Poroacoustic front propagation under the linearized Eringen–Cattaneo–Christov–Straughan model

Vittorio Zampoli; Pedro M. Jordan

doi:10.1515/jnet-2023-0121

Article Open Access

Poroacoustic front propagation under the linearized Eringen–Cattaneo–Christov–Straughan model

Vittorio Zampoli and Pedro M. Jordan

Published/Copyright: March 19, 2024

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From the journal Journal of Non-Equilibrium Thermodynamics Volume 49 Issue 2

Abstract

Employing the Laplace transform and its properties, we investigate the evolution of the coupled thermal and fluid-acoustic waveforms that arise in a signaling problem under the Eringen–Cattaneo–Christov–Straughan model of poroacoustic phenomena. Assuming a Heaviside temperature input, we determine the effects of what we term the “Eringen coefficient” by examining the propagation and evolution of the resulting shock and/or acceleration waveforms. Special/limiting cases are also discussed and, in the final section of this Note, possible follow-on studies are suggested.

Keywords: Eringen–Cattaneo–Christov–Straughan model; Laplace transform; poroacoustics; singular surfaces

1 Introduction

Theoretical poroacoustics, like many sub-fields of acoustics, is one that can trace its origins back to the work of Rayleigh, who first published on the topic in 1883; see, e.g., Lamb [1, §360b] and the references cited therein. Because it is a phenomenon that impacts so many applications, ranging from mundane to exotic, researchers have, since at least the 1930s, derived a variety of poroacoustic models – notable among them being those of Muskat [2], Biot [3], Beavers & Sparrow [4], and Eringen [5]; see also Refs. [6], [7].

In 2020, Straughan [8, §7] extended the Eringen model to include the phenomenon of second sound (i.e., thermal waves). Assuming the solid phase to be a rigid, stationary porous matrix, Straughan replaced the Fourier-based flux relation, which describes the heat flux within the fluid phase in Ref. [5], with the Cattaneo–Christov relation (see, e.g., Refs. [9, §3.1.2] and [10] and those cited therein):

(1) τ q t + ( v ⋅ ∇ ) q − ( q ⋅ ∇ ) v + ( ∇ ⋅ v ) q + q = − k ∇ ϑ .

Here, v denotes the usual velocity vector; it is related to V , the Darcy velocity, via V = χ v [6, §1.3], where χ is defined below. Moreover, ϑ is the absolute temperature of the fluid; q is the heat flux vector; k is the thermal conductivity of the fluid; and τ(>0) is the Cattaneo relaxation time of the fluid. Straughan [8, §7] went on to test the predictive efficacy of the ensuing model, which we term the Eringen–Cattaneo–Christov–Straughan (ECCS) poroacoustic model, by performing an acceleration wave analysis – one that yielded a number of interesting, possibly testable, results.

In this Note we, like Straughan, carry out a singular surface-based analysis. Specifically, we employ the Laplace transform and its properties to investigate the evolution and propagation of the thermo-poroacoustic wave-fronts that arise under the linearized version of the ECCS model; here, however, the setting is that of a signaling problem involving a Heaviside (i.e., jump) temperature input, and the permeating fluid is taken to be a perfect gas [11].

Our aims are three-fold: (I) derive/analyze the expressions for the amplitudes and speeds of the shock waves that can arise in the type of initial-boundary value problem (IBVP) formulated below; (II) take note of a number of interesting special/limiting cases, two of which yield pressure profiles exhibiting only acceleration waves; and (III), determine the effects of varying what we term the “Eringen coefficient”, which is represented by “γ” in both Refs. [5] and [8, §7], but which we denote by σ so as to avoid confusion with the symbol traditionally used to denote the ratio of specific heats.

To these ends, our exposition commences in the next section, wherein the governing system that is the focus of the present investigation is derived.

2 Mathematical formulation

2.1 System of fundamental equations

Under linearization, and limiting our attention to 1D propagation along the x-axis of a Cartesian coordinate system, the ECCS model can, assuming the absence of all body forces and that the material parameters of both phases are constant-valued, be expressed as

(2a) Ψ t = − u x ,

(2b)

(2c) c v ρ 0 [ ϑ t − ϑ 0 ( γ − 1 ) Ψ t ] = − q x + ρ 0 r ,

(2d) p = p 0 ( Ψ + ϑ / ϑ 0 ) ,

(2e) q + τ 0 q t = − k 0 ϑ x ,

where we recall our assumption that the permeating fluid is a perfect gas. In Sys. (2), p = p(x, t) denotes the thermodynamic pressure; ϑ = ϑ(x, t) is the absolute temperature of the gas; Ψ(x, t) = (ρ − ρ ₀)/ρ ₀ is known as the condensation, where ρ = ρ(x, t) is the mass density of the gas; the velocity and heat flux vectors have reduced to v = (u(x, t), 0, 0) and q = (q(x, t), 0, 0), respectively; and r, which carries (SI) units of W/kg, represents the external rate of supply of heat per unit mass. Moreover, γ = c _p/c _v, where c _p > c _v > 0 represent the specific heats at constant pressure and constant volume, respectively, of the gas; μ ₀ the shear viscosity of the gas; χ ∈ (0, 1) and are the porosity and permeability, respectively, of the porous matrix; and we list for later reference the following additional parameters:

(3)

which denote the kinematic viscosity of the gas, the thermal diffusivity of the gas, the isothermal sound speed in the gas, the adiabatic sound speed in the gas, and the (dimensional) Darcy coefficient, respectively.

Lastly, a zero (“0”) subscript attached to a dependent variable, or other quantity that appears in this section, identifies the uniform ambient state value of that variable or quantity, where u ₀ = q ₀ = 0 is assumed; i.e., the gas is assumed to be homogeneous and quiescent when in its ambient state.

2.2 Equations of motion and thermal transport

Hereafter focusing our attention on propagation in the half-space x > 0, and setting^[1] r = 0 so as to simplify the analysis to follow, we now introduce the dimensionless acoustic pressure and dimensionless excess temperature variables, viz.,

(4) P = ( p − p 0 ) / p 0 and Θ = ( ϑ − ϑ 0 ) / ϑ 0 ,

where we observe that, in terms of P and Θ, Eq. (2d) becomes

(5) P = Ψ + Θ .

We begin our investigation by eliminating q, Ψ, and u between the equations of Sys. (2); this yields, after simplifying, the following two-equation system in terms of P and Θ:

(6a) P t t + δ D P t − b 0 2 P x x = δ D Θ t + Θ t t + b 0 2 ( σ 0 ϑ 0 / p 0 ) Θ x x , ( x , t ) ∈ ( 0 , ∞ ) × ( 0 , ∞ ) ,

(6b) γ Θ t + γ τ 0 Θ t t − τ 0 a 0 2 Θ x x = ( γ − 1 ) ( P t + τ 0 P t t ) , ( x , t ) ∈ ( 0 , ∞ ) × ( 0 , ∞ ) ;

here, we have introduced a 0 = γ κ 0 / τ 0 , the speed of purely thermal waves.

On the other hand, eliminating q, u, and p between the equations of Sys. (2), yields the alternative, i.e., in terms of Ψ and Θ, two-equation system:

(7a) Ψ t t + δ D Ψ t − b 0 2 Ψ x x = b 0 2 ( 1 + σ 0 ϑ 0 / p 0 ) Θ x x , ( x , t ) ∈ ( 0 , ∞ ) × ( 0 , ∞ ) ,

(7b) Θ t + τ 0 Θ t t − τ 0 a 0 2 Θ x x = ( γ − 1 ) ( Ψ t + τ 0 Ψ t t ) , ( x , t ) ∈ ( 0 , ∞ ) × ( 0 , ∞ ) ;

here, we observe that Sys. (7) can also be derived directly from Sys. (6) with the aid of Eq. (5).

Lastly, eliminating q, Ψ, and p between the equations of Sys. (2) yields the second alternative, i.e., in terms of u and Θ, two-equation system:

(8a) u t t + δ D u t − b 0 2 u x x = − b 0 2 ( 1 + σ 0 ϑ 0 / p 0 ) Θ x t , ( x , t ) ∈ ( 0 , ∞ ) × ( 0 , ∞ ) ,

(8b) Θ t + τ 0 Θ t t − τ 0 a 0 2 Θ x x = − ( γ − 1 ) ( u x + τ 0 u x t ) , ( x , t ) ∈ ( 0 , ∞ ) × ( 0 , ∞ ) ;

the reader should compare Sys. (8) with the ν ≔ 0 special case of the system consisting of Eqs. (26) and (27) in Ref. [1, p. 649], wherein “η” is used to represent (our) Θ.

2.3 Issues from kinetic theory and the question of coupling

Based on the following results from kinetic theory:

(9) a 0 ≈ c 0 , γ = 5 / 3 , b 0 ( 9 γ − 5 ) / 4 + ζ 1 ( 5 − 3 γ ) / 2 , γ ∈ [ 9 / 7,5 / 3 ) ,

which are derivable using the expressions in Refs. [12, pp. 247–252] and [13, §3.8.2], we shall, in all that follows, assume:

(10) a 0 > b 0 > 0 .

Here, we observe that a ₀ ≥ c ₀ is, of course, consistent with a ₀ > b ₀ and that c ₀ > a ₀ > b ₀ > 0 is also possible. It should be noted, moreover, that γ = 5/3 corresponds to monatomic gases (e.g., Ar, He) [11, p. 80]; the range γ ∈ [9/7, 5/3) includes important gases such as CO₂ [11, p. 640] and air^[2] under NTP/STP conditions, but not polyatomic gases at high temperatures [14, p. 8]; and ζ ₁ is a gas-specific parameter [13, Table 3.1].

We will also assume that the kinetic theory-based relation (see, e.g., Refs. [13, §3.3.3] and [15])

(11) τ 0 = 3 τ 1 / 2 ,

where Maxwell termed τ ₁ = μ ₀/p ₀ the “modulus of the time of relaxation of rigidity”, holds true for all gases considered.

Finally, from a strictly mathematical standpoint the equations ^[3] of Systems (7) and (8) are seen to uncouple for both σ 0 = σ 0 * and γ ≔ 1; here, σ 0 * ( < 0 ) is defined as σ 0 * ≔ − p 0 / ϑ 0 = − c v ( γ − 1 ) ρ 0 . From the physical standpoint, however, the latter special case is clearly inadmissible, at least in the case of perfect gases. Hence, the question of uncoupling comes down to the following: Is σ 0 * an admissible value of σ ₀? Apart from the obvious σ ≡ 0 case, we have only Straughan’s [8, p. 10] statement that “We expect p _T + γ > 0 … ”, where we recall that σ is denoted by “γ” in Ref. [8, §7].

In what follows, we seek to shed light on this question by assuming σ 0 ≥ σ 0 * , which corresponds to p _T + γ ≥ 0 in Ref. [8, §7]; we have rejected the case σ 0 < σ 0 * since it reverses the sign of the temperature coupling terms in Systems (7) and (8) vis-à-vis their classical (i.e., σ ₀, τ ₀ ≔ 0) versions, and thus can be expected to yield physically unrealistic results.

2.4 Formulation of initial-boundary value problem

We complement the above Sys. (6) with the following boundary conditions (BC)s:

(12) Θ ( 0 , t ) = Θ • H ( t ) , Θ ( ∞ , t ) = P x ( 0 , t ) = P ( ∞ , t ) = 0 , t ∈ 0 , ∞ ,

along with the following initial conditions (IC)s:

(13) Θ ( x , 0 ) = Θ t ( x , 0 ) = P ( x , 0 ) = P t ( x , 0 ) = 0 , x ∈ 0 , ∞ ;

here, Θ^• = ϑ ^•/ϑ ₀, where ϑ ^• is a strictly positive constant, and H(t) denotes the Heaviside unit step function.

We close this section by introducing the dimensionless independent variables: x ^# = δ _D x/b ₀ and t ^# = δ _D t, along with the parameter m = b 0 − 1 γ κ 0 δ D = ( a 0 / b 0 ) τ 0 δ D , and rewriting Sys. (6) in the completely dimensionless form:

(14a) P tt + P t − P xx = Θ t + Θ tt + ( σ 0 ϑ 0 / p 0 ) Θ xx , x , t ∈ 0 , ∞ × 0 , ∞ ,

(14b) γ Θ t + τ 0 δ D γ Θ tt − m 2 Θ xx = ( γ − 1 ) ( P t + τ 0 δ D P tt ) , x , t ∈ 0 , ∞ × 0 , ∞ ;

together with the dimensionless BCs and ICs:

(15a) Θ ( 0 , t ) = Θ • H ( t ) , Θ ( ∞ , t ) = P x ( 0 , t ) = P ( ∞ , t ) = 0 , t ∈ 0 , ∞ ,

(15b) Θ ( x , 0 ) = Θ t ( x , 0 ) = P ( x , 0 ) = P t ( x , 0 ) = 0 , x ∈ 0 , ∞ .

Here and henceforth, the "#" superscripts shall be suppressed for typographical simplicity.

3 Application of the Laplace transform

3.1 Exact transformation domain solutions: the case σ 0 > σ 0 *

We apply the (temporal) Laplace transform [16] to the equations of Sys. (14) and the BCs (see Eq. (15a)). Doing so yields, after employing the ICs (see Eq. (15b)) and simplifying, the system of subsidiary equations:

(16a) λ 2 + λ P ̄ − P ̄ ′ ′ = λ 2 + λ Θ ̄ + ( σ 0 ϑ 0 / p 0 ) Θ ̄ ′ ′ , x ∈ 0 , ∞ ,

(16b) γ λ τ 0 δ D λ + 1 Θ ̄ − m 2 Θ ̄ ′ ′ = ( γ − 1 ) τ 0 δ D λ 2 + λ P ̄ , x ∈ 0 , ∞ ,

which in the present subsection is to be solved subject to

(17) Θ ̄ ( 0 , λ ) = Θ • / λ , Θ ̄ ( ∞ , λ ) = P ̄ ′ ( 0 , λ ) = P ̄ ( ∞ , λ ) = 0 .

Here, λ denotes the Laplace transform parameter [16], a superposed bar indicates the image of the (dependent) variable that appears under it in the Laplace domain, and a prime denotes d/dx.

Now employing the operator technique for solving such systems (see, e.g., Hetnarski [17] and Jordan & Puri [18] as examples of the utilization of said approach), the following exact expressions for P ̄ ( x , λ ) and Θ ̄ ( x , λ ) are readily obtained:

(18) P ̄ ( x , λ ) = λ 2 + λ + σ 0 ϑ 0 / p 0 ξ 1 2 λ 2 + λ + σ 0 ϑ 0 / p 0 ξ 2 2 λ 2 + λ 2 + σ 0 ϑ 0 / p 0 ξ 1 2 + ξ 2 2 + σ 0 ϑ 0 / p 0 + 1 ξ 1 ξ 2 λ 2 + λ − σ 0 ϑ 0 / p 0 ξ 1 2 ξ 2 2 × Θ • ξ 2 − ξ 1 λ ξ 2 ⁡ exp − ξ 1 x − ξ 1 ⁡ exp − ξ 2 x ,

(19) Θ ̄ ( x , λ ) = 1 λ 2 + λ 2 + σ 0 ϑ 0 / p 0 ξ 1 2 + ξ 2 2 + σ 0 ϑ 0 / p 0 + 1 ξ 1 ξ 2 λ 2 + λ − σ 0 ϑ 0 / p 0 ξ 1 2 ξ 2 2 × Θ • ξ 2 − ξ 1 λ λ 2 + λ − ξ 1 2 λ 2 + λ + σ 0 ϑ 0 / p 0 ξ 2 2 ξ 2 ⁡ exp − ξ 1 x − λ 2 + λ + σ 0 ϑ 0 / p 0 ξ 1 2 λ 2 + λ − ξ 2 2 ξ 1 ⁡ exp − ξ 2 x ,

where

(20) ξ 1,2 = 1 2 γ p 0 κ 0 δ D 2 Q 0 λ ± Q 0 2 λ − 4 γ b 0 2 p 0 2 κ 0 δ D 3 λ 3 + λ 2 τ 0 δ D λ + 1 ,

and where

(21) Q 0 λ = L 0 δ D λ 2 + M 0 δ D λ , L 0 = τ 0 b 0 2 σ 0 ϑ 0 γ − 1 + γ p 0 τ 0 b 0 2 + κ 0 , M 0 = b 0 2 σ 0 ϑ 0 γ − 1 + γ p 0 b 0 2 + δ D κ 0 .

3.2 Exact transformation domain solutions: the case σ 0 = σ 0 *

Under the assumption σ 0 = σ 0 * , Sys. (16) simplifies to

(22a) λ 2 + λ Ψ ̄ − Ψ ̄ ′ ′ = 0 , x ∈ 0 , ∞ ,

(22b) τ 0 δ D λ 2 + λ Θ ̄ − m 2 Θ ̄ ′ ′ = ( γ − 1 ) τ 0 δ D λ 2 + λ Ψ ̄ , x ∈ 0 , ∞ ,

where we have made use of Eq. (5).

On solving this system subject to the conditions given in Eq. (17), which are readily enforced with the aid of Eq. (5), it is not difficult to establish that

(23) P ̄ ( x , λ ) = Θ • λ 1 − ( γ − 1 ) f 1 ( λ ) K 1 ( λ ) exp − x λ 2 + λ + 1 + ( γ − 1 ) f 1 ( λ ) K 1 ( λ ) exp − ( x / m ) τ 0 δ D λ 2 + λ ,

(24) Θ ̄ ( x , λ ) = Θ • λ 1 + ( γ − 1 ) f 1 ( λ ) K 1 ( λ ) exp − ( x / m ) τ 0 δ D λ 2 + λ − ( γ − 1 ) f 1 ( λ ) K 1 ( λ ) exp − x λ 2 + λ ,

(25) Ψ ̄ ( x , λ ) = Θ • / λ K 1 ( λ ) exp − x λ 2 + λ ;

here, taking note of the fact that δ _D = 1/τ ₀ is a critical value, we have

(26) f 1 ( λ ) = τ 0 δ D λ + 1 m 2 ( λ + 1 ) − τ 0 δ D λ − 1 , δ D ≠ 1 / τ 0 , b 0 2 a 0 2 − b 0 2 , δ D = 1 / τ 0 ,

(27) K 1 ( λ ) = − τ 0 δ D λ + 1 m λ + 1 − m ( γ − 1 ) f 1 ( λ ) λ + 1 + ( γ − 1 ) f 1 ( λ ) τ 0 δ D λ + 1 , δ D ≠ 1 / τ 0 , b 0 a 0 + b 0 a 0 2 + b 0 a 0 + b 0 − c 0 2 , δ D = 1 / τ 0 .

3.3 Large-λ expansions

So that we might determine the amplitudes of the singular surfaces exhibited by our time-domain solution profiles (see Sections 4.2 and 4.3 below), in this section, we present expansions of the expressions for P ̄ and Θ ̄ about λ = ∞.

3.3.1 The case σ 0 > σ 0 *

For this case we begin by noting that

(28) ξ 1,2 ≈ λ / υ 1,2 + β 1,2 + γ 1,2 1 λ ,

where terms of O λ − 2 have been neglected. Here,

(29) 1 υ 1,2 = L 0 ± N 0 2 γ p 0 κ 0 , β 1,2 = 2 M 0 N 0 ± S 0 4 δ D N 0 2 γ p 0 κ 0 L 0 ± N 0 , γ 1,2 = β 1,2 δ D 2 M 0 N 0 ± S 0 ± 4 N 0 T 0 − S 0 2 4 N 0 − 2 M 0 N 0 ± S 0 2 8 N 0 L 0 ± N 0 , N 0 = L 0 2 − 4 τ 0 γ b 0 2 p 0 2 κ 0 , S 0 = 2 L 0 M 0 − 4 γ b 0 2 p 0 2 κ 0 1 + τ 0 δ D , T 0 = M 0 2 − 4 γ b 0 2 p 0 2 κ 0 δ D .

Here, we observe that σ 0 > σ 0 † implies that N ₀ > 0, where N ₀ is a concave-up quadratic in σ ₀ whose largest zero is

(30) σ 0 † = − | σ 0 * | ( a 0 / b 0 ) 2 − 2 ( a 0 / b 0 ) + γ γ − 1 .

Here, we observe that σ 0 † < 0 not only for γ ∈ [9/7, 5/3], but for all γ > 1; more specifically, however, our assumption a ₀ > b ₀ means that σ 0 † < σ 0 * , again for all γ > 1. Thus, our self-imposed restriction σ 0 > σ 0 * also implies that N ₀ > 0.

In order to get sufficiently compact notation, it is convenient to define the additional (dimensionless) constants:^[4]

(31) V 0 = b 0 2 τ 0 δ D a 0 1 + τ 0 δ D , W 0 = 1 + σ 0 ϑ 0 γ p 0 2 κ 0 L 0 + τ 0 σ 0 ϑ 0 b 0 2 , X 0 = 1 + b 0 a 0 σ 0 ϑ 0 p 0 + 1 + σ 0 ϑ 0 γ p 0 2 κ 0 L 0 − p 0 τ 0 b 0 2 , Y 0 = 1 + V 0 σ 0 ϑ 0 p 0 + 1 + σ 0 ϑ 0 γ p 0 2 κ 0 δ D M 0 − p 0 b 0 2 , Z 0 = 2 + σ 0 ϑ 0 γ p 0 2 κ 0 δ D δ D L 0 + M 0 + σ 0 ϑ 0 b 0 2 1 + τ 0 δ D = 1 + W 0 + σ 0 ϑ 0 γ p 0 2 κ 0 δ D M 0 + σ 0 ϑ 0 b 0 2 ,

as well as

(32) ε 1,2 = 1 2 γ p 0 κ 0 δ D M 0 ± S 0 2 N 0 .

In this way we are able to obtain, after long but straightforward calculations, the following large-λ expansions of the expressions given in Eqs. (18) and (19), respectively:

(33) P ̄ ( x , λ ) ≈ Θ • γ p 0 κ 0 X 0 N 0 ∑ j = 1 2 − 1 3 − j ⁡ exp − λ x / υ 3 − j − β 3 − j x W 0 υ j − 2 + b 0 a 0 1 λ + W 0 ε j + V 0 + υ j − 2 + b 0 a 0 Z 0 − W 0 1 + γ 3 − j x + Y 0 X 0 + S 0 2 δ D N 0 1 λ 2 ,

(34) Θ ̄ ( x , λ ) ≈ Θ • γ p 0 κ 0 X 0 N 0 ∑ j = 1 2 − 1 3 − j ⁡ exp − λ x / υ 3 − j − β 3 − j x × 1 − υ 3 − j − 2 + σ 0 ϑ 0 p 0 υ j − 2 − b 0 2 a 0 2 υ j − 2 + b 0 a 0 1 λ + 1 − υ 3 − j − 2 + σ 0 ϑ 0 p 0 υ j − 2 − b 0 2 a 0 2 ε j + V 0 − γ 3 − j x + Y 0 X 0 + S 0 2 δ D N 0 υ j − 2 + b 0 a 0 + 1 − ε 3 − j + σ 0 ϑ 0 p 0 ε j − b 0 2 τ 0 δ D a 0 2 υ j − 2 + b 0 a 0 1 λ 2 .

Noteworthy here is the following: Because the assumption a ₀ > b ₀ means that

(35) − | σ 0 * | ( a 0 / b 0 ) 2 + ( a 0 / b 0 ) γ − 1 < − | σ 0 * | ( γ ≤ 5 / 3 ) ,

the restriction σ 0 > σ 0 * assumed in this section ensures not only that N ₀ > 0, but also that X ₀ > 0; here, we observe that X ₀ is a concave-up quadratic, in σ ₀, whose zeros appear on the left- and right-hand sides of the inequality in Eq. (35).

3.3.2 The case σ 0 = σ 0 *

The large-λ expansions of the transform domain solutions given in Section 3.2 can easily be constructed from the following:

(36) exp − x λ 2 + λ = exp ( − λ x ) exp − x / 2 1 + x 8 1 λ + O ( λ − 2 ) ,

(37) exp − ( b 0 / a 0 ) x λ 2 + λ / ( τ 0 δ D ) = exp [ − ( b 0 / a 0 ) λ x ] exp − ( b 0 / a 0 ) x / ( 2 τ 0 δ D ) × 1 + ( b 0 / a 0 ) x 8 ( τ 0 δ D ) 2 1 λ + O ( λ − 2 ) ,

(38) f 1 λ = b 0 2 a 0 2 − b 0 2 + W 1 1 λ + O ( λ − 2 ) ,

(39) K 1 λ = − b 0 a 0 + b 0 a 0 2 + b 0 a 0 + b 0 − c 0 2 1 + 1 2 τ 0 δ D − Y 1 1 λ + O ( λ − 2 ) ;

here, W 1 = m 2 1 − τ 0 δ D m 2 − τ 0 δ D − 2 and

(40) Y 1 = 1 2 m + γ − 1 2 τ 0 δ D − m W 1 + τ 0 δ D − m τ 0 δ D m 2 − τ 0 δ D × m − γ − 1 τ 0 δ D τ 0 δ D + m − 1 .

4 Analytical results

4.1 Wave speed results

On expanding the expressions given in Eqs. (29) ₁ for small-τ ₀ one finds, after simplifying, that

(41) υ 1 = v 1 b 0 ≃ 1 − b 0 2 ( γ − 1 ) ( 1 − σ 0 / σ 0 * ) τ 0 2 γ κ 0 ( τ 0 → 0 ) ,

(42) υ 2 = v 2 b 0 ≃ a 0 b 0 1 + b 0 2 ( γ − 1 ) ( 1 − σ 0 / σ 0 * ) τ 0 2 γ κ 0 ( τ 0 → 0 ) ,

where v 1,2 denote the dimensional forms of υ _1,2, respectively, and terms of O τ 0 2 have been neglected.

For σ 0 > σ 0 * , the former and latter expansions make clear the fact that v 1,2 are, respectively, modifications of the mechanical (i.e., strictly acoustic) isothermal wave speed, b ₀, and the strictly thermal (i.e., second-sound) wave speed, a ₀. As will be shown in the next subsection, v 2 > v 1 are, respectively, the speeds at which the predominantly acoustic and the predominantly thermal wave-fronts propagate when σ 0 ≥ σ 0 * . What is more, υ _1,2 are, respectively, decreasing/increasing functions of σ 0 > σ 0 * ; taking σ 0 = σ 0 * , however, yields the exact, uncoupled, expressions υ _1,2 = {1, a ₀/b ₀}, which can be determined by first applying Ref. [16, Thm. V] to Eqs. (36) and (37), respectively.

4.2 Shock results

4.2.1 The case σ 0 > σ 0 *

On applying the theorem presented in Ref. [19, §4] to Eq. (33), it is readily established that

(43a) [ [ P ] ] 1 = − Θ • W 0 γ p 0 κ 0 X 0 N 0 υ 2 − 2 + b 0 / a 0 exp − α 1 t ( a 0 ≥ c 0 ) ,

(43b) [ [ P ] ] 2 = Θ • W 0 γ p 0 κ 0 X 0 N 0 υ 1 − 2 + b 0 / a 0 exp − α 2 t ( a 0 ≥ c 0 ) ,

while the application of this theorem to Eq. (34) yields

(44a) [ [ Θ ] ] 1 = − Θ • γ p 0 κ 0 X 0 N 0 1 − υ 1 − 2 + σ 0 ϑ 0 p 0 υ 2 − 2 − b 0 2 / a 0 2 υ 2 − 2 + b 0 / a 0 exp − α 1 t ( a 0 > b 0 ) ,

(44b) [ [ Θ ] ] 2 = Θ • γ p 0 κ 0 X 0 N 0 1 − υ 2 − 2 + σ 0 ϑ 0 p 0 υ 1 − 2 − b 0 2 / a 0 2 υ 1 − 2 + b 0 / a 0 exp − α 2 t ( a 0 ≥ c 0 ) .

Here, we have set

(45) α 1,2 = υ 1,2 β 1,2 = 2 M 0 N 0 ± S 0 4 δ D N 0 ( L 0 ± N 0 ) ,

respectively, and we have imposed the restriction a ₀ ≥ c ₀ to ensure that both W ₀ ≠ 0 and [[Θ]]₂ ≠ 0 for all σ 0 > σ 0 * .

The jumps with a “1” subscript occur across x = Σ₁(t), where Σ₁(t) = υ ₁ t, which in the present sub-subsection is an (predominantly) acoustic shock wave; those with a “2” subscript occur across x = Σ₂(t), where Σ₂(t) = υ ₂ t, which in the present sub-subsection is a (predominantly) thermal shock wave.

4.2.2 The case σ 0 = σ 0 *

The application of the theorem presented in Ref. [19, §4] to the large-λ expressions corresponding to Eqs. (23) and (24) (see Section 3.3.2), yields the following shock amplitudes:

(46a) [ [ P ] ] iso = − Θ • b 0 a 0 2 − c 0 2 a 0 − b 0 a 0 2 + b 0 a 0 + b 0 − c 0 2 exp − t / 2 ( 0 < b 0 < a 0 ≠ c 0 ) ,

(46b) [ [ P ] ] th = Θ • a 0 a 0 2 − c 0 2 a 0 − b 0 a 0 2 + b 0 a 0 + b 0 − c 0 2 exp − t / 2 τ 0 δ D ( 0 < b 0 < a 0 ) ,

(47a) [ [ Θ ] ] iso = Θ • b 0 3 γ − 1 a 0 − b 0 a 0 2 + b 0 a 0 + b 0 − c 0 2 exp − t / 2 ( 0 < b 0 < a 0 ) ,

(47b) [ [ Θ ] ] th = [ [ P ] ] th ( 0 < b 0 < a 0 ≠ c 0 ) ,

(48) [ [ Ψ ] ] iso = − Θ • b 0 a 0 + b 0 a 0 2 + b 0 a 0 + b 0 − c 0 2 exp − t / 2 ( 0 < b 0 < a 0 ) .

Here, the jumps with subscript “iso” occur across x = Σ_iso(t), where Σ_iso(t) = t, which in the present sub-subsection is a purely acoustic shock wave; those with subscript “th” occur across x = Σ_th(t), where Σ_th(t) = (a ₀/b ₀)t, which in the present sub-subsection is a purely thermal shock wave.

Additionally, we observe that the Ψ versus x profile admits only a single wave-front – the shock wave x = Σ_iso(t) – when σ 0 = σ 0 * holds, and we note the simplification [ [ Ψ ] ] iso = − Θ • ⁡ exp − t / 2 that follows on taking a ₀ = c ₀.

4.3 Acceleration wave case

4.3.1 The case σ 0 > σ 0 *

If c ₀ > a ₀ > b ₀, then there exists a value of σ ₀, call it σ 0 ‡ , such that setting σ 0 = σ 0 ‡ yields W ₀ = 0; here, we define σ 0 ‡ ≔ − ϖ 2 | σ 0 * | , where we have introduced the (dimensionless) parameter ϖ ≔ a ₀/c ₀, and we observe that − | σ 0 * | < σ 0 ‡ < − γ − 1 | σ 0 * | < 0 follows from the fact that ϖ ∈ (γ ^−1/2, 1).

On applying the theorem presented in Ref. [19, §4] to the σ 0 = σ 0 ‡ special case of Eq. (33), it is readily established that, while [[P]]₁ = [[P]]₂ = 0, we now have

(49a) [ [ P t ] ] 1 = − Θ • Z 0 γ p 0 κ 0 X 0 N 0 υ 2 − 2 + b 0 / a 0 exp − α 1 t σ 0 = σ 0 ‡ , δ D τ 0 ≠ 1 ,

(49b) [ [ P t ] ] 2 = Θ • Z 0 γ p 0 κ 0 X 0 N 0 υ 1 − 2 + b 0 / a 0 exp − α 2 t σ 0 = σ 0 ‡ , δ D τ 0 ≠ 1 ,

and x = Σ_1,2(t) have become pressure acceleration wave-fronts. Here, we observe that [ [ P x ] ] 1,2 are readily determined from the above expressions using the well known Maxwell compatibility condition, which gives

(50) [ [ P x ] ] 1,2 = − υ 1,2 − 1 [ [ P t ] ] 1,2 ,

and the restriction δ _D τ ₀ ≠ 1 has been imposed to ensure that the σ 0 = σ 0 ‡ special case of Z ₀ is nonzero.

In the case of the temperature field, setting σ 0 = σ 0 ‡ ⇒ 1 − υ 2 − 2 + σ 0 | σ 0 * | − 1 υ 1 − 2 − ( b 0 / a 0 ) 2 = 0 in Eq. (34). Consequently, not only is [[Θ]]₂ = 0, but it also follows that x = Σ₂(t) has become a temperature acceleration wave-front, across which the Θ_x versus x profile exhibits a jump whose amplitude is given by

(51) [ [ Θ x ] ] 2 = − Θ • γ p 0 κ 0 υ 2 X 0 N 0 1 − ε 2 − a 0 2 c 0 2 ε 1 − ( b 0 / a 0 ) 2 τ 0 δ D υ 1 − 2 + b 0 / a 0 exp − α 2 t σ 0 = σ 0 ‡ ,

assuming the expression within the large “{ }” is nonzero.^[5] Here, we note that [ [ Θ t ] ] 2 was first determined (from the σ 0 = σ 0 ‡ special case of Eq. (34)) using the theorem presented in Ref. [19, §4]; the expression for [ [ Θ x ] ] 2 given in Eq. (51) was then obtained from the former using (once again) the Maxwell compatibility condition, i.e., via [ [ Θ x ] ] 2 = − υ 2 − 1 [ [ Θ t ] ] 2 .

4.3.2 The case σ 0 = σ 0 *

The application of the theorem presented in Ref. [19, §4] to the a ₀ = c ₀ special cases of the large-λ expressions corresponding to Eqs. (23) and (24) (see Section 3.3.2), yields, after simplifying, the following acceleration wave amplitudes:

(52a) [ [ P t ] ] iso = − [ [ P x ] ] iso = Θ • γ 1 − δ D τ 0 δ D τ 0 γ − 1 exp − t / 2 ( a 0 = c 0 , δ D τ 0 ≠ 1 ) ,

(52b) [ [ P t ] ] th = − γ [ [ P x ] ] th = − Θ • 1 − δ D τ 0 γ 3 / 2 δ D τ 0 γ − 1 exp − t / 2 τ 0 δ D ( a 0 = c 0 , δ D τ 0 ≠ 1 ) ;

(53) [ [ Θ t ] ] th = − γ [ [ Θ x ] ] th = − Θ • 1 − δ D τ 0 γ 3 / 2 δ D τ 0 γ − 1 exp − t / 2 τ 0 δ D ( a 0 = c 0 , δ D τ 0 ≠ 1 ) .

The surfaces Σ_iso(t) = t and Σ th ( t ) = t γ are now acceleration wave-fronts, where we note that a 0 = c 0 ⇒ a 0 / b 0 = γ . Here, δ _D τ ₀ ≠ 1 has been imposed to ensure that the present acceleration wave amplitudes are not identically zero.

5 Numerical results

We now present time-domain solution profiles plotted from data sets generated by numerically inverting our transform domain solutions using the modified Riemann sum formula given in Ref. [20, Eq. (42)], wherein “M” denotes the upper bound of summation. In each graph presented, p ₀ = 1 atm and ϑ ₀ = 300 K ( ⇒ σ 0 * = − 337.75  Pa/K) were taken, the τ ₀ values were computed using Eq. (11), Θ^• = 1 was assumed, “time” was fixed at t = 0.1, we took M = 500 in Ref. [20, Eq. (42)], and 1001 points were plotted. Here, we observe that every profile presented exhibits two wave-fronts, where the slower/faster ones are the acoustic/thermal based fronts, respectively, and that in each case the imposed BC at x = 0 is satisfied.

In Figures 1 and 2, each wave-front depicted is a shock wave. Qualitatively, the behaviors seen in Figures 1 and 2 are also observed when a ₀ > c ₀ is taken under the respective σ 0 ≥ σ 0 * cases.

$Figure 1: P(x, 0.1) versus x (Blue) and Θ(x, 0.1) versus x (Red) profiles for σ 0 = σ 0 * / 2 > σ 0 * ${\sigma }_{0}={\sigma }_{0}^{{\ast}}/2{ >}{\sigma }_{0}^{{\ast}}$ , with a 0 = c 0, obtained from Eqs. (18) and (19), respectively, using Ref. [20, Eq. (42)]. Assuming the gas phase is Ar, the following parameter values were used [21]: ρ 0 = 1.6238 kg/m3, c 0 ≈ 322.67 m/s, μ 0 ≈ 2.278 × 10−5Pa s, along with γ = 5/3, τ 0 ≈ 3.367 × 10−10s, κ 0 ≈ 2.106 × 10−5 m2/s, and δ D = 4/τ 0. Here, the broken purple and orange lines correspond to Σ1(0.1) ≈ 0.0858 and Σ2(0.1) ≈ 0.1505, respectively. The listed value of μ 0 lies within the error bounds for μ 0 stated in Ref. [21] and was taken to achieve a 0 = c 0; otherwise, a 0/c 0 ≈ 1.00075 would have resulted.$

Figure 1:

P(x, 0.1) versus x (Blue) and Θ(x, 0.1) versus x (Red) profiles for σ 0 = σ 0 * / 2 > σ 0 * , with a ₀ = c ₀, obtained from Eqs. (18) and (19), respectively, using Ref. [20, Eq. (42)]. Assuming the gas phase is Ar, the following parameter values were used [21]: ρ ₀ = 1.6238 kg/m³, c ₀ ≈ 322.67 m/s, μ ₀ ≈ 2.278 × 10⁻⁵Pa s, along with γ = 5/3, τ ₀ ≈ 3.367 × 10⁻¹⁰s, κ ₀ ≈ 2.106 × 10⁻⁵ m²/s, and δ _D = 4/τ ₀. Here, the broken purple and orange lines correspond to Σ₁(0.1) ≈ 0.0858 and Σ₂(0.1) ≈ 0.1505, respectively. The listed value of μ ₀ lies within the error bounds for μ ₀ stated in Ref. [21] and was taken to achieve a ₀ = c ₀; otherwise, a ₀/c ₀ ≈ 1.00075 would have resulted.

$Figure 2: P(x, 0.1) versus x (Blue) and Θ(x, 0.1) versus x (Red) profiles for σ 0 = σ 0 * ${\sigma }_{0}={\sigma }_{0}^{{\ast}}$ , with c 0 > a 0, obtained from Eqs. (23) and (24), respectively, using Ref. [20, Eq. (42)]. Assuming the gas phase is air, the following parameter values were used (see the url in Footnote 2): γ = 1.400, ρ 0 = 1.177 kg/m3, μ 0 = 1.846 × 10−5Pa s, κ 0 ≈ 2.218 × 10−5 m2/s, along with τ 0 ≈ 2.733 × 10−10s, c 0 ≈ 347.26 m/s, a 0/c 0 ≈ 0.9707, and δ D = 4/τ 0. Here, the broken purple and orange lines correspond to Σiso(0.1) = 0.1 and Σth(0.1) ≈ 0.1149, respectively.$

Figure 2:

P(x, 0.1) versus x (Blue) and Θ(x, 0.1) versus x (Red) profiles for σ 0 = σ 0 * , with c ₀ > a ₀, obtained from Eqs. (23) and (24), respectively, using Ref. [20, Eq. (42)]. Assuming the gas phase is air, the following parameter values were used (see the url in Footnote 2): γ = 1.400, ρ ₀ = 1.177 kg/m³, μ ₀ = 1.846 × 10⁻⁵Pa s, κ ₀ ≈ 2.218 × 10⁻⁵ m²/s, along with τ ₀ ≈ 2.733 × 10⁻¹⁰s, c ₀ ≈ 347.26 m/s, a ₀/c ₀ ≈ 0.9707, and δ _D = 4/τ ₀. Here, the broken purple and orange lines correspond to Σ_iso(0.1) = 0.1 and Σ_th(0.1) ≈ 0.1149, respectively.

The left panels of Figures 3 and 4 illustrate the fact that, under the indicated conditions, both fronts exhibited by the respective pressure profiles are acceleration wave-fronts; the right panels of these figures, in contrast, show that, with regard to the respective temperature profiles, the acoustic based wave-fronts are always shock waves, while the thermal based ones have become acceleration wave-fronts. Figures 3 and 4 also illustrate the effect of taking δ _D τ ₀ greater/less than one, respectively, when acceleration waves occur.

$Figure 3: P(x, 0.1) versus x (Blue) and Θ(x, 0.1) versus x (Red) profiles for σ 0 = σ 0 ‡ ≈ − 318.324 ${\sigma }_{0}={\sigma }_{0}^{{\ddagger}}\approx -318.324$  Pa/K, with c 0 > a 0, obtained from Eqs. (18) and (19), respectively, using Ref. [20, Eq. (42)]. Again assuming the gas phase is air, we employ the same parameter values listed in the caption of Figure 2. Here, the broken purple and orange lines correspond to Σ1(0.1) ≈ 0.0971 and Σ2(0.1) ≈ 0.1183, respectively, and we note that ϖ ≈ 0.9707.$

Figure 3:

P(x, 0.1) versus x (Blue) and Θ(x, 0.1) versus x (Red) profiles for σ 0 = σ 0 ‡ ≈ − 318.324  Pa/K, with c ₀ > a ₀, obtained from Eqs. (18) and (19), respectively, using Ref. [20, Eq. (42)]. Again assuming the gas phase is air, we employ the same parameter values listed in the caption of Figure 2. Here, the broken purple and orange lines correspond to Σ₁(0.1) ≈ 0.0971 and Σ₂(0.1) ≈ 0.1183, respectively, and we note that ϖ ≈ 0.9707.

$Figure 4: P(x, 0.1) versus x (Blue) and Θ(x, 0.1) versus x (Red) profiles for σ 0 = σ 0 * ${\sigma }_{0}={\sigma }_{0}^{{\ast}}$ , with a 0 = c 0, obtained from Eqs. (23) and (24), respectively, using Ref. [20, Eq. (42)]. Again assuming the gas phase is Ar, we employ the same parameter values listed in the caption of Figure 1, except now δ D = 0.25/τ 0 has been taken. Here, the broken purple and orange lines correspond to Σiso(0.1) = 0.1 and Σth(0.1) ≈ 0.1291, respectively.$

Figure 4:

P(x, 0.1) versus x (Blue) and Θ(x, 0.1) versus x (Red) profiles for σ 0 = σ 0 * , with a ₀ = c ₀, obtained from Eqs. (23) and (24), respectively, using Ref. [20, Eq. (42)]. Again assuming the gas phase is Ar, we employ the same parameter values listed in the caption of Figure 1, except now δ _D = 0.25/τ ₀ has been taken. Here, the broken purple and orange lines correspond to Σ_iso(0.1) = 0.1 and Σ_th(0.1) ≈ 0.1291, respectively.

6 Final remarks

In this, our preliminary study of the linearized ECCS model, we have shown, among other things, that taking σ 0 ≥ σ 0 * yields physically plausible results for the class of gases considered; that the pressure profile can exhibit only acceleration wave-fronts; that σ 0 = σ 0 * , σ 0 = σ 0 ‡ , a ₀ = c ₀, and δ _D = 1/τ ₀ are all critical values; and that jumps in the second derivatives of P and Θ may be possible in the context of the IBVP considered. Apart from gaining a fuller understanding of the implications of the special case δ _D = 1/τ ₀, and determining if the second derivatives of P, Θ can, in fact, be made to exhibit jumps, one quite intriguing follow-on investigation involves re-working the present IBVP under the assumption 0 < a ₀ ≤ b ₀; it must be noted, however, that the question of if this inequality can be satisfied in the case of real gases appears to be an open one; recall Eq. (9), as well as Refs. [12, pp. 247–252] and [13, §3.8.2].

Corresponding author: Vittorio Zampoli, Dept. of Information and Electrical Engineering and Applied Mathematics, University of Salerno, 84084, Fisciano, Italy, E-mail: vzampoli@unisa.it

Distribution Statement A: Approved for public release. Distribution is unlimited.

Funding source: Gruppo Nazionale per la Fisica Matematica

Funding source: Ministero dell'Università e della Ricerca

Award Identifier / Grant number: 2022P5R22A - NextGenerationEU PRIN2022

Award Identifier / Grant number: P2022PE8BT - NextGenerationEU NRRP PRIN2022

Funding source: Office of Naval Research

Acknowledgments

All numerical results presented were computed/plotted using the software package Mathematica (ver. 11.2).

Research ethics: Not applicable.
Author contributions: The authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Competing interests: The authors state no competing interests.
Research funding: V.Z. was supported by funding provided by the Italian National Group of Mathematical Physics (GNFM – INdAM), by the NextGenerationEU PRIN2022 research project The Mathematics and Mechanics of Nonlinear Wave Propagation in Solids (grant no. 2022P5R22A), and by the National Recovery and Resilience Plan (NRRP) NextGenerationEU research project SUSTBUILD – SUSTainable composite structures for energy-harvesting and carbon-storing BUILDings (grant no. P2022PE8BT). P.M.J. was supported by funding provided by the U.S. Office of Naval Research (ONR).
Data availability: Not applicable.

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Received: 2023-12-12

Accepted: 2024-02-29

Published Online: 2024-03-19

Published in Print: 2024-04-25

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/jnet-2023-0121

Keywords for this article

Eringen–Cattaneo–Christov–Straughan model; Laplace transform; poroacoustics; singular surfaces

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