Shock waves of viscoelastic Burgers equations

International Journal of Engineering Science 149 (2020) 103226

Contents lists available at ScienceDirect

International Journal of Engineering Science journal homepage: www.elsevier.com/locate/ijengsci

Shock waves of viscoelastic Burgers equations J.I. Ramos Escuela de Ingenierías Industriales, Universidad de Málaga, Doctor Ortiz Ramos, s/n, Málaga 29071, Spain

a r t i c l e

i n f o

Article history: Received 2 November 2018 Revised 26 November 2019 Accepted 18 January 2020 Available online xxx Keywords: Viscoelastic Burgers equation Gordon–Schowalter time derivative Jaumann’s time derivative Upper– and lower–convected Maxwell Time derivatives

a b s t r a c t Travelling wave solutions of the front, kink or shock wave type are obtained analytically for one–dimensional viscoelastic Burgers equations with an objective Gordon–Schowalter time derivative in the constitutive law by means of a novel formulation based on changes of the independent and dependent variables that reduces the number of parameters to three. It is shown that, for the upper– and lower–convected Maxwell and Jaumann time derivatives, the thickness of the shock wave decreases as a nondimensional relaxation parameter is increased and that the velocity profiles for the Jaumann and lower–convected derivatives evolve from a sigmoid shape to linear piecewise ones as the relaxation parameter is increased. It is also shown that stress is a negative, even function that reaches a nil value at the upstream and downstream boundaries, and its width decreases as the relaxation parameter is increased for the Jaumann and upper–convected Maxwell derivatives. It is also reported that, at each location, the stress is larger for the lower–convected Maxwell derivative than for the Jaumann and upper–convected Maxwell derivatives. © 2020 Elsevier Ltd. All rights reserved.

1. Introduction The (Newtonian) Burgers equation with unity density is

ut + uux = σx , where σ = μux is the stress, u is the velocity, t and x denote the time and spatial coordinate, respectively, the subscripts denote partial differentiation, and μ is the dynamic viscosity, has received a great deal of attention in fluid dynamics (Burgers, 1948), nonlinear acoustics (Crighton, 1979; Rudenko & Soluyan, 1977), stochastic processes, traffic flow modelling (Lighthill & Whitham, 1955; Richards, 1956), number theory, etc. In Newtonian fluid dynamics, Burgers’ equation is the simplest, one–dimensional, parabolic equation that it contains a quadratic nonlinear convective flux and a linear diffusive one (Bateman, 1915), was originally employed as a model of turbulence because it contains the nonlinear terms associated with steepening and energy transfer and a linear dissipative one (Burgers, 1948), and has appeared in many models dealing with micro–fluid dynamics, elastic turbulence, dusty plasmas, sedimentation, interface dynamics (Kardar, Parisi, & Zhang, 1986), gas dynamics, dispersive water waves (Whitham, 1974), etc. The (Newtonian) Burgers equation has also appeared in the modelling of heat conduction, propagation of elastic waves in isotropic solids, Brownian motion, random walks, etc. In the (Newtonian) Burgers equation, the stress is linearly proportional to the strain rate ux with a constant viscosity, and this equation has a travelling wave solution of the kink, shock wave or front type for u(t, −∞ ) = uL and u(t, ∞ ) = uR , uL > uR , where uL and uR are constant, that is referred to as Taylor’s shock (Hamilton & Blackstock, 2008; Taylor, 1910). E-mail address: [email protected] https://doi.org/10.1016/j.ijengsci.2020.103226 0020-7225/© 2020 Elsevier Ltd. All rights reserved.

2

J.I. Ramos / International Journal of Engineering Science 149 (2020) 103226

Very few studies on the one–dimensional Burgers equation with a non–Newtonian constitutive equation have been reported in the literature. Ciancio and Restuccia (1987) considered the propagation of waves in compressible, visco–anelastic media with a constitutive law that, for one–dimensional problems, may be written as



σ



Dσ Dux D2 u x +λ = μ ux + λ1 + λ2 , Dt Dt Dt 2

using the nomenclature used in this paper, where DDtσ = σt + uσx is the material or substantial derivative, λ is the (stress) relaxation time, λ1 and λ2 are a retardation and the square of a retardation time, respectively, and showed that their thermodynamic theory of mechanical relaxation results in a one–dimensional, generalized Burgers equation. A generalized Burgers’ equation was also derived by Sugimoto and Kakutani (1985) in their studies of propagation of plane longitudinal waves in bounded regions in non–linear viscoelastic media with a hereditary constitutive law. Lasseigne and Olmstead (1990) analyzed the stability and bifurcations of a one–dimensional, viscoelastic generalization of the system of equations proposed by Burgers for the modelling of turbulent flows in channels that includes the pressure gradient that drives the mean velocity and a Volterra’s integro–differential equation for the velocity perturbation, and showed that the state after the bifurcation is periodic for highly elastic fluids. More recently, Ramos (2019) has studied the kink solutions of a generalized, super–diffusive Burgers equation with memory, which, in the absence of memory, is a model for the flow of droplets in microchannels, flows in unsaturated soils and infiltration flows. In this paper, we shall be concerned with the following one–dimensional equation

ut + uux = σx ,

(1)

which we shall refer to as the one–dimensional viscoelastic Burgers equation, where the stress σ satisfies

σ +λ

δσ = μu x , δt

(2)

λ is a constant relaxation time,

δσ Dσ = − 2aux σ , δt Dt

(3)

and a is a constant. The values of a = 1 and a = −1 in the time–derivative δσ δt correspond to the upper– and lower–convected Maxwell derivatives, respectively, while a = 0 corresponds to the material derivative. For the one–dimensional equation considered in this study, the material derivative, i.e., a = 0, coincides with the Jaumann’s derivative (Jaumann, 1911). δσ in Eq. (3) coincides with the Gordon–Schowalter time–derivative (Gordon & Schowalter, 1972), as shown in the Apδt pendix where, for completeness, a brief review of constitutive laws for incompressible, viscoelastic fluids is also presented. The co–rotational Jaumann’s derivative and the co–deformational lower– and upper–convected Maxwell derivatives are objective or frame–indifferent (Truesdell & Noll, 2004). Eq. (1) is identical to Eqs. (1.13) of Camacho, Guy, and Jacobsen (2008), whereas Eq. (1.14) of that reference may be written as

σ+

1

β

(σt + uσx − σ ux ) =

α u , β x

which, upon taking β1 = λ and βα = μ differs from Eqs. (2) and (3) in the coefficient that multiplies the term σ ux and the fact that these authors considered only an upper–convected Maxwell time derivative. Camacho et al. (2008) studied the phase plane of their Eqs. (1.13) and (1.14) to determine the heteroclinic connections between the upstream and downstream boundary conditions for both uL > uR and uR > uL , and performed numerical simulations. By way of contrast, the study presented here (1) considers lower– and upper–convected Maxwell and Jaumann’s derivatives, (2) employs novel dependent and independent variables that provide symmetry to the equations and boundary conditions, (3) uses simple geometrical arguments to assess the existence or lack thereof of travelling wave solutions of the front type for Eqs. (1)–(3), and (4) provides analytical solutions for a = 1, 0 and −1, only for uL > uR , although the study can be readily extended to uR > uL . Note that, as stated above, a = 0 corresponds to the Jaumann time derivative. The paper has been arranged as follows. In the next section, travelling wave solutions of the kink, front or shock wave type are determined analytically for a = 1, 0 and -1 using new dependent and independent variables. In Section 2, the existence of solutions for a > 1 and a < 1 is also considered. Some sample results illustrating both the wave’s velocity and stress profiles are shown for the lower– and upper–convected and Jaumann’s time derivatives. A summary of the major findings reported in the paper concludes the paper. 2. Travelling wave solutions Upon substituting u(t, x ) = F (ξ ) and σ (t, x ) = G(ξ ) where ξ = x − ct and c is the wave speed, into Eqs. (1)–(3), one obtains

G = ( F − c )F  ,

(4)

J.I. Ramos / International Journal of Engineering Science 149 (2020) 103226

3

and

G + λ((F − c )G − 2aGF  ) = μF  ,

(5)

where the prime denotes derivative with respect to ξ . Integration of Eq. (4) yields

1 1 2 1 F − cF + A ≡ P (F ) = (F − c )2 + A − c2 , 2 2 2

G=

(6)

where A is an integration constant, and substitution of Eq. (6) into Eq. (5) yields

P ( F ) = Q ( F )F  ,

Q (F ) ≡ μ + aλ(2A − c2 ) + λ(a − 1 )(F − c )2

(7)

which implies that F is given by the ratio of two second–order degree polynomials if a = 1 and the ratio of a second– to a zeroth–degree polynomial, otherwise. In this paper, we shall be interested in travelling wave solutions of the front, kink or shock wave type, i.e., solutions for which F (−∞ ) = FL and F (∞ ) = FR , where FL and FR are constants, FL > FR , and F (n ) (−∞ ) = F (n ) (∞ ) = 0 for n ≥ 1, where the superscript (n) denotes the nth derivative. For these conditions, Eq. (4) implies that G (−∞ ) = G (∞ ) = 0, while Eq. (5) implies that G(−∞ ) = G(∞ ) = 0, and successive differentiation of Eqs. (4) and (5) yields G(n ) (−∞ ) = G(n ) (∞ ) = 0 for n ≥ 2. Using the above boundary conditions for F, one can show that

c=

1 (FL + FR ), 2

A=

1 FL FR . 2

Moreover, by first introducing f ≡

D( f¯ )

F Fr

(8)

, where Fr = 0, and then f¯ = f − c¯, Eq. (7) becomes

d f¯ 1 = N ( f¯ ) dη 2

(9)

where η = μR ξ is a Reynolds number, and F

N ( f¯ ) = f¯2 − γ , D( f¯ ) = (a − 1 )τ f¯2 − aτ γ + 1, 1 γ = c¯2 − 2A¯ = ( fL − fR )2 > 0, 4 fL + fR fL fR A¯ = , . c¯ = 2 2 and fL =

FL Fr

and fR =

(10) (11)

λF 2 τ = r > 0, μ

(12) (13)

FR Fr .

Eqs. (10) and (11) indicate that D( f¯ ) and N ( f¯ ) are even functions of f¯, so their ratio is an even function of the same vari√ √ F −F able. Moreover, f¯ (−∞ ) = L2Fr R = γ ≡ f¯N+ and f¯ (∞ ) = − f¯ (−∞ ) = − γ ≡ f¯N− , and f¯N± are the zeroes of N ( f¯ ). This means that f¯(η ) = − f¯ (−η ) and, therefore, it suffices to consider f¯ ≥ 0. However, hereon, we may consider positive and negative values of f¯, and we will use this variable as well as f = f¯ + c¯ and F = f FR . Since, as stated above, our interest is in travelling wave solutions of the front type with f ∈ [fR , fL ], i.e., f¯ = [ f¯− , f¯+ ], N

N

FL > FR or fL > fR , N ( f¯ ) ≤ 0 for f¯ = [ f¯N− , f¯N+ ] and N ( f¯N± ) = 0. The zeroes of D( f¯ ) depend on (a − 1 )τ and aτ γ as shown in Eq. (11). Moreover, for a = 1, i.e., the upper–convected Maxwell time derivative, D( f¯ ) becomes a constant. For this reason, we shall consider the cases a = 1 and −1 ≤ a < 1 separately in the next two subsections. 2.1. a = 1 ¯ For a = 1, D( f¯ ) = 1 − τ γ , and, since, in this paper, we consider FL > FR , a shock wave exists if ddηf < 0 and this demands that D( f¯ )N ( f¯ ) < 0, i.e., (1 − τ γ )( f¯2 − γ ) < 0. Therefore, for 1 − τ γ > 0, f¯2 − γ < 0 which implies that fL ≥ f ≥ fR , i.e., the condition that we imposed at the beginning. On the other hand, no monotonic shock wave with FL > FR exists for 1 − τ γ < 0 because this condition would require that f¯2 − γ > 0, i.e., f > fL and f < fR , which do not fall within the domain of f considered above. For 1 − τ γ > 0, the solution to Eq. (9) is

√ f¯(η ) = − γ tanh or, alternatively,



 γ η , 2 (1 − τ γ ) √

   √ γ γ fL − fR fL + fR f (η ) = c¯ − γ tanh η = − tanh η , 2 (1 − τ γ ) 2 2 2 (1 − τ γ ) √



(14)

√

(15)

4

J.I. Ramos / International Journal of Engineering Science 149 (2020) 103226

which is identical to that of a Taylor’shock (Hamilton & Blackstock, 2008; Taylor, 1910). √ √ For 1 − τ γ = 0, D( f¯ ) = 0, and Eq. (9) requires that f¯2 − γ = 0, i.e., f¯ = ± γ or f = c¯ ± γ , where the plus and minus √ √ signs correspond to γ and − γ for f¯ or to fL and fR for f, respectively, and there is jump from fL to fR . Note that the μ 1 condition 1 − τ γ = 0 is satisfied if τ = ( f −4f )2 which corresponds to λ = 2 (FL − FR ), whereas the condition 1 − τ γ ≥ 0 demands that

μ

L

R

1 λ ≤ 2 (FL − FR ) which coincides with Eq. (3.10) of Camacho et al. (2008).

2.2. a < 1 For a < 1, the curvature of D( f¯ ) is negative, i.e., D( f¯ ) is concave downwards, and the existence  a shock wave of negative

slope with FL > FR requires that N ( f¯ )D( f¯ ) < 0. Since γ > 0, the zeroes of D( f¯ ) are f¯D± = ±

aτ γ −1 τ (a−1 ) which are real if the

radicand is equal to or greater than zero. For the lower–convected Maxwell and material time derivatives, a = −1 and 0, respectively, and, therefore, a − 1 < 0, so the existence of real roots of D( f¯ ) = 0 demands that aτ γ − 1 < 0. However, note that, for aτ γ − 1 > 0, the roots of D( f¯ ) ¯ are complex conjugate; therefore, D( f¯ )N ( f¯ ) ≥ 0 and ddηf ≥ 0; therefore, no shock wave solution exists for fL > fR . √ √ For aτ γ = 1 which is satisfied for 0 < a < 1, f¯D± = 0 is a double root, D( f¯ ) ≤ 0, and N ( f¯ ) ≤ 0 for f¯ ∈ [− γ , + γ ], i.e., f ∈ [fR , fL ], and, therefore,

d f¯ dη

≥ 0, i.e., there is not a monotonously decreasing function connecting f (−∞ ) = f L with

¯ f (∞ ) = fR with FL > FR . Note also that | ddηf | = ∞ at f¯D± . For aτ γ < 1, the following three cases have to be considered. ¯ √ √ Case 1: f¯D+ > f¯N+ . In this case, N ( f¯ )D( f¯ ) ≤ 0 and ddηf ≤ 0, for f¯ ∈ [− γ , γ ] or f ∈ [fR , fL ], there is a monotonously decreasing

function f(η) that connects fL with fR with fL > fR , and the solution to Eq. (9) is

(a − 1 )τ f¯ −

√ f¯ (η ) + γ 1−γτ 1 ln √ = η, √ ¯ 2 γ γ − f (η ) 2

which coincides with Eq. (14) for a = 1. Case 2: f¯+ = f¯+ . In this case, N ( f¯ )D( f¯ ) ≤ 0,

(16)

d f¯ dη

≤ 0 and the equality is only satisfied for fD+ = fN+ . The zeroes of both N ( f¯ ) and D( f¯ ) are located at fL and fR , and, therefore, N ( f¯ ) = (a − 1 )τ D( f¯ ), so that, integration of Eq. (9) yields D

N

1 2

(a − 1 )τ f¯ = η + C,

(17)

where C is an integration constant, but this solution is a straight line with negative slope that cannot satisfy the boundary conditions at η = ±∞ specified above. Therefore, this case does not result in a shock wave or kink, but its linear character would be of interest in the next section. For both a = 0 and a = −1, the condition fD+ = fN+ yields γ τ = 1, while fD+ > fN+ is satisfied for γ τ < 1. Therefore, the term T = 1 − γ τ that appears in D( f¯ ) and Eq. (11), plays a key role in determining the relative location of the zeroes of D( f¯ ) with respect to those of N ( f¯ ) for a = 0 and -1, and, in fact, for a < 1. Note that Eq. (11) may be written as

D( f¯ ) = (a − 1 )τ ( f¯2 − γ ) + 1 − τ γ .

(18)

¯ ¯ Case 3: f¯D+ < f¯N+ . In this case, N ( f¯ )D( f¯ ) < 0 and ddηf < 0 for f < fD+ , N ( f¯ )D( f¯ ) = 0 and | ddηf | = ∞ at f = fD+ , and N ( f¯ )D( f¯ ) > 0 ¯

and ddηf > 0 for f > fD+ ; therefore, there is not a monotonically decreasing velocity profile for fL > fR in this case. 2.3. a > 1 Although a > 1 may not be relevant under a rheological viewpoint, we consider it here for the sake of completeness. ¯ For a > 1, the curvature of D( f¯ ) is positive. If aτ γ < 1, the roots of D( f¯ ) are complex conjugate, D( f¯ )N ( f¯ ) < 0, d f < 0 dη

and a travelling wave solution of the shock wave exists for fL > fR ; this solution is given by Eq. (16). ¯ ¯ For aτ γ = 1, D( f¯ ) ≥ 0 has a double root at f¯ = 0, D( f¯ )N ( f¯ ) ≤ 0, ddηf ≤ 0, | ddηf | = ∞ at f¯ = 0, and the solution to Eq. (9) is also given by Eq. (16) and corresponds to a sharp front (Sánchez-Garduño & Maini, 1994). Finally, if aτ γ > 1, the roots of D( f¯ ) are real, and three cases analogous to those discussed above have to be considered ¯ depending on the locations of the zeroes of D( f¯ ) and N ( f¯ ). For f¯+ ≥ f¯+ , D( f¯ )N ( f¯ ) ≥ 0 and d f ≥ 0; therefore, no shock D

N

dη

¯ wave solution exists for fL > fR . On the other hand, for f¯D+ < f¯N+ , D( f¯ )N ( f¯ ) > 0 and ddηf > 0 for 0 ≤ f¯ < f¯D+ , D( f¯ )N ( f¯ ) = 0 at ¯ f¯ = f¯+ , and D( f¯ )N ( f¯ ) < 0 and d f < 0 for f¯+ ≤ f¯ < f¯+ ; therefore, there is not a monotonically decreasing velocity profile D

for fL > fR .

dη

D

N

J.I. Ramos / International Journal of Engineering Science 149 (2020) 103226

2

5

0 =0.05 =0.1 =0.5 =1.0

1.5 1 0.5

−0.2 −0.4 −0.6

a=+1

0 −2

−1

0

1

2

−0.8 −2

2

0

1.5

−0.2

1

−0.4

0.5

−1

0

1

2

−0.8 −2

2

0

1.5

−0.2

1

−0.4

0.5

1

2

−1

0

1

2

−1

0

1

2

−0.6

a=−1

0 −2

0

−0.6

a=0

0 −2

−1

−1

1

2

−0.8 −2

Fig. 1. Velocity (left column) and stress (right column) profiles for f L = 2.0, f R = 0.01 and four values of τ . (Top row: a = 1 (left); middle row: a = 0, a = −1). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

3. Results Fig. 1 illustrates f(η) and g(η) where g =

G Fr2

for fL = 2.0 and fR = 0.01 for four different values of τ . For a = +1, i.e.,

the upper–convected Maxwell time derivative, Fig. 1 indicates that the steepness of the nondimensional velocity profile f(η) increases as τ is increased and f(η) is nearly independent of τ for τ ≤ 0.05. These results are consistent with Eq. (14) which 2(1−τ γ ) √ indicates that the characteristic thickness of the tanh –profile, i.e., γ , decreases as τ is increased. Recall that, for a = 1, 1 − τ γ > 0. The top right graph of Fig. 1 also shows that the nondimensional stress g(η) is an even function of η, is semi–negative, reaches its largest absolute value at η = 0, and increases in a smooth manner towards a nil value at η = −∞ and ∞; its thickness decreases as τ is increased. Note that, after using the change of variables reported in Section 2, Eq. (4) may be written as

dg d f¯ = f¯ , dη dη

(19)

whose solution, after making use of the boundary conditions g(−∞ ) = g(∞ ) = 0 is

g( η ) =

1 ¯2 ( f ( η ) − γ ), 2

(20)

which confirms the results presented in the top right graph of Fig. 1. For a = 0, i.e., the material derivative in the constitutive law, the middle row of Fig. 1 indicates that the steepness of the velocity profile increases as τ is increased, and f(η) is independent of τ for τ ≤ 0.05. This middle row also indicates that the slope of the velocity profile for a = 0 is much smaller than for a = +1, and that, for τ = 1, the velocity profile is an almost piecewise linear function characterized by almost constant values of f(η) equal to fL and fR for η > 1 and η < −1, respectively, and a straight line of negative slope between η = −1 and η = 1. The graph on the right of the middle row of Fig. 2 shows the same qualitative features as the top right one, i.e., the width of the stress profile increases as τ is decreased. For a = −1, i.e., the lower–convected Maxwell time derivative in the constitutive equation, the results presented in the bottom row of Fig. 1 indicate that the velocity profile is an almost linear function of η for τ = 1, is almost independent of τ for τ ≤ 0.05, and its curvature decreases as τ is decreased. Moreover, the velocity value at, say, η = 1, is larger for τ = 1 than for τ = 0.01, for a = −1, whereas just the opposite trend is observed for a = 0 and 1.

6

J.I. Ramos / International Journal of Engineering Science 149 (2020) 103226

2

2 =0.1

=0.05 1.5 ( )

( )

1.5

1

0.5

1

0.5

0 −2

−1

0

1

0 −2

2

−1

0

1

2

a=1 a=0 a=−1 2

2 =0.5

=1.0 1.5 ( )

( )

1.5

1

0.5

1

0.5

0 −2

−1

0

1

0 −2

2

−1

0

1

2

Fig. 2. Velocity profiles for f L = 2.0, f R = 0.01 and four values of a. (Top left: τ = 0.05 (left); top right: τ = 0.1; bottom left: τ = 0.5; bottom right: τ = 1.0). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

0

0

−0.1

−0.1

−0.2

−0.2

−0.3

−0.3

−0.4 −0.5 −2

−0.4

=0.05 −1

0

1

=0.1 −0.5 −2

2

−1

0

1

2

a=1 a=0 a=−1 0

0

−0.1

−0.1

−0.2

−0.2

−0.3

−0.3

−0.4 −0.5 −2

−0.4

=0.5 −1

0

1

=1.0 2

−0.5 −2

−1

0

1

2

Fig. 3. Stress profiles for f L = 2.0, f R = 0.01 and four values of a. (Top left: τ = 0.05 (left); top right: τ = 0.1; bottom left: τ = 0.5; bottom right: τ = 1.0). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

J.I. Ramos / International Journal of Engineering Science 149 (2020) 103226

2 1.5

0 =0.8 =0.9 =1.0 =1.1

a=1

1

−0.2 −0.4

0.5 0

7

−0.6 −0.6

−0.4

−0.2

0

0.2

0.4

−0.8

0.6

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.2

3

0

2

−0.2 1

−0.4

a=0 0 −2

−1

0

1

2

−0.6 −2

2

0

1.5

−0.2

1

−0.4

0.5 0 −2

−1

0

1

2

−1

0

1

2

−0.6

a=−1 −1

0

1

2

−0.8 −2

Fig. 4. Velocity (left column) and stress (right column) profiles for f L = 2.0, f R = 0.01 and four values of τ . (Top row: a = 1 (left); middle row: a = 0, a = −1). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Figs. 2 and 3 illustrate the velocity and stress profiles, respectively, as functions of a for four values of τ . For τ = 0.01, there are very small differences between the velocity profiles corresponding to a = 1 and a = −1; this is in accord with the fact that as τ → 0, λ → 0 and the term λ δσ δt → 0 (cf. Eqs. (12) and (2), respectively.) The differences between the velocity profiles corresponding to a = 1, 0 and −1 increase as τ is increased, as the results for τ ≥ 0.05 shown in Fig. 2 indicate. This figure indicates that the slope of the velocity profile increases as τ is increased. Fig. 2 also shows that, for τ = 1, the f(η) profile is almost a step function for a = 1, exhibits a piecewise linear shape for a = 0 and is almost a linear function for a = −1. Fig. 3 indicates that the absolute value of the stress is greater for a = −1 than for a = 0, and that for a = 0 is larger than the stress for a = 1, for the same value of η. Moreover, since c¯ (cf. Eq. (13)) is the same for the tree different values of a shown in Fig. 3, it may be concluded that the above statement also holds for the same values of ξ . Fig. 3 also shows that the width of the stress profile increases as τ and a are decreased; the stress is negative except at the boundaries where it is nil, and its smoothness near the boundaries decreases as τ is increased for a = 0 (cf. the stress profiles for τ = 1). Fig. 4 shows the velocity and stress profiles for fR = 0.01 and fL = 2, i.e., γ = 0.995, and four values of τ which have been selected to illustrate the effect of the term T ≡ 1 − aτ γ discussed in Section 2 as a function of τ . For a = 1, 0 and -1, D( f¯ ) = 1 − τ γ , 1 − τ f¯2 and 1 + τ γ − 2τ f¯2 , respectively. Since the largest value of f¯2 is γ , T = 1 − τ γ and D( f¯ ) = T , T = 1 and D( f¯ ) ≤ T , and T = 1 + τ γ and D( f¯ ) ≤ T , for a = 1, 0 and −1, respectively. Moreover, T = 0 for a = 1 and τc = γ1 which corresponds to 1.005 for the value of γ considered here. The top left graph of Fig. 4 indicates that the steepness of the velocity profile for a = 1 increases as τ is increased from 0.8 to 1.0 which correspond to a shock wave of negative slope. For τ = 1.1 which is larger than τ c , a shock wave with fL > fR is not possible; instead, a shock wave of positive slope may be obtained for fR > fL as illustrated in Fig. 4. For a = 0, Fig. 4 shows a piecewise linear velocity profile characterized by f = f L for −∞ < η < −1, f = fR for 1 < η < ∞ and a straight line for −1 < η < 1. Fig. 4 also shows that smoothness of the velocity profile for a = 0 increases as τ is decreased from 1, and that the influence of τ on the velocity profiles is small for a = −1, in accord with discussion on D( f¯ ) above. The right column of Fig. 4 shows that the thickness of the stress profile decreases as τ is increased, especially for a = 1; for this value, the stress is a smooth function of η. For a = 0, however, the width of the stress profile decreases slightly as τ is increased, but its slope near the locations where the stress approaches a nil value increases as τ is increased; this is especially noticeable for τ = 1 and 1.1, whereas for τ = 0.8 and 0.9, the nil values of the stress at the upstream and downstream boundaries are reached very smoothly.

8

J.I. Ramos / International Journal of Engineering Science 149 (2020) 103226

For a = −1, the stress is not a very sensitive function of τ for τ ∈ [0.8, 1.1], although its absolute value increases very slightly as τ is increased. An analogous analysis to the one reported here has been performed for fR > fL , i.e., shock waves of positive slope or kinks, but is not reported here. 4. Conclusions Analytical solutions of the shock wave type to a one–dimensional, viscoelastic Burgers equation where the constitutive law contains objective or frame–indifferent derivatives of the co–rotational Jaumann’s and co–deformational Maxwell’s lower– and upper–convected types have been determined as functions of a nondimensional relaxation time, a parameter that identifies the type of time derivative in the constitutive law, and the upstream and downstream boundary conditions. The analytical solutions are based on the introduction of independent and dependent variables that result in a nonlinear, first–order, ordinary differential equation where the derivative of the velocity in a frame of reference moving with the wave is the ratio of two even, second–degree polynomials and the upstream boundary condition is identical to but has the opposite sign to the downstream one. It has been shown that for the upper– and lower–convected Maxwell and Jaumann’s time derivatives in the constitutive law, the steepness of the velocity profile increases as the relaxation time is increased. For the upper–convected Maxwell derivative, it has been shown that the velocity profile exhibits a shock wave profile analogous to that of a Taylor’s shock wave in Newtonian fluid dynamics, whereas that corresponding to a material derivative is piecewise linear with constant values equal to those at the upstream and downstream boundaries connected by a straight line of negative slope, for large values of the relaxation parameter. It has also been shown that, for the lower–convected Maxwell time derivative, the velocity profile evolves from a sigmoid shape to an almost straight line that connects the upstream and downstream boundary values as the relaxation parameter is increased. It has also been found that the width of the stress profile increases for both the upper–convected Maxwell and the material derivatives and decreases for the lower–convected one as the relaxation parameter is decreased. At each location in a frame of reference moving with the wave speed, it has been shown that the absolute value of the stress increases as the time relaxation parameter is decreased for both the material and the upper–convected Maxwell derivatives, whereas the opposite behavior has been observed for the lower–convected Maxwell time derivative. Acknowledgements The author is very grateful to the reviewer for his comments and suggestions on the original manuscript. The research reported in this paper was partially supported by Project UMA18–FEDERJA–248 from the Conserjería de Economía y Conocimiento, Junta de Andalucía, Spain. Declaration of Competing Interest The author declares that there is no conflict of interests regarding the publication of this article. Appendix As stated in the Introduction, in this Appendix, a brief summary of the governing equations and constitutive laws for incompressible, viscoelastic fluids is presented. The conservation of mass and linear momentum for an incompressible fluid may be written as

∇ · v = 0,

(A-1)

and

ρ

Dv = ρ (vt + v · ∇ v ) = −p + ρ fb + ∇ · σ , Dt

(A-2)

where ρ is the density, v, p and fb are the velocity vector, pressure and body force per unit mass, respectively, and σ is the deviatoric stress tensor. For a Newtonian fluid,

σ = 2 μD , 1 2 (∇ v

(A-3)

( ∇ v )T )

where D = + is the strain rate tensor, and the superscript T denotes transpose. The following constitutive law

δσ Dσ = + θ ( σ W − Wσ ) − a ( σ D + Dσ ) , δt Dt

(A-4)

J.I. Ramos / International Journal of Engineering Science 149 (2020) 103226

9

where W = 12 (∇ v − (∇ v )T ), includes the Gordon–Schowalter time derivative for θ = 1 (Gordon & Schowalter, 1972; Lee & Xu, 2006), the upper and lower–convected Maxwell time derivatives for θ = 0 and a = +1 and a = −1, respectively, and the Jaumann or Zaremba–Jaumann co–rotational time derivative for a = 0 and θ = 1 (Jaumann, 1911; Joseph, 1990; Zaremba, 1903). The objective Gordon–Schowalter derivative of a symmetric tensor is objective, but, only for the Jaumann’s derivative, the derivative of the unit tensor is nil and the objective derivative of a symmetric tensor is symmetric (da Costa Mattos, 2013). For one–dimensional flow problems, Eq. (A–4) becomes Eq. (2). It is interesting to notice that there are several objective or material–frame indifferent time derivatives, and the choice of one of them may be interpreted as a constitutive assumption (Christov, 2009; Joseph, 1990; Truesdell & Noll, 2004). It is also interesting to note that a constitutive equation for a Burgers’ fluid (not to be confused with Burgers’ equation) is (Burgers, 1935; Hayat, Fetecau, & Asghar, 2006)

  δ2σ δσ δR σ +α +ψ 2 =μ R+ξ , δt δt δt

(A-5)

where α is a relaxation time, ψ is the square of another relaxation time, ξ (ξ < α is a retardation time), and R is the Rivlin–Eriksen tensor (Joseph, 1990; Truesdell & Noll, 2004). Eq. (A.5) includes the Olroyd–B (Olroyd, 1950) (for ψ = 0), Maxwell (for ψ = ξ = 0) and linear viscous (for α = ψ = ξ = 0) models. For one–dimensional motions of incompressible fluids, Eq. (A–1) becomes ux = 0, i.e., u(t, x) is only a function of time, the right–hand sides of Eqs. (A–3) and (A–4) are zero, and, therefore, Eq. (A–2) reduces to ut = 0, i.e., u(t, x) is constant. This means that Eqs. (1)–(3) do not correspond to one–dimensional motions of incompressible fluids. However, in this paper, these equations are considered as a simple model that includes the nonlinearities associated with both convection and viscoelastic stresses, in a similar fashion to what it occurs in Newtonian fluid dynamics where the viscous Burgers equation is the simplest one–dimensional equation that includes a linear diffusion term and a quadratic, nonlinear, convective flux. They may also be considered as a one–dimensional model of compressible viscoelastic fluids. Most polymers may be considered as incompressible materials. Therefore, most rheological models available in the literature have been developed for incompressible polymeric solutions. However, compressible effects have been observed to play a paramount role on the breathing modes of isolated metal nanowires in water–glycerol mixtures (Galstyan, Pak, & Stone, 2013; Pelton, Chakraborty, Malachosky, Guyot-Sionnest, & Sader, 2013; Yu et al., 2015), bubble collapse (Lind & Phillips, 2013), injection/compression molding of center–gated disks (Kim, Park, Chung, & Kwon, 1999), high–speed extrusion, etc. (White & Metzner, 1965). Compressible and non–isothermal effects are expected to be important in flow processes involving high pressure, temperature and/or flow rates because the fluid properties and flow dynamics depend on the thermodynamic relationships between state variables (Mackay & Phillips, 2019). Many of the constitutive equations for compressible models of viscoelastic fluids developed to–date have been based on separating the contributions of the solvent and polymer, introducing compressibility through a pressure– density–temperature equation of state (Brujan, 1999), use of the constitutive law for compressible Newtonian fluids (Thompson, 1972) with or without relaxation times for the shear and compressional contributions to the deviatoric stress tensor (Chakraborty & Sader, 2015; Yong, 2014), etc. Other constitutive models based on a generalized Poisson–bracket formulation (Mackay & Phillips, 2019) or the GENERIC (General Equation for Non–Equilibrium Reversible–Irreversible Coupling) formalism of non–equilibrium thermodynamics (Dressler, Edwards, & Öttinger, 1999; Huo & Yong, 2016) result in a transport equation for the conformation tensor (Beris & Edwards, 1994; Edwards & Beris, 1990). In many of these models, issues related to their structural stability (Huo & Yong, 2016), change of equation type (Edwards & Beris, 1990), convergence to their Newtonian counterparts as the relaxation times tend to zero (Yong, 2014), definition of pressure, e.g., static, augmented, or thermodynamic (Bollada & Phillips, 2012), objectivity and objective time derivatives (Dressler et al., 1999), initial and boundary conditions for the conformation tensor (Dressler et al., 1999), integral or differential constitutive equations (Bollada & Phillips, 2012), etc., arise and have not yet been fully resolved in a satisfactory manner. In fact, some authors, e.g., (Bollada & Phillips, 2012), have proposed an integral Maxwell model that has been used to develop an equivalent density–dependent differential constitutive equation, and emphasized the important role played by volume elements in the modelling of compressible fluids. References Bateman, H. (1915). Some recent researches on the motion of fluids. Monthly Weather Review, 43(4), 163–170. Beris, A. N., & Edwards, B. J. (1994). Thermodynamics of flowing systems. New York: Oxford University Press. Bollada, P. C., & Phillips, T. N. (2012). On the mathematical modelling of a compressible viscoelastic fluid. Archives of Rational Mechanics and Analysis, 205(1), 1–26. Brujan, E. A. (1999). A first–order model for bubble dynamics in a compressible viscoelastic liquid. Journal of Non–Newtonian Fluid Mechanics, 84(1), 83–103. Burgers, J. M. (1935). Mechanical considerations–model systems–phenomenological theories of relaxation and viscosity. First report on viscosity and plasticity, committee for the study of viscosity of the academy of sciences at Amsterdam, second edition. New York: Nordemann Publishing Company, Inc.. Burgers, J. M. (1948). A mathematical model illustrating the theory of turbulence. In R. von Mises, & T. von Kármán (Eds.), Advances in applied mechanics 1 (pp. 171–199). New York: Academic Press, Inc.. Camacho, V., Guy, R. D., & Jacobsen, J. (2008). Traveling waves and shocks in a viscoelastic generalization of Burgers’ equation. SIAM Journal of Applied Mathematics, 68(5), 1316–1332.

10

J.I. Ramos / International Journal of Engineering Science 149 (2020) 103226

Chakraborty, D., & Sader, J. E. (2015). Constitutive models for linear compressible viscoelastic flows of simple liquids at nanometer length scales. Physics of Fluids, 27(5), 052002. 14 pages. Christov, C. I. (2009). On frame indifferent formulation of the Maxwell–Cattaneo model of finite–speed heat conduction. Mechanics Research Communications, 36(4), 481–486. Ciancio, V., & Restuccia, L. (1987). The generalized Burgers equation in viscoanelastic media with memory. Physica A, 142A(1–3), 309–320. Crighton, D. G. (1979). Model equations of nonlinear acoustics. Annual Review of Fluid Mechanics, 11, 11–33. da Costa Mattos, H. S. (2013). Some remarks about the use of Gordon–Schowalter time derivative in rate–type viscoelastic constitutive equations. International Journal of Engineering Science, 73, 56–66. Dressler, M., Edwards, B. J., & Öttinger, H. C. (1999). Macroscopic thermodynamics of flowing polymeric liquids. Rheological Acta, 38(2), 117–136. Edwards, B. J., & Beris, A. N. (1990). Remarks concerning compressible viscoelastic fluid models. Journal of Non–Newtonian Fluid Mechanics, 36, 411–417. Galstyan, V., Pak, O. S., & Stone, H. A. (2013). A note on the breathing mode of an elastic sphere in newtonian and complex fluids. Physics of Fluids, 27(3), 032001. 13 pages. Gordon, R. J., & Schowalter, W. R. (1972). Anisotropic fluid theory: A different approach to the dumbbell theory of dilute polymer solutions. Transactions of the Society of Rheology, 16(1), 79–97. Hamilton, M. F., & Blackstock, D. T. (2008). Nonlinear acoustics. New York: Acoustical Society of America. Hayat, T., Fetecau, C., & Asghar, S. (2006). Some simple flows of a Burgers’s fluid. International Journal of Engineering Science, 44(18–19), 1423–1431. Huo, X., & Yong, W. A. (2016). Structural stability of a 1d compressible viscoelastic fluid model. Journal of Differential Equations, 261(2), 1264–1284. Jaumann, G. (1911). Geschlossenes system physikalischer und chemischer differentialgesetze. In I. Mitteilung (Ed.), Sitzungsberichte der kaiserliche akademie der wissenschaften in wien (mathematisch IIa) CXX (pp. 385–530). Joseph, D. D. (1990). Fluid dynamics of viscoelastic liquids. New York: Springer–Verlag. Kardar, M., Parisi, G., & Zhang, Y. C. (1986). Dynamic scaling of growing interfaces. Physical Review Letters, 56(9), 889–892. Kim, I. H., Park, S. J., Chung, S. T., & Kwon, T. H. (1999). Numerical modeling of injection/compression molding for center–gated disk: Part I. Injection molding with viscoelastic compressible fluid model. Polymer Engineering and Science, 39(10), 1930–1942. Lasseigne, D. G., & Olmstead, W. E. (1990). Stability of a viscoelastic Burgers flow. SIAM Journal of Applied Mathematics, 50(2), 352–360. Lee, Y.-J., & Xu, J. (2006). New formulations, positive preserving discretizations and stability analysis for non–newtonian flow models. Computer Methods in Applied Mechanics and Engineering, 195(9–12), 1180–1206. Lighthill, M. J., & Whitham, G. B. (1955). On kinematic waves. II. A theory of traffic flow on long crowded roads. Proceedings of the Royal Society (London), Series A, 229(1178), 317–345. Lind, S. J., & Phillips, T. N. (2013). Bubble collapse in compressible fluids using a spectral element marker particle method. Part 2. Viscoelastic fluids. International Journal for Numerical Methods in Fluids, 71(9), 1103–1130. Mackay, A. T., & Phillips, T. N. (2019). On the derivation of macroscopic models for compressible viscoelastic fluids using the generalized bracket framework. Journal of Non–Newtonian Fluid Mechanics, 266, 59–71. Olroyd, J. G. (1950). On the formulation of rheological equations of state. Proceedings of the Royal Society (London), Series A, 200(1063), 523–541. Pelton, M., Chakraborty, D., Malachosky, E., Guyot-Sionnest, P., & Sader, J. E. (2013). Viscoelastic flows in simple liquids generated by vibrating nanostructures. Physical Review Letters, 111(24), 244502. 5 pages. Ramos, J. I. (2019). Kink solutions of the generalized, super–diffusive burgers equation with memory. Communications in Nonlinear Science and Numerical Simulation, 76, 25–44. Richards, P. L. (1956). Shock waves on the highway. Operations Research, 4(1), 42–51. Rudenko, O. V., & Soluyan, S. I. (1977). Theoretical foundations of nonlinear acoustics. New York: Consultants Bureau/Plenum Press. Sánchez-Garduño, F., & Maini, P. K. (1994). An approximation to a sharp front type solution of a density dependent reaction–diffusion equation. Applied Mathematics Letters, 7(1), 476–551. Sugimoto, N., & Kakutani, T. (1985). ’Generalized Burgers’ equation’ for nonlinear viscoelastic waves. Wave Motion, 7(5), 447–458. Taylor, G. I. (1910). The conditions necessary for discontinous motion in gases. Proceedings of the Royal Society (London), Series A, 84(571), 371–377. Thompson, P. A. (1972). Compressible fluid–Dynamics. New York: McGraw–Hill Book Company. Truesdell, C., & Noll, W. (2004). The non-linear field theories of mechanics, third edition. Berlin–Heidelberg–New York: Springer–Verlag. White, J. L., & Metzner, A. B. (1965). Constitutive equations for viscoelastic fluids with application to rapid external flows. AIChE Journal, 11(2), 324–330. Whitham, G. B. (1974). Linear and nonlinear waves. New York: John Wiley & Sons. Yong, W. A. (2014). Newtonian limit of Maxwell fluid flows. Archives of Rational Mechanics and Analysis, 214(3), 913–922. Yu, K., Major, T. A., Chakraborty, D., Devadas, M. S., Sader, J. E., & Hartland, G. V. (2015). Compressible viscoelastic liquid effects generated by the breathing modes of isolated metal nanowires. Nano Letters, 15(6), 3964–3970. Zaremba, S. (1903). Sur une généralisation de la théorie classique de la viscosité. Bulletin International de l’Académie des Sciences de Cracovie, 380–403.