constructing poisson and dissipative brackets of mix ...

CONSTRUCTING POISSON AND DISSIPATIVE BRACKETS OF MIXTURES BY USING LAGRANGIAN-TO-EULERIAN TRANSFORMATION K.-C. Chen * Institute of Applied Mechanics National Taiwan University Taipei, Taiwan 10617, R.O.C.

ABSTRACT This paper aims to construct the bracket formalism of mixture continua by using the method of Lagrangian-to-Eulerian (LE) transformation. The LE approach first builds up the transformation relations between the Eulerian state variables and the Lagrangian canonical variables, and then transforms the bracket in Lagrangian form to the bracket in Eulerian form. For the conservative part of the bracket formalism, this study systematically generates the noncanonical Poisson brackets of a two-component mixture. For the dissipative part, we deduce the Eulerian-variable-based dissipative brackets for viscous and diffusive mechanisms from their Lagrangian-variable-based counterparts. Finally, the evolution equations of a micromorphic fluid, which can be treated as a multi-component mixture, are derived by constructing its Poisson and dissipative brackets. Keywords : Noncanonical poisson bracket, Dissipative bracket, Lagrangian-to-Eulerian transformation, Micromorphic fluid.

1.

INTRODUCTION

To obtain the evolution equations of a system, two frameworks are generally adopted. The framework of Newtonian mechanics begins with the formulation of various balance equations, and the framework of Lagrangian mechanics starts from the integral or differential form of variational principles. In addition to the extensive applications of the two types of mechanics, Hamiltonian mechanics provides an efficient alternative to determine a system evolution by the Hamilton canonical equations. The Poisson bracket formalism manifests the mathematical beauty and elegance of Hamiltonian mechanics in that the formalism implies conservation of energy and that the Poisson bracket links the quantum commutator. Besides, Hamiltonian mechanics is more effective and convenient in analyzing the control and stability of a nonlinear system due to the fact that the trajectory of a Hamiltonian system lies in a phase space rather than in a configurational space [1]. However, compared with Newtonian and Lagrangian mechanics, the application of Hamiltonian mechanics to a continuous system is limited [2]. The reasons for the limitation may come from the follows. First, there is a misunderstanding that the Poisson bracket can not be applied to a field theory. Second, it *

is relatively difficult to obtain the associated Poisson bracket that is expressed by Eulerian field state variables rather than by Lagrangian canonical variables. Third, there is no systematical way to construct the dissipative bracket of a system. The bracket formalism of Hamiltonian mechanics was originally developed for discrete particle systems. Under the framework of Hamiltonian mechanics, the equations of motion for an N-particle system are expressed by Hamilton’s canonical equations with the Hamiltonian H ( x k , p k ) : (α ) xk =

∂H d x (kα ) = , dt ∂ p (kα )

(α )

pk =

d p (kα ) ∂H = − (α) , dt ∂x k

(1)

where x (kα )(t ) and p (kα )(t ) are the coordinate and momentum of particle α in the k-direction, and ( x k , 1 2 N p k ) denotes the set ( x1, x 2, , x N , p , p , , p ) . Provided that the Poisson bracket for the system is defined as

N ⎛ ∂F ∂G ∂G ∂F ⎞ {F , G} = ∑ ⎜ ( α ) − (α ) ⎟ , (α ) ⎜ ∂ x k ∂ p (kα ) ⎟⎠ α=1 ⎝ ∂ x k ∂ p k

(2)

Professor, corresponding author

Journal of Mechanics, Vol. 26, No. 2, June 2010

219

for any functions F ( x k , p k ) and G ( x k , p k ) , then combining Eqs. (1) and (2) generates d F ( x k, p k ) = dt

N

⎛ ∂F

∑ ⎜⎜ ∂ x

(α ) k

⎝ = {F , H } , α=1

∂H ∂F ∂H ⎞ − (α ) (α ) ⎟ (α ) ∂ pk ∂ p k ∂ x k ⎟⎠

(3)

which presents the general evolution of a conservative discrete system. It is noted that the Poisson bracket (2) is endowed with three properties: (i) it is bilinear, (ii) it is anti-symmetric in F and G, and (iii) it satisfies the Jacobi identity {{F, G}, H}} + {{H, F}, G}} + {{G, H}, F}} = 0. The application of the formulation (3) to a conservative continuous system is also availbale, and it can be traced back to the works of Arnold [3] on ideal fluids and of Marsden and Weinstein [4] on incompressible fluids. Its extension to a general nonconservative continuous system came later with the pioneering works of Kaufman [5], Morrison [6], and Grmela [7], who introduced the idea of the dissipative bracket to account for dissipative phenomena. These works stipulate that the time evolution of a system functional F can be expressed by dF = { F , E} + [ F , S ] , dt

(4)

where the functionals E and S are the total energy and entropy of a system. Here, [⋅,⋅] is the dissipative bracket, which can be represented by [8,9] [F , S ] =

with

δF δR , δa δ(δS / δa )

(5)

i, i , R, and a being the inner product, the dis-

sipation potential, and the state variables, respectively. The dissipation potential R is subject to the requirements that R(0) = 0, R has its minimum at 0, and R is convex in the neighborhood of 0. The significance of Eq. (4) lies in the two facts [8,10-12]. First, the dynamics of a system is separated into a reversible and conservative part, characterized by the Poisson bracket, and an irreversible and dissipative part, described by the dissipative bracket. Thus, the evolution of a dynamic system is fully determined by the two functionals and two brackets. A similar phenomenon also occurs in an elastic-perfectly plastic model, in which the plastic equations can be represented by a two-bracket formulation [13]. Second, because of the two properties that {F, F} = 0 and [F, F] ≥ 0, and because of the two degeneracy conditions that the entropy is a Casimir functional for the Poisson bracket and the total energy is a Casimir functional for the dissipative bracket, i.e., {F, S} = 0 and [F, E] = 0 for any functional F, the first and second law of thermodynamics, expressed by dE/dt = 0 and dS/dt ≥ 0, are naturally included in Eq. (4). The Poisson bracket for discrete systems has a canonical form due to the Lagrangian description; however, continuous systems usually adopt the Eulerian 220

description such that the canonical structure of the Poisson bracket is not preserved. Abarbanel et al. [14] use the LE transformation of state variables (called Marsden-Weinstein reduction [15,16]) for inviscid flows, followed by Edwards and Beris [17] for nonlinear elasticity, to derive the noncanonical Eulerian Poisson brackets from their canonical Lagrangian forms. Another useful approach to constructing the noncanonical Poisson bracket is to identify the underlying Lie algebraic structure of the state space expressed by the state variables of a given system [18]. To date, the procedure of former transformation is feasible in a few cases such as elastic fluids [17,19], rotational systems [20], and micromorphic elastic solids [21], and the possible expression of the noncanonical Poisson bracket is built mainly on the convective processes for the state variables and on one’s insight into the physics in question [11,22]. In a previous related study [21], we have used the LE transformation to present the bracket formalism of a micromorphic solid, where the consideration of dissipative effect is missing. This paper further applies this transformation to the construction of the noncanonical Poisson bracket of a two-component mixture system, the bracket of which has not yet been shown in this way. As for the dissipative bracket arising from viscous and diffusive dissipative processes, though its general expression has been studied from a continuum perspective [8,19], the linking between the dissipative bracket in Eulerian form and its Lagrangian counterpart has been left unaddressed. Thus, two simple dissipative mechanisms will be considered in Section 3 by modifying Hamilton’s canonical Eq. (1) and incorporating a dissipative bracket into Eq. (3). Section 4 derives the evolution equations of a micromorphic fluid, which can be treated as a multi-component mixture, through the bracket treatment. This treatment first constructs the noncanonical Poisson bracket of this fluid in a slightly different manner from that of a standard mixture and then proposes the Eulerian-type dissipative bracket by taking an analogy with its corresponding Lagrangiantype viscous bracket. Section 5 ends this paper with conclusions and remarks.

2. POISSON BRACKET FOR A TWO-COMPONENT FLUID MIXTURE

In the Hamilton’s setting, the first step to understanding the diffusion between different constituents of a mixture is to search for its reversible dynamics. This can be done by adopting the Lie group structure for the kinematics of the mass density and momentum of a one-component fluid, and then by using a one-to-one transformation to obtain the expression of the noncanonical Poisson bracket for a two-component fluid mixture [9,22,23]. Instead of the aforementioned treatment, this Poisson bracket can be systematically constructed by means of the LE transformation approach. To clarify various functional derivatives required in the following analysis, we first consider a continuous system. Let us assign a vector X to the position vector Journal of Mechanics, Vol. 26, No. 2, June 2010

of a material point X at time t = 0 and a function x ( X, t ) to the position function of X at time t. In other words, X is the coordinate of the point X before deformation in the Lagrangian description, and x ( X, t ) is the coordinate after deformation. The function x ( X, t ) with the initial condition x ( X, 0) = X specifies the motion of the continuum from the extent Ω with the boundary ∂Ω at time t = 0 to Ω′ with the associated boundary ∂Ω′ at time t. The dynamical parameters of a continuum in the Lagrangian description are the position x ( X, t ) and the momentum per unit volume u ( X, t ) = ρ0 ( X) ( ∂x ( X, t ) /∂t ) , where ρ0(X) is the mass density before deformation. Analogous to Eq. (2) in a discrete system, the Poisson bracket for a continuous system can be extended to ⎛ δF δG δF δG ⎞ 3 ⋅ − ⋅ {F , G}L = ∫ ⎜ ⎟d X . Ω δx δu δu δx ⎠ ⎝

(6)

Here, F = F [ x, u] and G = G[ x, u] represent two arbitrary functionals, and δ/δx denotes the Volterra functional derivatives [11]. The subscript “L” in the Poisson bracket indicates the Lagrangian description, which is used to trace the motion of a definite particle. In the current study, the Volterra functional derivatives are defined through the following two ways. (i) When a functional F represents the spatial integration of its density f ( x, ∇ X x, u, ∇ X u) such that F [ x, u] =

∫

Ω

f ( x, ∇ X x, u, ∇ X u) d 3 X , where ∇ X x = ∂x/∂X ,

the functional derivatives are defined as δF ∂f ∂f δF ∂f ∂f = − ∇X ⋅ , and = − ∇X ⋅ . δx ∂x ∂(∇ X x) δu ∂u ∂(∇ X u) (7) (ii) When a functional F is expressed as F [a, ∇a, b] =

∫

Ω′

f (a, ∇a, b) d 3 x , in which a = a(x, t) and b = b(x, t)

are the independent variables, the functional derivatives are defined as δF ∂f ∂f δF ∂f = −∇⋅ = . or δa ∂a ∂ (∇a) δb ∂b

(8)

We should emphasize that the boundary terms arising from integration by part are all discarded in this work, but not because of boundary effect insignificance. On the contrary, the boundary terms should be regarded as a disturbance or a source to a system that deserves further consideration. Now we consider a fluid mixture composed of two different continua, denoted by the superscripts “(1)” and “(2)”. The motions for the continua “(1)” and “(2)” are described by the two functions x ( X, t ) and y (Y, t ) , respectively. As Fig. 1 shows, two mass elements with their own position vectors X, Y, mass (2) densities ρ(1) 0 ( X) , ρ0 ( Y ) and momentum densities (1) (2) (1) (2) at u (Y, t ) = ρ0 (Y) y u ( X, t ) = ρ0 ( X) x ,

(

)

(

Journal of Mechanics, Vol. 26, No. 2, June 2010

)

Fig. 1

Kinematics of mass elements for a twocomponent mixture

time t = 0 will occupy the same physical space with position vector x at time t, i.e., x = x = y at time t. This treatment of two mass elements occupying the same space should be understood in a macroscopic sense. That is, it is compatible with the assumption taken in a general mixture theory that different constituents are allowed to occupy a common physical space [24]. In the Eulerian description, the time evolution of the mixture can be characterized by the state variables (ρ, c, u, w), where ρ = ρ(1) + ρ(2) is the total mass density, c = ρ(2) / (ρ(1) + ρ(2) ) is the concentration of the second continuum, u = u (1) + u (2) is the total momentum density, and w is the relative momentum density defined as w = (ρ(2) u (1) − ρ(1) u (2) ) / (ρ(1) + ρ(2) ) . The Lagrangian and Eulerian quantities can then be related to each other by accounting for the fact that the mass element X moves to the spatial point x at time t such that x = x ( X, t ) at time t. Accordingly, the LE transformation relations for the four variables (ρ, c, u, w) are ρ(x, t ) = ∫

3 3 ρ(1) 0 ( X) δ ( x − x) d X

ΩX

3 3 + ∫ ρ(2) 0 ( Y ) δ ( y − x) d Y ,

(9)

ΩY

(

)

c(x, t ) = ρ(2) (x, t ) / ρ(1) (x, t ) + ρ(2) (x, t ) , u(x, t ) = ∫

(10)

3 3 (1) u ( X, t ) δ ( x − x) d X

ΩX

+∫

ΩY

3 3 (2) u (Y, t ) δ ( y − x) d Y ,

w (x, t ) = cu (1) − (1 − c) u (2) ,

(11) (12)

where ΩX and ΩY are the domains occupied by the constituents “(1)” and “(2)” at time t = 0. From these relations it is straightforward to obtain the functional derivatives δα/δα , where α and α respectively denote one of the elements in the sets (ρ, c, ui, wi) and (2) ( x k , y k , u (1) k , uk ) . 221

Now, motivated by Eq. (3), the evolution equation of the two-component fluid mixture in the Lagrangian description is written as dF = {F , H } L = dt

∫

+

∫

ΩX

ΩY

⎛ δF δH δH δF ⎞ 3 − d X ⎜ (1) (1) ⎟ ⎝ δ x k δu k δ x k δu k ⎠ ⎛ δF δH δH δF ⎞ 3 − ⎜ ⎟ d Y , (13) (2) δ δ δ y y k δu (2) k ⎠ ⎝ k uk

which can be transformed, in terms of Eulerian state variables, to dF/dt = {F, H}E, where the subscript “E” indicates the Eulerian description and the two functionals F and H depend on (ρ, u, c, w). To be explicit, we aim to find the noncanonical Poisson bracket {F, H}E from its corresponding canonical bracket {F, H}L when the arguments of the system’s functional F are changed from the Lagrangian variables ( x, y, u (1), u (2)) to the Eulerian variables (ρ, u, c, w). Using the chain rule of differentiation, it then follows that δF = δα

⎛ δF δρ δF δu j δF δc δF δw j + + + ⎜ Ω ′ ⎜ δρ δα δu j δα δc δα δw j δα ⎝

∫

{F , H } E

δF ⎛ δH ⎞ δH ⎛ δF ⎞ ⎪⎧ δF ⎛ δH ⎞ uk − ρ − + wj ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎨ ⎜ ⎟ ⎜ ⎟ ′ Ω δwk ⎝⎜ δu j ⎠⎟, k ⎪⎩ δρ ⎝ δu j ⎠, j δuk ⎝ δu j ⎠, j ⎛ δF ⎞ δH δH ⎛ δF ⎞ δF δH ∂c + wj + wj ⎜ c ⎜ ⎟ − ⎜ ⎟ ⎜ δw j ⎟⎟ δwk δuk ⎝ δw j ⎠, k δc δu j ∂x j ⎝ ⎠, k

∫

⎛ δF + (1 − c)(cu j − w j )⎜⎜ ⎝ δw j

⎞ δH ⎟⎟ ⎠, k δwk

∂u j

⎞ ⎟⎟ , ⎠, j

⎛ δH δH δH ⎞ c, j − ⎜ wj + u j ⎟ δc δ w δ uk ⎠ , k ⎝ k ⎛ δH ⎞ ⎛ δH ⎞ ⎛ δH ⎞ − wk ⎜ ⎟ − ρ⎜ ⎟ , ⎟ − uk ⎜ ⎝ δwk ⎠ , j ⎝ δρ ⎠ , j ⎝ δuk ⎠, j

=

∂t

(18)

(19)

∂w j

⎛ 1 δH ⎞ u j ⎛ δH ⎞ = ρ c(1 − c)⎜ ⎟ ⎟ + ⎜ ρc(1 − c) ∂t ρ δ c ρ δ wk ⎠ , k ⎝ ⎠, j ⎝ ⎛ δH uk ⎞ ⎛ δH ⎞ + ρ c(1 − c)⎜ ⎟ − c ⎜ wj ⎟ ⎝ δwk ρ ⎠ , j ⎝ δwk ⎠ , k ⎛ δH ⎞ ⎡ δH ⎤ − wk ⎜ c ⎟ − ⎢(1 − c)(cu j − w j ) δwk ⎥⎦ , k ⎝ δwk ⎠ , j ⎣ ⎛ δH ⎞ ⎛ δH ⎞ ⎛ δH ⎞ − (1 − c) ( cuk − wk ) ⎜ ⎟ − wk ⎜ ⎟ − ⎜ wj ⎟ , w δ ⎝ k ⎠, j ⎝ δuk ⎠ , j ⎝ δuk ⎠ , k (20)

which govern the time evolutions for the total mass density, the concentration of the second continuum, the total momentum, and the relative momentum, respectively. With the specification of the Hamiltonian functional as H=

∫

Ω′

h(ρ, c, u, w ) d 3 x

⎛ u (1) ⋅ u (1) u (2) ⋅ u (2) ⎞ + + ϕˆ (ρ(1) , ρ(2) ) ⎟ d 3 x ⎜ (1) (2) Ω′ ρ ρ 2 2 ⎝ ⎠ ⎛ u ⋅u ⎞ w⋅w = ⎜ + + ϕ(ρ, c) ⎟ d 3 x , Ω′ ρ ρ − 2 2 c (1 c ) ⎝ ⎠

=

∫

∫

∂u j ∂t

⎫⎪ 3 δH ⎛ 1 δF ⎞ − ρc(1 − c) ⎜ ⎟ − (F ⇔ H )⎬ d x , δw j ⎝ ρ δc ⎠, j ⎪⎭

(21)

=−

⎞ ∂ ⎛ δH ⎞ ∂ ⎛⎜ δH ∂p ⎟ , wj ⎟ − ⎜ ⎜uj ⎟− ⎜ ⎟ ∂xk ⎝ δuk ⎠ ∂xk ⎝ δwk ∂x j ⎠

(22)

where the pressure p is given by (15)

where “( ),k” stands for ∂( ) / ∂xk. The notation (F⇔H) represents the above corresponding terms with the interchange of F and H, and adding the term −(F⇔H) in the above equation is for the skew-symmetry of the Poisson bracket {F, H}E. From Eq. (15) and the relation ⎛ δF ∂ρ δF ∂ u j δF ∂c δF ∂w j ⎞ 3 + + + ⎜ ⎟d x, Ω′ ⎜ δρ ∂t δ u j ∂t δc ∂t δw j ∂t ⎟⎠ ⎝ (16)

∫

the independence of the four variables (δF/δρ, δF/δc, δF/δuj, δF/δwj) yields the equations 222

(17)

Equation (19) can be reduced to a more familiar form as

δF u j ⎛ δH ⎞ + ⎜ ρ(1 − c)c ⎟ δw j ρ ⎝ δwk ⎠,k

dF = dt

⎞ ⎟⎟ , ⎠, j

1⎛ ∂c δH δH c, j + ⎜ ρ c(1 − c) =− ⎜ ∂t δu j ρ⎝ δw j

⎞ 3 ⎟⎟ d x . ⎠ (14)

Substituting those functional derivatives into Eq. (13), and taking several lengthen calculations lead to the noncanonical Poisson bracket in the form

=

⎛ δH ∂ρ = −⎜ ρ ⎜ ∂t ⎝ δu j

p = −h + ρ

δH δH δH +u⋅ +w⋅ . δρ δu δw

(23)

Since the Poisson bracket (15) is associated with the reversible transformation for the state variables of a system, these derived equations show the general forms for the reversible dynamics of a two-component fluid mixture. 3.

DISSIPATIVE BRACKETS WITH EULERIAN VARIABLES

The previous section uses the LE transformation method to generate the conservative bracket for a twoJournal of Mechanics, Vol. 26, No. 2, June 2010

component mixture. However, Eq. (4) has shown that a complete evolution of a system requires the second bracket. i.e., the dissipative bracket. According to the expression of dissipative bracket in Eq. (5), it can be simplified to δF δR , δa δ(δΦ / δa )

[F , S ] = −

δU ij (x, t )

(α ) xk =

∂H , ∂ p (kα )

(α)

pk = −

∂H ∂R + (α ) , (α ) ∂x k ∂x k

[ F , R ]L = −

∂F

N

∑ ∂p α=1

(α ) k

∂R . ∂ x (kα )

Based on this dissipative bracket of a discrete N-particle system, the analogous counterpart of a continuous fluid system, in which only the viscous dissipation is taken into account, can be written as

Ω

δF δR 3 d X . δu k δv k

(27)

For the sake of simplicity, assume that the state variables of the continuous fluid system are the mass density ρ and the linear momentum density ui (= ρvi). In other words, the density of system’s functional F, expressed by f, is a function of ρ and ui. Further assume that the density of dissipative functional, r, depends on the mass density ρ and the velocity gradient density Uij(= ρvi,j), where the transformation relation for the variable Uij is given by Journal of Mechanics, Vol. 26, No. 2, June 2010

(28)

with J = ρ0 /ρ.

⎞ ∂X K 3 ⎟ δ ( x − x) δik ⎟⎟ ⎟ ∂x j ⎠,K

∂ ⎛ 3 ⎞ ⎜ ρ( x, t ) δ ( x − x) δik ⎟ , ⎠ ∂x j ⎝

(29)

Substituting the relations

⎛ ∂f δρ ∂f δui ⎞ 3 δF = ⎜ + ⎟ d x, and δu k ∫Ω′ ⎝ ∂ρ δu k ∂ui δu k ⎠ δR ∂r δU ij 3 d x, =∫ Ω ′ ∂U δv k ij δv k

(30)

into Eq. (27) and accounting for the functional derivative (29) generates the dissipative bracket with Eulerian variables as ∂f Ω ′ ∂u i

[ F , R ]E = ∫

⎛ ∂r ⎜⎜ ρ ⎝ ∂U ij

⎞ 3 ⎟⎟ d x . ⎠, j

(31)

Now we adopt the simplest forms of the system functional F and the dissipative functional R for a simple fluid system, given by ⎛u u ⎞ F = ∫ f (ρ, u) d 3 x = ∫ ⎜ k k + ϕ(ρ) ⎟ d 3 x , Ω′ Ω′ ⎝ 2ρ ⎠

(32)

(26)

Comparing this dissipative bracket with Eq. (24) clearly reveals the correspondence between the two expressions as soon as (a, Φ, δΦ/δa) are replaced by ( p (kα ), H , x (kα )) .

[ F , R ]L = − ∫

⎜ ⎝

= −J

(25)

where the system’s dissipation is characterized by the velocity-dependent dissipative function R. This set of canonical equations implies the single evolution equation of the system dF/dt = {F, H}L + [F, R]L, where the dissipative bracket is

⎛ ⎜

= −⎜⎜ ρ0

δv k

Viscous Dissipation

Our analysis of the viscous dissipative bracket in terms of Eulerian variables begins with Hamilton’s canonical equations for an N-particle discrete system:

∂X K 3 δ ( x − x) d 3 X . ∂x j

This relation leads to the following functional derivative

(24)

while a system is kept at constant temperature T0, where Φ(a) = E(a) − T0S(a) is the Helmholtz free energy. For simplicity, we only concentrate on the viscous and diffusive dissipations of an isothermal system, and discusses the application of the LE transformation to constructing the dissipative bracket from Lagrangian variables to Eulerian variables. Dissipation due to heat conduction is not considered here because the unconserved entropy makes it infeasible to obtain the associated Lagrangian formulation of entropy equations. 3.1

U ij (x, t ) = ∫ ρ0 v i , K Ω

R = ∫ r (ρ, u) d 3 x = ∫ Ω′

Ω′

1 Cijkl (U ij + U ji )(U kl + U lk ) d 3 x , 8ρ (33)

where ϕ(ρ) is the potential energy density and Cijkl is the viscosity coefficient. With R in Eq. (33), a dissipative stress tensor Dτji(= ρ(∂r/∂Uij) = 1/2Cijkl(vk,l + vl,k)) will be generated by the dissipative bracket in the evolution equation of linear momentum. Clearly, adopting another expression of dissipative functional leads to a different constitutive relation between the dissipative stress and velocity gradient. However, regardless of which constitutive relations are selected, the LE transformation illustrates that the dissipative stress is directly related to the discrete form of the dissipative bracket in Eq. (26). 3.2

Diffusive Dissipation

The dissipative bracket of a two-component mixture from diffusive motion can be constructed as follows. The Hamilton canonical equations of this system are written as 223

∂H ∂H ∂R (1) , p k = − (1) + (1) , ∂ ∂ ∂ p (1) xk xk k ∂H ∂H ∂R (2) (2) , p k = − (2) + (2) , xk = ∂ ∂ ∂ p (2) x xk k k

4.

(1) xk =

(34)

where ∂R /∂ x (1) and ∂R /∂ x (2) are the interactive k k forces between the two components of the mixture and, (2) hence, satisfy the condition ∂R /∂ x (1) k + ∂R /∂ x k = 0 . It is possible to meet this condition if the dissipation (2) function R depends on the argument ( x (1) k − xk ) . Moreover, the equations of motion can be expressed in the general form of Eq. (4) with the dissipative bracket [ F , R ]L = −

∂F ∂R ∂F ∂R − . (1) (1) ∂ x (2) ∂ p k ∂ x k ∂ p (2) k k

(35)

The bracket in Eq. (35) sufficiently provides an analogous bracket, i.e., [ F , R ]L = −

∫

ΩX

δF δR 3 d X− (1) δu (1) k δv k

∫

ΩY

δF δR 3 d Y . (2) δu (2) k δv k (36)

for a two-component mixture continuum with the state variables (ρ, c, u, w). Let the dissipative functional for this continuum be dependent on the mass density ρ, concentration c, and relative momentum wk, i.e., R = ∫ r (ρ, c, wk ) d 3 x . From the transformation relaΩ′ tions in Eq. (12), the following functional derivatives (1) (2) (2) δρ(x, t ) /δv (1) k ( X, t ) = δc /δv k = δρ/δv k ( Y, t ) = δc /δv k (1) 3 (1) =0 , δw j /δv k = ρ0 ( X) c(x) δ ( x − x) δ jk , and 3 δw j /δv (2) ( Y , t ) = −ρ(2) can 0 ( Y ) (1 − c ( x) ) δ ( y − x) δ jk k be readily obtained. Thus, by taking into account the functional derivatives in the two integrands of Eq. (36), the dissipative bracket is found to be [ F , R ]E = −

∫

Ω′

ρc(1 − c)

∂f ∂r 3 d x. ∂w j ∂w j

(37)

Choosing a simple dissipative functional R of the type R=

∫

Ω′

rd 3 x =

∫

Ω′

Dw j w j 2ρc(1 − c)

d3x

(38)

will add a term −Dwj to the right-hand side of the evolution equation of relative momentum (20). This augmented equation had been used by Grmela et al. [9,22] to analyze non-Fickian diffusion. To summarize, the Eulerian-variable-based dissipative brackets (31) and (37) for viscous and diffusive mechanisms can be generaged from their Lagrangianvariable-based counterparts (27) and (36). Notice that the two forms of brackets for viscous and diffusive dissipations are almost the same under the Lagrangian description framework and both the two dissipations are derived from the velocity-dependent dissipation function R. However, R depends on self-velocity for viscous dissipation and on relative velocity for diffusive dissipation. 224

BRACKET FORMALISM FOR A MICROMORPHIC FLUID

The microcontinuum theory has been successfully developed to account for the internal degree of freedom of a material, where the internal characteristic length of the material is comparable to the length scale of the entire material body [25]. The standard approach to derive the evolution equations of a microcontinuum can be based on either Newtonian mechanics (i.e., to formulate the general balance laws) or Lagrangian mechanics (i.e., to look for a variational or energy principle) [26]. Different from Newtonian and Lagrangian mechanics, Hamiltonian mechanics provides a third method. In a previous work [21], we have proposed the noncanonical Poisson bracket of an elastic micromorphic solid without dissipation. To make a complete investigation of bracket formalism for a microcontinuum, this section will focus on the derivation of the evolution equations of a micromorphic fluid. While the Poisson bracket for the fluid shares the same expression as that for the elastic solid if the terms related to elastic contributions are disregarded, the major difference between the solid and the fluid in the Hamilton approach lies in the introduction of dissipative bracket required for describing the viscous behavior of the fluid. Thus, particular emphasis will be laid on the construction of the dissipation bracket, which is dependent on three types of deformation-rate tensors [25]. From the definition of a microcontinuum, which states that a microcontinuum is a continuous collection of deformable particles, a microcontinuum can reasonably be modeled as a multi-component mixture [27]. The components of this mixture occupy the same physical space and are distinguished by the relative position vector, ξ, connecting a microelement and the center of mass of its associated macroelement. In a microcontinuum, let ΔBx be the macroelement at the position x with volume Δvx and mass density ρ. The macroelement is composed of many microelements, distinguished by the ξ’s vectors, with volume dvξ′ and mass density ρ′. The counterparts of the seven quantities x, ξ, ΔBx, dvξ′ , Δvx, ρ, and ρ′ in the reference configuration are X, Ξ, ΔBX, dVΞ′ , ΔVX, ρ0, and ρ′0 , respectively. Furthermore, let f ′ be defined as ρ′/ρΔvx and its counterpart in the reference configuration be denoted by f 0′ . The conservation of mass is held in a macroelement such that we have f ′ dvξ′ = ρ′dvξ′ /ρΔvx = ρ′0 dVΞ′ /ρ0 ΔVX = f 0′ dVΞ′ . Note that quantities with the superscript “prime" are the ones for a microelement. For this multi-component mixture, the local centerof-mass coordinate system helps us to propose the canonical Poisson bracket as [21] Journal of Mechanics, Vol. 26, No. 2, June 2010

∫

{F , G}L =

+∫

Ω

+

∫

ΔBX

∫ ∫ Ω

ΔBX

Ω

{F , H }E

⎛ δF δG δF δG ⎞ 3 ⋅ − ⋅ ⎜ ⎟d X ⎝ δ x δu δu δ x ⎠

=

⎛ δF δG δF δG ⎞ 1 3 ⋅ − ⋅ ⎟ dVΞ′ d X 2 ⎜ f0′( X, Ξ)(ΔVΞ′ ) ⎝ δξ δp ξ δp ξ δξ ⎠

1 ⎛ δF δG δF δG ⎞ 3 ⋅ − ⋅ ⎜ ⎟ dVΞ′ d X , ΔVΞ′ ⎝ δp ξ δx δx δp ξ ⎠

(39)

ξ

with the micromomentum density p (= ρ0 ξ) . For a micromorphic fluid of grade one, the suitable state variables for describing the motion of this material are the mass density ρ, the momentum density u, the microinertia density ˆi , and the micromomentum density ˆ , where iˆ kl Δvx = ∫ ρ′(x, ξ, t ) ξk ξl dvξ′ and m ΔBx mˆ kl Δvx = ∫ΔB ρ′(x, ξ, t ) ξk ξ l dvξ′ . Note that the matex rial time rate of the vector ξ can be found to be ξ k = ν kl (x, t ) ξl , in which v is the gyration tensor. The independence of ξ in v is a linear approximation for a micromorphic medium of grade one [25]. With the four independent variables, we have δF = δα

∫

Ω′

⎛ δF δρ δF δu j δF δiˆ kl δF δmˆ kl ⎞ 3 + + + ⎜⎜ ⎟⎟ d x , ⎝ δρ δα δu j δα δiˆ kl δα δmˆ kl δα ⎠ (40)

where α stands for ( x n, u n, ξn , p ξn) . The four kinds of derivatives δρ/ δα , δu j / δα , δiˆ kl / δα , and δmˆ kl / δα can be evaluated from the LE transformation ˆ ) as relations for (ρ, u, ˆi, m

⎡ δF ⎛ δH ⎢− ⎜ρ Ω′ ⎢ δρ ⎜ δu j ⎝ ⎣

∫

⎞ δF ⎟⎟ − δ ⎠, j u j

⎛ δH ⎞ ⎜uj ⎟ ⎝ δuk ⎠ , k

−

δF ⎛ δH ⎜ iˆ kl δiˆ kl ⎜⎝ δu j

⎞ δF ⎛ δH ⎟⎟ − ⎜⎜ mˆ kl δ δ uj mˆ kl ⎝ ⎠, j

⎞ ⎟⎟ ⎠, j

+

δF δH δF δH iˆ lj + iˆ kj δiˆ kl δmˆ jk δiˆ kl δmˆ jl

+

⎤ 3 δF ⎛ δH ⎞ ⎜ mˆ jl ⎟ − (F ⇔ H )⎥ d x . ⎜ ⎟ δmˆ kl ⎝ δmˆ jk ⎠ ⎦⎥

(45)

Furthermore, based on the Hamilton canonical Eq. (25) for an N-particle discrete system and the form of its associated dissipative bracket in terms of the local center-of-mass coordinate system, we propose the dissipative bracket of a micromorphic fluid as ⎛ δF δR ⎞ 3 [ F , R ]L = − ∫ ⎜ ⋅ ⎟d X Ω δu δv ⎝ ⎠ ⎛ δF δR ⎞ 1 −∫ ∫ ⋅ ⎟ dVΞ′d 3 X ⎜ Ω ΔBX f ′( X, Ξ )( ΔV ′ ) 2 δp ξ δξ 0 Ξ ⎝ ⎠ 1 ⎛ δF δR ⎞ 3 +∫ ∫ ⋅ ⎟ dVΞ′d X . ⎜ Ω ΔBX ΔV ′ δp ξ δv Ξ ⎝ ⎠

(46) Assume that the density r of the dissipative functional depends on the three deformation-rates: aˆ kl , bˆ klm, and 3 cˆ kl , i.e., R = ∫Ω′ r (aˆ kl, bˆ klm, cˆ kl ) d x , where the LE transformation relations of the three rates are given by aˆ kl (x, t ) = ∫ ρ0 ( X) ( vl , k ( x, t ) − ν lk ( x, t ) ) δ3 [ x ( X, t ) − x ] d 3 X , Ω

ρ(x, t ) = ∫ ρ0 ( X) δ [ x ( X, t ) − x ] d X , 3

3

Ω

u(x, t ) =

ˆi (x, t ) =

∫

u ( X, t ) δ [ x ( X, t ) − x ] d X , 3

Ω

∫ (∫ Ω

ΔBX

f 0′( X, Ξ )ξ ⊗ ξ dVΞ′

3

∫ (∫ Ω

ΔBX

(41)

(48) (42)

)

f 0′( X, Ξ )ξ ⊗ p ξdVΞ′

3 3 bˆ klm(x, t ) = ∫Ω ρ0 ( X)ν kl , m ( x, t ) δ [ x ( X, t ) − x ] d X ,

2cˆ kl (x, t ) = ∫ ρ0 ( X) ( ν kl ( x, t ) + νlk ( x, t ) ) δ3 [ x ( X, t ) − x ] d 3 X . Ω

(49)

⋅ ρ0 ( X, t ) δ3 [ x ( X, t ) − x ] d 3 X ,

ˆ (x, t ) = m

(47)

(43)

The three relations generate the required functional derivatives which help to transform the dissipative bracket (46) into its Eulerian counterpart

(44)

[ F , R ]E =

)

δ3 [ x ( X, t ) − x ] d 3 X ,

where the operator “⊗” denotes the tensor product. Now substitute Eq. (40) and those derivatives into the bracket (39), and a slightly lengthy manipulation yields the noncanonical Poisson bracket of a micromorphic fluid as Journal of Mechanics, Vol. 26, No. 2, June 2010

⎧⎪ δF ⎛ δR ⎞ ⎜ρ ⎟ ⎨ Ω ′ δu ⎜ δ ˆ ⎟ ⎪⎩ i ⎝ a ij ⎠, j ⎫ δF ⎡⎢ δR δR ⎛ δR ⎞ ⎤⎥ ⎪ 3 + ρ −ρ + ⎜ρ ⎟ ⎬ d x, (50) δmˆ ij ⎢ δaˆ ij δcˆ ij ⎝⎜ δbˆ jik ⎠⎟ ⎥ ⎪ ,k ⎦ ⎭ ⎣

∫

whose explicit expression can be determined as soon as the density function r is specified. 225

Now let the Hamiltonian H of a micromorphic fluid be H=

∫

Ω′

hd 3 x =

∫

Ω′

⎛ u k u k 1 −1 ⎞ + iˆ pq mˆ pk mˆ qk + ε ⎟ d 3 x , (51) ⎜ ρ 2 2 ⎝ ⎠

where the first term represents the kinetic energy of the center of mass, and the second term characterizes the kinetic energy relative to the center of mass. The quantity ε stands for the internal energy density, and for a micromorphic fluid it can be assumed to be a function of mass density ρ and microinertia density iˆ kl . By virtue of the two above brackets and dF = { F , H } E + [ F , R ]E dt ⎛ δF ∂ρ δF ∂u j δF ∂iˆ kl δF ∂ mˆ kl ⎞ 3 = ⎜⎜ + + + ⎟d x, Ω ′ δρ ∂t δu j ∂t δiˆ kl ∂t δmˆ kl ∂t ⎟⎠ ⎝ (52)

∫

∂u j ∂t

= −(ρvk v j ), k + τkj , k ,

(56)

where the spin inertia per unit mass σkl, reversible stress tensor Rτij, dissipative stress tensor Dτij, dissipative microstress Dsij, dissipative stress moment Dγklm are defined as (57)

⎛ δH ⎞ δH δH δH τij = −δij ⎜ ρ + uk + iˆ kl + mˆ kl − h⎟ δuk δiˆ kl δmˆ kl ⎝ δρ ⎠ ⎛ ∂ε ⎞ ∂ε = −δij ⎜ ρ + iˆ kl − ε⎟ , ˆ ∂ρ ∂ i kl ⎝ ⎠

D

τij = ρ

∂r , ∂ aˆ ij

s =ρ

D ij

∂r , ∂ cˆ ij

(58)

γ

D klm

=ρ

∂r . ∂bˆ lmk

(59)

It can be verified that the derived four evolution equations are exactly the same as the balance equations of mass, linear momentum, microinertia, and momentum moment in the field theory of a micromorphic fluid [25]. In the Eringen theory, the specific energy ψE replaces the energy density ε in this study, so that it is easy to 226

(60)

Note that, since the internal energy density ε is a nonconservative quantity [21], the internal energy equation cannot be directly obtained from the bracket formalism. However, it can still be achieved by accounting for the partial time derivative of the energy function ε(x, t ) = ε ( ρ(x, t ), iˆ kl (x, t ) ) , i.e., ∂ε/∂t = (∂ε/∂ρ)(∂ρ/∂t ) + (∂ε/∂iˆ kl )(∂iˆ kl/∂t ) . Thus, accounting for the evolution Eqs. (53) and (55) leads to

(54)

⎛ ∂ε ∂ε ⎞ ρσlk = − ⎜ iˆ kj +ˆ + τ − s + γ , ⎜ ∂iˆ lj i kj ∂iˆ jl ⎟⎟ D kl D kl D mlk , m ⎝ ⎠

R

ρσlk − τkl + skl − D γ mlk , m = 0 .

which expresses the conservative part of energy balance equation.

(55)

∂ mˆ kl + (v j mˆ kl ) j − ν km iˆ mnν ln , ∂t

(

(53)

∂iˆ kl = −(iˆ kl vm ), m + iˆ kj ν lj + iˆ lj ν kj , dt

ρσlk =

)

⎛ ∂ε ∂ε ∂ε ⎞ = − (εvk ) , k + ⎜ ε − ρ − iˆ kl ⎟ v j, j ∂t ∂ρ ∂iˆ kl ⎠ ⎝ ⎛ ∂ε ∂ε ⎞ + ⎜ iˆ im + iˆ im ⎟ ν ji , ⎜ ∂iˆ jm ∂iˆ mj ⎟⎠ ⎝ = − (εvk ) , k + R τkk v j , j − ( R τij − R sij ) ν ji ,

the independence of the variables δF/δρ, δF/δuj, δF /δiˆ kl , and δF /δmˆ kl leads to ∂ρ = −(ρvk ), k , ∂t

find the one-to-one correspondences between the quantities used in [25], which are denoted by the superscript “E”, and the quantities adopted here. For example, E E −1 ρψ E (ρ, ikl ) = ε(ρ, iˆ kl ) , R τij = (∂ψ /∂ρ )δij E E −1 E = R τij , and R sij = (∂ψ /∂ρ )δij + ρ (∂ψ /∂iri )irj + (∂ψ E /∂iir )irj . Hence, Eq. (56) become

5.

(61)

CONCLUSIONS AND REMARKS

This paper illustrates a clear idea that the noncanonical Poisson brackets for the studied mixture systems and the dissipative brackets for viscous and diffusive processes can be directly constructed by the LE transformation. As soon as the two brackets of a mixture system are obtained, Eq. (4) clearly shows that the evolution of a system is fully determined by specifying the two system functionals, i.e., the energy and entropy (or dissipative) functionals. These brackets are derived through the three procedures. (i) We first determine the bracket formulation in Lagrangian description for the evolution of a system. (ii) We reach for the transformation relations of the Eulerian state variables in terms of the Lagrangian canonical variables. (iii) The bracket in Lagrangian form is then transformed to the Eulerian counterpart via the chain rule of differentiation. Different perspectives are adopted in dealing with the conservative part of the evolution for mixture continua. In the case of a two-component mixture, the state variables are ρ, c, u, and w. However, in the case of a micromorphic fluid, idealized as a multi- compoˆ . nent mixture, the state variables are ρ, u, ˆi , and m We note that, even though the two Poisson brackets (15) and (45) are not expressed in the canonical forms, the bilinearity and antisymmetry of these brackets still remain. In addition, the Jacobi identity for the brackets Journal of Mechanics, Vol. 26, No. 2, June 2010

(15) and (45) is also satisfied because they are derived from their canonical counterparts. As for the dissipative brackets under study, the procedure of the LE transformation necessitates a trace back to the Lagrangian formulation of a system, revealing that the diffusive mechanism resembles the viscous mechanism in the Lagrangian description. However, a small difference arising from the self-velocity-dependence or the relative-velocity-dependence in dissipation potentials causes the different Eulerian variable dependencies of the two mechanisms in the Eulerian description. Finally, let us briefly give two remarks on boundary conditions and future applications of this study. Remark 1. It is clear from the above derivation of the system evolutions that boundary conditions associated with the evolution equations cannot be directly obtained by the current LE transformation approach. This is due to the fact that the information about a possible discontinuity surface is missing in the process of performing the transformation of the equation: dF/dt = {F, H}L + [F, R]L. However, to complete the formulation of a system, the required boundary conditions should be introduced as auxiliary conditions. These conditions at the macroscopic point x on the boundary surface can be readily obtained by taking the integral form of the evolution equations over an infinitesimal material volume intersected by the boundary surface [24]. Comparing with other approaches in deriving field equations of a system, the Newtonian approach employs the balance laws on a material volume, which contains a possible discontinuity surface, to obtain the local balance equations as well as the associated boundary conditions. Similarly, the boundary conditions can be directly derived, accompanied by the field equations, under various frameworks of variational approach [26,28]. In spite of the slight limitation in deriving boundary conditions, the Hamiltonian approach has in particular shown its great advantage in being linked with thermodynamics. Remark 2. There are certainly some possible applications of the LE transformation. First, it should be stressed that the present derivation is definitely applicable to a saturated solid-fluid mixture, provided that additional state variables to account for elastic deformation of the composed solid continuum are included. As for application to a unsaturated mixture, involving voids with a negligible inertia with respect to that of solid or fluid media, a viable method to adopt is the volume-weighted approach [29,30], by which the porosity (or volume fraction) and the volume-weighted velocity could be chosen as the state variables in the Hamilton setting. Second, the formalism can also be applied to a microcontinuum of grade N, whose microstructure motion should be described by N different mappings, i.e., Ξi → ξi for i = 1, …, N. A simple example for a microcontinuum of grade N can be found in an N-component micromorphic mixture. Although a general theory of this continuum remains absent, it can be directly constructed in terms of the mixture theory. Third, the isothermal assumption adopted in this work Journal of Mechanics, Vol. 26, No. 2, June 2010

can be relaxed by treating heat conduction as transport of an entropy fluid [22,31]. In this treatment, the kinematics of the nondissipative entropy transport will be analogous to that of the mass transport. Accordingly, the evolution equation for the temperature- dependent entropy density can be achieved. These applications are interesting and their derivations await future research.

ACKNOWLEDGEMENTS The author is grateful for the financial support by the R.O.C. National Science Council under Grant NSC97-2221-E-002-126.

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(Manuscript received July 16, 2008, accepted for publication May 15, 2009.)

Journal of Mechanics, Vol. 26, No. 2, June 2010