continuum mech. thermodyn. 7 (1995) 387-414 continuum

Continuum Mech. Thermodyn. 7 (1995) 387-414 Continuum Mechanics T / and 1 I nerm0flynamlcs © Springer-Verlag 1995

Originals

Phase change interactions and singular fronts

L.W. Morland and J.M.N.T. Gray

The balances of mass, linear momentum and energy for a continuum provide jump relations between values of the physical variables on the two sides of a singular surface, either a boundary of the medium or an interior surface. In the case of a mixture, an overlap of interacting continua, there are jump relations for each constituent. While an elementary phase change front across which one phase of a constituent is transformed completely to a different phase can be treated as a single constituent, more general situations have co-existing phases on one side of the front, each with their own density, velocity, stress and internal energy fields, which must be treated as separate constituents. The phase change is now a mass transfer between constituents which becomes a surface production term in the mass balance jump relation for each constituent. In turn this implies surface production contributions to the momentum and energy relations associated with the surface mass transfer, including interaction body force and energy transfer contributions as well as the direct transfer terms. The general jump relations with such surface production contributions are formulated, and are illustrated for a number of situations arising in polythermal ice sheets and wet snow packs.

1 Introduction

The elementary view of a phase change surface is a surface which separates regions each containing only one phase of the given material; that is, different phases on the two sides. Examples are an advancing melt front into a block of ice, with ice ahead and water behind, or an advancing freezing front with water ahead and ice behind. The motion is therefore described in terms of a single set of variables: density, velocity, stress and internal energy, on each side, which are related by the jump relations of mass, momentum and energy for a single constituent. It can also be described in terms of different constituents on the two

388 L.W. Morland, J.M.N.T. Gray

sides with mass transfer at the surface, and it will be shown that this more general view leads to composite relations reproducing the elementary single constituent results. However, in polythermal ice sheets and wet snow packs there are regions of co-existing ice, water, water vapour and air, with external or internal surfaces at which phase change takes place in a singular manner, and here the phases must be distinguished as separate constituents. In the illustrations presented for ice sheets and snow packs, the single phase on one side of the front may be ahead or behind, so for generality and symmetry of results both phases are considered to exist on both sides. Furthermore, while an advancing front is the trigger for the singular mass transfer, it is shown that surface transfers ahead and behind are necessary in general to satisfy the jump relations for both phases. Particular phase change events are described by appropriate interpretations.

Phase change in continuous regions introduces a mass transfer rate contri- bution to the mass balance of each phase (constituent), with directly associated contributions to the linear momentum, angular momentum and energy balances. There are, however, further implicit contributions associated with the mass transfer, arising in the interaction body force and energy transfer terms which are necessary to maintain the composite balances for the mixture of the two or more phases. With appropriate postulates these mass transfer interactions have explicit forms. It is then possible to take limits of the physical balances for each phase through the continuous regions on each side of a singular surface, to determine the jump relations across the surface which include surface production terms arising from the surface mass transfers on both sides. These jump relations for each phase are necessary to correctly prescribe externally induced phase changes at a boundary, and to provide matching conditions for the motions across an internal phase change front. An entropy balance jump relation would determine whether a particular phase change front scenario is a thermodynamically admis- sible process, but this invokes further thermodynamic assumptions in regard to the adopted entropy production inequality and form of the entropy dependence on other variables, which are not proposed here,

A series of papers by Gurtin, [3], [4], [5], [6], develop a more sophisticated description of a singular surface between two phases of a material. The initial motivation is that a classical Stefan condition cannot induce the evolution of a tree-like structure of dendrite formation in a solidification process, and the generalisation is to endow the surface with its own capillary forces, both in- plane surface tension and normal shear force, energy and entropy. The treatments progress from rigid heat conductors to deformable solids, and consider capillary force balance both as a constitutive law and as an independent axiom. It is stated that mass accretion or loss requires additional surface forces. Such surface force and surface energy arise naturally in the present approach, defined by momentum and energy production in the surface associated with the surface mass transfer without further constitutive or balance postulates. Surface tension and surface shear force have been ignored here with the notion that they will not be significant in the evolution of smooth fronts in large-scale applications. This cannot be asserted for local small-scale phase change surface geometry. When appropriate they can be added directly to the momentum jump condition, but

Phase change interactions and singular fronts 389

would require further constitutive postulates. Gurtin's developments are much more general in that they incorporate non-equilibrium thermodynamics, ignored here, and address its implications; however, with further assumptions. There is no application to demonstrate the evolution of a tree-like structure.

An alternative concept of a surface driving traction is introduced by Abe- yaratne and Knowles [7], motivated by the non-uniqueness of equilibrium con- figurations of surfaces in non-elliptic elastic materials when it is absent, but developed generally for non-elastic materials. The driving traction induces dis- sipation which enters the energy and entropy jump relations, and prescription as a function of temperature and propagation speed yields a model for the rate of phase change at the surface. There is no correlation with the present derivation of jump conditions and the rates of mass transfer between phases.

The papers by Alts and Hutter, [8], [9], [10], [11], focus specifically on phase change surfaces between ice and water, seeking a description which allows the large curvature surfaces which arise in small-scale processes. The singular surface is endowed with its own mass, momentum, energy and entropy, and with a polar structure allowing a non-symmetric stress tensor and shear stress normal to the surface. Comparisons and interpretations are made by analysis of an alternative description of the phase change as rapidly varying fields through a thin boundary layer [8], thermodynamic restrictions on postulated constitutive laws for the surface variables are obtained [9], and their thermostatic conse- quences [10], and thermostatic stability is analysed [11]. Such local structure and thermodynamic aspects are lost in the present treatment.

In summary, the present paper provides a thermomechanical description of co-existing phases of a material undergoing phase change (mass transfer), possibly in the presence of other inert constituents, both in regions of continuous phase change and across a singular phase change surface. The analysis is complete for two phases, but a complete solution of the phase changes between each of three or more co-existing phases requires further physical description of the processes. Explicit forms for the interactions associated with the mass transfers are derived by a symmetry postulate imposed on the composite mixture restrictions. The jump relations of mass, linear momentum and energy balance, including surface mass transfer contributions on both sides of a singular surface, are then determined by appropriate limits of the integral balances. Recognition that surface mass transfers must be partitioned to reflect phase changes taking place at the different conditions on the opposite sides of the surface has a physical appeal, though not included in the treatments discussed above, and is consistent with the derived system of mathematical relations. Angular momentum balance is automatically satisfied. These jump relations are interpreted for examples of phase change fronts arising in ice sheets and snow packs, which include an elementary case also derived by a single constituent approach. There is no consideration of thermodynamic restrictions, nor of direct surface properties, so inappropriate application may encounter the weaknesses and failures which motivated the more general treatments mentioned above. These further aspects can be added in the present framework.


2 Balance laws

Mixture or interacting continua theory describes the macroscopic motions of each constituent as if it were distributed continuously over the entire mixture domain; that is, in terms of overlapping density, velocity, stress, temperature and internal energy fields for each constituent. See, for example, Truesdell [1] and Morland [2]. These fields are considered to define mean values of the respective variables over a mixture element large compared to the dimensions of the individual constituent elements, so the mixture element contains such an abundance of each constituent that mean values over the element are smooth fields. In this sense the point differential equations derived from integral balances by the usual continuity arguments refer only to such mean fields, as will the jump relations between continuous fields across a singular surface.

Let the superscript v label a constituent variable associated with a given v-constituent in a mixture of n constituents (v = 1,2 ..... n). The partial density of the v-constituent, pU, is the mass of constituent v per unit mixture volume, which is related to its intrinsic density pV,, its mass per unit constituent volume, by

p~ = qS~p ~*, (2.1)

where the volume fraction ~b ~ is the volume of constituent v per unit mixture volume. The velocity field v ~ is defined so that p " v ~ is the mass flux of constituent v parallel to v ~ per unit mixture cross-section normal to v ~, but is also assumed by Truesdell [1] to be the intrinsic velocity needed in the momentum and energy fluxes. Consistency implies the identification of the cross-section fraction of the v-constituent with its volume fraction ~b ~ for all section normals n. This isotropy of structure can also be inferred from other standpoints; see the discussion by Morland [2], from which the present notation and formulation is adopted. The partial traction t e (n ) supported by the v-constituent on unit mixture cross-section with unit outward normal n is then related to the intrinsic traction t "* (n), the force per unit constituent area, by the same scaling, and hence also are the partial and intrinsic stresses o "~ and o "v*, thus

t ~ ( n ) = o.Vn, tV*(n) = ~r~*n,

t ~ = ¢b~t ~*, er ~ = 49~r ~*. (2.2)

Such isotropic scaling has appeal for mixtures of fluids or granular materials, but is less convincing when a solid constituent with a coherent matrix structure is present. Let %~ denote the internal energy of constituent v per unit constituent mass (an intrinsic variable), and T ~ denote its temperature. A common temperature assumption, identical T ~ ---- T for all constituents, is usually made when relative motions are very slow, which is the case in wet ice and wet snow. For channel flows through temperate glaciers in which the relative velocity of water and ice may be significant in regard to both mass and momentum balances, the water temperature may still be, or may not be, sufficiently close to the local ice temperature for a common temperature assumption. In general, non-equilibrium


phase changes may take place with the phases at different temperatures, but the later explicit results for continuous regions and singular surfaces suppose that phase change take place with each phase at a common temperature, TM say, which simplifies the internal energy change. This is expected to provide a good approximation for the energy change in many situations, but can be modified if not. Constitutive laws to describe given constituent properties are concerned with intrinsic variables, but in addition constitutive laws are required to determine mixture properties: the evolution Of the volume fractions ~b ~ and the various interaction terms entering the physical balance laws for each constituent.

In order to derive the physical balance laws both for smooth fields and for fields undergoing discontinuities at singular surfaces, integral balance laws for the constituent v-material instantaneously occupying at time t a fixed region of space are formulated, then point limits appropriate to the nature of the fields are taken. Let ~t have a bounding surface 5 ° with unit outward normal n. As time proceeds, the envelope of p-material contained in ~t at time t and moving with the velocity field v ~, defines a u-matrix which is an evolving region ~t - However, if phase change (or other chemical interaction) takes place, in which there is a rate of mass transfer m ~ per unit mixture volume to constituent v from other constituents, then ~t~ is not the envelope of the original fixed set of u- elements occupying ~t at time t. Let this latter envelope be denoted by ~t}, then the physical balance laws refer to rates of change of quantities associated with ~t}. If ¢r ~* denotes the intrinsic specific density of a given physical variable; that is, the amount of that variable per unit mass of constituent ~, then the amount per unit mixture volume is p~rr ~*, and hence

d = d d°V- f d°¢, (2.3)

where dCF is a volume differential, which extracts the increase carried with the newly created u-material.

First consider differentiable fields throughout ~ , then the Transport Theorem for the envelope ~ moving with the velocity field v v gives

d p~-~* d~V = ~ (p ~- ) d~V p . dS °, (2.4) --dt / ~ + -~¢r~*v ~ n

where dS ° is a surface differential. Eliminating the surface integral by the Di- vergence Theorem and the volume integral over ~t~ by the identity (2.3), shows that

d p~'w a'* d ~ 1, ,, ¢ / , , [ l~vp" vl, - - -= (p - ~ + + p ~ d i v - m"] )d~ , (2.5) d t ~ L D t

where

Dv 0 - - + v C g r a d (2.6)

D t 3t

is the convected time derivative moving with the p-velocity field v ~. Mass conservation of the fixed set of v-elements in the envelope ~ } is simply the vanish-


ing of the left-hand rate term in (2.5) when ~r ~* = 1 , and hence the vanishing of the right-hand side which implies that

" [DvP v - - ~ + p~div v ~ - m ~] d ~ = 0 (2.7)

for all regions ~t in the mixture. Thus, in regions where the integrand is continuous, mass balance takes the point form

Dvp v - - + pVdiv v v = m y, (2.8)

Dt

so the term in (2.5) enclosed by the brackets [ ] vanishes to yield the general rate result

d p~cr"* J ~ P Dt v (2.9) dt

for continuously differentiable fields.

Table 1. Integrands for the physical constituent balances.

~'~* g~ h~

v p p linear momentum v i PVb~ + fii o-~

angular m o m e n t u m cZijkXjVVk eijkxj(pVbV k ~- p~vk) aijkXjO'Vkl

energy %~ + ½ v~.vÈ P~( b~.vÈ + r") + P(fl È.v~ + 4#) vs°'~I - q7

The linear momentum, angular momentum and energy integral balances for constituent v can now be expressed in a common form

f~t u Dv~rv* P Dt d o g = J s ° h V ' n d ~ f + . / ~ g V d ' ' (2.10)

once the shrinking tetrahedron argument has been used in the linear momentum balance to deduce the existence of a partial stress tensor satisfying (2.2). Table 1 lists the respective ~-~*, g~ and component hi of the vector h ~ for the ith components of the linear and angular momentum balances and for the energy balance, where b ~ is the external body force per unit mass of constituent v, fl~ is the interaction body force per unit mass of mixture due to other constituents, r ~ is the rate of external energy supply per unit constituent mass, 4, ~ is the rate of interaction energy supply per unit mixture mass from other constituents, and q~ is the energy flux through constituent v per unit mixture cross-section. •ijk are the components of the alterating tensor. Eliminating the surface integral by the Divergence Theorem and invoking continuity of the volume integral integrands, leads to the point form

p~ D~v~ = div ~ + p~b ~ + pl3 ~ (2.11) Dt


for the linear momentum balance, symmetry of the partial stress tensor by the angular momentum balance (in the absence of body couples and couple stresses), and the point form

DÈ%È q~ p 'r ~ p -~ -- div + +p~" +tr(~r~D ~) (2.12)

for the energy balance, where

t2

P = Z P " (2.13) V=l

is the total mixture density, and

Oij = 2 (~xj + ~xi )

are the components of the (partial) strain-rate D v of constituent v.

(2.14)

3 Interactions

Constitutive laws are required for the stress, internal energy and energy flux of each constituent, and for the evolution of its volume fraction to prescribe how the constituents share the mixture space under changing stress and temperature. In addition, constitutive laws are required for the interaction body force and interaction energy transfer, and these must distinguish the contributions from mass transfer. Kelly [12], in an early formulation of a reacting continuum theory, formally partitions supply terms into those associated with the constituent viewed as a single continuum and those associated with reactions and diffusion, but does not proceed to explicit results. Following Morland [2], introduce the decompositions

/ 3 ~ : / ) ~ + / ) ", 4 F : ~ + ~ ", q"=O"+gT", (3.1)

where /)~ and ~" are the parts due solely to the mass transfer m y, and /)~ and ~ are independent of the mass transfer. /)" is commonly defined as a drag due to the motion of constituent v relative to other constituents in slow flow, or as the constraint necessary to prevent relative motion when a common velocity approximation is made, and ~ is assumed to describe the working of interaction forces not associated with mass transfer. Here, however, an energy flux decomposition has been incorporated to allow a transfer q~ which reflects energy transfer to constituent v by conduction across its boundaries with other constituents, in addition to the flux 0 ~ within constituent u. This transfer will provide a necessary interaction flux when thermal constraints are imposed on co-existing phases, akin to the role of/~" when a common velocity is imposed. Formally, such contributions could be included in the bulk transfer ~ , or by a


further partition of ~b ~, which would lead to the same final results. Summing the constituent momentum balances (2.11) over all constituents subject to

~ B ~ = o , (3.2) v= l

since there is no net interaction force in the absence of momentum transfer associated with mass transfer, and comparing with the corresponding balance for the material composed of all the constituents which is subject only to the external body forces and surface tractions, shows that

n

p K = -

v=l v= l

(3.3)

Similarly, summing the constituent energy balances (2.12) over all constituents and comparing with the balance for the material composed of all the constituents which is subject only to the external energy Supplies and surface fluxes, shows that

n n n

p(9 ~ - div ~ gl ~ = - ~ p,8 .v , v=l v= l v= l

(3.4)

and

n n 1 .

= - + + ) } ,

v= l v---1

(3.5)

when mass transfer dependent and independent contributions are separated. The sums (3.3)-(3.5) are restrictions on the individual interaction terms/)~,

~}~, ~ and }~, but do not determine them, even in the case of a binary mixture (n = 2). However, with the further assertion, Morland [2], that contributions from each binary interaction between a constituent v and a constituent/z have the same form for all pairs v and /~, a symmetry postulate, explicit relations can be derived from the mixture restrictions (3.3)-(3.5). This is a significant reduction which avoids the need for more tenuous constitutive postulates for the various interaction terms. It is the reduced interaction model adopted for the present analysis. The following derivations echo the arguments given by Morland [2]. First make the decompositions into binary interactions:

m " = ~ m " ~ , i~"= il "t*, ~b"= (b"**, /z=l /*=1 /~=1

(3.6)

n n q ,

/1,=1 /*=1 /z=l

(3.7)


where m ~ , f i ~ , ~ , fi~u, ~ and O ~ define the effects on constituent u from constituent/x, and

m ~ = - m ~ , g/~ = - q ~ , m ~ = 0, q~ = 0. (3.8) l,=1 ~'=1

Note that when there is no phase change between constituents u and /x, the binary contributions m ~ , /)~u and ~ " to the decompositions (3.6) all vanish. The mixture restriction (3.3) suggests that, at least, the binary interaction f i ~ should be linear in the binary mass transfer m "~ and in the relative velocity v ~ - v ~, the latter to satisfy frame indifference, so with the symmetry postulate proposed above its most simple form is

p f i ~ " = km~'~(v ~ - v " ) , (3.9)

where k is a constant independent of ~, and #. Substitution in the restriction (3.3) shows that k = -½, and hence

1 m ~ ( v ~ - v~) . (3.10) o K =

The mixture restriction (3.5) now reduces to

Z P ~ " = m"~%", (3.11) v=l p=l /x=l

which suggests that the binary interactions ~"~ should be linear in rn "~ and in the internal energy difference %" - %~, so with the symmetry postulate its most simple form is

p ~ = l m ~ ( % ~ - %~), (3.12)

where l is a constant independent of p and/z. Substitution in (3.11) shows that l = -½, and hence

t2

l ~ m~U(%~ _ %~). (3.13) /x=l

Similarly, in view of the flux restriction (3.8), the mixture restriction (3.4) suggests that the mechanical transfer @~ is linear in the working f i " ~ . ( v ~ - v ~)

of constituent/z on constituent ~,, and then the symmetry postulate leads to

n

p ~ = -21 ~ / ) ~ U . ( v ~ _ v~ ). (3.14) /x=l

This reduced model represents the most simple set of symmetric relations for the interaction terms consistent with the mixture restrictions, but may exclude


contributions yet to be shown essential. If there is no phase change between constituents v and p and they have independent temperatures T" and T @ , and hence independent internal energies %" and g p , the two corresponding constituent energy balances (2.12) are independent equations for the two temperatures once the flux transfers q" and q p are prescribed. In the absence of phase changes or temperature constraints between constituents, there is neither basis nor neces- sity for such flux transfers, and a complete theory is obtained with the simple prescription

However, when a common temperature TV = T p = T is imposed, with or without phase change, a flux transfer is necessary to ensure that both energy balances determine the same temperature. The sum and difference of the energy balances (2.12) for each pair of constituents v and p, incorporating the interaction relations (3.6)-(3.8) and (3.13), are

D"%" D,%@ P"T f pp- + div (q" IL qp)" - (pur" f ppup) - p($" f $")" Dt

where the superscript " implies that contributions from interactions between the v and p constituents are excluded, these being carried to the right-hand side expressions. That is, the " terms include all interaction contributions to constituent v from all constituents other than p, and to constituent p from all constituents other than v, which arise both from mass transfers and from cross- boundary fluxes. The mass transfer contributions to the " terms vanish if there is phase change only between the v and p constituents, but not necessarily the fluxes qvo and qpW (w # v, p ) . If the common temperature is imposed on only two constituents v and p , then there is only a single flux 9°F and a single mass transfer rnVp (possibly zero). If there is no phase change, mv* = 0, then the sum equation determines the single temperature T independent of the flux transfer q"p, and then the difference equation determines the necessary bulk transfer div qVp. When there is phase change, the common temperature is a prescribed phase equilibrium temperature T M , in general a function of a common mean intrinsic pressure of the two phases, or more generally the energies %" and %p are prescribed, and the sum equation then determines the mass transfer m"p and the difference equation determines div q"p. The interaction flux enters only in the bulk (non-directional) form div qVp, which could have been represented by a further scalar energy transfer partition of *"p.

Now consider the situation when the common temperature is imposed on more than two constituents, N say, then there are N ~ 2 binary flux transfers qVp and N ~ 2 binary mass transfers mv@, but only N independent energy balances. If there are no phase changes, so each m"p is zero, then reduction to a single balance for the common temperature imposes N - 1 relations between the N ~ 2

divij"p, which are therefore not determined when N > 2, but they have no


explicit role in the remaining balances. However, when there are mass transfers between the N constituents and a common temperature (or each internal energy) is prescribed, the N energy balances are coupled equations for the NC2 div rl ~ and NC2 m ~, and so the m ~ are not determined when N > 2. Thus, phase changes between more than two co-existing phases require further postulates about the physical processes. In the case of three phases there are three coupled energy equations for three q ~ and three m ~ . A simple assumption is to assert that the cross-boundary flux constraints vanish since there are sufficient mass transfers to satisfy the constraint, but this is not consistent with the binary interaction prescription, and mass transfers resulting from this assumption must be assessed in any given situation.

4 Singular surfaces

When the region ~t contains a moving singular surface b °* separating ~ into changing regions ~t- and ~ + in which p"~-~* is differentiable, but p~Tr ~* is discontinuous across 5e*, then the Transport Theorem (2.4) is modified. If S v* moves with speed un along its normal in the direction ~ - ---> ~+, a limit analysis for the rate of change of the 1,-material integral, Morland [2], shows that

- j ; , [p~¢r"*]un dO °, (4.1)

where

[f] = ( f+ - f - ) so , (4.2)

is the jump in f crossing ~* from ~ to ~+. The general integral balance law now has the form

0

/ ~ (g~" + m ~ ~'*) d~, (4.3)

where cr ~*, g~ and component h~ of the vector h ~ for the i th components of the linear and angular momentum balances and for the energy balance are listed in Table 1, and the mass balance is obtained with ~-~* = 1 and g~ and h ~ both zero.

If ~ - and ~ + are bounded by S °- USe* and Se + USe* respectively, consider the limit as S °- -+ ~e* with outward normal n ---> -n* and ~+ -+ 5 c* with outward normal n --> n*, when the regions ~ , ~ - , ~ + --> 0. By supposition, p~r ~* and its time derivative, and v ~ and h ~, are bounded in ~ - and ~ + separately for


each balance law, and hence

0 f ~ ( p " ~ " * ) d ~ V + j ~ p U ~ ' * v " . n d 5 0 - / 5 0 h l ' . n d 5 0

--+ f50, [p~' cr~'*vY.n * - h~'.n *] d50. (4.4)

Further, it is asserted that the external body force b y and energy supply r ~ remain bounded in ~ - and ~+ , and also the interaction force/)P and interaction energy transfer t} ~ which are determined by the continuous bounded motions in ~ - and ~+ . However, a phase change (mass transfer) taking place at the singular surface 50* implies surface mass productions or losses for the interacting constituents, which are defined by allowing m y, and in consequence the interaction terms ~)~ and ~P defined by (3.10) and (3.13), to become unbounded in the vicinity of 50", so that the limit process for the volume integral of (g~ ÷ m~r ~*) yields a corresponding surface production integral. Recall that capillary forces such as surface tension have been excluded, but could be introduced in the momentum balance and energy balance by a further surface integral over 50* in (4.1), leading to a further term in (4.4). More general surface productions can be included in similar manner. The present analysis focuses on the contributions arising from mass transfer at the surface.

To allow surface mass transfers to take place between the phases in their states on both sides 50'- and 50"+, denoted by superscripts - and + respectively, consider the limits in ~ - and ~t + separately. Let M "~+ and M ~ - be the rates of mass transfer per unit area of 50* from constituent/x to constituent u on the sides 50"- and 5 °*+ respectively, then

~_ m y~ d ~ --+ J ~ , M y~- d50,

and hence

n

,/~+ m y~ d~" ~ f50, M y~+ d50, (4.5)

/x=l

Thus, for mass balance, the relation (4.5) and limits (4.4) and (4.6) show that

n

Since the integrand terms are continuous on their respective sides of 50", and the relation holds on all sections of 50', the mass jump condition has the point form

n

[pY(v~'.n * - un)] ----- m~ + - m~- = Z ( M y~- + MY~+), (4.8) /~=1


~+ and ~- are the mass fluxes of constituent ~, relative to 5e*. where m/7 m n

Now for any field s continuous and bounded separately in ~'t- and ~ + ,

./ + M' +s+)d5 °. (4.9)

Thus, in the general balance (4.3),

/7

~ .'~-'* d°V --~/~o. ~ (M"-~'"*- + M"+~-'*+) d., (4.10) Ix,=1

and

/7

/L=I

(4.11)

where ~r ~* and g are given in Table 1 for the respective balances, and the K ~ are determined from the transfer interaction terms (3.10) and (3.13), namely

l-(v - linear momentum • K"~ = -- 2

1 ~)k ) ' angular momentum" K ~u - 2 e i j ~ x j ( v ~ - ~

1 ~ 1 energy" K ~ = - ~ (v - v U ) . v ~ - -~(% - %~) .

(4.12)

(4.13)

(4.14)

Since each term of the angular momentum balance is just the moment about the origin of the corresponding term in the linear momentum balance, the resulting jump conditions are identical. The balance (4.3) with limit results (4.4), (4.10) and (4.11), invoking continuity of the integrands on each side of 5e*, now yields the point form

n

[ f f ~r"*(v~.n * - u/7) - h " . n *] = ~ ( M ~ - ~ c ~ - + M ~ + ~ + ) ,

i x = l

(4.15)

where

1 . linear momentum" ~ = g ( v i +

energy • ~ = ½(%~ + %~ + v ~ . v ~ ) , (4.16)

are symmetric in p and/x. The mass jump relation (4.8) is obtained by setting ~"g = 1 with ~.v. = 1 and h ~ = 0. The remaining details, from Table 1, are

linear momentum " ¢#* = v~, h L n * = t~,

energy • ~r v* = %~ + ½v~.v ~, h ~ . n * = F . v ~ - q~.n* . (4.17)


Explicit relations for the linear momentum and energy jumps of constituent u, incorporating the mass flux notation (4.8), then become

[F] - m nz'+ v ~+ + m ~ - v ~'- 1 n

~-~[M~/~- (v~' + v~ ) - + M~U+ (v ~' + v~)+1,(4.18) 2

/z=l

1 ~ v [t~.v ~ - q~.n*] - m~,+(% ~ + v~.v~) + + m~-(% ~ + ~ v .v ) - =

1 ~ [ M ~ _ ( % v + %Iz + vV.v~) - + M~,,~+(%v + %~ + vV.v~)+]. 2

/z=l

(4.19)

Note that (4.19) is invariant under the addition of a common arbitrary constant % to the internal energy of each phase, using (4.8) to eliminate the term (m~ + - m~-)%. Restriction to a common additive constant is necessary to leave the internal energy jump with change of phase invariant. Hutter [13,14] postulates further that the tangential velocity of each constituent is continuous at a phase change front. This would preclude mechanically driven flow of an inviscid fluid phase tangential to a singular front at which it is lost by a thermally dominated phase change, for which a continuously zero tangential traction would seem more appropriate. Tangential restrictions can be imposed as appropriate in applications, and the linear momentum jump in the vector form (4.18) is retained here and in the later illustrations. If constituent v is undergoing no phase change, so M ~ is zero for all #, then the mass, linear momentum and energy jump relations (4.8), (4.19) and (4.20) reduce to the standard forms

m n~+ = mn~- = {p~(vV.n * - u n ) } - = {p~(v~ .n * - Un)} + = m n,* (4.20)

* p [t ~] = rnn[v ], (4.21)

[ F . v ~ - q~.n*l = m~,[~ ~ + ½v~.v~]. (4.22)

Whenever constituent u is undergoing phase change with constituent/x on one or both sides, accompanied by mass transfer M "~± = -M ~"±, the internal energies %~ and %~ are linked by a phase change latent heat criterion, and in addition, the constituents co-existing ahead or behind the front have a common intrinsic mean pressure PM, which is in general related to a phase equilibrium temperature TM. Thus the surface temperature for both constituents is a given function of the common intrinsic pressure, which provides the necessary thermal boundary condition on 5f* for the energy balances of both constituents in both regions ~ - and ~+. The common pressure is automatically maintained throughout ~ - and ~ + when both phases co-exist. A common approximation is to assume TM is constant, independent of the pressure PM. Recalling the common temperature implications (3.16) for continuous regions, allowing m ~ to become unbounded in the vicinity of 5f* implies that the rates of change of the internal energies become unbounded, and in turn that d i v q ~ becomes unbounded. In


the limit process above this is reflected by a discontinuity [c]~] in the normal flux, where the normal is taken in the positive n* direction. While [~ ] is governed by a prescribed constitutive laws, [c]Jq arises as a constraint flux between constituents v and/z necessary to maintain the common temperature through the jump, and has no constitutive prescription. Thus, each interacting pair introduces surface terms M ~+, M ~ - and [q~], in addition to the propagation speed un.

If there are N constituents on one side, and M _< N on the other side, then the N momentum jump relations (4.21) relate 3 N traction or velocity components on the first side to the tractions and velocities, and other variables, on the other side. When evaluated, these are the required 3 N mechanical boundary conditions on the first side. The second side requires 3 M mechanical boundary conditions, and the total 3 (N + M) conditions are in principle determined by the 3 N jump relations and 3 M displacement component continuity conditions for the M constituents existing on both sides of the surface. The thermal boundary conditions are given by the phase change criterion. There remain 2 N mass and energy jump relations, with the internal energy jumps now prescribed by phase change conditions, but 2Nc2 surface transfers M ~ + and M ~ - , NC2 flUX discontinuities [q~u], and the propagation speed u~. For two interacting phases these four jump relations determine the four surface quantities, completing the system of required matching conditions across the surface. If N > 2 however, additional physical postulates are required, analogous to the continuous region situation. The system is completed by the constitutive equations for the stress and internal energy of each constituent. The jump relations are independent Of the constitutive laws, but the phase change condition depends on the internal energy law.

Specific illustrations will be presented in the next section for phase changes arising between just two constituents, with examples of ice and snow phase change fronts for which simple internal energy laws are adopted.

5 Binary transfer illustrations

The foregoing jump relations will now be applied to a number of phase change examples for ice sheets and snow packs in which the phase change occurs only between two specific constituents, labelled v and #, with accompanying surface mass transfers

M - = M ~/~- = - M ~ - , M + = M ~ + = _ M ~ + , (5.1)

on the two sides, and a discontinuity in the normal constraint flux

[qn] = [ ~ ] = - [ q ~ q , (5.2)

triggered by the passage of the front. The absence of one constituent on either side implies vanishing partial density and traction, and velocity and energy flux, for that constituent. In general, the internal energy %~ will depend on temperature and intrinsic deformation, and while the temperature is continuous at TM, the deformation will in general be discontinuous. A common approximation is that


the internal energy for both the water and the ice depends only on T , and hence is continuous across the front, permitting the simplification

%~+ = %~- = %", %~+ = %~- = %~. (5.3)

During phase change from constituent/z to constituent u, the internal energies are related by

%~_ %~ = L ~ - p~ + - - ptX '

where, for each constituent,

p p~ = - ~tr (r = 49~ p ~*,

(5.4)

= 4,"p (5.5)

L ~ is the latent heat per unit mass absorbed in the phase change, and it is supposed that the phases remain close to thermodynamic equilibrium (see, for example, Hutter[3]). A further common approximation is that the work associated with the density change is negligible compared to the latent heat, and then

%~ - %~ = L ~g. ( 5 . 6 )

Finally, it is supposed that the energy flux within each constituent is governed by a linear heat conduction law

g l ~ = - K V g r a d T v, K v =qSVK v*, ? 1 ~ = - K t Z g r a d T ~, K ~ =~b~K/x*, (5.7)

where K ;* and K ~* are the intrinsic conductivities, so with the postulate (5.2), and noting the possibilities of discontinuities in ~b ~ and q5/~, the normal flux discontinuities have the decompositions

[q~] [0nl+[qn] ~ aT~ - = = - [ K -gh-]+[qn], [q~] = [0n]-[c~] = - [K~n" ] - [q~] , (5 .8 )

where the normal direction and derivative refer to the positive n* direction, which expresses the flux jump in terms of the normal temperature gradients on the two sides and the constraint flux jump. The thermal boundary conditions for both regions, ahead of and behind the front, is simply T = TM for both constituents.

5.1 Advancing melt or freeze front

An elementary situation is when the front advances into a medium of a single constituent (phase) which is converted completely to the other constituent (phase) immediately behind the front. This is a melt front when the medium ahead, say constituent u, is ice, and the constituent behind, /z, is water, and vice-versa for a freeze front. Here there is no continuous phase change in the regions ahead and behind the front, where the temperature fields are determined by the single constituent energy balances, and the entire phase change takes place by surface mass transfer. Figure 1 illustrates this situation.


pP- = 0 t x - --+V+un pU+ = 0 v v- = 0 V ix+ = 0

P- 0 m~ + 0 m n ~

F - = 0 t ~+ = 0 gt ~- = 0 q~+ = 0

Fig. 1. Advancing melt or freeze front

403

Applying the mass jump relations (4.8) with the conditions given in Fig.1 to both constituents ~, and /z shows that

pq- m,, = m n ~ = M - + M + , = M (5.9)

say, the net surface mass transfer; that is,

pP+(v ~+~ - u~) = p ~ - ( v ~ - - un) = M. (5.10)

In turn, from the momentum jump relations (4.18) for constituents z, and/x,

1 p+ 1 ~+ 1 M v ~ _ 1 + .v+ t p+ ~ M v ÷ ~ M - ( v - v U - ) , t ~- = - - = 2 ~ M (,, vU-),(5.11)

and the difference determines the total traction-velocity jump relation

[t] = [F + t u] = F + - t ~- = M ( v p+ - v ~- ) = M[v ~ + v ~] = M[v]. (5.12)

The latter is the standard momentum jump relation obtained when a single constituent is assumed with the labels ~, and /x used to identify the material ahead and behind the front, possible here since only one constituent is present on each side, but (5.11) gives the individual constituent tractions required in the individual energy jump relations.

The energy jump relations (4.19) for constituents u and /z now become

[ tCvp - q~n] = ( F'vp - Cl~) + - [On] =

1 ~ 1 + p 1Mv~+'v~+2 + M%~+ - -~M-(% + %~)- - -~M (% + %~)+, (5.13)

- - [ t a . v a - - q ~ ] = ( t ~ . v ~ - ?7,~)- - - [qn] =

1 ~ 1 p 1 M v ~ - . v ~- + M~g ~- - ~ M - ( % + %~)- - -~M+(% + %~)+, (5.14)

which, using the momentum jump relations (5.11), have the alternative forms

1M-[v] .v~+ - ~M(% p - %~)+ - 1M-[%P + %~], (5.15) q'~+ + [q~] = 2

1 + 1 v 1 + C)n~- + [qn] = - ~ M [v].v ~- + ~ M ( ~ - %~)- + ~ M [% + %~], (5.16)


where [%~+ %/~] is zero if the approximations (5.3) hold. Differencing (5.13) and (5.14) gives the total energy jump relation

(F.vV - ~ - ~ M v ; . v ; ) + - ( t~.v~ - O~ - ~ M v ~ . v ~ ) - = M(%~+ - % ~ ),(5.17)

which is the standard single constituent result independent of the partitioning (5.9) of the surface mass transfer. The surface mass partitioning and constraint flux discontinuity are related by (5.15) and (5.16). While this partitioning is not required in the jump relations between single phases on each side of the singular front, it is necessary to relate velocity and traction jumps when two (or more) phases occur on one or both sides of the front, as shown in the more general illustrations to follow. Applying both the common approximations (5.3) and (5.6) to (5.12)-(5.15) gives the results

( t v . vv ^v 4- 1 - qn) -- ( t~ 'v~ -- q ~ ) - = M{ L~u + ~ ( v~+'v~+ - v~-'v~*-)}, (5.18)

1 v+ Cl~ v + O~ + 2 [qn] = ~ [ v ] . ( M - v - M + v ~ - ) . (5.19)

If the plied, to the dition

approximation commonly made in thermally dominated situations is ap- namely, that the stress working and kinetic energy are negligible compared heat conduction and the latent heat, then (5.18) reduces to the Stefan con-

c ~ + -

w h i c h

+

~ - = - M L " ~ ,

determines M, and (5.19) reduces to

~ - = - 2 [qn],

(5.20)

(5.21)

which determines the required constraint [qn]. The surface propagation speed is then given by (5.10) in terms of the velocity ahead or behind the front.

5.2 Phase creation front

More general is the situation when the front advances into a medium of a single constituent (phase), but behind the front there are both constituents (phases), as illustrated in Fig.2. That is, a second phase appears after the passage of the front through the single phase region, which is the basis for the term "creation". However, the total mass transfer from constituent v to constituent/x may com- prise both mass transfer across the front and mass transfer immediately behind

~+ does not necessarily equal m~-. Contrast the previous the front; that is, m n example in section 5.1. In the case of a polythermal ice sheet this configuration describes the case of a temperate zone expanding into a cold ice region, where constituent v is ice and constituent/x is water, and the front is a cold-temperate transition surface (Hutter [13,14]). Hutter [14] discusses a reduced model in which the separate ice and water velocities in the temperate zone are recognised


only in the mass conservation laws, and momentum and energy balances are applied only to the mixture, which eliminates the need for distinct i/:e and water momentum and energy conditions at the front. In addition the acceleration is neglected in the very slow flow. He notes, however, that separate momentum balances are required in wholly temperate ice sheets to describe the water movement through the ice matrix, when there is no containment by cold regions, and further that separate energy balances are required to describe water percolation through snow. The general treatment below which incorporates the distinct ice and water motions will allow systematic approximations of the jump conditions to reflect the above reductions. A second example arises in water percolation through snow-a porous matrix of ice crystals with the pore space also containing inert air which is free to be displaced. The front then defines the limit of the region with sufficient heat to maintain a liquid water content. This investigation of general phase change fronts was prompted by the need to define interface conditions on both constituents at such a percolation front, as well as at a melt surface. A recent detailed one-dimensional analysis of water movement through an isothermal snow pack by Gray [15] replaces the percolation front by a far field zero saturation condition, both in numerical solutions and in the derivation of a travelling wave solution which exhibits a front ahead of which the saturation is zero. The possible influence of more general conditions at the front is lost.

p - I x - --~U+un pg+ = 0 v ~+ = 0 m~ + = 0 t ~+ = 0 0 u+ = 0

Fig. 2. Phase creation front

Applying the mass jump relations (4.8) with the conditions given in Fig.2 to both constituents u and # shows that

m~ + - m ~ - = m ~ = M - + M + , = M , (5.22)

the net surface mass transfer. The momentum jump relations (4.18) for constituents l, and #, eliminating mE + by (5.22), give the traction jump relations

= ~ Mv~+ + ~ M - ( v~+ - v~- - v ~ - ) + m~-Vv~ln L J, (5.23) tv+

z 1 v t~- 1Mv~-2 - 1M+(v"+2 - v¢ - ) - ~ M - v - . (5.24)

The total traction jump is then given by

t ~+ - ( F - + t ~- ) = M ( v ~+ - v~ - ) + m~-[v~], (5.25)

but a total velocity jump has no meaning.


The energy jump relations (4.19) for constituents ~ and/z now become

1 . ._ /? . . / s . . /J1 [ t ' . vv -- OK - - 2 r r t n V . u ] - - [ q n ] :

1 + u - -1 -M-v~- ' v~ -2 + [m~%~] -- m-(%~ +%~) - - ~M (% +%~)+, (5.26)

1 (t~ v ~ - O~n - ~ M v ~ . v ~ ) - - [tin ] =

1 1 ~ 1 + - - M - v ~ - . v ~ - + M % ~ - - ~ M - ( % + % ~ ) - - ~ M (% + %~)+

2

giving a total energy jump relation

(5.27)

1

[ t ' . v ~ - ~ - ½m~v~.v ~1 - (tU.v ~ - ~ - i M v a . v ~ ) -

= M(~ ~+ - ]g~-) + m~-[}gu]. (5.28)

Eliminating F + and t ~- by (5.23) and (5.24) gives the alternative energy jump relations

1 1 [c)~ + m~% v] -q- [On] - ~-M-( ~v + ~ ) - - 2M+( %" + %~)+ =

2 L

[v~l.{t u- -t- ~ M - ( v - v a- ) ÷ ~ m , - [ v l}, (5.29)

1 + - v 4 ~ - + [ q , ] + M % ~ - - M - ( % ~ + % ~ ) - - ~ M (% +%~)+

= - 1 m + v ~ - . ( v ~ + - v ~ - ) , (5.30) 2

which are coupled relations between the mass transfer partitions M + and M- and the flux constraint discontinuity [qn]" Applying the internal energy approximations (5.3) and (5.6), and noting that ~ - and ~ - both vanish since the phase equilibrium temperature is uniform (or very closely uniform) in the region of co-existing phases behind the front, simplifies (5.29) and (5.30) to

1 + /x-- ~,-F ~ + = - M L ~ + ~ M v .(v - v " - )

1 ~+ +[v~l.{t ~- + - ~ M - ( v - v ~ - ) + 2m~-[v~]},

2 [ttn] = M L TM - M+vtL- . (v"+ -- v " - ) .

(5.31)

(5.32)


These combine with the two mass jump relations (5.22) to provide four relations coupling M +, M - , [q~] and u , .

The temperate ice zone approximation proposed by Hutter [13,14] describes only the total momentum and energy balances of the ice and water, treated as a single constituent moving with the barycentric velocity 0-, defined by

p - O - = pV-v~ ' - + p ~ - v ~ - , p - = p " - + p ~ - , (5.33)

in the expanding temperate zone behind the front. Distinct ice and water velocities are allowed in the mass fluxes, which are related by a diffusion law to the concentration gradient. The corresponding traction jumps are obtained from (5.23), (5.24) by replacing v ~- and v ~'- by 0-, yielding

1 + 1 + _ [ t q = t ~ + - t ~- = ~M o- - ~M [v] + m~+[0], (5.34)

[t~] = - t ~ - = - 1M+0-2 + 1m+[° ] ' (5.35)

where v ~+ is interpreted as 0 +, and the cold ice mass flux relative to the front, equal to the temperate ice and water mass flux relative to the front, becomes

m~ + = p - ( 0 - - u,). (5.36)

The total traction jump is then

[F + t ~] = rn~+[0]. (5.37)

The total energy jump (5.31) and constraint flux discontinuity (5.32) reduce to

1 ~+ ~,+ = M L + [9].(t ~+ - ~m, [9]), 2 [c7, ] = - M L - M+9-[9], (5.38)

where

L = - L ~ > 0 (5.39)

is the latent heat of phase change from ice to water. Further approximations of the energy jump depend on relative magnitudes of the stress working and kinetic energy flux.

Next consider a phase change front advancing into a stationary v-constituent, so v v+ = 0, then by (4.8) and (5.22),

- u , , [ p q ~ - v - - p v n = p ~ - ( v ~ - - u,,) = M , (5.40)

which gives the front speed expression

plZ-V~n - + pU-v~n- p Vn = > 0 (5.41) Un = p~-- __ [pV] p-- __ pV+'

by definition, and hence

p - - p~+ > 0 +-~ O n > 0, p - - p~+ < 0 ~ ~- < 0. (5.42)


Further,

v~- - un > 0 +-~ M > 0, v~- - un < 0 +-~ M < 0, (5.43)

so, for example, there is a water flux from a wet snow zone into the front where freezing (M > 0) occurs, and a water flux into a temperate ice zone from the front where melting (M < 0) has occurred. There is necessarily a flux of constituent i, into the front from ahead, but the direction of the relative u-flux behind is not simply determined by the sign of M. The constraint energy jump (5.32) reduces to

M + IvY-12 = 2 [qn] - M L ~ , (5.44)

and the total or constituent u energy jump (5.31) becomes

1 12 1 ~ 1 OK + = - M L ~ - -~M+lv " - + v ~ - . ( - t ~- + ~ M - v - + ~ m ~ - v ~ - ) . (5.45)

Finally, the traction jumps (5.23) and (5.24) become

1 F + - F - - M - ( v ~- + v ~ - ) - m n~- v~- , (5.46)

2

1 t ~ - = m v ~ - - ~ M - ( v - + v ~ - ) . (5.47)

For water percolation through a snow pack, but not generally in a temperate ice sheet zone, the approximation Iv"-[ >> I v~-I may be appropriate. Then the traction jumps (5.46), (5.47), and energy jump (5.45), become

1 F + - F - = - ~ ( M - M + ) v ~-, t ~ - =

~ + = - M L ~ - 2 M + l v ~ - I 2, > 0

1 M ~ ( + M+) v " , [t] = - M v ~ - , (5.48)

(5.49)

since there is a decrease of temperature from T M at the front into the cold ice region ahead, and the constraint relation (5.44) is unchanged. Now the mass balance relations (5.40) reduce to

p~- v~- M un P~- + P~'- _ p,~+ P " - _ p~,+, > O, (5.50)

where M > 0 at a percolation front. While p~- + p~- - p~+ > 0 if the volume fraction of air is less behind than ahead of the front, displaced by the percolating water, and v~- > 0, consistent with positive un, p~- - p~+ < 0 since there is less ice behind the front, which is a contradiction. Thus the non-zero ice velocity v ~- must be retained in the front speed expression (5.41) following from the mass jump conditions, which is consistent with the approximation Iv~-I >> I vy [ provided that

p~- >> p~- ~ 05 ~- << ~b ~-. (5.51)


Otherwise a stationary ice approximation cannot be valid. The assumption that both constituents are intrinsically incompressible, which

would be a good approximation for water and ice in a glacier and water and ice grains in a snow pack, does not change the above conclusions, nor simplify the general relations.

5.3 Phase annihilation front

The alternative situation is when the front advances into a medium of both constituents (phases), but behind the front there is a single constituent (phase), as illustrated in Fig.3. That is, the second phase disappears after the passage of the front, but not necessarily by mass transfer across the front to the constituent behind. One example is a contracting temperate ice zone when v and # denote ice and water respectively, and a second example is an advancing melt surface into a wet snow pack when p and /z denote water and ice respectively.

p~- = 0 v- v + #+ V/x- = 0

m ~ - = O -+ un t/x- = 0 O/x- = 0

Fig. 3. Phase annihilation front

For the conditions given in Fig.3, the mass jump relations become

m~ + - m,~- = - m ~ + = M - + M +, = M, (5.52)

and the momentum jump relations become

1 1 + ~ - _ v V + _ v / x + ) ~+ v F+ - F - = - M v v - + ~ M (v + r n n [v] , (5.53) 2

1 1M+v"+ (5.54) t / x + - ~Mv/x+ + ~ M - ( v - - v/x+) + 2

F + + t ~+ - F - = M ( v ~- - v/x+) ÷ m~+[v~]. (5.55)

The energy jump relations become

[t~.v ~ _ q~ - lm~v~.v~] - IcOn ] =

1 + ,, 1M+v~'+.vg+ + [ r n ~ ;] - M-(% ~ + %~)- - ~ M (% + %g)+, (5.56) 2 z

( t~'v~ - q~ + 1 M v ~ . v ~ ) + + [qn] =

1-M+vV+'v~+2 - M%~+ + ~ M - ( % + %~)- + M+(% ~ + %~)+, (5.57)

410

^/,,' 1 ,b' /,, /~ [ F ' v v - q n - ~ m n v ' v ] - ( t ~ . v ~ - 2 t ~ + Mv~.v~) +

= M(% ~- _ %~+) + mV+r%~l n i_ .IT

o r

[0: + m ~ %~] + [~]n]- 1 M - ( %~ + %~)- - 1 M + ( %v + %~)+ =

[v~].{F+ - ~ M + ( v ~ _ v ~ + ) _ - 2 mnl v+[v~]},

L.W. Morland, J.M.N.T. Gray

(5.58)

(5.59)

1 M q~+ - [qn] - M%~+ + ~ -(% + %~)- + ~M+( %~ + %~')+

= 1 M - v ~ + . ( v ~ - - v~+). (5.60)

These traction and energy jump relations can also be obtained from (5.23)- (5.25) and (5.26)-(5.28) by interchanging the - and + superscripts, noting that this interchange in (5.22) implies the replacement of M, M + and M - by their negatives to retain the interpretation of M as the net mass transfer from constituent/x to constituent v with the superscript + denoting conditions ahead of the front. With the energy approximations (5.3) and (5.6), and the vanishing heat fluxes in both constituents co-existing ahead of the front, the energy jump relations become

1 + ~ - = M L ~ _ ~ M - v ~ .(v ~- _ v~+)

1 v + v -[v~].{F + - 1M+(vV- - v ~+) - ~ m n [v ]}, (5.61)

2 [qn] = M L ~ - M - v ~ + . ( v ~- - v~+). (5.62)

For a contracting temperate ice sheet zone, when v denotes ice, L v~ = - L < 0, and M > 0 for freezing at the front, introduce the barycentric velocities 9- and 9 + by

O- = v ~-, p+9 + = p~+v ~+ + p~+v ~+, p+ = p~+ + p~+. (5.63)

Then, replacing constituent velocities by the appropriate barycentric velocities,

[ tq = t ~+ - t ~- = M - 9 + + M - D ] + m, [9], (5.64)

[ t~] = t~ + 1 __+ 1 = - ~ M v - ~M-[9] , (5.65)

where

m~,- = p~-(9- - un), (5.66)


and the total traction jump is

[ t ~ + t**l = m ~ - [ 9 1 .

The energy jumps (5.61) and (5.62) become

1 ~v- = - M L - [O].(F- + 2mn-[9]), > O,

2 [qn] = - M L + M-9+.[9],

and the front speed is given by

P + v + - P ~ vn > 0. /~n = p+ _ pV-

411

(5.67)

(5.68)

(5.69)

(5.70)

An approximation of stationary cold ice behind the front is consistent with the above relations.

For a melt surface advancing into a wet snow pack, when v denotes water, L ~** = L > 0, and M > 0 for melting at the front, so the ice mass jump relation (5.52) implies that

mn ~+ < 0, v~ + < un. (5.71)

With the approximations

Iv**+[ << IvY+l, IvY-I, un, (5.72)

which suppose that the ice grains move very slowly in comparison with the percolating and surface water, and front, the mass jump conditions (5.52) reduce to

[p~v~n] - [p~]un = M, un = M / p **+, (5.73)

the energy jumps to

1 v ~ 1 + ~ 1 ~+ ~ - = M L - ~ M v * * + . v - + [ v ] . { ~ M v - + ~ r n ~ [ v ] - F + } , (5.74)

2 [q,] = M L - M-v**+.v ~-, (5.75)

and the traction jumps to

1 ~ 1 + . . . . m~ [v ], (5.76) [t v] t ~+ t ~- ~ M v - ~ M [v ] + ~+ ~

1 v 1 + v [t**] = t **+ = ~ M v - + ~M [v ], (5.77)

v + [t ~ + t**] = M v v+ + m ~ - [ v v] = M y ~- + m~ [v~]. (5.78)


In the case of a thin melt water layer percolating slowly into a snow matrix, plausible simplifications are

v ~- = unn*, t ~- = 0. (5.79)

Then

m v+ = pV+(vVn+ - Un) = M, un = m / p ~+, (5.80)

1 I_M+v~+ t~+ M v r + t~+, t ~+ = : M - u ~ + = - (5.81) 2 Z

~ = M L + t/*+.(v ~+ - unn*) - ~M(v~+) 2

+ ~ u n n * . ( M + v v+ + M - u ~ n * ) , (5.82)

and the constraint flux is given by (5.75). These simplify considerably when kinetic energy and stress working contributions to the energy jumps are neglected, giving

M L = 2 [qn] = qn^~-, M ~+ p + M (5.83) un -- p~+ , vn pV+ p~+ ,

t~ + 1 1 v2+, = ~M-un + ~M + t n~+ = M v ~ + - t~ +. (5.84)

Tangential conditions must also be prescribed. The surface mass partitioning required to determine the separate ice and water tractions depends on the surface conditions of the inert air, which is displaced as water enters the snow, and the stress coupling between the air and water.

6 Conclusion

The essential ingredient of phase change is the mass transfer between the constituents defining the separate phases. Such transfers give rise to direct momentum and energy transfers and implicit interaction body force and energy transfer contributions. Explicit forms for these interaction terms have been derived on the basis of a symmetry postulate: that the binary interactions between each pair of interacting phases have a common form. A reduced interaction model is proposed and adopted, in which the binary forms are the most simple consistent with composite mixture restrictions. At a singular phase change front the mass transfer occurs as a surface production term, and transfers on opposite sides of the surface define changes between different states of the phases. This is reflected by introducing a partition of the transfer into distinct transfers on


the two sides, together with an associated normal constraint energy flux discontinuity. The general jump conditions of mass, momentum and energy for each constituent have been derived, and provide a complete set of matching conditions for each constituent across a phase change front involving two interacting phases. Further postulates are required to describe the physical processes when there are more than two interacting phases. The theory is illustrated for a variety of binary transfers at a plane phase change front. Examples related to polythermal ice sheets and wet snow packs are presented, and consistencies with conventional approximations are discussed.

Acknowledgements. This research was pursued in association with a NERC Research Grant in the School of Mathematics at UEA on Snowmelt Hydrology. Two constructive reports on the original manuscript have improved both the presentation and the accuracy of some of the assertions.

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L.W. Morland J.M.N.T. Gray School of Mathematics University of East Anglia Norwich, NR4 7TJ United Kingdom

Received June 6, 1994

continuum mech. thermodyn. 7 (1995) 387-414 continuum

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