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Continuum Mech. Thermodyn. 8 (1996) 171-187 © Springer-Verlag 1996

Original Article

Balance relations for classical mixtures containing a moving non-material surface with application to phase transitions

B. Svendsen, J.M.N.T. Gray

Institute for Mechanics, Technische Hochschule Darmstadt, D-64289 Darmstadt, Germany

Received June 6, 1995

This work is concerned with an extension of classical mixture theory to the case in which the mixture contains an evolving non-material surface on which the constituents may interact, as well as be created and/or annihilated. The formulation of constituent and mixture jump balance relations on/across such a non-material surface proceed by analogy with the standard "volume" or "bulk" constituent and mixture balance relations. On this basis, we derive various forms of the constituent mass, momentum, energy and entropy balances assuming (1), that the constituent in question is present on both sides of the moving, non-material surface, and (2), that it is created or annihilated on this surface, as would be the case in a phase transition. In particular, we apply the latter model to the transition between cold and temperate ice found in polythermal ice masses, obtaining in the process the conditions under which melting or freezing takes place at this boundary. On a more general level, one of the most interesting aspects of this formulation is that it gives rise to certain combinations of the limits of constituent and mixture volume fields on the moving mixture interface which can be interpreted as the corresponding surface form of these fields, leading to the possibility of exploiting the surface entropy inequality to obtain restrictions on surface constitutive relations.

1 Introduction

Broadly speaking, a mixture is a "material" which actually consists of a number of different "pure" materials (occupying in general one or more phases) which interact with each other in various ways. When a continuum model is appropriate, a mixture is usually modeled with the help of continuum mixture theory. In the standard form of this theory (e.g., Truesdell, 1984, Lecture 5), such interactions are represented as volume interactions, appropriate for the case in which the constituents are interacting "microscopically" and distributed more or less uniformly throughout the mixture. On the other hand, a mixture may also contain constituents which interact with each other "macroscopically" on moving, in general non-material surfaces or interfaces separating pairs and/or groups of such constituents from each other. In particular, such interfaces can arise when one or more of the constituents are present only in one part, or certain parts, of the mixture; such may be the case, for example, when the interactions between constituents take the form of phase transitions. An example of this would be a glacier or ice sheet, in which water "appears" in temperate regions, and "disappears" in cold regions, of the ice mass. Another arises in the case of snow, in which ice, water, vapor and air interact on such transition interfaces. Interactions of this nature cannot really be well-idealized by classical "microscopic" volume interactions alone, or at all. In fact, since a "real" mixture will in general contain constituents interacting on various length- and timescales, any theory for such mixtures should then include a model for both classical volume interactions as well as surface interactions between constituents. The purpose of this work is to formulate one possible extension of classical mixture theory to deal with this case.

172 B. Svendsen, J.M.N.T. Gray

The model of a material undergoing a phase change as (1), a "mixture" of the phases involved, or (2), as an evolving, moving, non-material or "singular" surface or front between the phases, has a long history, going back at least to the time of Gibbs (see Gibbs, 1961). The advantage of (1) over (2) is that multiple phase changes can be modelled simultaneously, while that of (2) over (1) is a more detailed description of the phase change process itself. By combining these two points of view, we also obtain a model which can deal with a mixture containing a variable number of constituents, something that standard mixture theory cannot handle. Such a situation is encountered in a number of applications. Kelly (1964) took the first step in this direction, extending the classical mixture theory of Truesdell (e.g., Truesdell & Toupin, 1960, §158) to the case in which the mixture contains a moving non-material surface S. Much more sophisticated extensions of standard mixture theory using averaging procedures which include models for surface interactions between constituents or mixture regions can be found in Hassanizadeh & Gray (1990), or Dobran (1991). The model we formulate here is much less sophisticated than such mixture models as well as the detailed thermodynamic models of a phase boundary as a moving, non-material surface such as Alts & Hutter (1988), or Gurtin & Struthers (1990); in particular, we do not model the phase boundary as a distinct thermodynamic entity, as is done in these works. Rather, we focus here solely on constituent interactions on S which must occur if mass, momentum, and energy are to be transferred across S . This compares in fact to the work of Morland & Gray (1995), who developed a model for such a mixture in which volume mass interactions are extended to a moving, non-material surface in the mixture on which one of the constituents may appear or disappear. Such a formulation, however, does not account for true surface processes, i.e., processes which result in an exchange of additive thermodynamic quantities across the moving, non-material surface. Here we formulate the general balance relations for such a case; one possible constitutive model for such interactions is discussed elsewhere (Gray & Svendsen, 1995). In the process, we extend and generalize the formulation of Kelly (1964) for such a continuum mixture.

After some mathematical preliminaries (Sect. 2), we extend the standard mixture balance relations to the case in which the mixture contains a moving, non-material surface (Sect. 3). Standard continuity arguments then yield the corresponding local forms of these; in particular, unlike Morland & Gray (1995), we assume that constituent volume interactions are bounded. As such, the corresponding local constituent jump balance relations on S are affected only by true surface interactions. Using the general constituent mixture and jump balance relations so obtained, we derive their specific forms for mass, momentum and energy balance as usual (Sect.4); the mixture forms of these include a number of results from Fried (1995) as special cases. In doing this, we are assuming that the constituent in question exists on both sides of S. On the other hand, when one or more constituents vanish and/or appear on S, the corresponding "jump" balance relations must be formulated somewhat differently (Sect. 5). To gain further insight into the models so obtained, we finally apply these to the case of (1), a binary mixture containing S, and (2), such a mixture in which one of the constituents disappears or appears on S (Sect. 6).

2 Mathematical preliminaries

Let E represent 3-dimensional Euclidean point space and ~ " its oriented linear translation space. Classical or standard mixture theory (e.g., Truesdell, 1984, Lecture 5) is formulated on the basis of the notion of superposed constituents. As such, all constituents and mixture fields are defined on the common mixture region M C E in E. As discussed in the introduction, the extension of classical mixture theory in this work is based on assuming that M contains a non-material surface S moving with a spatial velocity w. Such a surface induces the following structure on any subset R c M that it intersects in M:

R = R + U S R U R - ,

bR = b M R + U b S R U b M R - ,

bR + = bM R=~ U sR ,

(2.1)

SR := R N S ,

Balance relations for classical mixtures containing a moving non-material surface 173

with bR the boundary of R, bR + that of R ±, and bM R± its the material part. As usual, we work with the convention

n + = q: n ( 2 . 2 )

for the orientation of the outward unit normals n ± to bR ± with respect to that n of S. For any vector field z defined on S, let

z := n . z ~ (2.3)

zll := [ I - n ® n ] z ,

represent its normal component, and normal and tangential parts, respectively, where I C Lin(~/', ~ ) represents the identity linear transformation on ~ ' .

Let I C ~ represent a time interval, ~b:I × R \ SR ---+ °7/// a time-dependent, different±able ~Y-valued spatial field, v: I × R \ SR ~ ~v" the spatial velocity field of some material (e.g., constituent or mixture), and w:I x S --~ ~ that of S. For R +, we have as usual the extended transport relation 1

' ~ ± ~ b = f R ± ( O ¢ ) + f b u R + ~ b v . n + f s R ~b±w± - n ± (2.4)

(see, e.g., Truesdell & Toupin, 1960, §198; Chadwick, 1976, Ch. 3), where ~b + ( ¢ - ) represents the extension of the ¢ to the + ( - ) side of S, and w + the spatial velocity of S expressed relative to the orientation of bR ±. In addition, Stokes' theorem can be written in the extended form

fR±(d ivCP)=~ uR± q a ' n + ~ q ° ± ' n ± (2.5)

for any time-dependent, different±able ~ ' -va lued vector field qa: I × R ± ~ Lin(~ ' , ~ ) . From (2.5), we obtain

Cv " n +/s. ¢±w± ' n± = f ± div(~bv) + fsR ~b±(w~: - v±) . n± , (2,6)

which when substituted into (2.4) yields its alternative form

' f R ± ¢ = fR±[(0~)+ div(~bv)] + f~ ~b±(w± - v±) . n ± . (2.7)

From this last result, then, we obtain

6]R\SR (2.8)

via (2.3), qo[~:, n ± = 0, and the fact that ¢±(w ± - v ±) points out of R ± by convention, where

[[ff]] := ~+ - ~ - (2 .9 )

represents the "jump" of the quantity ff across S. Lastly, (2.5) yields

fbR\bS R q° ' n = fe\sR div qo + fsR[[~]] , (2.10)

again with (2.3), qol~= • n ± = 0, and the fact that cP± points out of R ± by convention.

1 In this work, 6 represents the absolute time derivative operator, and we drop the dv and da notation in volume and surface integrals, respectively, for simplicity.


3 General balance & sum relations

Consider now a mixture of n constituents. The spatial form of the general balance relation for the time- dependent field density ~b: I x R \ SR ~ ~ of some thermodynamic quantity associated with a constituent

Ct

with spatial velocity v : I x R \ SR ~ ~ / i n the mixture can be written in the form 2 O

\s~ \s~ °v oS \bs~ a \s~ a \s~ °v aS (3A)

Here ~ ~v : I x R \ SR ~ fY/Y, 6-*:~ I × R \ Sn ~ ~ and ~ v ~* : I × R \ Sn ~ ~ / represent the internal production,

external supply, and interaction supply, rate densities per unit mixture volume, respectively, of the constituent thermodynamic quantity whose density per unit mixture volume is represented by ~ (see Table 1 below for

a

the standard forms of these). Likewise, (b¢:I x R \ Sn ~ L i n ( ~ , ~/Y) represents the constituent flux density G

per unit mixture area of this quantity. Constituent fields per unit mixture volume (area for ~¢) are related to a

those per unit constituent volume (or area: ~b ¢) via the constituent volume fraction aUv:I x R \ Sn ~ (0, 1) ~t

in the mixture, i.e.,

f f , o ,oVj-- v ,ov, o ,oV

holds; again, the constituent fields on the right-hand side are those per unit constituent volume (or area). By analogy, the surface fields appearing on the right-hand side of (3.1) have the same interpretation with respect to SR as their "volume" counterparts appearing in (3.11) do with respect to the "bulk" material. Further, we have

by analogy with (3.2), where v s represents the area fraction of the corresponding constituent on SR, is in

general different from the limits v + of its volume counterparts, being, for example, a function of these. av We have the following standard special cases of the fields appearing in (3.1):

Table 1. Constituent densities

Balance ~ ~ ~* o* ft i1 ft

mass o 0 0 o i1

momentum Qv 0 T b a f t a tt

m o m e n t o f m o m e n t u m r A f l y 0 r A T r A b a f t fl ft

total energy 0e 0 TTv -- q r + b • v aft a ft a ft ft

entropy Qn ~ ¢pon ~ron a f t ft ft

where e := (a~ + ½ v a • ~) represents the constituent specific total energy. In general, the constituent partial

density ~ and spatial velocity v a represent independent variables. On the other hand, the constituent partial

stress tensor "i', specific internal energy 6, heat flux density fl, specific entropy ~, entropy flux density n on a a a I1 Q

- O r / and intrinsic entropy production rate density ~ v , as well as the mass a5°' momentum a ~y' energy ~ and

entropy act interaction supply rate densities, all represent dependent variables, i.e., constitutive quantities, in

general. The momentum b, internal energy r~ and entropy ~0n supply rate densities depend in general on the O

environment of the mixture under consideration.

2 Kelly's (1964) surface quantity s ~s s(C, ) is equivalent here to - .


The spatial form of the mixture general global balance relation is as usual that for a single material, i.e.,

= 7r ~ + + .n + fn \ s e ~r ~ (3.4)

where the mixture fields have the same interpretation as their counterparts in the constituent general local balance relation (3.1). As for each constituent, we have the following standard special cases of the fields appearing in (3.4):

Table 2. Mixture densities

Balance ~ *r*v 4~* a*

mass 0 0 0 0

momentum Ov 0 T b

moment of momentum r A go 0 r A T r A b

total energy ~oe 0 T T v - - q r + b • v

entropy Or/ 7r ~,7 ~ n o~n

where e := (e + ½ v • v ) represents the mixture specific total energy. The constituent (3.1) and mixture (3.4) balance relations are related to each other by the sum relations

(e.g., Truesdell, 1984, Lecture 5) and

cl

-- 2 *v, (3.5)

7r ~s = ~ ~s ~ , (3.6)

t t with ~ := ~ a = l , which hold if and only if the sum of the constituent interaction supply rate densities vanishes, i.e.,

~ = 0 (3.7)

and

is* = o , (3.8)

representing generalizations of Newton 's mechanical action-reaction principle to general interactions. From the point of view of the sum relations (3.5) and constraint (3.7), the mixture balance relation (3.4) represents in a sense a "linear combinaion" of the constituent balance relations (3.1) (a = 1, . . . , n).

On the basis of (2.8) and (2.10), (3.1) can be rewritten in the form

fn \ s R { 0 ~ - - d i v ( q S e - ~ ® v ) - g - ~ - # ~ - - g ~ } + J s R { l [ ~ ( v - w ) ~ - - ~ q ~ - ~ ' ~ - - ~ } = 0 , av a av a as as (3.9)

In a similar fashion, we obtain the alternative form

of (3.4). To obtain local balance relations from these global statements, we note that all quantities appearing in the integrands of the volume integrals in (3.9) and (3.10) may be considered to be continuous in R \ SR, such that (3.9) and (3.10) reduce to


~¢v + av ~ = 0¢ ~ - div ( ~ - ~ ® va) - ~ in R \SR (3.11)

7r¢ v 0¢ div (~b ¢ - ¢ ® v) - a ¢

respectively, in R \ SR. In the limit as we shrink R down to SR, then, (3.9) and (3.10) yield

+ _- - - } as on SR. (3.12)

~r¢ s ~¢(v w)~ ~¢~11

As in the standard jump balance relations, note that (3.12) contain no surface supply rate densities corresponding to the external (volume) supply rate densities cr¢ and a¢, respectively, appearing in (3.11).

a

4 Volume and surface balance relations in a classical mixture

Using the results from the last section, we first examine the case of a classical mixture containing SR, i.e., all constituents exists on both sides of SR, and interact in R \ SR and on SR with each other. In terms of the

-

material time derivatives := O~ + (V~)v a and ~ := O~ + (V~b)v, (3.11) take the forms

~ ¢ v + ~ = ¢ + ¢ d i v v - d i v ~ ¢ - ~ ¢ a a a a a (4.1)

7rv~ = ~ ) + ¢ d i v v - div~b ¢ - a ¢

In particular, for mass balance, (4.1)1 and (3.12) 1 reduce to

= ~ + 0 d i v ~ , a a (4.2)

for each constituent, and while (4.1)2 and (3.12) 2 reduce to

0 = O + o d i v v ,

0 = l[m]] , (4.3)

via Tables 1 and 2, as well as (3.7) and (3.8), where we have introduced the constituent and mixture mass fluxes

~ ± := ~=L(~± _ w)

m + := Q±(v i - w )

mass fluxes relative to SR as it is approach from the + sides. In terms of the constituent mass flux

(4.4)

J := Q ( v - v ) (4.5) a a a

relative to the barycentric mixture velocity v, we also have the form

m :L = J± + ~± ~m a a a

(4.6)

for m±. Multiplying (4.6) by au and summing the result over all constituents yields

± ± (4.7)


since J+ sums to zero via the form of (3.5) 1 for momentum balance from Tables 1 and 2, and since ~± sums t l 0.

to 1 via the form of (3.5) 1 for mass balance. To develop particular constituent and mixture surface jump balance relations in a way that facilitates

comparison with their volume conterparts, it is useful to work with the surface quantity

1 ((~)) := ~ (~+ + ~- ) (4.8)

(e.g., Fried, 1995), representing the "average" of the volume field ( on SR. The angle bracket is complementary to the jump bracket in the sense that the relations

IF~ e ~11 = ((@) e ~ /~ + ~ e ((~)) (4.9)

((o, e 9)) = ((,~)) e ((/~)) + ¼ ll:oa e ~:~]~

hold for any type of distributive product ® (e.g., scalar multiplication, or an inner product)• This comple- mentarity is also reflected by the relation ~+ = ((o~)) + ½ I[~]. As we will see in what follows, use of both brackets in a judicious manner yields constituent and mixture jump balances possessing a structure analogous to their volume counterparts, facilitating the interpretation of the terms arising in these jump balances. In particular, (4.9)~ yields the form

~e = ~]] + ((m )) [l~]] (4.10) a S a

for the constituent mass interaction supply rate density via (4.3)2 and (4.9)a, where ~ := g/~ represents the a a

constituent true mass fraction. In addition,

((m~)) = <~)) + ((~))((m)> (4.11) fl 11

holds via (4.5) and (4.3) 2 , with

((m)) = E ((~)) (4.12)

from (4.7) and (4.8). Since m + = m - follows from (4.3), we have ((m)) = m ±. Note that the quantity m defined in (2.3) in Fried (1995) is equal to -((rn)) in this work. Another consequence of the constituent (4.2)2 and mixture (4.3)2 jump mass balances are the expressions

II-~]l = I[m~/a~]] = ((m~))l[1/Oll + ((1/a~)}Z ~ , ( 4 . 1 3 )

I[v]] = [[m/ff[] = ((m))[[1/p]],

for the normal constituent and mixture velocity jumps, respectively, across SR, obtained using (4.9) 1 and fact that w is continuous across SR.

Introducing the specific form ~ of tb, i.e., ~b = Q~, as well as that tb = Q~ for tb, (4.1) 1 and (3.12)1

become := ~ _ -o - ~-~ d i v e * - 6-¢ ov ov ~ , ~ = O ~ ov - f l f l a O.

aS aS aS a

(4.14)

via (4.2) and (4.9) l Apparently, the interaction supply rate densities ~ and ~ represent that part of the • av aS

total interaction supply rate densities ~-~v and aZs ~, respectively, due to processes other than mass exchange.

Likewise, (4.1)2 and (3.12) 2 transform to

o = Q~ - ~ v ~ - d i v &

0 : ((m))[[~p]l - 7rs~ - I[~b~]], (4.15)


respectively, via (4.3) and (4.9)~. For momentum balance, (4.14) and (4.15) imply

5 a v a a a a

as = - I [ i i i ,

(4.16)

and 0 = Q~O - divT - b ,

0 = ( ( m ) ) [ [ v ] - ~ t ] ] , (4.17)

respectively, via Tables 1 and 2, as well as (4.9) 1. In (4.16), Zv represents the Euclidean frame-indifferent a

(EFI) part of Zpv. As for moment of momentum balance, we have the forms a

~v-rAv = 2skw(T)

~rAv = r A ~v a S aS

(4.18)

for the constituents via Table 1, (4.14), (4.16) and the fact that I[r~ = 0, as well as that

0 = skw(T) (4.19)

for the mixture volume from (4.15); the mixture moment of mometum jump balance is satisfied identically. With the help of the results

aS

(4.20) [ [ t - v l l - ((m}}l[ I v . v ] ] = ((t}} . l [vl l ,

from (4.9)1, (4.16)2 and (4.17) 2, the constituent (4.14) and mixture (4.15) specific balance relations for the case of energy balance reduce to the forms

~o - - -V ae ~" ~ = 0 ~ + div fl - T- (Vv) - ,

a a a a a

(4.21)

and 0 = O~ + divq - T . ( V v )

0 = ((m}}[[e]] + l[q]] - ((t}).[[v]],

- - r

(4.22)

respectively. Since all other terms in (4.21)2 are EFI, note that the combination fs - ((va)) " Z"as is as well.

An alternative form of the energy jump balance relations results when we split the constituent and mixture surface mechanical dissipations into their normal and tangential parts relative to SR, i.e.,

((t)) - [[vii = ( ( t ) ) [ [v~ + ((tll)) . ~ v t l ] , (4.23)

with

((t))[v]] = ((m))((t))l[1/e]],

+ <<{>><<1@ o¢, (4.24)

holding for the normal parts from (4.13). Note that the average quantities (<!11// and ((tll)) can be associated

with constituent and mixture frictional forces on SR (e.g., Fried, 1995). Using (4.23)1,2 with (4.24)1,2 in (4.21) 2 and (4.22) 2 , respectively, then yields the alternative forms


f 1 ~es - ((~)) • ~;Vas + <{ )){{1/ )) [ 0 asEco: ({@)> ~1[~]]- {(~}}l[l/O]]jt, + [[g/]]- ((~N)) • [[vii] ]

a (4.25)

0 = ((m)) {[[¢]] - ((t))[[1/Q]]} + [[q]] - ((tll)) .[[vll]] ,

for the constituent and mixture energy jump balance relations, respectively. Note that the result (4.25) 2 is equivalent to (2.11) in Fried (1995). In terms of the specific enthalpies

h := 11

(4.26) h := e - t i p ,

we have the alternative forms

[[a~]] - ((t~))[]_l/~]] = [[ha]] + ((1/a~))[[~]] ,

[[e]] - ( ( t ) ) [ [ 1 / p ] ] = [ [h~ + ( ( 1 / g ) ) [ [ t ] , (4.27)

for the coefficients of ((m~)) and ((m)), respectively, in (4.25).

The constituent interaction terms appearing on the left-hand side of (4.25) 1 can be rewritten in the form

= - . - { } ~o (4.28) ~v ~ b~ e - - <(V)) ~Osv - - k S - 1 ((/aTa)) . ((,~a)) + I [[,O11]] , [[,aOa] ] a S - ((7)) . s + ( ( t ) ) ( ( l@) os

via Table 1, (4.9)2 and (4.14), where

h s := ((~)) - ((t~))((1/a~))= ((h))+ ¼ [It, ll 1[1/~]] (4.29)

represents a constituent specific enthalpy-like surface field, i.e., its the specifc enthalpy as a constituent of the surface mixture, where the second form follows from (4.9) 2 and (4.26) 1 . As shown in (4.29), has is equal

to ((h)) only if the normal stress 7 or density 0 are continuous across SR. In an analogous fashion, it might be a a 11

reasonable to interpret ((v)) as the constituent velocity on SR, and ½ ((va) } - ((va} } as its surface specific kinetic

energy. Other such fields will arise in what follows. Thermodynamic processes are constrained in general by the second law of thermodynamics. A formulation

and exploitation of this constraint in the current case is based on the constituent

- co. ~V ~ ~ v + = 0¢/ - div(~) °" - #coo 11 I1 0. 11

~es" + ,~ = ((m~))[~]] - [[~co"]], (4.30)

and mixture 7r ~" = QT) V - div(,b Q" - c r e " ,

7rOs" = ((m))lDT]] - [[q~O,]], (4.31)

entropy balances, respectively, from Table 1, Table 2, (4.14) and (4.15), respectively, where ~ a~ = 0 and

~ " = 0 hold via Table 1, (3.7) and (3.8), while aS

on = V" 5 -07 ~V Z.-~ a V

-- S t " " S :

(4.32)

follow from Table 1, Table 2 and (3.6). The standard mixture form of the second law (e.g., Truesdell, 1984, o,7 to be greater than or equal to zero; in the current context, this requirement extends to Lecture 5) requires 7r v

7r es" as well. Somewhat more unconventional is the notion that, at the constituent level, the total constituent


entropy interaction rate density -on -eo ~v + ~v in R \ SR (and in the current formulation, ~r a'7 + ~e'~ on SR) be a S aS

greater than or equal to zero (e.g., Svendsen, 1996); this does not of course require ~-env (or #eras ) to be greater

than or equal to zero, something considered in earlier works (e.g., Sampaio & Williams, 1979) and generally considered controversial.

Alternative forms of (4.30)~ and (4.31)~ may be obtained with the help of corresponding energy balances by introducing the non-thermal entropy fluxes

k := ~b °n + q / 0 , 0, 0. fl 0.

k := ~°O+q/O, (4.33)

where O and 0 represent the constituent and mixture absolute temperatures, respectively. The relation (4.9)1 I1

and definitions (4.33) yield

= l[kl l - + ( ( 0 / 0 ) ) f r 0 ] , a a a ¢1 a o. a

I[~en]] = I[k] - I[q]ll((O)) + ( ( q / O ) ) l [ O ] . (4.34)

From this last result, we see that the jump of the entropy flux is influenced by that of the heat flux, the temperature, and the extra entropy flux. Introducing next the constituent and mixture specific free energies

tt a ~ a

(4.35)

(4.9)1 likewise yields

a

((0)) vll = I t , l l - II 11 -

Substituting (4.34)1, 2 and (4.36)1, 2 into (4.30) 2 and (4.31)2, respectively, rearrangement leads to

+

<(m~)) {I[~]1 + ((~))IIGO]I- <<~))lll/f]l}

+ os

(4.36)

(4.37)

and ((0)) 7r °sn = ((tll)). ~vl111

- ((0)) {[[k]] + ((q/O))l[O]]} (4.38)

- ( (m}) {[[~b]] + ((r/))[[O]] - ( ( t ) ) [ [ 1 / O ] }

for the constituent and mixture surface entropy production rate densities, respectively via (4.23). The mixture quantity limb]] - ((t))l[1/Q]] appearing in (4.38) is referred to by Fried (1995) as the (specific) energy release. If k and O are in fact continuous across SR, (4.38) reduces to a form equivalent to the result (3.8) in Fried (1995).

Another form of (4.37) and (4.38) can be obtained in terms of the specific free enthalpies


:= ~ - ~ / f , fl ¢t

-/ := ~ - t/~, (4.39)

i.e.,

and

I/o)> : : ' + : 1 = /<~,// t . s .s j I[v.ll~

(4.40) - ((m~)){l[aT]l + ((1/~))l[t~]l + ((~))l[aO]] }

+ { / : oS + q)>//~/~>} + i/o)>//,>> °s

((0}} 7r ° so = ((tll }}. I[vll ]1

- ((0}) {[[k]] + ((q/O))l[O]]} (4.41)

- ((m}) { ~ 7 ] + ( {1/O)}~t] + ((~7))~0~}

using (4.9) 1. As with the constituent interaction terms in (4.25)i, those appearing in (4.40) can be rewritten in the form

a~- v((°)) 4, vas + { ((t~))((1/~))+ ((0)} ((~))}as:= ac~- ((v))'~sV- { : s - ½ ( ( v ) ) - -((v))+ ~ l Iv ] - l[v]l )a~ (4.42)

via Table 1, (4.9) 2 and (4.14), where

1 i[ar]] ] 1[0]1 + 1 [[~] ° (4.43) ~s := ((~)) - ((~)) ((0}} - ((t~)) ((1/a~)) = ((a7}) + g g [II/O]]

represents a constituent specific free enthalpy-like, or constituent chemical potential-like, surface quantity, analogous to (4.29), where the second form is obtained from (4.9)2 and (4.39) 1. Clearly, aTs reduces to ((a7))

only when ~/and 7, or 0 and 0, are in fact continuous across SR. O ¢1 fl O

Yet another form of the entropy results is based on the relations

((m))[[lb]] - ((t))[[vJ] = ((m))[[ 7 + ½(v - w)2]] , (4.44)

obtained using (4.9)1, (4.13), (4.24), (4.39) and the fact that w is continuous across SR. Substituting (4.44)1 into (4.37), and (4.44) 2 into (4.38), and taking (4.24) into account, yields

//0)/ to sS:' + oS:} = I/i, >> ~ 1 ~

-<<V +


via (4.42) and (4.43), and

((0)} 7r esO =(( t l l }}. [[vii ]]

- <<o>> { l e ~ + <<q/O>>l[o~} (4.46)

Again, assuming k and 0 are in fact continuous across SR, (4.46) reduces to a form equivalent to (3.8) and (3.10) in Fried (1995).

The above formulation is appropriate only to the classical case when all constituents of the mixture exist on both sides of SR. More interesting and perhaps relevant cases arise, however, when one or more constituents vanish and/or appear on SR, representing then a (perhaps generalized) phase change surface for these. A model for this case is our next task.

5 Creation or annihi lat ion of a const i tuent on SR

In this section, we consider the case that one of the constituent in the mixture exists only on one side of SR, i.e., up + = 0, or up - = 0, which we summarize as up + = 0. Since SR moves into R + by convention, the case

a a

up + = 0 corresponds to the creation, and up- = 0 to the annihilation, of the corresponding constituent, in the

mixture. In either case, all corresponding extensive constituent quantities vanish identically. As such, (4.2) 2 and (4.6) reduce to

a~s = -4-~+ = 4 - a ~ + ( v + - w ) = J=]+ -4-(±m + , (5.1) a a

while (4.14) 2 takes the form

~.~ + as ~ = ~±rOa os ~: ~++ , (5.2)

where we have accounted for (5.1) in (5.2). On the basis of Table 1, we can, as before, obtain the particular forms taken by (5.2) in the remaining cases of momentum, energy and entropy balance. In particular,

~°s" = v + ~ 7= i + (5.3) {1 l1 11

follows from Table 1 and (5.2) for the form of the constituent momentum jump balance on SR when the corresponding constituent is created (i.e., up + = 0), or annihilated (up- = 0). Similarly,

(e a ' va+ " v + ) r ° ± 0 4- 7: i + "v ± (5.4) a b~e : : : t : ' t ' 2 a " a ° a a a

holds in this case for the energy jump balance of such a constituent, and

~ 7 + r+" = ,7 ± ro m +;o.± (5.5) a S a a s a

for its entropy jump balance, on SR, via Table 1 and (5.2). Using (4.33)1 and (4.35) 1, (5.4) and (5.5) can be combined to yield the alternative form

0 +/"#o, + ~0,7 ~, - t v + ~e i + I. a S a S J = ab°s e - ( e + + 2 a " ~ + ) a S q= 0 + ~ : + -4- . ~ : t : ( 5 . 6 ) ¢~ ¢t a a a

of the constituent entropy jump balance in terms of ~p+ in the current context. tl

As done in the last section in the general case, the basic constituent energy (5.4) and entropy (5.5) jump balances can be recast into yet other, and perhaps more useful, forms. In particular, the result

v += w ± ~o/~+ (5.7) a aS la

from (5.1) yields the relation


T C-.v~= , ~ -h , * :F i ± .v~: ( ~ / p ~o ~ ~± .~:~ o o . o oll ~11 = - ) ~ T ~ :Fo, o i l ' (5.8)

and so the alternative form

F; (h (5.9)

of (5.4) in terms of the constituent specific enthalpy (4.26)1 via the normal part of (5.3). Further, this last form of the constituent energy jump balance leads to that

o : ~ I , ~ - = - ~v + . ~ ± , ~ ) - ~ g o, o, :Fo~:U (5.10) a [ a s + aLT} ~ e _ (7:k + 2 o ,~a-4- _ ~ -- :E ~4-..UZk a °

for constituent entropy via (4.33)1, (4.35)1 and (4.39)1.

6 Two-constituent mixture

To gain further insight into the constituent jump balance relations obtained on SR in the last two sections, consider the simple case of a mixture of two constituents a = 1,2. Adding the momentum jump balances (4.16) 2 for these two together, we obtain the expression

[[~ + ~]l = ((~))[[y] + ((~))ll-~]l + (((~)) - ((~))) ~ (6.1)

for the jump of the total constituent traction via (3.8) as determined by the momentum flux of each constituent relative to SR, as well as any mass exchange between the two constituents on SR. Doing the same in the case of energy balance yields

0 = ((r~)) (1[~] - ((~))1[1/~ll) + ((r~)) (l[e]l -/(~))ll'l/2~]] )

+ [[c/]l + l I0] 1 2

- ((}11))'[[711]] -((~[i)) ' [[~/11 ]] (6.2)

+ {(h~ - ~ ) - ~- (<<y>> <<y>>- <<~2 >> <<7>>/. ~ ([[y~ ~y]]- ]]~]]. [[~)} (~

+ (<<,i,>> - <<7>>) .~7 via (3.8) for energy, (4.9)2, (4.25)1 and (4.28). For the case in which the specific kinetic-energy-like terms are small, then, the mass exchange contribution to the energy balance is mediated principally by the difference h s - h s in the constituent surface specific enthalpies, i.e., a latent-heat-like quantity. Lastly, assuming for 1 simplicity that

(( f )) = (( 0 )) = ((0)) (6.3)

holds on SR, we have

l 1 1 2 2 2 J

- ( (~) ){ [ [~] ]+( (~) ) [ [O]] - ( ( t~) ) [ l -1 /~ , ) - ( (r~) ){ [ [~] ]+( (~) ) ] ]O]] - ( (~) ) [ [1 /~] ] ) (6.4)

+ {(>->/-½(<<~>>.<<7>>-<<y>>.<<y>>/+½(~9-[[~,~-~y~.~y~)}~°,s + (<<7>>- <<y>>) -~7


in the case of entropy via (3.8) for entropy, (4.25) 1 and (4.28). In this case, then, the mass exchange contribution is determined predominantly by the difference 72 s - 17 s in the constituent surface specific free

enthalpies, or chemical potentials, when the specific kinetic energy-like terms are negligible. Analogous results to those just obtained arise in the case that one or the other of the two constituents

under consideration vanish or appear on SR. For example, assume that the first constituent vanishes on either the + or - side of SR, i.e., v Q:F = 0. Adding the momentum jump balances (5.3) for 1 and (4.16)2 for 2

together, we obtain ± il ~ +1[~]1 = ((r~))[[ff~ + (v± - ( (~) ) ) ~ , (6.5)

again via (3.8) for momentum. As for energy, adding (5.9) for 1 and (4.25) 1 for 2 together yields

+ ~ +~,~11 2

T i-'- ,o-'- <<~11>> "1~11 n 1 II " 1 II + (6.6)

from (3.8) for energy. Assuming once again for simplicity that the surface temperatures are equal, i.e.,

0 f = << 2o >> -- <<0>> (6.7)

on SR, (5.10) for 1 and (4.25) 1 for 2 combine to give the expression

+ { (ys -,.r~) - -~ (<<g >> • </g ?/+,,..,f .,oF) +-~rfT:~ • ~:.~u + ~F ~ } t g

(6.8)

for the entropy production rate density 7r esn in the surface mixture from (3.8) for entropy and (4.43). These latter results can be applied to the so-called cold-temperate transition surface (CTS) present in

polythermal glaciers (e.g., Hutter, 1993; Greve & Hutter, 1995; Svendsen et al., 1996), representing the interface between pure "cold" ice and temperate ice, which consists of a mixture of water and ice on the ice-water phase boundary. To do this, consider the simplest non-trivial senario for this latter mixture in which water and ice coexist on the phase boundary in equilibrium, i.e.,

0ps = 0 = 0 ( 6 . 9 ) w i

and PPB = Pw = Pi • (6.10)

Then

PB

(6.11)


holds in the temperate region, where ~5p8 represents the Clayperon slope of the ice-water phase boundary (which is negative in this case). Assuming then that Fourier conduction applies, i.e.,

q = - ~ ( V O ) , (6.12) gt

we have

q - = - ~ ; - ~PB (~7p) . (6.13)

Let cold ice be on the +, and temperate ice on the - , side of the CTS (i.e., SR), so that z/÷ = 0. Since ice w

and water coexist in the temperate region essentially in equilibrium, the transformation of ice to water or vice-versa in the entire system takes place primarily on the CTS. Assuming next that the ice mass flux (( m~ ))

across the CTS, ~ °sV, as well as all velocity-squared and dissipation terms, are negligible, that k and £" are w i

zero, and that 0 is continuous across SR, we obtain the form i

r =i- +~- + (w'-- <<y>>) wS ~ i i w (6.14)

for momentum balance from (6.5), that

: ( v o ) + = ( ~ - + ~-) e~ t ( V p ) , n+] + L g i i

(6.15)

for energy balance from (6.6), (6.12) and (6.13) (note L ,+ = 1), and that i

e ~ 7 = ( T s - 7 - ) ~ = <<7>>-w~-+~ i o g (6.16)

for entropy balance from (4.43) and (6.8), with

L := w h - - ihs = w h - - << h >> - 1 ~ ~ r l / ~ 1 1 i

(6.17)

the latent heat associated with the water-ice phase transition, where (6.17)2 follows from (4.29). The entropy result (6.16) yields

aa < 0 freezing conditions 7s > 7 - ¢==> wS l w ae > 0 melting conditions (6.18) rr~ '7 > 0 ~ 7s < 7 - '*=::> wS i w

in the non-equilibrium case, i.e., the so-called melting and freezing conditions at the CTS. Combining (6.15) and (6.16) with rr~sn > 0, we obtain the alternative form

~ v [ ( V . 0 ) + - n ] > (~--+~--)~SPBt(~7p)'n] '- ~ < 0 freezing conditions 1

. ~o > 0 melting conditions t ~ + t ( V q ) + i . n] < ( ~ - + ~ - - ) ~ P B [ ( V p ) - n ] ~ wS (6.19)

of the melting-freezing conditions (6.18). In the case when the pressure-melting term is neglected, (6.19) reduced to the form of these conditions used by Greve & Hutter (1995) to model the evolution of the CTS in the Greenland ice sheet.


7 D i s c u s s i o n

Consider once again the constituent

//oil + 5" } - -

via (4.42) and

+

(4.37) and mixture (4.38) entropy jump balance relations, i.e.,

/<o>> {@+

/(0}) 7r ~s ~ = ((tll)). [[vii ]]

- ( (0)) {~k]] + ((q/O))~O]]} - ( (m) ) {[[~b]] + @l ) ) [ [0 ] ] - ( ( t ) ) [ [ 1 / e ] ] } ,

respectively. These can be compared with the corresponding volume versions, i.e.,

OtavJ'~o. + ~7 } = s.a (Vv)a

}

a all

+ a b%e ~ ~ ~Vv- - - I v a a S '

from (4.2)1, (4.21)1, (4.30)1, (4.33)1, (4.35)1 and (4.39)1 for each constituent, and that

0~r~ = s . ( V v )

(7.1)

(7.2)

(7.3)

- 0 { ( d i v k ) + q / 0 . ( V 0 ) }

- ~ {¢ + 70} + (p/~)~ (7.4)

+ r - Oa on ,

from (4.3)1, (4.22)1, (4.31)1, (4.33)2 and (4.35)2, for the mixture, where the spherical-deviatoric split

]" = - / S I + S , G a Q

T = - p I + S , (7.5)

of the corresponding stress tensors was used. Comparing (4.38) and (7.4), for example, we see that ((0)) is analogous to 0, ((tll)) • [[vii]] to S . (Vv) , [[kl] to divk, ((q/O))[[O] to q / 0 - ( V 0 ) , ((rl))[[01] to r/0, ((m))[[~l]

to Q~), ((t)} to - p , and so on. On this basis of this analogy, then, it is reasonable to interpret ((tll)), [[k]], [[~0]], {(r/)) and ((t)) as constitutive quantities on SR dependent on ((0)), ((m)), [[vii]], [[0]] and Ill/Q], i.e., the independent variables on SR in this case. The same comparison, yielding analogous results, can be made of course in the constituent case.


Analogous to their volume conterparts (7.3) and (7.4), one would like to utilize the surface entropy

relations (7.1) and (7.2), respectively, to obtain restrictions on the constitutive form of dependent surface

quantities such as those just discussed. In the absolute simplest case, for example, we could assume that

((0)) 7r ~s ~ as given in (7.2) is linear in the independent variables ((0)), {(m)) and [[vlt ~, yielding the necessary

and sufficient conditions ((tll)) : 0

[k ]l : - ( ( q /O))[O~ (7.6)

for ((0)) 7r esO _> 0 not to be violated. In turn, (7.6)2 implies the "classic" form

[ [ ( 9 ~ = - ~ q ] l / ((O)) (7.7)

for the jump in the normal entropy flux density via (4.34), and (7.6)2 the form

~e]] = (( 0))[[~7~ + ((t ))~ l / Q] (7.8)

for the jump of the mixture specific internal energy across SR from (4.36). In general, of course, such

constitutive dependences do not apply; more realistic cases involving (7.2) with [[0]] = 0 and [[k]] = 0

have been discussed by Fried (1995). Nevertheless, it is interesting to note the complete formal analogy

between (7.6)2, (7.8), and their classical thermodynamic volume counterparts when jumps are replaced by

differentials, and surface averages by the variables themselves. The same simplest case can be considered

for each constituent in the context of (7.1), yielding analogous, but in this case only necessary, restrictions,

because the resulting reduced form of (7.1) is non-trivial for non-trivial interactions; these and other more

realistic constitutive cases for the constituents will be the subject of future work.

Acknowledgements. We thank Leslie Morland, Kolumban Hutter and Eliot Fried for useful discussions and comments on many aspects of this work.

References

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3. Chadwick P (1976) Continuum mechanics, George Allen & Unwin Ltd. 4. Dobran F (1919) Theory of structured multiphase mixtures, Springer Lecture Notes in Physics 372 5. Fried E (1995) Energy release, friction, and supplemental relations at phase boundaries, Continuum Mech. Thermodyn. 7, 111-121 6. Gibbs JW (1961) On the equilibrium of heterogeneous substances, in: The Scientific Papers ofJ. Willard Gibbs, Dover 7. Gray JMNT, Svendsen B (1995) Interaction models for mixtures with application to phase transitions, submitted to Int. J. Eng. Sci. 8. Greve R, Hutter K (1995) Numerical simulation of the Greenland ice sheet, Annals of Glaciology, in press 9. Gurtin ME, Struthers A (1990) Multiphase thermomechanics with interface structure, 3. Evolving phase boundaries in the presence

of bulk deformation, Arch. Rat. Mech. Anal 112, 97-160 10. Hassanizadeh SM, Gray WG (1990) Mechanics and thermodynamics of multiphase flow in porous media including interface

boundaries, Adv. Water Resources 13, 169-186 11. Hutter K (1993) Thermomechanically-coupled ice-sheet response - cold, polythermal, temperate, J. Glaciology 39, 65-86 12. Kelly PD (1964) A reacting continuum, Int. J. Eng. Sci. 2, 129-153 13. Morland LW, Gray JMNT (1995) Phase change interactions and singular fronts, Continuum Mech. Thermodyn. in press 14. Sampaio RW, Williams O (1979) Thermodynamics of diffusing mixtures, J. de Mgcanique 18, 19--45 15. Svendsen B (1996) On the constituent structure of a classical mixture, Meccanica, in press 16. Svendsen B, Greve R, Hutter K (1996) On the thermodynamics of a mixture of water and ice, with application to polythermal ice,

glaciers and ice sheets, in preparation 17. Truesdell C (1984) Rational Thermodynamics, Springer-Verlag, 18. Truesdell C, Toupin R (1960) The classic field theories, in Handbuch der Physik Ill/I, edited by S. FRigge, Springer-Verlag

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