The modified Jeffreys model approach for elasto-viscoplastic thixotropic substances

Physics Letters A 380 (2016) 585–595

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Physics Letters A www.elsevier.com/locate/pla

The modified Jeffreys model approach for elasto-viscoplastic thixotropic substances ✩ Hilbeth P. Azikri de Deus a,∗ , Cezar O.R. Negrão b , Admilson T. Franco b a b

CERNN-UTFPR/NuMAT-UTFPR, Curitiba, PR, Brazil CERNN-UTFPR, Curitiba, PR, Brazil

a r t i c l e

i n f o

Article history: Received 5 April 2015 Received in revised form 8 October 2015 Accepted 8 November 2015 Available online 21 November 2015 Communicated by F. Porcelli Keywords: Thixotropic fluids Constitutive equation Jeffreys model Thermodynamic consistency

a b s t r a c t In this work, a new constitutive model for thixotropic fluids is proposed via a formal analysis of their functional forms. The constitutive model for thixotropic substances is basically composed by a set of two equations: a constitutive equation based on viscoelastic models and a rate equation (an equation related to the evolution of the micro-structural character of the substance). The constitutive equations, in many works, do not have taken into account in the dynamical principles from which is developed the micro-structural dependence of shear modulus and viscosity. To aim this fault, in the current study a new constitutive model (based on Jeffreys model) is proposed, in coherence with expected behavior for thixotropic fluids. In addition a special emphasis has been given to the structural nature of substances (based on coagulation theory of Smoluchowski) and thermodynamic consistency. The proposed model for thixotropic fluids takes into account a simple isothermal laminar shear flows and some important aspects of that are explored. © 2015 Elsevier B.V. All rights reserved.

λ˙ = Gγ˙ (λ, γ˙ ).

1. Introduction The time-dependent rheological behavior of the thixotropic materials, as presented in numerous works [1–5], is descripted in terms of a set of equations that take into account the breakdown and build-up of their microstructure. In general, the equation associated to the micro-structural evolutionary process is justified by a qualitative explanation, without more details about the formal physical principles on which it is based. In large number of thixotropy models [2,5–10], the rheological properties are described in terms of scalar system of equations consisting of a constitutive relationship based on viscoelastic models and a rate equation (associated to the micro-structural evolution). One can write the constitutive equation, in a generic form, as

τ = τ (λ, γ˙ ),

(1)

where τ is the shear stress, γ˙ is the shear rate and λ is the structural parameter (a single variable that represents the structural level of substance). In a similar way, the rate equation is presented as ✩ This work was supported by CNPq (grant: 204175/2014-3) and PETROBRAS (grant: 0050.00645885.11.9). Corresponding author. E-mail address: [email protected] (H.P. Azikri de Deus).

*

http://dx.doi.org/10.1016/j.physleta.2015.11.016 0375-9601/© 2015 Elsevier B.V. All rights reserved.

(2)

However it is important to point out that the form of Eq. (1), which has been used in several works to approach the thixotropic phenomenon [5,11–14], does not take into account the influence of the temporal variation of the shear modulus and viscosity in their dynamical principles. They are directly based on linear viscoelastic models (Maxwell, Jeffreys or Kelvin–Voigt for instance) or some combination of these. This approach is not consistent with the expected behavior of thixotropic substances. In this paper, this fault is corrected in the constitutive model and some of their properties will be analyzed in rheological and thermodynamic context. The equations, (1) and (2), connect the structural and viscoelastic natures of these substances. In real thixotropic fluids these two phenomena are in a very close simultaneous connection. In this sense, the thixotropic time-dependent behavior described by the equations (1) and (2) must relate structural characteristic and viscoelasticity. Another physical interpretation for the fluid structure (thixotropic fluid) carries a question over the possible existence of more than one type of structure (structural parameters) in a fluid. In the specialized literature, different types of structures have been analyzed as polymers in the molten state, suspension of solid particles in a liquid (the particles may degrade under deformation and show irreversible time-dependent properties) and aqueous clay suspensions, for instance. The structure formation is deeply linked to chemical nature of the suspension liquid, and to pH if it is aque-

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H.P. Azikri de Deus et al. / Physics Letters A 380 (2016) 585–595

It is important to comment that the present model takes into account the time variation of the shear modulus that is not considered in the standard Maxwell model. This assumption is in agreement with the expected physical behavior of thixotropic fluids [23, 24]. On the other hand, from the viscous element (.μ ), it follows

γ˙μ =

τμ ∴ τ˙μ = η˙ μ γ˙μ + ημ γ¨μ . ημ

(6)

Taking into account that γ˙ = γ˙m = γ˙μ and τ˙μ , it can be written

Fig. 1. Sketch of the modified Jeffreys model.

ous. More than one type of structure can exist in a fluid, and each of them can be affected differently by the shear rate. The specialized literature shows a great variety of microstructure types [15–22]. In the current work a new constitutive model for thixotropic substances is proposed and some of their properties are analyzed in rheological and thermodynamic sense. It is important to stand out that the predictions reached from previous models [12–14], that were based on Maxwell-like constitutive equation, have shown a good agreement in relation to experimental observations. The present work proposes a new constitutive model for structured fluids which constitutive equation is developed from a mechanical element composed by a structuringlevel-dependent Maxwell-like element in parallel with a Newtonian element. If the structural level dependence is not considered, this mechanical element is known as Jeffreys element. The Newtonian element in parallel has the objective to enhance the quality of the predictions of the model. Another difference of this proposed model, in relation to the previous models based on the Jeffrey element, is the fact that the structural dependence of the shear modulus (G) and viscosity coefficients (ημ and ην ) has be taken into account in the constitutive equation development, as can be seen in the next section, what is more compatible with the expected behavior of a thixotropic fluid. Only one type of structure is assumed in this model, which can be described by a single scalar parameter. However, the developed analysis can be easily extended to an approach that includes more than one type of structure, i.e. more than one structural parameter.

τ˙ = G γ˙ − ∴

ην G

G



ην



τ˙ + 1 −

1−

ην G˙ G2

ην G˙

τ = τm + τμ ∴ τ˙ = τ˙m +



G2



τm + η˙ μ γ˙ + ημ γ¨ ; 



τ = ην + 1 − +

ην ημ G

ην G˙ G2

(7)



ημ +

ην η˙ μ G

γ¨ .



γ˙ (8)

It is important to take in mind the fact that the parameter λ can be used to characterize the structural level of the substance, or in other words it is related to some physical properties of structural nature. The structural parameter λ is a measure of build-up state of the structural nature of the fluid, i.e., large values of λ imply highly build-up state of structure, and when it is small the structure is in a broken-down state. The structural parameter can be represented mathematically in different ways. Finally, some constraints associated to the rheological behavior for functional forms of thixotropic fluids [25–29] should be taken into account

η(λ, γ˙ ) > 0;  ∂ τ  > 0; ∂ γ˙ λ

(9) (10)

 ∂Gγ˙  < 0; ∂ γ˙ 

(11)

Gγ˙ (λ, γ˙ ) = 0, if λ = λeq (γ˙ );

(12)

Gγ˙ (λ, γ˙ ) > 0, if λ < λeq (γ˙ );

(13)

2. The constitutive equation

Gγ˙ (λ, γ˙ ) < 0, if λ > λeq (γ˙ ),

(14)

The approach proposed in this paper takes into account the representation of the thixotropic system as a composition of a spring-damper element (dependent of the structural behavior) and a viscous element (with a similar structural dependence). In this sense, in the current study, a Jeffreys like fluid model is proposed (Maxwell element in parallel with viscous element) as presented in Fig. 1. Supposing a certain shear load (τ ), it can be written for the Maxwell element (.m ),

where (.)eq stands for the equilibrium condition. These restrictions aim to understand the expected behavior of the thixotropic model, that is better explored in the next section.

τm = ην γ˙mν ;

(3)

τ˙m = G γ˙me + G˙ γme .

(4)

Note that the total strain rate for the Maxwell element is γ˙m = γ˙mν + γ˙me , where γ˙mν is the viscous strain rate, γ˙me is the elastic strain rate, ην = ην (λ) is the fluid viscosity and G = G (λ), the elastic shear modulus for the Maxwell element. Differently from the standard Maxwell model, both G and ην are structural dependent properties. After some algebraic manipulation, one has

γ˙m =

τ˙m − G˙ γme

τm + ην ην

∴ ην γ˙m =

G

G

;



τ˙m + 1 −

ην G˙ G2



τm .

(5)

λ

3. Some important points

˙ = 0 (12) The equilibrium state can be defined as the state at λ for a given shear rate γ˙ . The steady state was carefully analyzed following a similar screenplay presented in [12,26], in order to verify the satisfaction of constraints (9)–(14). In this sense, it follows the rate equation (2) in terms of (λ, τ ) dλ dt

= Gγ˙ (λ, γ˙ ) = Gτ (λ, τ ),

(15)

in a similar to the restrictions (12), (13) and (14), one has

Gτ (λ, τ ) = 0, if λ = λeq (τ );

(16)

Gτ (λ, τ ) > 0, if λ < λeq (τ );

(17)

Gτ (λ, τ ) < 0, if λ > λeq (τ ).

(18)

Then, the equilibrium curve (EC) can be observed as a divisor line of the surface f (λ, τ , γ˙ ) = 0 in two distinct regions given by inequalities (13) and (14). Aiming a better illustration of the physical

H.P. Azikri de Deus et al. / Physics Letters A 380 (2016) 585–595

587

Another important question is about the influence of the constitutive equation over the rate equation. Taken into account a controlled shear strain rate,



γ˙ =

γ˙o to

t , if t ≤ to ;

γ˙o ,

(19)

if t > to ,

where γ˙o is a selected constant positive strain rate, t o is the instant related to get the constant strain rate γ˙o by rheometer and t ∈ [0, T ] is the time variable (T is the final time of the analysis). Thus, in accordance with above lines one has Fig. 2. Equilibrium curve with positive slope.

τ = τ (λ, γ˙ ) ∴ dτ =

∂τ ∂τ ∂τ dλ + dγ˙ ∴ dτ = dλ, ∂λ ∂ γ˙ ∂λ

(20)

since t > t o . Following the same main steps of [12], it can be shown in the first order analysis (from the Taylor series expansion about λ = λeq ), that

    G G˙ γ˙ Gγ˙ = G + − ημ + η˙ μ  

−

Fig. 3. Equilibrium curve with negative slope (snap-through S-shaped).

behavior, two kinds of test the controlled strain rate (CR) and the controlled stress (CS) are considered. On the neighbor of the EC positive slope point (Fig. 2), the structure would change in such a way that it tends towards the equilibrium point B. Imagine that point E (λ E > λ B ) is being tested at constant γ˙ (CR), the microstructure would tend to break down (a smaller stiff response) and move towards B by stress decreasing. On other side, suppose that D (λ B > λ D ) is being tested under constant τ (CS) and the tendency is to move towards B by shear rate decreasing. A similar reasoning is used to describe the behavior from A and C . Although it is important to point out  the possibility of nonmonotonic equilibrium curve, i.e.

 ∂Gτ  ∂λ 

τeq

> 0 ( ddtλ

 τ  ≈ ∂G ∂λ 

τeq

dτ  dγ˙ (λ ,τ ) eq eq

< 0 that implies

(λ − λeq )), without violate (17)–(18)

constraints. To aim the interpretation of this behavior it takes into account a snap-through S-shaped flow curve (Fig. 3). Suppose that E (λ D > λ E ) is being tested under constant τ (CS) and in this case instead of the structure increasing towards the equilibrium point D, a breakdown process of micro-structure originates the tendency of this point go towards F by shear rate increasing. A similar reasoning is used to describe the behavior from C (λC > λ D ), a build up process of micro-structure motivates the tendency of this point go towards B by shear rate decreasing. If one supposes M (λ M > λ D ) is being tested at constant γ˙ (CR) the micro-structure would tend to break down (a smaller stiff response) and move towards D by stress decreasing. A similar behavior is applied in the case of K point. The behavior of points in neighbor of F and B points follows the same ideas presented in previous paragraph, i.e. positive slope equilibrium curve. In this case (snap-through flow curve), the equilibrium points B, D and F can be reached via constant shear rate test (CR) [30]. Although, for constant stress testing (CS) the point D cannot be accessed. It’s important to point out that a snap-back S-shaped flow curve presents an inaccessible region for the both tests, (CS) and (CR) [31]. The case of S-shaped (τ vs.γ˙ ) equilibrium curve is analyzed in the paper [32].

ην

G

ην

−

G

 τ (λeq , γ˙ ) G˙ G

c

c

 + λ − λeq ;

(21)

where c  is a positive constant. It is important to point out that these results are in agreement with the restrictions (12)–(14), ˙ < 0 and limλ→0+ λ˙ > 0 for appropriate choosing of G (λ), limλ→1− λ ημ (λ) and ην (λ). The thixotropic effects are arised from the network of long chains of particles. Under shear load, this network is broken into “flocs” and, if the breakdown continues, the process of flocs decomposition may result into individual particles. Due to Brownian motion and/or laminar regime, these flocs and/or particles can reconstruct the network. In this work the microstructural level of the thixotropic substance is approached by a single variable, the structural parameter λ (0 ≤ λ ≤ 1), as early explained. A continuous network corresponds to λ = 1, and the null value for λ represents a completed broken down structure composed of individual particles. It is important to point out that when λ = 1, the substance shows a pure viscoelastic behavior, by other side when λ = 0 the substance exhibits a pure viscous behavior. In context of modified Jeffreys con˙

= 0 for stitutive equation (8) limλ→0 G (λ) = ∞ and limλ→0 GG2(λ) (λ) bounded ην (λ) function. It is important to remark that G (λ) is the smallest when the material is fully structured (λ = 1) and increases monotonically as λ is decreased [11,33]. 4. Considerations on the structural nature of thixotropic substances This section presents some aspects on structural nature of the substance, and is divided in two subsections. The first subsection approaches the structural breaking and in the second subsection the building up process is analyzed. 4.1. The structural breaking process Initially the focus is the structural degradation process. Under shear loads the links between the fluid particles are solicited until their breaking. Considering a single linear chain composed in units of α -state (length lα ) and β -state (length lβ ) under a shear stress τ . In this sense, from generalized Gibbs equation one has

du = θ ds − τ dγ ∗ + ςα dN α + ςβ dN β ,

(22)

where u is the intensive internal energy, θ is the absolute temperature, s is the intensive entropy, γ ∗ is the strain, ςα is the chemical

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H.P. Azikri de Deus et al. / Physics Letters A 380 (2016) 585–595

infrequent [46] and are disregarded here. In this sense, the Smoluchowski coagulation theory [12] claims that

d

Fig. 4. Sketch of a chain.

dt potential for α -unit, ςβ is the chemical potential for β -unit, N α is the number of α -units and N β is the number of β -units. If the β -units are interpreted as the fluid particles (i.e. lβ ≈ 0 and lα lβ ) and the α -units as the linkages, one has the configuration presented in Fig. 4. For isothermic case, one has: (i) dγ ∗ = N α dγα = Ndγ (taking off the α -index for the sake of simplicity) and (ii) N β is constant. It is also reasonable to assume that Ndγ γ dN. These comments allow to rewrite the Gibbs equation as

(u˙ − θ s˙ ) + τ N γ˙ = ς N˙ ; ψ˙ + τ N γ˙ = N˙ ,

(23)

ς

where ψ˙ represents the Helmholtz free energy rate (for isothermic case). In a more realistic point of view, there are substances (with great industrial interest) that exhibit high concentration where the chains overlap each other. In this case, the motion of these chains (or macromolecules) cannot be considered in an isolated form and the interactions due to entangled motion must be considered [34–38]. The number of entanglements increases with concentration. Aiming to take into account this effect, even for a single chain, the reptation model [39–43] was used. Following the same steps presented in [43] (Ch. 5, Sec. 5.4), and assuming that the relaxation time is ∼ N 3 [35,41], after some algebraic manipulations one has

ψ=

7

˙2

Kψ N θ γ 2

∴ ψ˙ =

7 2

;

˙ θ γ˙ 2 + K ψ N 7 θ γ˙ γ¨ , Kψ N6 N

(24)

where K ψ is a constant that incorporates some bulk effect of the structural nature. If the same idea used above is considered (i.e. ˙ γ˙ ), one has N γ¨ N

ψ˙ = K ψ N 7 θ γ˙ γ¨ .

(25)

˙ = N

 =

ς

∞ − n(x, t ) 0

n(x, 0) = no (x),

(27)

R∗+ . In this equation n(x, t ) and n( y , t ) are the

on n(x, t ), for x ∈ continuous density distributions of probability for size particles x and y, respectively and the symmetric factor K (x, y ) (i.e. K (x, y ) = K ( y , x)) is called the coagulation factor. Additional details about the existence of solution can be gotten in [12]. In this context, the following theorem is enunciated: Theorem 1. Let n(x, 0) (x ≥ 0) be a continuous, non-negative, bounded and integrable function such that

∞ N (0) =

n(x, 0)dx. 0

Then, eq. (27), with homogeneous particles size in an isothermal motion with the coagulation kernel given by

ko

K (x, y ) =

tf

 e



× e

 − px β2 −1

− py



β 2

−1



n(x, t )





n( y , t )

β

2 −1

β 2



−1



,

has a unique solution n(x, t ), which is continuous, non-negative, bounded, integrable in x for each t, and analytic in t for each x. Furthermore the following integrals

∞ N (t ) =

n(x, t )dx; 0

∞ ( p , t ) =

e − px n(x, t )dx,

0



K ψ N 6 θ γ¨ + τ N γ˙

d

.

(26)

4.2. The structural building up process From this point forward the goal is to analyze the building up process of the structure. The flocs can join to form larger flocs and/or continuous network due to Brownian motion and laminar flow regime. This process can be understood as a coagulation process (is also often referred to as coalescence or clustering). Considering the coagulation reaction in association with the law of mass action [44,45], a floc of size1 x can stick to a floc of size y to form a floc of size x + y. In this work this process is symbolically represented as: x, y → x + y. In this case, reactions that involve three or more flocs sticking together at the same time are events extremely

1

K (x, y )n( y , t )dy ;

where p is any complex number satisfying Re ( p ) ≥ 0, converge and the equation

;

ς

K (x − y , y )n(x − y , t )n( y , t )dy

2 0

Returning to the equation (23), it follows

ψ˙ + τ N γ˙

x

1

n(x, t ) =

It can be interpreted in length, area or volumetric sense.

dt

N (t ) = −

1 ko 2 tf

N β (t ),

is held, where ko , β are positive constants, since

⎛

⎞2

∞ β ⎝ (n(x, s)) 2 dx⎠ ≈ N β (t ), 0

and N (t ) ∈ L 2 (0, t ), where t ∈ (0, T ). Proof. In the Appendix A.

2

Considering N (t ) = N o − N (t ) where N o is the initial number of links per volume unit and N is the actual number of links per volume unit, as have ever mentioned in [12], and the fact that x can be interpreted as lα (length of the α -units from previous subsection). In this sense, one has

H.P. Azikri de Deus et al. / Physics Letters A 380 (2016) 585–595

d dt

N (t ) =

1 ko 2 tf

( N o − N (t ))β ,

589

(28)

where ko can be interpreted as a kind of potential for some possible additional link. Effects of energy of activation and of the orientation were not explicitly considered here. In this sense, taking into account all phenomena presented in this section one has

d dt

N (t ) =

1 tf



1 2

ko ( N o − N (t ))

or

dλ dt

 =





β

1  ∗ ko (1 − λ)β −

−



K ψ ( N (t ))6 θ γ¨ + τ N (t )γ˙

ς

,



∗ λ6 θ γ¨ + τ λγ˙ Kψ

ς

tf

with structural parameter λ defined as λ = β−1

(29)

, N (t ) , No

∗ = K N 6 . Finally, the proposed and where ko∗ = N o 2 ko and K ψ ψ o model can be presented as follows

ην G



τ˙ + 1 −

ην G˙ G2







τ = ην + 1 − + 

1  ∗ = k (1 − λ)β − dt tf o

dλ





ην ημ G

ην G˙



G2

ημ +

γ¨ ;

ην η˙ μ G



γ˙ (30)



∗ λ6 θ γ¨ + τ λγ˙ Kψ

ς

;

Fig. 5. Sketch of the configuration spaces.

with 0 ≤ λ ≤ 1,

(31)

G (λ) = G o exp mλ−1 ;

(32)

ην (λ) = ηo {exp [(α1 + α2 ) λ] − exp (α2 λ)};

(33)

ημ (λ) = ηo exp (α2 λ) .

(34)

In this sense, there are the following positive parameters: ko∗ , β , t f , ∗ , η , α and α . The choice of G = G (λ) was based on G o , m, ς , K ψ o 1 2

Prony series, widely used in viscoelastic models [47–49]. The development of ην = ην (λ) equation had similar steps/ideas of [50]. The equations for G = G (λ), ημ = ημ (λ) and ην = ην (λ) are proposed in agreement with the comments presented in section 3. 5. Some issues on thermodynamic consistence The main goal of this section is to analyze the proposed model on entropy-producing processes, and its compatibility with Clausius–Duhem inequality. As demonstrated in [51] a class of constitutive functions can be made so that the state variables evolve in a way that maximizes the rate of entropy production. According to this procedure, once known the functional forms of the Helmholtz potential and the rate of dissipation, the rate of dissipation is maximized in agreement with any constraints that the thermodynamic process is subjected. In a strictly mechanical point of view, the material behavior is characterized by the constitutive informations contained in the Helmholtz potential and the rate of dissipation functions. Thinking in a closed system, the entropy can increase until to achieve its maximum equilibrium value, and the fastest form in which this maximum is reached is determined by maximizing of the rate of dissipation. Differently from the theorems of Orsanger [52] that are related to quadratic forms of entropy production, the presented tools in [51] can be used to obtain nonlinear laws and even yield-type phenomena. In this sense, the context of natural configuration [53–55] is took account. The time-dependent nature of a thixotropic substance is intimately related to its structural level, or in terms of model, related to the structural parameter. An important definition used in this

development is the concept of natural configurations. A natural configuration, associated to the actual/current configuration of the body, can be interpreted as the configuration that it would reach when the external stimuli are removed. It is important to stand out that the natural configuration, associated to a current configuration, is dependent of the manner in which the external loads are removed (very fast, quasi-statically, adiabatically, etc.), and in this sense is not unique [54]. In following figure (Fig. 5) is presented a sketch of the configurations spaces used in this approach for the body B , where kˆ o (B ) is the reference configuration, kˆ t (B ) is the current/actual configuration, kˆ n(t ) (B ) is a family of natural configurations. In isothermal process one supposes that kˆ n(t ) (B ) = kˆ n(t ) (B )(λ), where λ is a function of the flow history, or in other words it is supposed that the family of natural configurations can be parametrized by the structural level of the substance (λ). It is important to comment that this type of procedure can be in association with many particularities of the considered system (fibers alignment or distribuand tion, entanglements topology, etc.). In this context, F kˆ , F kˆ o

n(t )

F kˆ →kˆ represent the second order tensor gradient of the motion o n(t ) from reference configuration to current configuration, from natural configuration to current configuration and from reference configuration to natural configuration respectively. In this sense, denoting for ξ -th relaxation mechanism ξ ξ ξ ξ

B kˆ

= ξ F kˆ

ξ

n(t )

C kˆ

= ξ F Tˆ

ξ

n(t )

L kˆ

n(t )

= ξ F˙ kˆ

D kˆ

n(t )

F Tˆ

;

(35)

F kˆ

;

(36)

kn(t )

n(t )

kn(t )

n(t )

ˆ

o →kn(t )

= sym



ξ

ξ

1 F− ; kˆ o →kˆ n(t )

(37)



L kˆ

n(t )

(38)

,

since ξ B kˆ = ξ B kˆ (t ), one has n(t ) n(t ) ξ

B˙ kˆ

n(t )

1 = ξ F˙ kˆ ξ C − ˆ

ko →kˆ n(t )

o

ξ

1 ξ T F Tˆ + ξ F kˆ ξ C˙ − Fˆ o ko ko kˆ o →kˆ n(t )

1 + ξ F kˆ ξ C − ˆ o

ko →kˆ n(t )

ξ

F˙ Tˆ , ko

(39)

from the fact ξ

1 ξ C kˆ →kˆ = ξ F Tˆ ˆ ξ F kˆ →kˆ and ξ F kˆ →kˆ = ξ F − F kˆ . o o o o n(t ) n(t ) n(t ) ko →kn(t ) kˆ n(t )

(40) From Eq. (39) and noting that ξ

1ξ F− B kˆ ˆ ko

n(t )

1 = ξ C− ˆ

ko →kˆ n(t )

ξ

F Tˆ ; ko

(41)

590

ξ

H.P. Azikri de Deus et al. / Physics Letters A 380 (2016) 585–595

ξ

B kˆ

n(t )

T 1 F− = ξ F kˆ ξ C − , o kˆ o kˆ o →kˆ n(t )

(42)

it follows that ξ

B˙ kˆ

n(t )

1 ξ T L T + ξ F kˆ ξ C˙ − Fˆ , o ko kˆ o →kˆ n(t ) n(t )

= L ξ B kˆ

+ ξ B kˆ

n(t )

(43)

1 where L = F˙ kˆ F − . ˆ o k o

Then, from Clausius–Duhem inequality one has the specific rate of dissipation 

˙ =T : D −  r   = T− 2 ,Iξ +Iξ ,II ξ ξ B ˆ k ξ =1



At this point, taking account the “Oldroyd time derivative” (()=

˙ − L () − () L T ), one has () ξ

Bˆ

kn(t )

1 = ξ F kˆ ξ C˙ − ˆ

ko →kˆ n(t )

o

ξ

F Tˆ .

+ III ξ ,III ξ I

(44)

ko

+

1 1 1 ξ˙ C˙ − = −ξ C − C kˆ →kˆ ξ C − ; o n(t ) kˆ o →kˆ n(t ) kˆ o →kˆ n(t ) kˆ o →kˆ n(t ) 1 = −ξ F − ˆ

ko →kˆ n(t )

(46) where ξ C˙ kˆ →kˆ o n(t )

=

ξ F˙ T ξF kˆ o →kˆ n(t ) kˆ o →kˆ n(t )

ξ FT ξ F˙ . In kˆ o →kˆ n(t ) kˆ o →kˆ n(t )

+

this sense, it follows from (36) ξ

T 1 ξ˙ F− C kˆ →kˆ ξ F − = 2ξ D kˆ ; o n(t ) n(t ) kˆ o →kˆ n(t ) kˆ o →kˆ n(t )

1 ∴ C˙ − ˆ

ξ

ko →kˆ n(t )

1 T ξ = −2 F − D kˆ ξ F − . n(t ) kˆ o →kˆ n(t ) kˆ o →kˆ n(t )

(48)

2 ,Iξ +Iξ ,II ξ ξ B ˆ − ,II ξ ξ B 2ˆ kn(t ) kn(t ) n(t )

 kˆ = T − t

r  

ξ =1

2 ,Iξ +Iξ ,II ξ ξ B ˆ − ,II ξ ξ B 2ˆ kn(t ) kn(t )

Then from eq. (44) ξ



Bˆ

kn(t )

= −2ξ F kˆ

ξ

n(t )

ξ

D kˆ

n(t )

F Tˆ

kn(t )

Returning to eq. (43), one has ξ

B˙ kˆ

n(t )

= L ξ B kˆ

n(t )

+ ξ B kˆ

n(t )

+ III ξ ,III ξ I

(49)

.

L T − 2ξ F kˆ

ξ

n(t )

ξ

D kˆ

F Tˆ

kn(t )

n(t )

(50)

.

(52)

n(t )

n(t )

ξ =1

(51)

where follows from chain rule , ξ B kˆ

n(t )

 = ,Iξ +Iξ ,II ξ I −

T , with Iξ = tr(ξ B ˆ ), II ξ = ,II ξ ξ B kˆ + III ξ ,III ξ ξ B − kn(t ) kˆ n(t ) n(t )   1 tr(ξ B kˆ )2 − tr(ξ B 2ˆ ) and III ξ = det(ξ B ˆ ). Without lost 2 k kn(t )

n(t )

n(t )

of generality it is chosen kˆ n(t ) such that ξ F kˆ

n(t )

ξB

=ξV

kˆ n(t )

(∴ ξ C

kˆ n(t )

=

), i.e. for an appropriate rotated natural configuration kˆ n(t ) , kˆ n(t )

n(t )

ξ

Bˆ

kn(t )

ξ

: ξ B˙ kˆ

n(t )

T B− : ξ B˙ kˆ ˆ kn(t )

n(t )

(54)

r   2 ,Iξ +Iξ ,II ξ ξ B ˆ k

n(t )

ξ =1

  + III ξ ,III ξ I : D − ξ D kˆ

n(t )

kn(t )

≥ 0,

(60)

kˆ = 0 for no changes in kˆ n(t ) (B ), one has t

T=

r  

2 (,Iξ +Iξ ,II ξ )ξ B ˆ − ,II ξ ξ B 2ˆ kn(t ) kn(t )

 + III ξ ,III ξ I , 

and denoting ξ τ = 2[ ,Iξ +Iξ ,II ξ

III ξ ,III ξ I ], it can be written T=

r 

ξ

τ.

(61)

ξ

Bˆ

kn(t )

− ,II ξ ξ B 2ˆ

kn(t )

+

(62)

ξ =1

− ,II ξ ξ B 2ˆ

The rate of dissipation due to each relaxation mechanism can be written as

kn(t )

.

− ,II ξ ξ B 2ˆ

is the rate of dissipation due to changes in kˆ n(t ) (B ). In this context, it is suitable imagine that each one rate of dissipation associated to a specific process has a non-negative value as a requirement, for each relaxation mechanism ξ . Supposing that any mechanical dissipation in the actual configuration (kˆ t (B )) implicates in a changing of the natural configuration (kˆ n(t ) (B )), or in other words

(55)

n(t )

(59)

where, kˆ is the rate of dissipation due to changes in kˆ t (B )), λ is t the rate of dissipation due to changes in micro-structure and kˆ n(t )

(53)

Returning to eq. (52), one has (details in the Appendix B – iv)

˙ = ,λ λ˙ + 

ξ =1

ξ =1

 = 2ξ B 2ˆ : D − ξ D kˆ ; n(t ) kn(t )  = 2I : D − ξ D kˆ .

(58)

n(t )

n(t )



I : ξ B˙ ˆ = 2ξ B kˆ : D − ξ D kˆ ; kn(t ) n(t ) n(t )

: D ≥ 0;

 + III ξ ,III ξ I : D kˆ

one has (details in the Appendix B – i , ii, iii)



 

λ = −,λ λ˙ ≥ 0; r   kˆ = 2 ,Iξ +Iξ ,II ξ ξ B ˆ k

In this point, it suppose that the extensive thermodynamical Helmholtz potential ( ) is such that

 = (λ, I1 , II 1 , III 1 , . . . , Ir , II r , III r ); r  ˙ = , λ λ˙ + ∴ , ξ B kˆ : ξ B˙ kˆ ,

(57)

,

where T is the Cauchy stress tensor and D is the symmetric part of velocity gradient (L). This approach makes possible the analysis of the constitutive law and the non-negativity of the entropy production in the context of irreversible thermodynamic processes. In this sense, it is assumed that total rate dissipation  can be divided in three parts, each one associated to a specific process, as follows

(47)

ξ

kn(t )

: D − ,λ λ˙

 + III ξ ,III ξ I : ξ D kˆ

(45)

T 1 ξ˙ ξ −T F− C kˆ →kˆ ξ F − Fˆ ˆ , o n(t ) kˆ o →kˆ n(t ) kˆ o →kˆ n(t ) ko →kn(t )

ξ

 

− ,II ξ ξ B 2ˆ

r  

ξ =1

Noting that ξ

n(t )

(56)

ξ

kˆ

n(t )

= ξ τ : ξ D kˆ

n(t )

.

(63)

H.P. Azikri de Deus et al. / Physics Letters A 380 (2016) 585–595

In agreement to the theorems 6.1 and 6.2 from [51], which attest existence, unicity and characterization to the maximum value of entropy production, the problem of maximum value for ξ kˆ can

Taking

be seen as a constrained problem of maximum value in following sense

Gξ

n(t )

ξ

∗ˆ

kn(t )

= ξ kˆ



− ℵ1

n(t )

ξ

∗ˆ

kn(t )

n(t )



τ=

ξ

kˆ

ℵ1

n(t )

, ξ D kˆ

(65)

.

n(t )

In this study are proposed the following  and kˆ

= ξ

⎧ 1⎨ 2⎩

kˆ

n(t )

where

n(t )

=

ξ =1

ξ =1

kˆ n(t )



2ηξ D kˆ : n(t )

ξ

ξ

B kˆ

D kˆ

n(t )

(67)

,

n(t )

:

ξB

kˆ n(t )

ξD

, G ξ and

kˆ n(t )

n(t )

definition of natural configuration [56]. From eq. (65) one has



ξ

τ = 4 ηξ

1 + ℵ1



ξ

ℵ1

ξ

B kˆ

n(t )

D kˆ

n(t )

(68)

,

and on the other side from eq. (62) ξ

τ = − A I + 2G ξ ξ B kˆn(t ) .

(69)

Thus, it can be written

− A I + 2G ξ B kˆ

n(t )

∴ ξ D kˆ

n(t )

=



Gξ



= 4 ηξ

ξ

1 + ℵ1



ℵ1

I−

1 + ℵ1

2 ηξ

 ξ

ℵ1 A

ξ

B kˆ

n(t )



D kˆ

n(t )



ℵ1

ξ

1 + ℵ1

4 ηξ

; 1 B− . ˆ kn(t )

(70)

Returning to Eqs. (50) and (39), one has (details in the Appendix B – vi)



ηξ



1 + ℵ1

ξ

ℵ1

Gξ



Bˆ

kn(t )

+ ξ B kˆ

n(t )

=

A ξ F kˆ

2G ξ

ξ

n(t )

1 ξ T B− Fˆ . kn(t ) kˆ n(t )

(71)

From eqs. (62) and (69), it follows ξ

τ = − A I + 2G ξ ξ B kˆ

n(t )

∴ ξ B kˆ

n(t )

=

1 ξ τ + AI , 2G ξ

(72)

for each relaxation mechanism. Substituting eq. (72) in eq. (71), one has (details in the Appendix B – vii)



ηξ



1 + ℵ1

ℵ1 

Gξ

=2



ξ

τ + 1−

ηξ

1 + ℵ1

Gξ

ℵ1

ηξ G˙ ξ



G 2ξ



1 + ℵ1



ξ

ℵ1

τ

AD

   ˙  ηξ 1 + ℵ1 ˙ − Gξ A I + −A − A Gξ ℵ1 Gξ + A ξ F kˆ

n(t )

ξ

1 ξ T B− Fˆ ˆ kn(t )

kn(t )

.

kn(t )

kn(t )

(74)

.



ξ

τ = ηξ D .

(75)





(76)

τ μ = ημ D .

(77)

G

τ = τm + τμ one has      ην η˙ μ ην η G˙ η G˙ τ + 1 − ν 2 τ = ην + 1 − ν 2 ημ + D 

(73)

G

G

+

ηξ are associated to

elastic and viscous effects, respectively. Note that, in this case  and kˆ n(t ) (where ξ F kˆ = ξ V kˆ ) are in agreement with the formal n(t )

1 ξ T B− Fˆ ˆ

ην η G˙ τ m + 1 − ν 2 τ m = ην D ;

G

n(t )

ξ

n(t )

Therefore, as



ξ

F kˆ

Considering two relaxation mechanisms (in agreement with Fig. 1),  = η , G  = G, η = η and G  → +∞, it follows where ηm ν μ m v v

G

(66)

τ

ξ

2

Gξ

f (λ) is a function of the  microstructural parameter,

= 2ηξ ξ D kˆ

ξ

⎫   ⎬ 2G ξ Iξ − 3 − A log |III ξ | ; ⎭

r  

f (λ) + r 

functionals

ξ

Remembering that ξ F kˆ = ξ V kˆ , and after some algebraic man(t ) n(t ) nipulation, for each ξ -relaxation mechanism, it can be shown that

Gξ



1 + ℵ1

2

and G ξ = 4G ξ = 2 A it can be written



G ξ

I+

ηξ ξ ηξ G˙ ξ τ + 1 −  2

one has (details in the Appendix B – v) ξ

G ξ



= 0,

, ξ D kˆ



ℵ1

Gξ

= ηξ D −

(64)

where ℵ1 is the Lagrange multiplier. Thus, claiming the necessary condition

1+ℵ1

ηξ ξ ηξ G˙ ξ τ + 1 −  2



τ : ξ D kˆn(t ) − ξ kˆn(t ) ,

ξ



ηξ = 4ηξ 

591

ην ημ G

D.

G

(78)

Note that in this analysis the isochoric motion constraint is not required, and in this sense, the proposed model is coherent with the fact of the thixotropic behavior be sensible to compressible effects [57,58]. 6. Conclusion The main objective of this paper is to propose a constitutive model for thixotropic fluids consistent with the expected rheological behavior for this type of substance and with some theoretical issues. In this context, its important to stand out some non-standard points of the approach presented in this work for thixotropic modeling in relation to the others models: i the micro-structural dependence of the shear modulus (G (λ)) and the viscosity coefficients (ημ (λ) and ην (λ)) are taken into account in the development of the new constitutive equation (Eq. (8)), and not only via external functionals. This is much more adequate in terms of the expected behavior of thixotropic substances; ii the development of the rate equation (Eq. (29)) was theoretically based in well-established physical principles as the Gibbs equation, the reptation model and the Smoluchowski theory of coagulation, making clear where the effect of the Brownian motion is taken into account; iii the development of a three dimensional version of the model, besides the proof of its thermodynamic consistence (Sec. 5). The proposed model was analyzed, in a theoretical sense, and shows coherence with physical behavior expected for a thixotropic fluid, as can be seen along this work. This present paper do not explain completely the set of characteristics involved with the thixotropy. There are aspects as the influence of the temperature and the existence of more than one type of micro-structure in a same substance, to cite some examples, that were not considered here. However, this work presents some new perspectives for thixotropic modeling that can be extended and generalized in future works.

592

H.P. Azikri de Deus et al. / Physics Letters A 380 (2016) 585–595

t ∞ ∞

Appendix A

−

The existence and uniqueness of a solution n(x, t ), which is continuous, non-negative, bounded, integrable in x for each t, and analytic in t for each x was first proven in [59]. From hypotheses, one can conclude that ∃C > 0, real constant, such that 0 ≤ n(x, t ) < C , ∀x ≥ 0 and ∀t ≥ 0. Due to the bounds of n(x, t ), one can write

0

2

0

≤ 0

0

xn(x, t )dx ≤ C + ρ ,

0

K (x, y ) =

where

0

e

tf

 − px β2 −1

( p , t ) = ( p , 0) + ko

t n(x, t ) = no (x) +

⎛

⎝1

2

∞ − n(x, s)

0

⎛ ⎝

tf

0

∞ e

− px

β

−1 2

⎞



t

2

− px

0

no (x)dx

=−

0

t ∞ x

2 0

0

− 0

e − px K (x − y , y )n(x − y , s)n( y , s)dydxds

0

t ∞ ∞

e − px K (x, y )n(x, s)n( y , s)dydxds;

0

= ( p , 0)

t ∞ x 1 e − px K (x − y , y )n(x − y , s)n( y , s)dydxds +

t ∞ |e 0

− 0

0

e − px K (x, y )n(x, s)n( y , s)dydxds;

2

0

tf

⎛∞ ⎞2

β ⎝ (n(x, s)) 2 dx⎠ ds ∴ d N (t ) dt

0

N β (t ).

0

t

0

( N (s))2 ds ≤ +∞;

n(x, s) N (s)|dxds ≤ 0

|e − px n(x − y , s)n( y , s)|dydxds

x

t ∞ ∞ ≤ 0

0

|e − p (x− y )n(x − y , s)||e − py n( y , s)|dydxds

x

t ( N (s))2 ds ≤ +∞.

≤ 0

0

= ( p , 0)

t ∞ ∞ 1 e − p (x+ y ) K (x, y )n(x, s)n( y , s)dydxds + 0

2 tf

0

0

t ∞ ∞

− px

t ∞ ∞

2

0

1 ko

1

Note that

∴ ( p , t )

0

β

n( y , s) 2 dy ⎠ ds;

∴ N (t ) = N (0) − ko

0

0

0



1

0



⎛∞ ⎞

β − px ⎝ (e n(x, s)) 2 dx⎠

0

e − px n(x, t )dx

1

−1

⎞

∞ β ⎝ (e − px n(x, s)) 2 dx⎠

1

− ko

K (x, y )n( y , s)dy ⎠ ds.

Then

e

2

⎛

t

⎞

0

∞

β

⎞

∞ β × ⎝ (e − px n( y , s)) 2 dy ⎠ ds

K (x − y , y )n(x − y , s)n( y , s)dy 0

1 tf

0

⎛

x

2

0

+

t

1

that was proven in [60], i.e. the solution conserves the density. Integrating over t > 0 and from (27), one has

=



n(x, t )

where ko carries information about to some power such as the third. In this sense, it follows

ρ (t ) = 0,

∞



ko

xn(x, t )dx,

satisfying

d

0

    β − py β2 −1 −1 , n( y , t ) 2 × e

∞

dt

K (x, y )e − px n(x, s)n( y , s)dydxds.

Considering a special type of coagulation kinetics

1

ρ (t ) =

0

−

∞ n(x, t )dx +

0

t ∞ ∞

n(x, t )dx 0

1

0

= ( p , 0)

t ∞ ∞ 1 K (x, y )e − px n(x, s)e − py n( y , s)dydxds +

∞ 0 ≤ N (t ) = (0, t ) =

0

e − px K (x, y )n(x, s)n( y , s)dydxds;

0

Appendix B Here are presented the intermediate steps of some results used in the section 5.

H.P. Azikri de Deus et al. / Physics Letters A 380 (2016) 585–595





T : ξ B˙ kˆ ∴ ξ B− ˆ

. i ) I : ξ B˙ ˆ = 2ξ B kˆ : D − ξ D kˆ kn(t ) n(t ) n(t )

kn(t )

  = I : L ξ B kˆ + ξ B kˆ L T − 2ξ F kˆ ξ D kˆ ξ F Tˆ ; n(t ) n(t ) n(t ) kn(t ) n(t ) n(t )    − 2ξ F kˆ ξ D kˆ ξ F Tˆ ; ∴ I : ξ B˙ kˆ = I : 2sym L ξ B kˆ n(t ) n(t ) n(t ) kn(t ) n(t )   ; ∴ I : ξ B˙ kˆ = I : 2D ξ B kˆ − 2ξ F kˆ ξ D kˆ ξ F Tˆ n(t ) n(t ) n(t ) kn(t ) n(t )     − tr ξ F kˆ ξ D kˆ ξ F Tˆ ; ∴ I : ξ B˙ kˆ = 2 tr D ξ B kˆ n(t ) n(t ) n(t ) kn(t ) n(t )     − tr ξ F kˆ ξ D kˆ ξ F Tˆ ; ∴ I : ξ B˙ kˆ = 2 tr D ξ B kˆ n(t ) n(t ) n(t ) kn(t ) n(t )     ; ∴ I : ξ B˙ kˆ = 2 tr ξ B kˆ D − tr ξ F Tˆ ξ F kˆ ξ D kˆ n(t ) n(t ) n(t ) kn(t ) n(t )   ; ∴ I : ξ B˙ kˆ = 2 ξ B Tˆ : D − ξ C Tˆ : ξ D kˆ n(t ) kn(t ) kn(t ) n(t )  ; ∴ I : ξ B˙ kˆ = 2 ξ B kˆ : D − ξ C kˆ : ξ D kˆ n(t ) n(t ) n(t ) n(t )  . ∴ I : ξ B˙ kˆ = 2ξ B kˆ : D − ξ D kˆ

 I : ξ B˙ kˆ

n(t )

n(t )

593

= 2I : ( D − ξ D kˆ

n(t )

n(t )

). 

r   2 (,Iξ +Iξ ,II ξ )ξ B ˆ k

˙ = ,λ λ˙ + iv) 

ξ =1

 + III ξ ,III ξ I : ( D − ξ D kˆ

n(t )

).

n(t )

ξ =1



T + III ξ ,III ξ ξ B − ˆ

kn(t )

: ξ B˙ kˆ

n(t )

;

r  

˙ = ,λ λ˙ + ∴

ξ =1

2(,Iξ +Iξ ,II ξ )ξ B kˆ

kn(t )

n(t )



: B˙ kˆ

= B kˆ

r   2 (,Iξ +Iξ ,II ξ )ξ B ˆ k

n(t )

 +III ξ ,III ξ I : ( D − ξ D kˆ



Bˆ

kn(t )

ξ

n(t )

: B˙ kˆ

∴ B kˆ

ξ

n(t )

n(t )

T B− kˆ n(t )

ξ

n(t )

kn(t )

: B˙ kˆ

=2

ξ

B 2ˆ kn(t )

kn(t )

: ( D − D kˆ ξ

ξ

∴

T B− kˆ n(t )

). 

n(t )

ξ

v) ξ τ =



ξ



1 + ℵ1 ξ kˆ , ξ D kˆ . n(t ) n(t )

ℵ1

∗ˆ , ξ D kˆ n(t ) kn(t )

= 0;

∴ 0 = ξ kˆ , ξ D kˆ n(t ) n(t )  − ℵ1 ξ τ : ξ D kˆ , ξ D kˆ n(t )

∴ 0 = ξ kˆ

n(t )

; − ξ kˆ , ξ D kˆ n(t ) n(t ) ξ − ℵ1 τ : ( I ⊗ I ) − ξ kˆ n(t )

, ξ D kˆ

n(t )

n(t )

, ξ D kˆ

n(t )

 ;

∴ 0 = ξ kˆ , ξ D kˆ − ℵ1 [ξ τ − ξ kˆ , ξ D kˆ ]; n(t ) n(t ) n(t ) n(t )   1 + ℵ1 ξ kˆ , ξ D kˆ .  ∴ ξτ = n(t ) n(t ) ℵ1

).  vi)



ηξ



1 + ℵ1 ξ Bˆ

ℵ1

Gξ

kn(t )

+ ξ B kˆ

n(t )

=

A ξ F kˆ

2G ξ

ξ

n(t )

1 ξ T B− Fˆ ˆ

kn(t )

kn(t )

.

n(t )

kn(t )

ξ

kn(t )

n(t )

= 2I : ( D − D kˆ ). n(t )  ξ ξ −T ξ ˙ ξ −T  B ˆ : B kˆ = B ˆ : L B kˆ + ξ B kˆ L T n(t ) n(t ) kn(t ) kn(t ) n(t ) ; − 2ξ F kˆ ξ D kˆ ξ F Tˆ n(t ) n(t ) kn(t )  T ξ˙ ξ −T ∴ ξ B− : B = B ˆ : 2sym( L ξ B kˆ ) ˆkn(t ) ˆkn(t ) n(t ) kn(t )  ; − 2ξ F kˆ ξ D kˆ ξ F Tˆ n(t ) n(t ) kn(t )  T ξ˙ ξ −T ∴ ξ B− : B = B ˆ : 2D ξ B kˆ − 2ξ F kˆ ξ D kˆ ξ F Tˆ ; kˆ n(t ) n(t ) n(t ) n(t ) kn(t ) kˆ n(t ) kn(t )   T : ξ B˙ kˆ = 2 tr ξ B −1 kˆ D ξ B kˆ ∴ ξ B− n(t ) kˆ n(t ) n(t ) n(t )   ; − tr ξ B −1 kˆ ξ F kˆ ξ D kˆ ξ F Tˆ n(t ) n(t ) kn(t ) n(t )   T ∴ ξ B− : ξ B˙ kˆ = 2 tr( D ) − tr(ξ D kˆ ) ; ˆ

iii)

ξ



n(t )

n(t )

n(t )

ξ

ξ

− ,II ξ ξ B 2ˆ

T

: L B kˆ + B kˆ L n(t ) n(t ) ; − 2ξ F kˆ ξ D kˆ ξ F Tˆ n(t ) n(t ) kn(t )  ξ ξ˙ ξ ∴ B kˆ : B kˆ = B kˆ : 2sym( L ξ B kˆ ) n ( t ) n(t ) n(t ) n(t )  ξ ξ ξ T ; − 2 F kˆ D kˆ Fˆ n(t ) n(t ) kn(t )  ; ∴ ξ B kˆ : ξ B˙ kˆ = ξ B kˆ : 2D ξ B kˆ − 2ξ F kˆ ξ D kˆ ξ F Tˆ n(t ) n(t ) n(t ) n(t ) kn(t ) n(t ) n(t )   ∴ ξ B kˆ : ξ B˙ kˆ = 2 tr ξ B Tˆ D ξ B kˆ n(t ) n(t ) kn(t ) n(t ) ξ T ξ  ξ ξ T − tr B ˆ F kˆ D kˆ Fˆ ; n(t ) n(t ) kn(t ) kn(t )     ; ∴ ξ B kˆ : ξ B˙ kˆ = 2 tr ξ B 2ˆ D − tr ξ B 2ˆ ξ D kˆ n(t ) n(t ) kn(t ) kn(t ) n(t )  ; ∴ ξ B kˆ : ξ B˙ kˆ = 2 ξ B 2ˆ : D − ξ B 2ˆ : ξ D kˆ ξ

)

 );

ξ =1

ξ

n(t )



n(t )

+ 2III ξ ,III ξ I : ( D − ξ D kˆ ˙ = ,λ λ˙ + ∴

: ( D − ξ D kˆ

n(t )

 : D − ξ D kˆ

− 2,II ξ ξ B 2ˆ

ii) ξ B ˆ : ξ B˙ kˆ = 2ξ B 2ˆ : ( D − ξ D kˆ ). kn(t ) n(t ) n(t ) kn(t )

kn(t )

r   (,Iξ +Iξ ,II ξ ) I − ,II ξ ξ B kˆ

˙ = ,λ λ˙ +  

n(t )

− ,II ξ ξ B 2ˆ

n(t )

n(t )

n(t )

: B˙ kˆ ξ

n(t )

= 2( I : D − I : D kˆ ξ

n(t )

);

 ξ B˙ kˆ

n(t )

∴ ξ B˙ kˆ

= L ξ B kˆ

n(t )

+ ξ B kˆ

n(t )

L T − 2ξ F kˆ

n(t )

ξ

D kˆ

n(t )

ξ

F Tˆ

kn(t )

;

− ξ B kˆ L T = −2ξ F kˆ ξ D kˆ ξ F Tˆ ; n(t ) n(t ) kn(t ) n(t )     Gξ ℵ1 ∴ ξ B ˆ = −2ξ F kˆ I ξ F Tˆ n(t ) 2η kn(t ) kn(t ) 1 + ℵ1 ξ     ℵ1 A ξ −1 ξ T + 2ξ F kˆ Bˆ Fˆ ; n(t ) 4η kn(t ) kn(t ) 1 + ℵ1 ξ   Gξ ℵ1 ξ ∴ ξ Bˆ = − B kˆ n(t ) kn(t ) ηξ 1 + ℵ1   ℵ1 A 1 ξ T ξ + F kˆ ξ B − Fˆ ; n(t ) kn(t ) kˆ n(t ) 2ηξ 1 + ℵ1   ηξ 1 + ℵ1 ξ A ξ ξ ∴ F ˆ ξ B −1 ξ F T .  B ˆ + B kˆ n(t ) = kn(t ) Gξ ℵ1 2G ξ kn(t ) kˆ n(t ) kˆ n(t ) n(t )

− L ξ B kˆ

n(t )

594

H.P. Azikri de Deus et al. / Physics Letters A 380 (2016) 585–595

vii)

ηξ



Gξ

=2





η G˙ 1 + ℵ1 ξ τ + 1 − ξ 2ξ ℵ1 Gξ

ηξ



1 + ℵ1

ℵ1 

Gξ

 ηξ + −A − ξ

n(t )



ηξ

ηξ

1 2G ξ



1 + ℵ1

ηξ

1



ℵ1

Gξ

τ

˙− A

G˙ ξ Gξ

 A

I



−

G˙ ξ ξ 1 ξ ( τ + AI) + ( τ + A˙ I + A I ) 2 2G ξ 2G ξ



A ξ F kˆ

2G ξ

ξ

1 ξ T B− Fˆ ˆ kn(t )

n(t )

kn(t )

; 

G˙ ξ  1 ξ − 2 ξ τ + AI + ( τ + A˙ I − 2 A D ) 2G ξ 2G ξ

(ξ τ + A I ) =

1 + ℵ1



ℵ1



ℵ1 2G ξ

ξ

ℵ1

AD

(ξ τ + A I ) =

1 + ℵ1



1 ξ T B− Fˆ . kn(t ) kˆ n(t )

ℵ1

Gξ

+ ∴

1 + ℵ1

Gξ

+ ∴



1 + ℵ1



Gξ

+ A ξ F kˆ



 −

A ξ F kˆ

2G ξ

G˙ ξ ξ Gξ

ξ

n(t )



1 ξ T B− Fˆ ; kn(t ) kˆ n(t )





τ + A I + (ξ τ + A˙ I − 2 A D )

1 ξ T Fˆ ; + ( τ + A I ) = A F kˆ ξ B − n(t ) kn(t ) kˆ n(t )      ηξ 1 + ℵ1 ξ ηξ G˙ ξ 1 + ℵ1 ξ τ + 1− 2 ∴ τ Gξ ℵ1 ℵ1 Gξ   ηξ 1 + ℵ1 =2 AD Gξ ℵ1    ˙  ηξ 1 + ℵ1 ˙ − Gξ A I + −A − A Gξ ℵ1 Gξ

ξ

ξ

+ A ξ F kˆ

n(t )

ξ

1 ξ T B− Fˆ . kn(t ) kˆ n(t )

References [1] X. Zhang, W. Li, X. Gong, Thixotropy of MR shear-thickening fluids, Smart Mater. Struct. 19 (125012) (2010) 1–6. [2] H. El-Gendy, M. Alcoutlabi, M. Jemmett, M. Deo, J. Magda, Rama Venkatesan, A. Montesi, The propagation of pressure in a gelled waxy oil pipeline as studied by particle imaging velocimetry, AIChE J. 58 (3) (2012) 302–311. [3] Q. Nguyen, D. Boger, Thixotropic behaviour of concentrated bauxite residue suspensions, Rheol. Acta 24 (1985) 427–437. [4] A. Mujumdar, A. Beris, A. Metzner, Transient phenomena in thixotropic systems, J. Non-Newton. Fluid Mech. 102 (2002) 157–178. [5] P.R. de Souza Mendes, Modeling the thixotropic behavior of structured fluids, J. Non-Newton. Fluid Mech. 164 (2009) 66–75. [6] J. Mewis, Thixotropy: a general review, J. Non-Newton. Fluid Mech. 6 (1979) 1–20. [7] H. Barnes, Thixotropy: a review, J. Non-Newton. Fluid Mech. 70 (1997) 1–33. [8] E. Toorman, Modeling the thixotropic behaviour of dense cohesive sediment suspensions, Rheol. Acta 36 (1997) 56–65. [9] R. Ritter, J. Batycky, Numerical prediction of the pipeline flow characteristics of thixotropic liquids, SPE J. 7 (1967) 369–376. [10] K. Dullaert, J. Mewis, A structural kinetics model for thixotropy, J. Non-Newton. Fluid Mech. 139 (2006) 21–30. [11] P.R. de Souza Mendes, Thixotropic elasto-viscoplastic model for structured fluids, Soft Matter 7 (2011) 2471–2483. [12] H.A. de Deus, G. Dupim, On behavior of the thixotropic fluids, Phys. Lett. A 337 (2013) 478–485. [13] H.A. de Deus, G. Dupim, Some aspects over thixotropic fluid behavior, Adv. Mater. Res. 629 (2013) 623–634. [14] H.A. de Deus, G. Dupim, Over structural nature of the thixotropic fluids behavior, Appl. Math. Sci. 6 (2012) 6871–6889. [15] H.A. Ardakani, E. Mitsoulis, S.G. Hatzikiriakos, Thixotropic flow of toothpaste through extrusion dies, J. Non-Newton. Fluid Mech. 166 (2011) 1262–1271. [16] G. Oliveira, L. Rocha, A. Franco, C. Negrão, Numerical simulation of the start-up of Bingham fluid flows in pipelines, J. Non-Newton. Fluid Mech. 165 (2010) 1114–1128.

[17] J. Mewis, N. Wagner, Thixotropy, Adv. Colloid Interface Sci. 147—148 (2009) 214–227. [18] K. Dullaert, J. Mewis, Thixotropy: build-up and breakdown curves during flow, J. Rheol. 49 (6) (2005) 1213–1230. [19] A. Potanin, 3D simulations of the flow of thixotropic fluids, in large-gap Couette and vane-cup geometries, J. Non-Newton. Fluid Mech. 165 (2010) 299–312. [20] J. Derksen Prashant, Simulations of complex flow of thixotropic liquids, J. NonNewton. Fluid Mech. 160 (2009) 65–75. [21] H. Barnes, The yield stress – a review or ‘panta rei’ – everything flows?, J. Non-Newton. Fluid Mech. 81 (1999) 133–178. [22] P. Patil, J. Feng, S. Hatzikiriakos, Constitutive modeling and flow simulation of polytetrafluoroethylene (PTFE) paste extrusion, J. Non-Newton. Fluid Mech. 139 (2006) 44–53. [23] G. Marrucci, The free energy constitutive equation for polymer solutions from the dumbell model, Trans. Soc. Rheol. 16 (1972) 321–330. [24] G. Marrucci, G. Titomanlio, G. Sarti, Testing of a constitutive equation for entangled networks by elongational and shear data of polymer melts, Rheol. Acta 12 (1973) 269–275. [25] D. Cheng, On the Behaviour of Thixotropic Fluids with a Distribution of Structure, Modern Mechanics and Mathematics Series, vol. 3, Chapman & Hall/CRC, 2000. [26] D. Cheng, On the behaviour of thixotropic fluids with a distribution of structure, J. Phys. D, Appl. Phys. 7 (1974) 155–158. [27] D. Cheng, F. Evans, Phenomenological characterization of the rheological behavior of inelastic reversible thixotropic an antithixotropic fluids, Br. J. Appl. Phys. 16 (1965) 1599–1617. [28] C. Truesdell, W. Noll, The Non-Linear Field Theories of Mechanics, 3rd edition, Springer, 2010. [29] M. Silhavy, The Mechanics and Thermodynamics of Continuous Media, Texts and Monographs in Physics, Springer, 1949. [30] E. Soares, R. Thompson, A. Machado, Measuring the yielding of waxy crude oils considering its time-dependency and apparent-yield-stress nature, Appl. Rheol. 26 (3) (2013) 62798.1–62798.11. [31] P. Olmsted, Perspectives on shear banding in complex fluids, Rheol. Acta 47 (2008) 283–300. [32] D. Cheng, Characterisation of thixotropy revisited, Rheol. Acta 42 (2003) 372–382. [33] P.R. de Souza Mendes, R.L. Thompson, A unified approach to model elastoviscoplastic thixotropic yield-stress materials and apparent yield-stress fluids, Rheol. Acta 53 (2013) 673–694. [34] M. Doi, S. Edwards, Theory of Polymer Dynamics, 1st edition, Clarendon, 1996. [35] M. Rubinstein, R. Colby, Polymer Physics, 1st edition, Oxford University Press, 2003. [36] V. Pokrovskii, A justification of the reptation-tube dynamics of a linear macromolecule in the mesoscopic approach, Physica A 366 (2006) 88–106. [37] H. Deutsch, K. Binder, Interdiffusion and selfdiffusion in polymer mixtures: a Monte Carlo study, J. Chem. Phys. 94 (1991) 2294–2304. [38] W. Paul, K. Binder, D. Heermann, K. Kremer, Crossover scaling in semidilute polymer solutions: a Monte Carlo test, J. Phys. II 1 (1991) 37–60. [39] P. de Gennes, Reptation of a polymer chain in presence of fixed obstacles, J. Chem. Phys. 55 (1971) 572–579. [40] I. Sanchez, R. Lacombe, Statistical thermodynamics of polymer solutions, Micromolecules 11 (1978) 1145–1156. [41] P. de Gennes, Scaling Concepts in Polymer Physics, 1st edition, Cornell University Press, 1979. [42] H. Öttinger, A thermodynamically admissible reptation model for fast flows of entangled polymers, J. Rheol. 43 (1999) 1461–1493. [43] D. Jou, J. Casas-Vásquez, M. Criado-Sancho, Thermodynamics of Fluids Under Flow, 2nd edition, Springer, 2011. [44] R. Drake, A general mathematical survey of the coagulation equation, in: International Reviews in Aerosol Physics and Chemistry, vol. 3, 1972, pp. 201–376. [45] N. Kokholm, On Smoluchowski’s coagulation equation, J. Phys. A 21 (1988) 839–842. [46] H. Freundlich, H. Hatfield, Colloid and Capillary Chemistry, Methuen, 1926. [47] K. Kowalczyk-Gajewska, H. Petryk, Sequential linearization method for viscous/elastic heterogeneous materials, Eur. J. Mech. A, Solids 30 (2011) 650–664. [48] A. Rekik, R. Brenner, Optimization of the collocation inversion method for the linear viscoelastic homogenization, Mech. Res. Commun. 38 (2011) 305–308. [49] M. Lévesque, K. Derrien, L. Mishnaevski Jr., D. Baptiste, M. Gilchrist, A micromechanical model for nonlinear viscoelastic particle reinforced polymeric composite materials – undamaged state, Composites, Part A, Appl. Sci. Manuf. 35 (2004) 905–913. [50] I. Krieger, T. Dougherty, A mechanism for non-Newtonian flow in suspension of rigid spheres, Trans. Soc. Rheol. 3 (1959) 137–152. [51] K. Rajagopal, A. Srinivasa, On thermomechanical restrictions of continua, Proc. R. Soc. Lond. A 406 (2004) 631–651.

H.P. Azikri de Deus et al. / Physics Letters A 380 (2016) 585–595

[52] L. Onsager, Reciprocal relations in irreversible thermodynamics. I, Phys. Rev. 37 (1931) 405–426. [53] K. Rajagopal, A. Srinivasa, Inelastic behavior of materials, I: theoretical underpinnings, Int. J. Plast. 14 (1998) 945–967. [54] K. Rajagopal, A. Srinivasa, Inelastic behavior of materials, II: inelastic response, Int. J. Plast. 14 (1998) 967–995. [55] K. Rajagopal, A. Srinivasa, A thermodynamic framework for rate-type fluid models, J. Non-Newton. Fluid Mech. 88 (2000) 207–227. [56] M. Gurtin, E. Fried, L. Anand, The Mechanics and Thermodynamics of Continua, Cambridge University Press, 2013.

595

[57] C. Negrão, A. Franco, L. Rocha, A weakly compressible flow model for the restart of thixotropic drilling fluids, J. Non-Newton. Fluid Mech. 166 (2011) 1369–1381. [58] L. Kuma, C. Lawrence, J. Sjöblom, Mechanism of pressure propagation and weakly compressible homogeneous and heterogeneous thixotropic gel breakage to study flow restart, RSC Adv. 406 (2014) 27493–27501. [59] Z. Melzak, A scalar transport equation, Trans. Am. Math. Soc. 85 (1957) 547–560. [60] I. Stewart, Density conservation for a coagulation equation, J. Appl. Math. Phys. (ZAMP) 42 (1991) 746–756.