Governing equations in non-isothermal diffusion

International Journal of Non-Linear Mechanics 55 (2013) 90–97

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Governing equations in non-isothermal diffusion A. Morro n University of Genoa, DIBRIS, 16145 Genoa, Italy

art ic l e i nf o

a b s t r a c t

Article history: Received 21 January 2013 Received in revised form 15 April 2013 Accepted 15 April 2013 Available online 29 April 2013

Generalizations of Fick's law for the diffusion flux are often considered in the literature by analogy with those for the heat flux. The paper reviews the balance equations for a fluid mixture and provides the equations for the diffusion fluxes. As a consequence, the mass densities are shown to satisfy a system of hyperbolic equations. Moreover, for a binary mixture of ideal gases in stationary conditions, Fick's law is recovered. Next, diffusion fluxes are regarded as constitutive functions and a whole set of thermodynamic restrictions are determined which account for diffusion, heat conduction, viscosity and inhomogeneities. Hyperbolic models for diffusion and heat fluxes are established which involve the co-rotational derivative. The driving term of diffusion turns out to be the gradient of chemical potential rescaled by the temperature. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Diffusion Fick's law Hyperbolic models Evolution equations Co-rotational derivative

1. Introduction Diffusion means the movement of a chemical species from a region of high concentration (of a solute) to a region of lower concentration. The seeming analogy between diffusion and heat conduction was taken over by Fick who, in essence, applied to diffusion the modeling of heat conduction, established by Fourier, by means of the correspondence between temperature and concentration (see [1]). Indeed, often the modeling of diffusion is shown to follow from the so-called Fick's laws. The first one is a constitutive equation for the diffusion flux, h say. If c is the concentration of the solute in a solvent or, e.g., impurities in a solid, then h ¼ −D∇c;

ð1Þ

where D is said to be the diffusivity. Fick's second law is a balance equation, namely ∂t c ¼ −∇  h: If D is constant then the two laws provide ∂t c ¼ DΔc;

ð2Þ

which is the classical parabolic equation modeling diffusion. Indeed, we are accustomed to the diffusion equation (2), along with the heat equation, as the prototype of parabolic equations. Fick's law (1) was then framed within non-equilibrium thermodynamics by Onsager [2,3] and, within ionized fluids, is generalized as the Nerst–Planck equation through an additive electric field effect [4]. Teorell formula n

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0020-7462/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijnonlinmec.2013.04.010

(see [5] and refs therein) states that the addition holds while the diffusivity is proportional to the concentration c. As with Fick's first law, the claimed, strict correspondence between the modeling of heat conduction and diffusion has been widely applied in the literature, sometimes within the joint topic of heat-mass transfer. The correspondence is emphasized also by the modeling of Soret and Dufour effects via the Onsager relations (see [6]). Hence, by appealing to extended thermodynamics (see [7,8] or other methods such as space–time non-locality [9] and the relaxational approach [7], various generalizations of (1) have been considered for h. The purpose is usually said to be the modeling of heat–mass transfer with finite speed of propagation or applicable at high frequencies. Among them, the so-called relaxational Maxwell–Cattaneo equation [10–12] τD ∂t h þ h ¼ −D∇c is most often applied. Borrowing from [13–15], in [16] an equation for the diffusion flux is considered with the time derivative replaced by the Oldroyd derivative. The purpose of this paper is two-fold. First, to re-visit the balance equations of fluid mixtures and to show that the evolution equations of the diffusion fluxes are in fact a standard consequence of the balance of mass and linear momentum. Both the Eulerian and the Lagrangian description are considered. The differential equations for the mass densities and the concentrations are derived explicitly by regarding the constituents as ideal gases. Second, to show that if the customary view is taken that the diffusion fluxes are variables to be framed within the constitutive theory then appropriate evolution equations follow from the thermodynamic requirements. The thermodynamic scheme so applied accounts for viscosity, heat conduction, diffusion. Spatial

A. Morro / International Journal of Non-Linear Mechanics 55 (2013) 90–97

inhomogeneity is incorporated through the gradient of temperature and concentrations. Notation. Let Ω be a time-dependent region occupied by a mixture and denote by x the position vector of a point relative to a chosen origin. The functions describing the evolution of the mixture have Ω  R as their common space–time domain. We denote by ∂t the derivative with respect to t∈R and by ∇ the gradient operator in Ω. The mixture is allowed to undergo chemical reactions. The subscripts α; β ¼ 1; 2; …; n label the quantities pertaining to the n constituents of the mixture. Hence ρα is the mass density, vα is the velocity, ϵα is the internal energy, θα is the temperature, ηα is the entropy, on Ω  R, of the αth constituent.

91

where qα is the heat flux, r α is the external heat supply, and eα is the growth of energy. Due to the internal character, the growths fmα g and feα g are subject to ∑mα ¼ 0; α

∑eα ¼ 0:

ð11Þ

α

The balance of entropy is written in the form ∂t ðρα ηα Þ þ ∇  ðρα ηα vα Þ þ ∇  ϕα −ρα sα ¼ sα ;

ð12Þ

where ϕα is the entropy flux and sα is the entropy supply. Also, sα is the entropy growth. The second law of thermodynamics is expressed by saying that ∑sα ≥0;

ð13Þ

α

2. Balance equations The mass density ρα and the velocity vα are required to satisfy the continuity equations ∂t ρα þ ∇  ðρα vα Þ ¼ τα ;

ð3Þ

where τα is the mass produced, per unit time and unit volume, of the αth constituent. The overall conservation of mass requires that ∑τα ¼ 0:

ð4Þ

α

The mass density of the mixture, ρ, and the baricentric velocity, v, are defined by ρ ¼ ∑ρα ; α

ρv ¼ ∑ρα vα :

Hence summation of (3) over α provides ð5Þ

Let a superposed dot stand for the derivative along the baricentric motion, ψ_ ¼ ∂t ψ þ v  ∇ψ for any function on Ω  R. Hence we can write (5) in the form ρ_ þ ρ∇  v ¼ 0:

ð6Þ

The concentration (or mass fraction) cα and the diffusion velocity uα are defined by ρ cα ¼ α ; uα ¼ vα −v: ρ Since ∑α cα ¼ 1 we may regard c1 ; …; cn−1 and ρ as n variables in 1–1 correspondence with ρ1 ; ρ2 ; …; ρn . Substitution of ρα with ρcα in (3) and use of (6) provides ρc_ α ¼ −∇  hα þ τα ;

hα ¼ ρα uα :

ϕ α ≔∂t Φα ¼ ∂t ϕα þ ðvα  ∇Þϕα : Hence ϕ α denotes the material derivative following the motion of the αth constituent. Since ϕα is peculiar [17], to the αth constituent, we can say that ϕ α is the αth peculiar time derivative. In terms of the peculiar time derivative, Eq. (5) can be written in the form ρ α þ ρα ∇  vα ¼ 0: The connection between balance equations in the two descriptions is made easy by the following. Proposition 1. If ϕα ; ρα ; vα are differentiable on Ω  R then ρα ϕ α ¼ ξα

⟺

∂t ðρα ϕα Þ þ ∇  ðρα ϕα vα Þ ¼ ξα þ τα ϕα :

ð14Þ

The proof is immediate. Since þ∇  ðρα ϕα vα Þ−ϕα ð∂t ρα þ ∇  ðρα vα ÞÞ; by means of (3) we have ð8Þ

The differential equations (3), (6)–(8) are equivalent forms of the continuity equations. The αth mass fluxes ρα vα and γ α vα in (3) and (8) involve the peculiar velocity vα whereas hα in (6) involves the diffusion velocity uα . Still within the Eulerian description we write the equations of motion in the form ð9Þ

where Tα is the stress tensor, f α is the body force, and mα is the growth of linear momentum, due to the action of the other constituents. Also, we write the balance of energy in the form ∂t ðρα ðϵα þ v2α =2ÞÞ þ ∇  ðρα ðϵα þ v2α =2Þvα Þ ¼ ∇  ðvα Tα Þ þρα f α  vα −∇  qα þ ρα r α þ eα ;

Since ∂t χα ¼ vα then we can write

ρα ϕ α ¼ ρα ð∂t ϕα þ vα  ∇ϕα Þ ¼ ∂t ðρα ϕα Þ

where M α is the molar mass. Hence by (3) we have

∂t ðρα vα Þ þ ∇  ðρα vα ⊗vα Þ ¼ ∇  Tα þ ρα f α þ mα ;

and regard Φα as the Lagrangian description of ϕα . Consequently

ð7Þ

The vector hα is termed the diffusion flux. In addition, we may use the molar concentration ρ γα ¼ α ; Mα ∂t γ α þ ∇  ðγ α vα Þ ¼ τα =M α :

ϕα ðx; tÞ ¼ ϕα ðχ α ðXα ; tÞ; tÞ≕Φα ðXα ; tÞ; ∂t Φα ¼ ∂t ϕα þ ∇ϕα  ∂t χα :

α

∂t ρ þ ∇  ðρvÞ ¼ 0:

has to hold for all admissible fields compatible with the balance equations (3), (9) and (10). The growths fmα g; feα g; fsα g are non-objective in that they are related to the particular reference chosen for the Eulerian description. Objective quantities are provided in a while within the Lagrangian description. Let Rα be the region occupied by the αth constituent in the reference configuration R of the mixture. Hence x ¼ χα ðXα ; tÞ on Rα  R is the motion of the αth constituent. For any field ϕα on Ω  R we can write

ρϕ α ¼ ∂t ðρα ϕα Þ þ ∇  ðρα ϕα vα Þ−τα ϕα and the result follows. As a consequence, Eq. (9) becomes ρα v α ¼ ∇  Tα þ ρα f α þ p^ α ; where p^ α ≔mα −τα vα : By (10) we have ∂t ðρα ϵα Þ þ ∇  ðρα ϵα vα Þ þ τα v2α =2 þ ρα ðv2α =2Þ¼ ∇  ðvα Tα Þ þρα f α  vα −∇  qα þ ρα r α þ eα : As a consequence of (15) it follows that:

ð10Þ

ð15Þ

∂t ðρα ϵα Þ þ ∇  ðρα ϵα vα Þ þ τα v2α =2 ¼ Tα  Lα −∇  qα þ ρα r α þ eα :

ð16Þ

92

A. Morro / International Journal of Non-Linear Mechanics 55 (2013) 90–97

Hence, applying again (14) we find that ρα ϵ α ¼ Tα  Lα −∇  qα þ ρα r α þ ϵ^ α ;

where ð17Þ

where ϵ^ α ≔eα −p^ α  vα −τα ðϵα þ v2α =2Þ:

ð18Þ

∑½^ϵ α þ p^ α  uα þ τα ðϵα þ u2α =2Þ ¼ 0:

α

ð19Þ

α

Eq. (19) involves only objective vectors and scalars. Look now at the entropy inequality (13). It is convenient to express ϕα and sα in the form ϕα ¼

qα þ kα ; θα

sα ¼

rα þ κα ; θα

Upon replacing ∇  qα −ρα r α from (17) we have α

α

w≔∑½−uα Tα þ ρα ðϵα þ u2α =2Þuα ; α

and ρr ¼ ρr I þ ∑ρα f α  uα ;

ρr I ≔∑ρα r α : α

The sums of peculiar terms like qI and TI are called the inner, or intrinsic, parts (of q and T). We now look at a formulation of the entropy inequality in the baricentric reference. As a preliminary step we establish the identity ρα ϕ α ¼ ρα ϕ_ α þ hα  ∇ϕα ;

ð23Þ

for any function ϕα on Ω  R. This is a direct consequence of

kα being called the extra-entropy flux and κ α the extra-entropy supply. Hence, by means of (14), inequality (13) becomes

 qα ρ rα ∑ρα η α þ τα ηα þ ∇  þ kα − α þ ρα κα ¼ ∑sα ≥0: θα θα α α

∑

qI ≔∑qα ;

α

By means of (4), the constraints (11) provide ∑½p^ α þ τα uα  ¼ 0;

q ¼ qI þ w;

ϕ α ¼ ϕ_ α þ uα  ∇ϕα and the definition of hα . Further, by (23) we can write ρα ϕ α þ τα ϕα ¼ ρα ϕ_ α þ hα  ϕα þ τα ϕα ¼ ρα ϕ_ α þ ∇  ðϕα hα Þ þ ϕα ðτα −∇  hα Þ whence, by the balance relation (7), we obtain

1 1 ½ρ ðθα η α −ϵ α Þ þ τα θα ηα þ Tα  Lα − qα  ∇θα þ ϵ^ α θα α θα

ρα ϕ α þ τα ϕα ¼ ρα ϕ_ α þ ∇  ðϕα hα Þ þ ρϕα c_ α :

þθα ∇  kα −ρα θα κα ≥0:

ð24Þ

Hence, letting ϕα ¼ ηα we have

This is the appropriate formulation whenever the constitutive properties of the single constituents are examined. The entropy inequality (3.3) in [18] amounts to assuming kα ¼ 0; κα ¼ 0. Instead, Eq. (3.6) of [19] shows that non-zero kα and κα are required to occur.

∂t ðρα ηα Þ þ ∇  ðρα ηα vα Þ ¼ ρα η α þ τα ηα ¼ ρα η_ α þ ∇  ðηα hα Þ þ ρηα c_ α : Let η ¼ ∑cα ηα α

whence ρ_η ¼ ∑½ρηα c_ α þ ρα η_ α :

3. Balance equations for a single body

α

For later use we derive the equation of motion in the baricentric reference. The definitions of ρ and v give

ρ_η þ ∑½∇  ðϕα þ ηα hα Þ−ρα sα ≥0:

ð25Þ

α

∑ρα vα ¼ ρv: α

In the baricentric reference, the effective entropy flux is ∑α ðϕα þ ηα hα Þ. As a simplification, the constituents are constrained to all have the same temperature

Also, by (6) _ ∂t ðρvÞ þ ∇  ðρv⊗vÞ ¼ ∂t ðρvÞ þ ρðv  ∇Þv þ v∇  ðρvÞ ¼ ρv: Hence, summation of (9) over α provides

θα ¼ θðx; tÞ;

∂t ðρvÞ þ ∇  ðρv⊗vÞ ¼ ∇  T þ ρf;

on Ω  R. Hence upon multiplying (25) by θ and letting ρκ ¼ ∑α ρα κ α we have

where T ¼ TI þ T u ;

As a consequence we can write (13) in the form

TI ≔∑Tα ; α

Tu ≔−∑ρα uα ⊗uα ;

1 ρθη_ þ ∑ð∇  qα −ρα r α Þ− qI  ∇θ þ θ∑∇  ðkα þ ηα hα Þ−ρθκ≥0: θ α α

α

and

ð26Þ

By (22) we have

ρf ¼ ∑ρα f α :

∑ð∇  qα −ρα r α Þ ¼ T  D−ρ_ϵ −∇  w þ ∑ρα f α  uα :

α

α

As a consequence we find the equations of motion in the baricentric reference ρv_ ¼ ∇  T þ ρf;

ð20Þ

the stress tensor T being symmetric. Since ρα ¼ ρcα , by (20) we have ρα v_ ¼ cα ∇  T þ ρα f:

ð21Þ

Let

α

Substitution in (26) gives 1 ρθη_ −ρ_ϵ þ T  D− qI  ∇θ−∇  w θ
 þθ∑∇  ðkα þ ηα hα Þ− ρθκ−∑ρα f α  uα ≥0: α

α

The external character of the forces ff α g suggests that we let

ρϵ ¼ ∑ρα ðϵα þ u2α =2Þ:

ρθκ ¼ ∑ρα f α  uα

α

α

By (10), replacing vα with v þ uα and summing over α we eventually obtain ρ_ϵ ¼ T  D−∇  q þ ρr;

ð22Þ

so that no definiteness is required on the power ∑α ρα f α  uα . Now let wα k^ α ¼ kα − ; θ

wα ¼ −uα Tα þ ρα ðϵα þ u2α =2Þuα :

A. Morro / International Journal of Non-Linear Mechanics 55 (2013) 90–97

93

1 ρðθη_ −_ϵ Þ þ T  D− q  ∇θ þ θ∇  ∑ðk^ α þ ηα hα Þ≥0: θ α

frame. Eq. (30) is consistent with the evolution equation determined by Müller in [21], Section 5, where ρα v_ occurs instead of cα ∇  T−ρα ðf α −fÞ. Now, in [21] f α ¼ 0 and hence, by (20), ρα v_ is simply cα ∇  T. None of Eqs. (28)–(30) is objective. Indeed, hα is objective but h_ α þ ðL þ ∇  v1Þhα is not. Curiously, upon a mere change of sign, h_ α þ ð−L þ ∇  v1Þhα would be objective and would be just the upper convected Oldroyd derivative [15]. Anyway, there is no surprise that the left-hand sides of (28)–(30) are not objective in that they are direct consequences of the equations of motion. Instead, the right-hand side of (29) and (30) is objective in that

In terms of the free energy ψ ¼ ϵ−θη we have

mα −τα v ¼ p^ α þ τα uα

Hence 1 1 − qI  ∇θ−∇  w þ θ∑∇  kα ¼ − ðqI þ wÞ  ∇θ þ θ∇  ∑k^ α : θ θ α α Since ∇  w ¼ θ∇ 

w 1 þ w  ∇θ θ θ

then the entropy inequality becomes

_ þ T  D− 1 q  ∇θ þ θ∇  ∑ðk^ α þ ηα hα Þ≥0: −ρðψ_ þ ηθÞ θ α

ð27Þ

is objective. Remark 1. The required objectivity of p^ α and the constraints (4) on fτα g and (11) on fmα g suggest that we let

By (27), the entropy balance for the mixture as a single body shows that the equivalent entropy flux is q þ w  α ∑ðϕα þ ηα hα Þ ¼ ∑ α þ k^ α þ ηα hα θ α α

mα −τα v ¼ ∑ M αβ ðvβ −vα Þ;

as though q ¼ qI þ w would be the effective heat flux. If f α is common to all constituents, f α ¼ f, then

M αβ ¼ M βα 4 0;

where α≠β;

and M αβ ¼ 0 if α ¼ β. Hence, by (16)

∑ρα f α  uα ¼ f  ∑ρα uα ¼ 0: α

α

In such a case we can take κ ¼ 0. This is consistent with the absence of the entropy supply e.g. in [7], Eq. (1.58) – see [20], Section 3 – and [21].

p^ α ¼ ∑ M αβ ðvβ −vα Þ−τα uα β≠α

is objective. It is worth pointing out that sometimes (see e.g. [22]) constitutive equations are considered for the interaction force p^ α ¼ mα −τα vα rather than mα −τα v. Since

4. Evolution equations for the diffusion fluxes

∑ðmα −τα vα Þ ¼ −∑τα uα ≠0

By means of the continuity equations (3) and the equations of motion (9) we can derive the balance equations for the diffusion flux ρα vα or ρα uα . By (15) and (3) we have

mα −τα vα ¼ ∑ M αβ ðvβ −vα Þ−τα uα :

α

ðρα vα Þ¼ −ρα ð∇  vα Þvα þ τα vα þ ∇  Tα þ ρα f α þ p^ α ð28Þ

Relative to hα ¼ ρα uα , by (28) we have h α ¼ ðρα vα Þ−ðρα vÞ¼ −ð∇  vα Þρα vα þ ∇  Tα þ ρα f α þ mα −ðρα vÞ: Now, by (3)

β≠α

Now that we know the evolution equation for hα we ask for the partial differential equations for the mass density ρα and the concentration cα . For simplicity, in this section we regard the temperature θ as a given field and let τα ¼ τα ðρα ; θÞ:

ð32Þ

Moreover we let f α and mα be known functions, on Ω  R. Hence (3) and (9) are a closed non-linear system for ρα and vα whereas (7) and (29), or (30), are a closed non-linear system for cα and hα . We now look for linearised versions of the differential equations. Since

Moreover ρα ðuα  ∇Þv ¼ Lρα uα : Hence (20) and (21) and some rearrangements yield h α þ ðL þ ∇  vα 1Þhα ¼ ∇  Tα −cα ∇  T þ ρα ðf α −fÞ þ mα −τα v:

∇  Tα ¼ −∂ρα pα ∇ρα −∂θ pα ∇θ:

By means of the identities

Eqs. (3) and (9) allow for an equilibrium solution

ð∇  vα Þhα ¼ ð∇  vÞhα þ ð∇  uα Þhα ;

ρα ¼ ρ α ;

ð∇  uα Þhα ¼ ∇  ðhα ⊗uα Þ−ðuα  ∇Þhα ;

θ¼θ

if τα ; vα ; f α , and mα are zero. The constants ρ α are determined by the whole masses of the components. Let

we can write

ϱα ¼ ρα −ρ α ; ð29Þ

Since h α ¼ h_ α þ ðuα  ∇Þhα then Eq. (29) gives h_ α þ ðL þ ∇  v1Þhα ¼ ∇  ðTα −hα ⊗uα Þ−cα ∇  T þ ρα ðf α −fÞ þ mα −τα v:

then the analogue of (31) should be

Tα ¼ −pα ðρα ; θÞ1;

ðρα vÞ¼ ð−ρα ∇  vα þ τα Þv þ ρα ½v_ þ ðuα  ∇Þv:

h α þ ðL þ ∇  v1Þhα −ðuα  ∇Þhα ¼ ∇  ðTα −hα ⊗uα Þ−cα ∇  T þ ρα ðf α −fÞ þ mα −τα v:

α

5. Partial differential equations modeling diffusion

whence ðρα vα Þþ ð∇  vα Þρα vα ¼ ∇  Tα þ ρα f α þ mα :

ð31Þ

β≠α

ϑ ¼ θ−θ

and assume mα in the form (31). In the linear approximation relative to ϱα ; vα , and ϑ we have ∂t ϱα þ ρ α ∇  vα ¼ ∂ρα τα ϱα þ ∂θ τα ϑ;

ð30Þ

Eq. (28) governs the evolution of ρα vα relative to the observer Oα that follows the motion of the αth constituent whereas (29) and (30) govern the evolution of hα relative to Oα and the baricentric reference

ð33Þ

ρ α ∂t vα ¼ −∂ρα pα ∇ϱα −∂θ pα ∇ϑ þ ρ α ∇  f α þ ∑ M αβ ðvβ −vα Þ;

ð34Þ

β≠α

where ∂ρα τα ; ∂θ τα ; ∂ρα pα , and ∂θ pα are evaluated at equilibrium. Partial time differentiation of (33), the divergence of (34) and elimination of

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A. Morro / International Journal of Non-Linear Mechanics 55 (2013) 90–97

∇  ∂t v yield
 Mα Mα −∂ρα τα ∂t ϱα −∂ρα pα Δϱα − ∂ ρ τ α ϱα ∂t2 ϱα þ ρα ρα α ¼ Gα ðϑ; fvβ g; f α Þ

If, further, the constants λα are nearly equal and we let λα ¼ λ; ð35Þ

α ¼ 1; 2; …; n;

then ∑λβ Δcβ ¼ 0; β

where Gα ¼ ∂θ pα Δϑ þ ∂θ τα ∂t ϑ þ

∑λβ ρβ ¼ ρ: β

In such a case (39) simplifies to

Mα ∂θ τα ϑ−ρ α ∇  f α − ∑ M αβ ∇  vβ ρα β≠α

∂t2 cα −λθΔcα ¼ 0:

and

ð41Þ

Eq. (41) is homogeneous and uncoupled. Moreover, they show that pffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi cα propagates with the speed λθ ¼ ∂ρα pα .

M α ¼ ∑ M αβ 4 0: β≠α

In non-reacting mixtures, τα ¼ 0, Eq. (35) takes the form of a telegraph equation. In addition, Eq. (35) becomes the hyperbolic equation of linear acoustics if interaction forces are taken to be zero, M αβ ¼ 0. Consistent with the classical acoustics, we might apply the adiabatic approximation which follows by letting pα ¼ pα ðρα ; ηÞ and taking η as homogeneous and constant. Things are more involved if we look at the differential equation governing the concentrations cα . The whole system of equations, in the unknowns ρ; fcα g; v; fhα g, consists of (6), (7), (20) and (30), the temperature θ being regarded as known. Denote by H ¼ ðρ; c1 ; …; cn−1 ; v; h1 ; …; hn−1 Þ the unknown vector field. Hence the system of equations can be written in the form H_ ¼ FðH; ∇HÞ:

ð36Þ

The explicit expression of (36) shows that the differential equations are a coupled system. This in turn indicates that the system (35), α ¼ 1; 2; …; n, in the unknown mass densities is handier than the system for ρ and the concentrations fcα g. Nevertheless, in Section 6 we show that, subject to appropriate assumptions, Fick's law (1) follows from (30) and the classical diffusion equation is then recovered. For definiteness and a more direct comparison with other approaches we now let τα ¼ 0, f α ¼ f, and mα ¼ 0. Moreover, since we are mainly interested in the diffusion effects, we let the baricentric velocity be zero, v ¼ 0; L ¼ 0. Hence ∇  v ¼ 0 and ρ is regarded as constant. The linear approximation in hα , and uα , of Eqs. (7) and (30) then becomes ρ∂t cα ¼ −∇  hα ;

ð37Þ

∂t hα ¼ −∇pα þ cα ∑∇pβ :

ð38Þ

β

Now, by (32)

6. Fick's law as a limit case Fick's law is now obtained within a simple model where the growths fτα g and fmα g are allowed to be non-zero. Look at a binary mixture in the approximation (stationary conditions) h_ α ¼ 0;

α ¼ 1; 2:

Also, let the overall motion be uniform, L ¼ 0. Moreover, let f α ¼ f and disregard the non-linear terms uα ⊗uα . With these assumptions, Eq. (30) reduces to mα −τα v ¼ cα ∇  ∑Tβ −∇  Tα ;

α ¼ 1; 2:

β

We now apply (31) and let M 12 ¼ kρ1 ρ2 ; k being any positive-valued function of ρ1 ; ρ2 , and θ. Hence we have m1 −τ1 v ¼ −kρ1 ρ2 ðv1 −v2 Þ;

m2 −τ2 v ¼ −ðm1 −τ1 vÞ:

Because v1 −v2 ¼ u1 −u2 ;

h2 ¼ −h1 ;

we find that m1 −τ1 v ¼ −kðρ2 h1 −ρ1 h2 Þ ¼ −kρh1 ; and m2 −τ2 v ¼ −kρh2 . Hence mα −τα v ¼ −kρhα ;

cα ∇  ðT1 þ T2 Þ−∇  Tα ¼ cα ½−λ∇ðρθÞ þ λ∇ðρα θÞ ¼ λρθ∇cα : As a consequence, (42) and (43) provide

Apply ∂t to (37) and the divergence ∇ to (38). Eliminate ∇  ∂t hα and keep only the linear terms to obtain

hα ¼ −

∂t2 cα −ð1−cα Þ∂ρα pα Δcα ¼ −cα ∑ ∂ρβ pβ Δcβ þ

ð39Þ

Remark 2. Let pα be given by the equation of state of an ideal gas ð40Þ

where λα is the gas constant over the molecular weight. Hence ∑∂θ pβ ¼ ∑λβ ρβ : β

β

ð44Þ

ρ∂t cα ¼ −∇  hα þ τα ; provides the classical, parabolic diffusion equation.

As with (35) for ρα , the differential equations (39) for cα are hyperbolic. However, any equation (39) contains Δcβ , β≠α, thus making equations (39) for fcα g a coupled system. Instead, Eq. (35) for fρα g is an uncoupled system.

pα ¼ λα ρα θ;

λθ ∇cα : k

Eq. (44) is just the first Fick law (1) with diffusion constant D ¼ λθ=k. Substitution in the linearized version of (7)

β≠α

1 ∂θ pα −cα ∑∂θ pβ Δθ: ρ β

∂ρα pα ¼ λα θ;

ð43Þ

which shows that mα −τα v and the diffusion flux hα have antiparallel directions. Still within the scheme (32) we consider the equation of state of ideal gases (40) with equal values of λα . It follows that

pα ¼ pα ðρα ; θÞ ¼ pα ðρcα ; θÞ:

!

ð42Þ

7. A rate-type thermodynamic scheme Thermodynamic restrictions on constitutive equations follow by exploiting the entropy inequality. In this section we allow for viscosity and heat conduction and show how a constitutive equation for the diffusion fluxes fhα g can be derived in analogy with the evolution equation (30). By this approach, rate-type equations are shown to be compatible with thermodynamics. Accordingly, we account for the balance equations (7) but disregard the definition

A. Morro / International Journal of Non-Linear Mechanics 55 (2013) 90–97

hα ¼ ρα uα and (30), and look for a simpler evolution equation. We regard the mixture as a single body and hence consider the entropy inequality in the form (27). This view is consistent with recent models of diffusion phenomena (see [23]) and allows us to generalize results on constitutive equations for the diffusion fluxes fhα g. Let Γ ¼ ðθ; ρ; fcα g; q; fhα g; f∇cα g; ∇θ; DÞ as the set of independent variables. Also let ψ be a continuously _ fh_ α g are differentiable function of Γ whereas η; T; q; fk^ α g, and q; continuous functions of Γ. Substitution of ψ ðΓÞ in (27) provides

 −ρ ð∂θ ψ þ ηÞθ_ þ ∂ρ ψ ρ_ þ ∂q ψ  q_ þ ∑ ∂cα ψ c_ α þ ∑∂hα ψ  h_ α þ ∂∇cα ψ  ∇c_ α α

α

_ þ ∂ ψ  DÞ _ þ T  D− 1 q  ∇θ þ θ∇  ∑ðk^ α þ η hα Þ≥0: −ρð∂∇θ ψ  ∇θ D α θ α ð45Þ The functions θ and D on Ω  R allow for arbitrary values of _ , and D, _ at any point x and time t, while Γ is kept fixed. _ ∇θ θ, Hence inequality (45) holds only if ψ is independent of ∇θ and D whereas η ¼ −∂θ ψ: Also, upon replacing ρ_ with −ρ tr L and dividing throughout by θ (45) becomes

 ρ −∂ρ ψρ tr L þ ∂q ψ  q_ þ ∑ ∂cα ψ c_ α þ ∑∂hα ψ  h_ α þ ∂∇cα ψ  ∇c_ α − θ α α þ

1 1 T  D− 2 q  ∇θ þ ∇  ∑ðk^ α þ ηα hα Þ≥0: θ θ α

ð46Þ

Since

95

The possible dependence on D is now exploited to model viscosity by letting T, and hence T , depend linearly on D. A sufficient condition for the validity of (47) is that T ¼ 2μD þ λðtr DÞ1;

ð48Þ






ρ 1 k^ α ¼ ∂∇cα ψ c_ α − μ þ ηα hα ; θ θ α

ð49Þ

∑α τα μα ≤0;

ð50Þ


 1 1 μα þ ∂hα ψ  h_ α ≤0; hα  ∇ θ θ

ð51Þ

1 q  ∇θ þ ∂q ψ  q_ ≤0; ρθ

ð52Þ

μ and λ being the viscosity coefficients, possibly dependent on Γ, subject to μ≥0; 2μ þ 3λ≥0. Consistent with the separate requirements (48)–(52), we set aside cross-coupling terms and let ψ depend on wq ¼ 12 q2 ;

2

wα ¼ 12hα :

Hence (52) becomes
 1 ∂wq ψ q_ þ ∇θ  q ≤0; ρθ which is equivalent to
 1 ∂wq ψ q_ þ b  q þ ∇θ  q ≤0; ρθ

ð53Þ

for any vector field b. We take this advantage to let b ¼ −∂wq ψω;

∇c_ α ¼ ∇c_ α −LT ∇cα

ω being the vorticity of the baricentric motion

then ρ  ρ  ρ ρ ∂∇cα ψ  ∇c_ α ¼ ∇  ∂∇cα ψ c_ α −∇  ∂∇cα ψ c_ α − ð∇cα ⊗∂∇cα ψÞ  L: θ θ θ θ Hence (46) can be written in the form 1 1 ρ ρ _ ∂hα  h_ α Þ− 2 q  ∇θ ðT þ ρ2 ∂ρ ψ1 þ ρ∑∇cα ⊗∂∇cα ψ Þ  L−∑ μα c_ α − ð∂q ψ  q−∑ θ θ α θ α α θ ρ þ ∇  ∑ðk^ α − ∂∇cα ψ c_ α þ ηα hα Þ≥0; θ α

ωi ¼ 12 ϵijk W kj ;

ω ¼ 12∇  v:

Hence (53) becomes
 J 1 ∂wq ψq þ ∇θ  q ≤0; ρθ

ð54Þ

where J

_ q q ¼ q−ω

where

is the co-rotational derivative of q. Inequality (54) holds if and only if

ρ  θ ∂∇cα ψ : μα ≔∂cα ψ− ∇  ρ θ

∂wq ψq þ

We can view μα as the αth chemical potential. Since L ¼ D þ W then the linearity in the spin tensor W implies that

ξ being a positive-valued function of Γ. Hence we have the constitutive equation for q in the form

∑∇cα ⊗∂∇cα ψ∈Sym; α

Sym being the set of symmetric tensors. Substitution of c_ α from (7) gives

 ρ 1 1 1 ∑μα hα þ ∑hα  ∇ μα þ ∑ μα τα : ∑ μα c_ α ¼ −∇  θ α θ α θ α α θ Hence it follows that:
 1 1 ρ 1 1 ρ T  D−∑hα  ∇ μα − ∑∂hα ψ  h_ α − ∑μα τα − 2 q  ∇θ− ∂q ψ  q_ θ θ θ α θ α θ θ α
  ρ 1 ∂c ψ þ ηα hα ≥0 þ ∇  ∑ k^ α − ∂∇cα ψ c_ α þ ð47Þ θ θ α α where

J

1 1 ∇θ ¼ − q; ρθ ξ

J

ξ∂wq ψ q þ q ¼ −

ξ ∇θ: ρθ

By the same token, observe that ∂hα ψ ¼ ∂wα ψhα and then (53) becomes
 μ 1 ∇ α þ ∂wα ψ h_ α  hα ≤0 θ θ or
 J μ 1 ∇ α þ ∂wα ψhα  hα ≤0 θ θ in that

2

T ≔T þ ρ ∂ρ ψ1 þ ρ∑∇cα ⊗∂∇cα ψ: α

J

hα  hα ¼ ðh_ α −ω  hα Þ  hα ¼ h_ α  hα :

ð55Þ

96

A. Morro / International Journal of Non-Linear Mechanics 55 (2013) 90–97

Hence J μ 1 1 ∇ α þ ∂wα ψ hα ¼ − hα ; θ ζα θ

ζ α being a positive-valued function of Γ. As a consequence of (52) the constitutive equation for hα takes the form J ζα μ ∂wα ψhα þ hα ¼ −ζ α ∇ α : θ θ

ð56Þ

and (59) follows if ρ; θ and α are uniform. The thermodynamic requirement (50) then avoids any appeal to the hypothesis about relaxation toward equilibrium. For non-reacting mixtures we have ρc_ ¼ −∇  h: In the sense of Section 5, with pα ¼ λρθcα and f α ¼ f; mα ¼ 0 then h ¼ −λρθ∇c. If, further, λ; ρ and θ are uniform then ρc_ ¼ −λρθΔc: J

If, instead, we apply (56) with h ¼ 0 then (7) becomes 7.1. Remarks on the constitutive equations

ρc_ ¼

It is natural to contrast Eqs. (30) and (56) as alternate constitutive equations for the diffusion fluxes hα . In (30), h_ α þ ðL þ ∇  v1Þhα is non-objective. This should come as no surprise because (30) has been derived by evaluating the derivative of hα ¼ ρα uα via the balance equations (3) and (9), which are non-objective. Eq. (56) has been derived by regarding h_ α as a constitutive function to be compatible with thermodynamics, subject to (7) only. It is objective in that it involves the co-rotational derivative and this is allowed by the identical vanishing of ω  hα  hα . Eq. (30) involves the term ðL þ ∇  v1Þhα induced by the baricentric motion while (56) involves ω  hα . Assume that fTα g are pressure tensors and the baricentric motion is negligible. Moreover, disregard the non-linear term ∇  ðhα ⊗uα Þ in (30) and also let f α ¼ f. As a consequence, (30) and (56) become, respectively h_ α ¼ mα −τα v þ cα ∇p−∇pα ; h_ α ¼ −

θ θ μ hα − ∇ α: ζ α ∂w α ψ ∂w α ψ θ

ð57Þ ð58Þ

Both (57) and (58) are first-order evolution equations. In addition, for binary mixtures mα −τα v ¼ kρhα . So the main difference occurs between cα ∇p−∇pα , as provided by the balance equations, and ðθ=∂wα ψÞ∇ðμα =θÞ, as given by the thermodynamic scheme. The evolution equation for q has been derived in the form (55). The present approach is similar but more general than that in [23,10] where the continuum under consideration is taken to be rigid. As an aside it is worth observing that the constitutive equation for the stress T might be of rate type thus providing a hyperbolic model also in viscous materials. We merely should regard ψ as a function of T  T and follow the same procedure as we have done for hα and q. 7.2. Differential equations of two-phase systems The Ginzburg–Landau equation c_ ¼ −βðf c −αΔcÞ;

ð59Þ

and the Cahn–Hilliard equation c_ ¼ κΔðf c −αΔcÞ;

ð60Þ

where α; β; κ 40 and f c ¼ ∂c f , are taken to model qualitative features of two-phase materials. Eq. (59) models relaxation toward equilibrium whereas (60) describes the diffusion process (see [24] and refs therein). By (50), with n ¼ 2, t ¼ t1 , c ¼ c1 , μ ¼ μ1 μ2 , we have

ζ Δðf c −αΔcÞ; θ

namely the Cahn–Hilliard equation.

8. Conclusions The paper shows that the evolution (or constitutive) equation for the diffusion flux takes the form (30) or (28) according as we look at ρα uα or ρα vα as the appropriate diffusion flux. Both equations are non-linear and involve the constitutive function for the peculiar stress Tα . For definiteness, in Section 5 Tα is taken as a pure-pressure tensor dependent on (the peculiar) mass density and temperature. Hence the differential equations for the mass densities fρα g are established in the form (35). They are hyperbolic and exhibit the effects of the growths of mass fτα g and the coefficients fM α g of the growths fmα g. Eq. (57) is linear in hα and is likely to be a good approximation, of (30), whenever we can disregard the underlying baricentric motion of the mixture. The differential equations for the concentrations fcα g prove to be more involved. If the growths fτα g and fmα g are disregarded and the baricentric velocity is assumed to be zero then the equations have the form (39). Fick's law for the diffusion flux hα is shown to hold within severe assumptions. In addition to neglecting the non-linear term uα ⊗uα in the stress tensor Tα , we restrict attention to a binary mixture with a constant diffusion flux, h_ α ¼ 0. Hence we find the collinearity condition (43) between mα −τα v and hα . Next, letting the constituents be ideal gases with (nearly) equal molecular weights we find that the first Fick law hα ¼ −D∇cα holds. Eqs. (51) and (52) show that, if we regard hα as entering (7) as an unknown variable then there is a close analogy between hα and q, as is the case in standard thermodynamic schemes (see [6,7]). Eqs. (56) and (55) are objective evolution equations compatible with thermodynamics. It is worth emphasizing that this is so because the entropy inequality produces (52), as well as (51), and not merely the heat conduction inequality q  ∇θ ≤0: Otherwise, as is the case e.g. in [25,26], only the classical Fourier law is allowed without constraints. Of course we might consider
 1 ρ 1 ρ −∑hα  ∇ μα − ∑∂hα ψ  h_ α − 2 q  ∇θ− ∂q ψ  q_ ≤0 θ θ α θ θ α as a sufficient condition instead of (51) and (52) separately. In such a case also the cross-coupling terms (Dufour, Soret) may be incorporated in the model.

τ ¼ −βμ where β is positive valued. If diffusion is disregarded, h ¼ 0, then (7) provides ρc_ ¼ −βμ:

 If ψ ¼ f ðρ; c; θÞ þ 12α∇cj2 then ρ  θ α∇c μ ¼ f c− ∇  ρ θ

References [1] J. Crank, The Mathematics of Diffusion, Clarendon, Oxford, 1964. (Chapter 1). [2] L. Onsager, Reciprocal relations in irreversible processes, I, Physical Review 37 (1931) 405–426. [3] L. Onsager, Reciprocal relations in irreversible processes, II, Physical Review 38 (1931) 2265–2279. [4] J.O'M. Bokris, A.K.N. Reddy, Modern Electrochemistry, Plenum, New York, 1997, p. 405.

A. Morro / International Journal of Non-Linear Mechanics 55 (2013) 90–97 [5] A.N. Gorban, H.P. Sargsyan, H.A. Wahab, Quasichemical models of multicomponent non-linear diffusion, Mathematical Modelling of Natural Phenomena 6 (2011) 184–262. [6] S.R. De Groot, P. Mazur, Non-Equilibrium Thermodynamics, Dover, New York, 1984. (Chapter 11). [7] D. Jou, J. Casas-Vazquez, G. Lebon, Extended Irreversible Thermodynamics, Springer, New York, 2010. [8] I. Müller, T. Ruggeri, Rational Extended Thermodynamics, Springer, Berlin, 1998. [9] S.L. Sobolev, Space-time nonlocal model for heat conduction, Physical Review E 50 (1994) 3255–3258. [10] D. Jou, P. Galenko, Fluctuations and stochastic noise in systems with hyperbolic mass transport, Physica A 366 (2006) 149–158. [11] P.K. Galenko, D.A. Danilov, Local nonequilibrium effect on rapid dendritic growth in a binary alloy melt, Physics Letters A 235 (1997) 271–280. [12] P. Galenko, S. Sobolev, Local nonequilibrium on undercooling in rapid solidification of alloys, Physical Review E 55 (1997) 343–352. [13] C.I. Christov, On frame indifferent formulation of the Maxwell–Cattaneo model of finite-speed heat conduction, Mechanics Research Communications 36 (2009) 481–486. [14] A. Morro, Evolution equations and thermodynamic restrictions for dissipative solids, Mathematical and Computer Modelling 52 (2010) 1869–1876. [15] A. Morro, Evolution equations for non-simple viscoelastic solids, Journal of Elasticity 105 (2011) 93–105.

97

[16] M. Ciarletta, B. Straughan, V. Tibullo, Christov–Morro theory for non-isothermal diffusion, Nonlinear Analysis: Real World Applications 13 (2012) 1224–1228. [17] C. Truesdell, Rational Thermodynamics, Springer, New York, 1984. (Chapter 5). [18] R.M. Bowen, J.C. Wiese, Diffusion in mixtures of elastic materials, International Journal of Engineering Science 7 (1969) 689–722. [19] I. Müller, Thermodynamics of mixtures of fluids, Journal de Mécanique 14 (1975) 267–303. [20] S. Whitaker, Mechanics and thermodynamics of diffusion, Chemical Engineering Science 68 (2012) 362–375. [21] I. Müller, Thermodynamics of mixtures and phase field theory, International Journal of Solids and Structures 38 (2001) 1105–1113. [22] I. Müller, Thermodynamic theory of mixtures of fluids, Archive for Rational Mechanics and Analysis 28 (1968) 1–39. [23] P. Galenko, D. Jou, Diffuse-interface model for rapid phase transformations in nonequilibrium systems, Physical Review E 71 (2005) 046125. [24] M.E. Gurtin, Generalized Ginzburg–Landau and Cahn–Hilliard equations based on a microforce balance, Physica D 92 (1996) 178–192. [25] N. Fox, Generalised thermoelasticity, International Journal of Engineering Science 7 (1969) 437–445. [26] H.S. da Costa Mattos, On thermodynamics, objectivity and the modeling of non-isothermal viscoelastic fluid behavior, Nonlinear Analysis: Real World Applications 13 (2012) 2134–2143.