈ 1, × [0, +∞);
5
max ξ1 (γ , λ) < lim ξ1 (γ , λ) = 0, λ→∞
(45)
3
γ ∈ 1,
5 3
(46)
.
Then, with the aid of numerically generated surface plots of ξ1 vs. (γ , λ) (not shown), it is a straightforward matter to establish that, in the case of a perfect gas,
ξ1
5 3
, 0 ξ1 (γ , 0) < ξ1 (γ , λ) < 0 (λ > 0).
(47)
Thus, ξ1 (γ , λ) is found to be not only a strictly negative quantity but also an increasing (resp. decreasing) function of λ (resp.
γ ).
Remark 5. Since β = 12 (γ + 1) in the case of a perfect gas, the three amplitude expressions presented in Ref. [15, Eq. (4.11)] are equivalent to their counterparts given in (44), where β(> 1) is the coefficient of nonlinearity [3]. Remark 6. Under Darcy’s law, ξ1 = ξ1 (γ , 0) admits the smallest possible upper bound, specifically,
−
1.454
δ
≈ ξ1
1 + ln(4) 1.193 , 0 ξ1 (γ , 0) < − ≈− , 3 2δ δ 5
γ ∈ 1,
5 3
.
(48)
6. Possible follow-on studies Using (17), it is not difficult to show that | g | → ∞, as ξ → ∞, for all λ 0, an unfortunate consequence4 of the fact that our model allows for only one of two asymptotic conditions to be enforced. ˜ u xx , where the positive constant Hence, the most obvious follow-on to the present study is to return to Section 2 and add the term μ μ˜ is an effective viscosity coefficient [18, §1.5.3], to the right-hand side of (5). Generalized in this way, our system is now based on the Brinkman–Forchheimer model [6,18,23]; and it can be expected to yield bounded TWSs for f , g, etc. Of course, other modeling approaches that introduce higher-order spatial derivatives do exist. Of these, we note the following: (a) treat the gas phase as a type of generalized continua, e.g., as a perfect gas which admits one or more material length scale parameters (see, e.g., Refs. [16,22] and those therein); and (b), modify the system in Section 2 according to what has been termed “finite-scale” theory, a recently-introduced extension of classical compressible flow theory originating in the field of turbulence modeling [17]. Finally, it is noteworthy that generalizations and extensions of (1), itself, have been proposed; see, e.g., Refs. [12,14,23] and those therein. 4
Physically, the blow-up of g is a result of the breakdown of our (continuum-based) model as the wave propagates towards the vacuum state.
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Acknowledgements The author thanks Prof. Brian Straughan for his valuable comments. All figures appearing in this Letter were generated using the software package Mathematica (ver. 8). This work was supported by ONR funding. References [1] J. Angulo, Nonlinear Dispersive Equations: Existence and Stability of Solitary and Periodic Travelling Wave Solutions, American Mathematical Society, Providence, RI, 2009. [2] G.S. Beavers, E.M. Sparrow, Int. J. Heat Mass Transfer 14 (1971) 1855. [3] R.T. Beyer, in: M.F. Hamilton, D.T. Blackstock (Eds.), Nonlinear Acoustics, Academic Press, San Diego, CA, 1997, pp. 25–39. [4] K. Boomsma, D. Poulikakos, ASME J. Fluids Eng. 124 (2002) 263. [5] I. Christov, P.M. Jordan, J. Thermal Stresses 33 (2010) 1109. [6] M. Ciarletta, B. Straughan, V. Tibullo, Nonlinear Anal.: Real World Appl. 12 (2011) 2839. [7] M. Ciarletta, B. Straughan, V. Tibullo, Mech. Res. Commun. 37 (2010) 137. [8] M. Ciarletta, B. Straughan, V. Zampoli, Int. J. Non-Linear Mech. 52 (2013) 8. [9] M. Ciarletta, B. Straughan, V. Zampoli, Int. J. Eng. Sci. 45 (2007) 736. [10] C. Di Nucci, A. Russo Spena, Meccanica 48 (2013) 15. [11] M. Fabrizio, F. Franchi, B. Straughan, Int. J. Eng. Sci. 46 (2008) 790. [12] M. Firdaouss, et al., J. Fluid Mech. 343 (1997) 331. [13] L. Fusi, et al., Meccanica 48 (2013) 57. [14] M. Hassanizadeh, W.G. Gray, Transp. Porous Media 2 (1987) 521. [15] P.M. Jordan, J.K. Fulford, Appl. Math. 56 (2011) 99. [16] P.M. Jordan, G. Saccomandi, Proc. R. Soc. A 468 (2012) 3441. [17] L.G. Margolin, D.E. Vaughan, Mech. Res. Commun. 45 (2012) 64. [18] D.A. Nield, A. Bejan, Convection in Porous Media, second ed., Springer-Verlag, New York, 1999. [19] P. Paoletti, Discrete Continuous Dyn. Sys., Ser. B (DCDS-B) 17 (2012) 637. [20] A.D. Pierce, Acoustics: An Introduction to Its Physical Principles and Applications, Acoustical Society of America, Woodbury, NY, 1989. [21] A.E. Scheidegger, The Physics of Flow Through Porous Media, third ed., University of Toronto Press, Toronto, 1974. [22] B. Straughan, Ann. Mat. Pura Appl. 126 (1980) 187. [23] B. Straughan, Stability and Wave Motion in Porous Media, Springer-Verlag, New York, 2008. [24] D. Ruth, H. Ma, Transp. Porous Media 7 (1992) 255. [25] P.A. Thompson, Compressible-Fluid Dynamics, McGraw–Hill, New York, 1972. [26] G.B. Whitham, Linear and Nonlinear Waves, Wiley, New York, 1974.
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Physics Letters A www.elsevier.com/locate/pla
A note on poroacoustic traveling waves under Forchheimer’s law P.M. Jordan ∗ Acoustics Div., U.S. Naval Research Laboratory, Stennis Space Ctr., MS 39529, USA
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 13 December 2012 Accepted 29 March 2013 Available online 6 April 2013 Communicated by R. Wu
Acoustic traveling waves in a gas that saturates a rigid porous medium is investigated under the assumption that the drag experienced by the gas is modeled by Forchheimer’s law. Exact traveling wave solutions (TWS)s, as well as approximate and asymptotic expressions, are obtained; decay rates are determined; and acceleration wave results are presented. In addition, special cases are considered, critical values of the wave variable and parameters are derived, and comparisons with predictions based on Darcy’s law are performed. It is shown that, with respect to the Darcy case, most of the metrics that characterize such waveforms exhibit an increase in magnitude under Forchheimer’s law. Published by Elsevier B.V.
Keywords: Nonlinear poroacoustics Forchheimer’s law Traveling wave solutions Acceleration waves
1. Introduction In situations involving high-velocity flow through porous materials, Darcy’s law has been shown to fail [18,21]. In such cases, the consensus in the literature is that Forchheimer’s equation, the most common version of which is [18, §1.5.2]
μχ C f χ 2 |v|v, ∇P = − v− √ K
K
(1)
then becomes the appropriate form of the resistance law. Here, P is an intrinsic pressure; (> 0) is the mass density of the gas; the positive constants μ, K , and χ (< 1) denote, respectively, the shear viscosity, permeability, and porosity; the constant C f ( 0) is the formdrag (or inertia) coefficient; and the intrinsic (i.e., volume-averaged over a volume element consisting of gas only) velocity vector v is related to the filtration velocity vector V via the Dupuit–Forchheimer relationship, namely, V = χ v. Since its introduction in 1901, (1) has been the topic of a great many investigations, both theoretical and experimental; see, e.g., Refs. [6,13,18,21,23] and those therein. After more than a century of effort, however, the question of when to use (1) instead of Darcy’s law, which the former reduces to on setting C f = 0, is one that is still debated. The main difficultly centers on quantifying the transition region between the Darcy and Forchheimer flow regimes, where the former and latter correspond to low and high values, respectively, of the associated Reynolds number; again, see Ref. [18, §1.5.2]. According to Beavers and Sparrow [2], Forchheimer’s law should be used in situations where the following inequality is not satisfied:
(Re K )−1 C f ,
(2)
where Re K , known as the permeability-based Reynolds number, is defined below in (11). It should be noted that this criterion is equivalent to Ruth and Ma’s [24, p. 261] finding that “[Forchheimer’s law should be used] when Fo becomes experimentally significant with respect to 1.” In Ref. [24], Fo := C f Re K defines the Forchheimer number. Our primary aim here is to compare/contrast the Forchheimer and Darcy formulations in the traveling wave context, the results of which we hope sheds light on the transition region issue. Specifically, we consider poroacoustic traveling waves, in one-dimension (1D), for the case in which (1) is the resistance law and the permeating fluid is a perfect gas. As such, the present study extends the findings of Jordan and Fulford [15], who modeled the impact of the porous matrix on the acoustic field using Darcy’s law, to the high Reynolds number regime (i.e., the regime in which the inequality in (2) is not satisfied). To this end, the present Letter is arranged as follows. In Section 2, the governing equations and constitutive relations are stated. In Section 3, a traveling wave analysis is carried out and exact
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P.M. Jordan / Physics Letters A 377 (2013) 1350–1357
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TWSs are presented. In Section 4, a variety of asymptotic/approximate results are derived and special/limiting cases are noted. Then, in Section 5, acceleration wave results are presented and analyzed. Lastly, in Section 6, possible extensions of the present investigation are proposed. Remark 1. It is noteworthy that a resistance law in the form of (1) also arises when modeling compressible flow in deformable pipes; see, e.g., Ref. [10] and those therein. 2. Constitutive assumptions and balance laws Consider a perfect gas1 that both permeates and is flowing through a fixed and rigid, non-thermally conducting, homogeneous and isotropic porous medium, the drag of which on the gas is described by Forchheimer’s law. Assuming the flow can be regarded as homentropic [25], the equation of state for such a gas becomes
℘ = ℘e (/e )γ .
(3)
where ℘ (> 0) is the thermodynamic pressure, γ = c p /c v is a positive constant known as the adiabatic index, and the positive constants e and ℘e denote the equilibrium state values of and ℘ , respectively. In the case of plane waves propagating along the x-axis, to which we henceforth restrict our attention, v = (u (x, t ), 0, 0), while and ℘ are both functions of x and t only. Thus, the velocity field is irrotational and the continuity and momentum equations become
t + u x + u x = 0,
(4)
(ut + uu x ) = −℘ x − au − b|u |u ,
(5)
respectively, where the absence of all external body forces in (5) has been posited. Here, we have set a = μχ K −1 and b
= Cfχ
2
K −1/2 for
convenience; from (3) we find that
℘ = c e2 (/e )γ −1 ,
(6)
√
where c e = γ ℘e /e denotes the sound speed in the undisturbed gas; and by equilibrium state we mean the unperturbed state characterized by u = 0, = e , and ℘ = ℘e . Eliminating ℘ from (5) using (6) and then introducing the following non-dimensional quantities:
u◦ = u/V ,
◦ = /e ,
℘ ◦ = ℘/℘e ,
x◦ = x/ L ,
t ◦ = cet /L ,
(7)
our system (3)–(5) is reduced to
℘ = γ ,
(8)
t + (u )x = 0,
ut +
2
(9)
uu x = −x γ −1 − δ u 1 + Re K C f |u | .
(10)
Here, = V /c e denotes the Mach number, where the positive constant V represents a characteristic speed; δ = χ / Re is the dimensionless Darcy coefficient, where Re = c e ( K / L )/ν is a Reynolds number, the positive constant L represents a characteristic length, and ν = μ/e is the kinematic viscosity; the permeability-based Reynolds number [4] is defined as
Re K := (χ V
√
K )/ν ;
(11)
and all circle superscripts (◦ ) have been suppressed but are to remain understood. Remark 2. The continuum assumption, on which the present poroacoustic model is based, requires Kn 1, where Kn = Knudsen number of the flow. Thus, δ can be re-expressed as
δ = χ 2 / Kn / Re K .
√
K / L is the
(12)
3. Traveling wave analysis 3.1. Ansatzs and associated ODE Further restricting our attention to only those plane waves that are right-running, we set = f (ξ ), u = g (ξ ), and ℘ = p (ξ ), where ξ := x − ct is the wave variable and c, the speed of the traveling waveform, is a positive constant. Now, substituting the first and last ansatzs into (8) gives
p (ξ ) = f γ (ξ ),
(13)
where for future reference we note that
p (ξ ) = γ f γ −1 (ξ ) f (ξ ). 1
By which we mean an ideal gas for which c p > c v > 0, i.e., the specific heats at constant pressure and volume, are constants; see Thompson [25, p. 79].
(14)
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P.M. Jordan / Physics Letters A 377 (2013) 1350–1357
Next, we substitute the first and second ansatzs into both (9) and (10). This yields, after integrating the former with respect to ξ once and simplifying, the following system:
f ( g − c ) = K1 ,
(15)
g f (−c + g ) + f γ −1 f = − δ g 1 + Fo | g | f ,
(16)
where K1 is the constant of integration, a prime denotes d/dξ , and we recall that Fo = Re K C f is the Forchheimer number. Solving for K1 using the equilibrium state conditions f = 1 and g = 0, (15) can be recast as
g = c ( f − 1)/ f
( g < c / ),
(17)
where the restriction now imposed on g ensures f > 0. Finally, after eliminating g between (16) and (17), using the fact that
g = c f / f 2,
(18)
and simplifying, we obtain the associated ordinary differential equation (ODE) for the density field, namely,
c 1 − c −2 f γ +1 f = −δ f (1 − f ) 1 + c λ|1 − f | , where λ := −1 Fo, i.e.,
(19)
λ denotes the -scaled Forchheimer number.
3.2. Phase plane analysis and stability of equilibria An analysis of (19) in the ( f , f )-plane reveals that it admits the equilibrium solutions ¯f = {0, 1}. If f w ∈ (0, 1), where f w := f (0) denotes the value of f at the wavefront x = ct, then f ∈ (0, 1) for every (positive) c = 0, while f is always unbounded for f w > 1. More significantly, however, (19) is found to describe two distinct flow regimes, corresponding to the two cases of c = 1, separated by the degenerate case c = 1. For c > 1, the equilibrium solutions ¯f = {0, 1} are stable and unstable, respectively, and, provided f w ∈ (0, 1), there exists a unique, strictly decreasing kink-type [1] integral curve of (19), such that f ∈ (0, 1) for every ξ ∈ R. For c < 1, it is easily established that ¯f = {0, 1} are both stable and that | f | = ∞ at f = f s , where f s ∈ (0, 1) is given by f s := 2/(γ +1) c . In turns out that, just as in the Darcy case treated by Jordan and Fulford [15], f is dual-valued when c < 1 and f w ∈ (0, 1) hold simultaneously. As these authors went on to show, the physical interpretation of this loss of uniqueness is a propagating shock wave [26, p. 77]. For c = 1, the equilibrium ¯f = 0 is once again stable, with f < 0 for f ∈ (0, 1); however, the stability/instability of ¯f = 1 cannot be determined. To understand how this degeneracy impacts the integral curves, we first expand f γ +1 about ¯f = 1 and then re-express the c = 1 case of (19) as
1 1 (1 − f ) (γ + 1) 1 − γ (1 − f ) + γ (γ − 1)(1 − f )2 + · · · f + δ f 1 + λ|1 − f | = 0 (c = 1), 2
6
(20)
from which it is obvious that f = 1 is a solution; however,
lim
f →1
δ −δ f (1 − f )(1 + λ|1 − f |) =− γ +1 1 − f γ +1
(c = 1).
(21)
Since the right-hand side of (21) is clearly not zero, it follows that taking c = 1 has caused f to suffer a jump discontinuity at ¯f = 1. 3.3. Exact solutions Since our interest lies only with those solutions for f that are both bounded and non-increasing, f w ∈ (0, 1) and c > c s are henceforth posited. Noting that the former assumption implies f ∈ (0, 1), we return to (19), from which the absolute value bars can now of course be omitted, and separate variables. Integrating now subject to f (0) = f w , we find ourselves faced with the task of evaluating
f c fw
(1 − c −2 fγ +1 ) df = −δξ f(1 − f)[1 + c λ(1 − f)]
(0 < f < 1).
(22)
Although its integrand is somewhat more complicated than that of Ref. [15, Eq. (3.11)], which corresponds to the Darcy (i.e., λ = 0) case, this integral can, like the former, be evaluated in closed form with the aid of special functions. Using the symbolic routine Integrate, which is provided in the software package Mathematica (ver. 8.0), it is readily established that the exact, albeit implicit, solution of (19) is given by the following. For c = 1 (c > c s ):
δξ = Z ( f , c , λ) − Z ( f w , c , λ),
f ∈ (0, 1).
(23)
For c = 1:
f (ξ ) = 1,
ξ ∈ (−∞, ξ1 (γ , λ)];
δξ = Z ( f , 1, λ) − Z ( f w , 1, λ),
f ∈ (0, 1).
(24)
P.M. Jordan / Physics Letters A 377 (2013) 1350–1357
1353
γ = 1.4, δ = 1.0, and f w = 0.5. Blue: λ = 0 (Darcy’s law). Red: λ = 0.5. Gold: λ = λ∗ ≈ 1.461 (see Section 4.3). Green: λ = 2.0. (Color graphics only
Fig. 1. f vs. ξ for c = 1.1, in electronic version.)
Here,
Z ( f , c , λ) :=
2 F 1 (1,
+
γ + 1; γ + 2; f ) f γ +1 λ{2 F 1 [1, γ + 1; γ + 2; c λ f (1 + c λ)−1 ] f γ +1 } − c (γ + 1) (γ + 1)(1 + c λ)
c 1 + cλ
ln
1− f f
− c λ ln
1 + c λ(1 − f ) 1− f
(25)
,
where 2 F 1 denotes the Gauss hypergeometric series;
ξ1 (γ , λ) := −
1
δ
ϑ + ψ(γ + 1) +
2 F 1 (1,
γ + 1; γ + 2; f w ) f wγ +1 γ +1 γ +1
λ{2 F 1 [1, γ + 1; γ + 2; λ(1 + λ)−1 ] − 2 F 1 [1, γ + 1; γ + 2; λ f w (1 + λ)−1 ] f w (γ + 1)(1 + λ) 1 1 − fw 1 + λ(1 − f w ) − λ ln , ln + 1+λ fw 1 − fw
+
}
(26)
where ψ(·) is the digamma function and ϑ ≈ 0.5772 is the Euler–Mascheroni constant; and we observe that f ∈ (0, 1) implies ξ ∈ (ξ1 (γ , λ), ∞) for the case c = 1. To simplify the forthcoming analysis, which will focus primarily on the density2 field and c 1, we take advantage of the fact that (19) is invariant under ξ → ξ + const. and assume henceforth, without loss of generality, f w = 1/2, a consequence of which is that the lower bound on c now becomes c s = 2−(γ +1)/2 . Remark 3. It should be noted that ξ1 (γ , 0) = ξ1 ( f w ), where ξ1 ( f w ) denotes the c = 1 special case of the quantity defined in Ref. [15, Eq. (3.14)]. 3.4. Shock thickness Based on the definition given in Ref. [20, p. 591], the shock thickness, (> 0), associated with our kink-type TWSs for f is given by
:=
4(c 2 − 2−(γ +1) ) c δ(1 + c λ/2)
( f w = 1/2, c 1),
(27)
from which it is clear that is a decreasing function of λ. Thus, for given values of c, γ , etc., Forchheimer’s law predicts a steeper slope at (0, f w ), which is the center point of the transition region (i.e., |ξ | < /2) of the kink’s shock-front, than does Darcy’s law; see Fig. 1, all the curves of which were plotted for the case of a diatomic gas (i.e., γ = 7/5).
2
Of course, corresponding results for the pressure and velocity fields can be derived using (13) and (17), respectively.
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P.M. Jordan / Physics Letters A 377 (2013) 1350–1357
3.5. Special cases of perfect gases In the case of both monatomic (i.e., γ = 5/3) and diatomic (i.e., γ = 7/5) gases under ordinary conditions, the integral in (22) can be evaluated exclusively in terms of elementary functions. Unfortunately, due to the length and complexity of the resulting expressions, neither of these cases can be treated here in detail; however, as shown in Ref. [15, §4.2], the special case λ = 0 (i.e., the Darcy case) does results in analytically tractable solutions for both γ = 5/3 and γ = 7/5. √ It is also noteworthy that the evaluation of (22) for γ = 7/5 yields an expression which contains the famous number ϕ , where ϕ = 12 (1 + 5 ) denotes the Golden ratio; again, see Ref. [15, §4.2]. 4. Analytical results 4.1. Results for small- f Expanding (23) and (24) about f = 0, neglecting terms O ( f 2 ), and then solving for f , the following approximations are readily obtained. For c = 1 (c > c s ):
f (ξ ) ≈
1 + cλ
1 + 2c λ
W 0 exp −
(1 + c λ)( Z ( 12 , c , λ) + δξ )
c
1 + 2c λ
(1 + c λ)1+c λ
,
ξ ξ˜ (c , λ).
(28)
For c = 1:
f (ξ ) ≈
1+λ
W0
1 + 2λ
(1 + 2λ) exp[−(1 + λ)( Z ( 12 , 1, λ) + δξ )] , (1 + λ)1+λ
ξ ξ˜ (1, λ).
(29)
Here, W 0 denotes the principal branch of the Lambert W -function,3
ξ˜ (c , λ) :=
1
c
δ 1 + cλ
−Z
1 2
, c, λ
(30)
,
and it should be noted that, in the case of a perfect gas, ξ˜ (1, λ) > ξ1 (γ , λ) for all λ 0. From these approximations and Ref. [5, Eq. (B.4)], it is easily established that f (ξ ) ∼ A(ξ, λ) as ξ → ∞, where
(1 + c λ) Z ( 12 , c , λ) exp[−δ(1 + c λ)ξ/c ] A(ξ, λ) := exp − c (1 + c λ)c λ
(c > c s ).
(31)
4.2. Asymptotic rate of decay: Forchheimer vs. Darcy In carrying out this phase of our analysis it is useful to introduce the function
(ξ, λ) := A(ξ, λ) − A(ξ, 0) (λ > 0).
(32)
Observing that (ξ , λ) = 0, where ξ = ξ (c , λ) is given by †
ξ † (c , λ) :=
†
†
{2 F 1 [1, γ + 1; γ + 2; c λ(2 + 2c λ)−1 ] − 2 F 1 [1, γ + 1; γ + 2; 12 ]}( 12 )γ +1 1 (λ > 0), + c δ −1 ln 1 + c δ(γ + 1) 1 + cλ
(33)
we recast (32) in the more useful form
(η, λ) = exp −λξ † (η − 1) − 1 A(η, 0) (λ > 0),
(34)
where we have set ξ = ξ † η . By plotting the surface ξ † vs. (c , λ) for different values of γ (not shown), it can be numerically established that ξ † > 0 for all parameter values of interest. It follows, therefore, that if c > c s , then (ξ, λ) < 0 for ξ ∈ (ξ † , ∞). Thus, if c > c s , Forchheimer’s law yields a faster asymptotic rate of decay than Darcy’s, a fact clearly illustrated by the curves of Fig. 1. 4.3. A critical value of λ and small-|ξ | approximations The curves plotted in Fig. 2, which exactly correspond to the f vs. ξ profiles appearing in Fig. 1, illustrate the impact of varying λ on −c f (ξ ) (i.e., t ). Clearly, ξ¯ , where f (ξ¯ ) = 0, and maxξ ∈R [−c f (ξ )] are strictly increasing functions of λ, and thus minλ0 (ξ¯ ) and minλ0 [maxξ ∈R [−c f (ξ )]] both occur under Darcy’s law (i.e., for λ = 0). The gold curve that appears in Fig. 2 was plotted for λ = λ∗ , where ξ¯ = 0 when λ = λ∗ . To learn more about this critical value of the -scaled Forchheimer number, we start by differentiating (19) with respect to ξ , the result of which can be expressed as
3 For a summary of the major properties of this relatively new addition to the family of special functions, see, e.g., Christov and Jordan [5, Appendix B], as well as the references cited therein.
P.M. Jordan / Physics Letters A 377 (2013) 1350–1357
Fig. 2. −c f vs. ξ for c = 1.1, version.)
1355
γ = 1.4, δ = 1.0, and f w = 0.5. Blue: λ = 0 (Darcy’s law). Red: λ = 0.5. Gold: λ = λ∗ (≈ 1.461). Green: λ = 2.0. (Color graphics only in electronic
[γ (1 + c λ) − (γ − 1)(1 + 2c λ) f + c λ(γ − 2) f 2 ] f γ +1 (c 2 − f γ +1 )2 2 c [1 + c λ − 2(1 + 2c λ) f + 3c λ f 2 ] f (1 − f )[1 + c λ(1 − f )] + . (c 2 − f γ +1 )2 c 2 − f γ +1
2 2
f =c δ
(35)
On setting f = 1/2 and λ = λ∗ , where
λ∗ =
2(γ + 1)
(c > 1),
c 3 2γ +1 − c (γ + 2)
(36)
the sum within the large braces vanishes and (35) becomes f = 0. Thus, with little additional effort, the following can be established:
max −c f (ξ ) = −c f (0) = ξ ∈R
δ c 2 2γ −1 c 2 2γ +1 − (γ + 2)
λ = λ∗ , c > 1 ,
(37)
where we observe that 2γ +1 > γ + 2 for 1 < γ 53 . Evidently, letting λ → λ∗ increases the symmetry of the f vs. ξ curve with respect to the f -axis. In the remainder of this subsection we show how the symmetry introduced by the special case λ = λ∗ allows for relatively simple, but quite accurate, explicit small-|ξ | approximations to be obtained. As the first step in this process, we set f = w + 1/2 and then expand about w = 0; as a result, (19) becomes, on neglecting terms O( w 3 ), a special case of Riccati’s well known equation, namely,
w = f (0) 1 − 4σ w 2 where
| w | 1 , λ = λ∗ ,
(38)
σ (> 0) is given here by
σ := 1 −
γ (γ + 1) 2 |1 − c 2 2 γ + 1 |
(c > 1).
(39)
Separating variables and integrating, it is not difficult to show that
f (ξ ) ≈
1 2
1 + σ −1/2 tanh 2σ 1/2 f (0)ξ
|ξ | 1, λ = λ∗ ,
(40)
from which it follows that
f (ξ ) ≈ f (0) sech2 2σ 1/2 f (0)ξ
|ξ | 1, λ = λ∗ .
(41)
In the vicinity of ξ = 0, (40) and (41) provide very good approximations of the gold (i.e., λ = λ∗ ) curves shown in Figs. 1 and 2, respectively. Remark 4. When λ = λ∗ , Forchheimer’s law yields TWSs for f that are, qualitatively, similar to those of the classic (i.e., parabolic) Burgers’ equation, at least for small-|ξ |.
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P.M. Jordan / Physics Letters A 377 (2013) 1350–1357
5. Acceleration waves: Forchheimer vs. Darcy As a review of the literature will reveal, interest in poroacoustic acceleration waves has been on the rise in recent years; see, e.g., Refs. [7–9,11,15,19,23] and those therein. In this subsection we investigate this special class of singular surface phenomenon in the context of the present traveling wave analysis; in particular, we compare/contrast the two cases of λ 0, i.e., the Forchheimer and Darcy cases, respectively, both of which are capable of admitting acceleration waves. To this end, we introduce the following concepts and formulae from singular surface theory [23]: (i) The plane x = Σ(t ) represents a singular surface (i.e., a wavefront), which propagates along the x-axis with constant speed, across which some function, say, F = F(x, t ), suffers a jump discontinuity. (ii) The amplitude of the jump in F is defined here as
[[F]] := F− − F+ ,
(42)
where F± := limx→Σ ± F are assumed to exist and the ± superscripts correspond to the regions ahead of and behind Σ , respectively. (iii) The jump in the product of two functions, e.g., F and G, across x = Σ(t ) is given by
[[FG]] = F+ [[G]] + G+ [[F]] + [[F]][[G]].
(43)
In the present study, the jumps in f , g , etc., which occur across the plane ξ = ξ1 (γ , λ) when c = 1 (recall Section 3.2), are the acceleration wave amplitudes. Making the identification Σ(t ) = t + ξ1 (γ , λ) and noting that ( f )− = 0, it can be shown that
J f K =
δ
γ +1
JgK =
,
δ
(γ + 1)
,
J pK =
γδ (c = 1), γ +1
(44)
all of which we observe are independent of λ. Here, [[ f ]] follows immediately from (21) and the definition given in (42); [[ g ]] and [[ p ]], on the other hand, were determined using (18) and (14), respectively, in conjunction with (44)1 and (43). While the acceleration wave amplitudes are independent of λ, the location of their common wavefront, i.e., x = t + ξ1 (γ , λ), is not. To understand the nature of the dependence of ξ1 on λ, as well as on γ , we first observe that
min ξ1 (γ , λ) = ξ1
1.454 , ,0 ≈ − 3 δ
5 (γ , λ) ∈ 1, × [0, +∞);
5
max ξ1 (γ , λ) < lim ξ1 (γ , λ) = 0, λ→∞
(45)
3
γ ∈ 1,
5 3
(46)
.
Then, with the aid of numerically generated surface plots of ξ1 vs. (γ , λ) (not shown), it is a straightforward matter to establish that, in the case of a perfect gas,
ξ1
5 3
, 0 ξ1 (γ , 0) < ξ1 (γ , λ) < 0 (λ > 0).
(47)
Thus, ξ1 (γ , λ) is found to be not only a strictly negative quantity but also an increasing (resp. decreasing) function of λ (resp.
γ ).
Remark 5. Since β = 12 (γ + 1) in the case of a perfect gas, the three amplitude expressions presented in Ref. [15, Eq. (4.11)] are equivalent to their counterparts given in (44), where β(> 1) is the coefficient of nonlinearity [3]. Remark 6. Under Darcy’s law, ξ1 = ξ1 (γ , 0) admits the smallest possible upper bound, specifically,
−
1.454
δ
≈ ξ1
1 + ln(4) 1.193 , 0 ξ1 (γ , 0) < − ≈− , 3 2δ δ 5
γ ∈ 1,
5 3
.
(48)
6. Possible follow-on studies Using (17), it is not difficult to show that | g | → ∞, as ξ → ∞, for all λ 0, an unfortunate consequence4 of the fact that our model allows for only one of two asymptotic conditions to be enforced. ˜ u xx , where the positive constant Hence, the most obvious follow-on to the present study is to return to Section 2 and add the term μ μ˜ is an effective viscosity coefficient [18, §1.5.3], to the right-hand side of (5). Generalized in this way, our system is now based on the Brinkman–Forchheimer model [6,18,23]; and it can be expected to yield bounded TWSs for f , g, etc. Of course, other modeling approaches that introduce higher-order spatial derivatives do exist. Of these, we note the following: (a) treat the gas phase as a type of generalized continua, e.g., as a perfect gas which admits one or more material length scale parameters (see, e.g., Refs. [16,22] and those therein); and (b), modify the system in Section 2 according to what has been termed “finite-scale” theory, a recently-introduced extension of classical compressible flow theory originating in the field of turbulence modeling [17]. Finally, it is noteworthy that generalizations and extensions of (1), itself, have been proposed; see, e.g., Refs. [12,14,23] and those therein. 4
Physically, the blow-up of g is a result of the breakdown of our (continuum-based) model as the wave propagates towards the vacuum state.
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Acknowledgements The author thanks Prof. Brian Straughan for his valuable comments. All figures appearing in this Letter were generated using the software package Mathematica (ver. 8). This work was supported by ONR funding. References [1] J. Angulo, Nonlinear Dispersive Equations: Existence and Stability of Solitary and Periodic Travelling Wave Solutions, American Mathematical Society, Providence, RI, 2009. [2] G.S. Beavers, E.M. Sparrow, Int. J. Heat Mass Transfer 14 (1971) 1855. [3] R.T. Beyer, in: M.F. Hamilton, D.T. Blackstock (Eds.), Nonlinear Acoustics, Academic Press, San Diego, CA, 1997, pp. 25–39. [4] K. Boomsma, D. Poulikakos, ASME J. Fluids Eng. 124 (2002) 263. [5] I. Christov, P.M. Jordan, J. Thermal Stresses 33 (2010) 1109. [6] M. Ciarletta, B. Straughan, V. Tibullo, Nonlinear Anal.: Real World Appl. 12 (2011) 2839. [7] M. Ciarletta, B. Straughan, V. Tibullo, Mech. Res. Commun. 37 (2010) 137. [8] M. Ciarletta, B. Straughan, V. Zampoli, Int. J. Non-Linear Mech. 52 (2013) 8. [9] M. Ciarletta, B. Straughan, V. Zampoli, Int. J. Eng. Sci. 45 (2007) 736. [10] C. Di Nucci, A. Russo Spena, Meccanica 48 (2013) 15. [11] M. Fabrizio, F. Franchi, B. Straughan, Int. J. Eng. Sci. 46 (2008) 790. [12] M. Firdaouss, et al., J. Fluid Mech. 343 (1997) 331. [13] L. Fusi, et al., Meccanica 48 (2013) 57. [14] M. Hassanizadeh, W.G. Gray, Transp. Porous Media 2 (1987) 521. [15] P.M. Jordan, J.K. Fulford, Appl. Math. 56 (2011) 99. [16] P.M. Jordan, G. Saccomandi, Proc. R. Soc. A 468 (2012) 3441. [17] L.G. Margolin, D.E. Vaughan, Mech. Res. Commun. 45 (2012) 64. [18] D.A. Nield, A. Bejan, Convection in Porous Media, second ed., Springer-Verlag, New York, 1999. [19] P. Paoletti, Discrete Continuous Dyn. Sys., Ser. B (DCDS-B) 17 (2012) 637. [20] A.D. Pierce, Acoustics: An Introduction to Its Physical Principles and Applications, Acoustical Society of America, Woodbury, NY, 1989. [21] A.E. Scheidegger, The Physics of Flow Through Porous Media, third ed., University of Toronto Press, Toronto, 1974. [22] B. Straughan, Ann. Mat. Pura Appl. 126 (1980) 187. [23] B. Straughan, Stability and Wave Motion in Porous Media, Springer-Verlag, New York, 2008. [24] D. Ruth, H. Ma, Transp. Porous Media 7 (1992) 255. [25] P.A. Thompson, Compressible-Fluid Dynamics, McGraw–Hill, New York, 1972. [26] G.B. Whitham, Linear and Nonlinear Waves, Wiley, New York, 1974.