G E N E R A L BALANCE EQUATION FOR A P H A S E INTERFACE JOHN C. S L A T T E R Y Department of Chemical Engineering, Northwestern University, Evanston, Ill.
A phase interface is viewed as a three-dimensional region which separates the bulk portions of two phases .) may differ from those applicable and in which the constitutive equations (for mass flux, stress, energy flux,
..
in the bulk portions of each phase. The interfacial region may include a singular surface; otherwise, all quantities are continuous functions of position. Two forms of the general balance equation or conservation law (for mass, momentum, energy, ...) at a phase interface are derived. The rigorous form requires knowledge of the constitutive equations in the interfacial region. The approximate form follows from the general viewpoint of Buff; to use it one must make what amounts to a constitutive assumption for a distributed source term which accounts for ‘‘surface behavior.”
HE purpose of this paper is to derive the general balance T e q u a t i o n or general conservation law for some quantity (such as mass, momentum, . . .) at a deforming phase interface. At least one of the problems is to decide how the interfacial region is to be described. Should it be regarded as a threedimensional region of some thickness, or as a two-dimensional surface, singular with respect to one or more quantities? If the limiting value of a quantity a t a point on a surface as obtained by approaching the point along a path restricted to one side of the surface differs from that obtained by approaching this point from the other side of the surface, the surface is said to be singular with respect to this quantity at this point. A surface which is singular with respect to one or more quantities is referred to as a singular surface in what follows (75, p. 492). One might choose among a t least three possible models.
I t might be a singular surface containing a distributed source. It might be a three-dimensional region in which all quantities are continuous, but in which the constitutive equations for the mass flux vector, the stress tensor, the energy flux vector, . (such as Fick‘s first law, Newton’s law of viscosity, Fourier’s law, . . .) differ from those applicable outside the interfacial region. I t might be a three-dimensional region which surrounds a surface singular with respect to density, specific internal energy, . . . , and in which the constitutive equations differ from those appropriate at some distance from the interface.
..
Gibbs (9) and Buff (2-5) viewed the equilibrium interface as a three-dimensional region in which all quantities are continuous (though perhaps rapidly changing) functions of position. This viewpoint is tentatively confirmed by experimental evidence which indicates a diffuse interfacial region, particularly near the critical state (70, p. 373). For convenience in calculations Gibbs suggested that all surface effects be taken as originating a t a “dividing surface,” the location of which would be sensibly coincident with the physical interface, but otherwise arbitrary. H e recommended a specific dividing surface, the “surface of tension,” which when used in the calculation of thermodynamic properties of a static droplet leads to simpler results (70, p. 349). Another useful possibility in some cases is the equimolecular dividing surface (70, p. 339). Scriven (72) [see also Aris (7)] wrote mass and momentum balances at a deforming interface under the assumption that there was no mass transfer between the surface and the adjoin108
I&EC FUNDAMENTALS
ing phases. The phase interface was visualized as a twodimensional surface (possibly) containing mass, which meant that the surface represented an interfacial region of some thickness. Subsequently, Eliassen (6) and Eliassen and Scriven (7) explicitly viewed the interfacial region as threedimensional in deriving these balances, though for convenience the results were presented in terms of quantities defined as fields over a surface contained within the interfacial region. The relation between the approach of Eliassen and Scriven and that of this paper is discussed below. I n his initial development Scriven (72) wrote for a phase interface a mass balance which accounted for possible mass transfer with the surrounding phases. His approach was followed in an attempt (73, 74) to write momentum and moment-of-momentum balances, which included the effects of interphase mass transfer. These relations were constructed by making balances on a deforming region of material moving within the interface. The derivations are unsatisfactory, in that the concept of following a region of material within the interface contradicts the assumption of simultaneous interphase mass transfer. Recognizing that in our present state of knowledge the use of Kotchine’s theorem the general balance at a singular surface, in solving a problem is not practical, we extend to the dynamic problem the approach which Buff (3) developed for the static equilibrium situation. One of the results is the basis for more detailed discussion of mass and momentum balances. Relation between Model for Interface and Balance Equations
The purpose of the balance equations for mass, momentum, energy, at an interface is to provide connecting conditions between the solutions to the mass, momentum, energy, ., balances applicable in the separate phases. I n thinking about the phase interface for the first time, a singular surface may be visualized. On reflection, it may seem more reasonable that from the continuum viewpoint all quantities should be taken as continuous functions of position. As pointed out above, many workers have followed Gibbs in adopting the latter viewpoint; there is even some experimental evidence to substantiate this view. O n the other hand, it is not inconceivable that phases should sometimes be viewed as meeting at singular surfaces. Since it adds no appreciable difficulty and since the results are readily simplified when all quantities are everywhere continuous functions of position,
...
..
we assume here that the interfacial region may include a singular surface. When a singular surface is not present, we follow Gibbs (9) in introducing a “dividing surface,’’ w,hich artificially separates the phases. The position of the dividing surface must be defined, but for our purposes here it is not necessary to adopt a specific definition. If we think in terms oi’ what must be done to solve a problem of simultaneous mass, momentum, and energy transport involving two phases (we will not concern ourselves here with a possible third phase in the form of a container or conduit in which the first two phases move), the equations to be satisfied are the balance equations in the two phases, the balance equations at the phase interface, other boundary conditions and continuity requirements, and finally the constitutive equations for the mass flux vector, the stress tensor, the energy flux vector, . . ., which describe the behavior of the materials involved. I t is important to remember that our knowledge of the constitutive equations for materials comes from experiments which describe the behavior of materials at a distance from the interface or in the absence of an interface. We have no right to assume that the constitutive equations in the neighborhood of an interface are the same as those of the bulk fluid, as illustrated below. Because the constitutive equations applicable in the neighborhood of the phase interface may vary from those applicable at a distance from the phase interface, analysis of the transport of mass, momentum, energy, . . in the presence of a deforming fluid-fluid phase interface is particularly difficult. It appears that there are three general approaches.
.
The most satisfying procedure is to describe precisely the material behavior everywhere. This leads to the particularly simple result of Kotchine’s theorem below. T o our knowledge no one has taken this approach in solving a problem, the general belief apparently being that this Tvould be too difficult. I t is not entirely clear that this is correct, though it is true that at present we have no experiments by means of which we can study the constitutive equations in the neighborhood of a phase interface. Eliassen (6) and Eliassen and Scriven (7) enclose by two parallel surfaces the interfacial region or the region in which the constitutive equations differ from those applicable in the bulk flow. They adopt the term “stratum” for this region. Mass and momentum balances written for the stratum are expressed in terms of a dividing surface which is taken to be one of the surfaces in the stratum parallel with respect to the bounding surfaces. In this way they avoid in a dynamic situation the problem of in some way extrapolating bulk quantities into the interfacial region in an effort to follow the approach which Buff took in the equilibrium situation. All of this has the disadvantage that the bulk flow variables are matched at the edges of the stratum rather than at the dividing surface. At this time it is not clear whether the disadvantage is real or illusory. After developing Kotchine’s theorem, we focus our attention on an approximate solution to whatever problem we are considering. The approximate solution is obtained by assuming that bulk constitutive equations are applicable in each phase up to the singular or dividing surface, and that the singular or dividing surface contains a distributed source for each quantity being balanced. The source terms are constructed in such a way that the approximate solution agrees with the true solution to the problem outside some stratum [of possibly different dimensions than that of Eliassen and Scriven (6, 7)]. The object of the Approach of Buff is to obtain an expression for the distributed source term in the singular or dividing surface in terms of differences between the true and approximate solutions. As with the approach of Eliassen and Scriven, the assumption is that one is generally interested not in the precise details of the interfacial region, but rather in how the interfacial region affects ihe bulk flows.
Preliminaries
Let u stand for a pair of surface parameters or surface coordinates ua (a = 1, 2) identifying by definition the surface point. Point u is not to be identified with a material particle in the motion under consideration. A set of three coordinates x i (i = I , 2,3) in an arbitrary coordinate system fixed in space is represented by x. The family of surfaces given by x = x (u,t )
(3.1)
describes the motion of a surface by specifying the places x occupied by the surface point, u, as time t progresses. The velocity, V, of a surface point, u, is defined by (3.2)
If we eliminate the parameters u, we may write Equation 3.1 in the form
f
(x, t ) = 0
(3.3)
Differentiating this with respect to time following surface point u, we have
3 +v at
*
vf
= 0
(3.4)
Let us introduce C; for the unit normal to the surface,
E
=
Vf/IVf/
(3.5)
to obtain from Equation 3.4
(3.6) This means that all possible V corresponding to different parameterizations u of the moving surface have the same normal component, V,,,, which is defined to be the speed of displacement of the surface (75, p. 499). General Balance at Singular or Dividing Surface of Phase Interface
The general balance or general conservation law for some quantity associated with a material volume R of arbitrary size is of the form (75, p. 468) :
+
where p indicates the mass density, the density of the quantity per unit mass, 4 the flux of the quantity through the bounding surface S of R, n the unit vector outwardly directed with respect to the closed surface S, and the rate of production of the quantity per unit mass a t each point throughout the material volume R. Equation 4.1 implies [Latin indices indicate tensors with respect to coordinate transformations of 3-space; Greek indices denote tensors with respect to surface coordinate transformations. Comma notation stands for covariant differentiation ( 7 7, p. 197), semicolon notation stands for a total covariant derivative of a double tensor field (8, p. 811), and the summation convention is employed throughout. Subscripts or superscripts enclosed by parentheses have no tensorial significance] :
when each term in this equation is continuous. Referring to VOL. 6
NO. 1
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109
Figure 1 and the discussion above, we ask here what Equation 4.1 implies for a material volume, R, which intersects a phase interface during the period of time under consideration. I n Figure 1 we sketch a material volume, R, which, during the period of time under consideration, intersects a phase interface. The interfacial region discussed above is represented by regions R(D+) and R(D-1; outside this interfacial region in R(+)and R ( - ) the bulk constitutive equations apply. The singular surface (75, p. 492) within the interfacial region is S(D), If no singular surface is present, we follow Gibbs (9) in taking S ( D ) to be a “dividing surface”; a specific definition for the dividing surface need not be adopted for our purposes (see Introduction). The displacement velocity, Equation 3.6, of the singular or dividing surface is Vet). The unit vector, F, normal to the singular surface is defined such that the vectors [(dxz/dul), (dx2/du2), ti] have the same orientation as tangent vectors to the x- coordinate curves-i.e., the orientation is positive ( 7 7 , p. 197); it is assumed in what follows that points from R(D-) into R(D+). The bounding surfaces S(D+) and S(D-) of the interfacial region are defined to be parallel [a surface S whose points are at a constant distance along the normal E; from a surface S(D)is said to be parallel to S(D) (76, p. 116)] with respect to S(D)and move with the same displacement velocity, V(E). $0) is not necessarily a material surface, independent of whether it is to be interpreted as a singular surface or a “dividing surface.” Physically this means that we are allowing mass transfer across the moving and deforming phase interface. The discussion which follows is based upon a similar development presented by Truesdell and Toupin (75, p. 526). The term on the left of Equation 4.1 may be expressed as
Figure 1. A material volume R which intersects a R ( D - ) of interfacial region portion R(Df)
+
Substitution of Equations 4.3,4.6, and 4.7 into 4.1 yields
where by dv(+)/dtand dv(-)/dt we mean time derivatives following (not necessarily material) regions, the closed surfaces of which move with the respective velocity distributions (4.4)
P
(4.8) where [,,$I‘D)
and (4.5) The speed of displacement of the surface, V,,), is defined by Equation 3.6. The transport theorem, generalized for nonmaterial regions (75, p. 347) may be applied to obtain
a P(D+)
$(D+)
-
,,(D-)
+(D-)
(4.9)
The quantity $ ( D + ) , for example, is the limiting value of $ at a point on S(D) as this point is approached from the interior of R(+) R(D+)while remaining within R(+) R(D+). Equation 4.8 may be simplified by remembering that Equation 4.2 applies everywhere within each phase. Integration of Equation 4.2 over the region R(+) R(D+)gives, after an application of the divergence theorem,
+
+
+
n
(4.6) Similarly, for R(-) f R(D-) and
P
l ( D , { p ( D - ) +(D-)
u([)(~-)
+
$J(,~(D-)}
dS = 0
Subtracting Equations 4.10 and 4.11 from 4.8, we have 110
I&EC FUNDAMENTALS
(4.11)
observation ; to bring theory and experiment into agreement, we artificially introduce a surface source in Equation 4.1. Since the size of system R is arbitrary, this implies that the integrand itself is zero: [P$
4
D
)
- [ P W ) "(6)
+
[4(t.)l(D'
=
(4.13)
0
This is known as Kotchine's theorem (75, p. 527). To use this equation properly in solving a problem or analyzing an experiment it is necessary that the constitutive equations (for the mass flux, stress, the energy flux, .) used to describe the behavior of the material be accurate in the immediate neighborhood of the phase interface. At the present time, for lack of more accurate information we use the bulk constitutive equations in the neighborhood of the phase interface. I t is for this reason that Equation 4.13 appears to be inadequate when it is used in the analysis of experimental results.
Here we have expressed the surface source in terms of a surface flux tensor, e", inwardly directed through the boundary curve, C ( D )of , S ( D ) , using Green's theorem for surfaces (77, p. 189). n
n
..
General Balance for Computations Valid beyond Immediate Neighborhood of Phase Interface
Equation 4.13 is of little practical value a t the present time, since we know so little about the constitutive equations for materials within the interfacial region. Fortunately, we are usually content to be able to describe concentration, velocity, temperature, , distributions outside the interfacial region. Here we formulate the general balance at the singular or dividing surface of a phase interface with the understanding that when this general balance is used in analyzing a problem the bulk constitutive equations will be assumed to apply in each phase up to the dividing surface. Simplest Approach. The most expedient approach to the general balance under these circumstances is to replace by Figure 2 the representation of a material region in Figure 1 which intersects the interfacial region over the period of time under consideration. In this view we apply the bulk constitutive equations in each phase up to the dividing surface, S ( D ) , which we are now forced to recognize under all circumstances as a singular surface. We realize that Equation 4.13 does not allow for an accurate description of experimental
..
The outwardly directed unit first-order surface tensor normal to the curve O D ) and tangent to the surface S ( D ) is indicated by p a . Repeating the development in Section 4 through Equation 4.13, we have
where
[4(t,l = 4 w + - 4 w -
(5.4)
The obvious difficulty with Equation 5.3 is that we have thrown all of our ignorance into ea,a. We partially correct this situation below. Approach of Buff. In this section we follow the general approach of Buff ( 3 ) ,although the details of his treatment are limited to the static situation. The purpose is to obtain an explicit expression for Ossa in Equation 5.3. I n what follows quantities with the superscripts and are to be associated with an approximate solution of a problem in which the balance equations a t the dividing surface are taken to be of the form of Equation 5.3, and in which the constitutive equations for the bulk phases are assumed to be valid up to the dividing surface. Variables with the superscripts (D+) or (D-) indicate for regions R ( D + ) and W - ) in Figure 3, respectively, the exact solutions to the problem in which variations in the form of the constitutive equations are properly taken into account in the neighborhood of the dividing surface, and in which the balance equations a t the dividing surface are of the form of Equation 4.13. The region R(D+) R(D-)in Figure 3 is defined such that these two sets of solutions coincide outside this region. Boundaries S ( D + ) and S ( D - ) are surfaces constructed so as to be parallel (see definition of parallel surfaces following Equation 4.2) to the dividing surface, S ( D ) , and such that they move with the same displacement velocity, Vo), as the dividing surface. Surfaces SR+) and S('--) are constructed so as to be normal to all surR(D-) which are parallel with faces within the region R ( D + ) respect to the dividing surface, S ( D ) . The region under consideration is not necessarily a material region.
+
-
+
+
Figure 2. A mlaterial region R which intersects a phase interface, represented here by geometrical surface S(D)
Figure 3. A (not necessarily material) portion of interfacial region VOL. 6
NO. 1
FEBRUARY 1967
111
In both the exact and approximate solutions the balance equations to be satisfied at each point of the continuous regions are of the form of Equation 4.2. Let us take the difference between the balance equations associated with the exact and approximate solutions and integrate them over the region R ( D + ) to obtain
Our next concern is to reformulate the third and fourth terms on the left of Equation 5.9 in terms of integrals over the dividing surface. If in SR+)and S(R--)we take as orthogonal coordinates X and the arc length, s, measured along the bounding curves of surfaces parallel with respect to the dividing surface, we may write for an integrand of the form A ( n ):
S
A ( , ) dS =
S ( R +)
S
ni =
+ Js(j,lp$
7qe)](-) f [4(E)](-)) dS
= 0
(5.7)
n(D)’ 4(o(D-)
- +(e)-
(5.12)
(5.8)
(5.13)
n(D)’‘
But since n ( D ) is tangent to the dividing surface, we have (17, p. 197):
where
[4(e)l‘-’ =
Ain‘ dX ds
0
For simplicity let us restrict ourselves to rectangular Cartesian coordinates. [The alternative is to introduce Euclidean shifters (8, p. 806).] The unit vectors, n, which are normal to ScR+) or and which lie along any normal to S(D) passing through C ( D ) , are all parallel, and their rectangular Cartesian components are identical. Let us identify ni with the rectangular Cartesian components, n ( ~i, of ) the corresponding unit normal vector tangent to S ( D ) ,
I n the same way we can construct for R ( D - )
[4cn)1(-))d s
[A(D+)
C(D)
=
n ( D ) j x’;a(D)
(5.14)
a(D)a@ xi;@(D)
where
Adding Equations 5.5 and 5.7 and applying a t the dividing surface Equation 4.13, we have
X‘;@(D)
=
bx‘/bu’I
(5.15)
SCD)
Equations 5.12 to 5.14 together with an application of Green’s theorem for surfaces (7 7, p. 189) yield
=
The first two terms on the left of Equation 5.9 may be expressed in terms of integrals over the dividing surface by means of a result derived in the Appendix, Equation A. 19:
Li,,
u ( D ) ~ ’ B dX dS
A*a dX),a d S
(5.16) (5.17)
Here we denote the surface tensor corresponding to a shifted spatial tensor by A*a = A i
(5.18)
x‘;@(D) a(D)a@
The third and fourth terms on the left of Equation 5.9 may now be rewritten with the help of Equation 5.17 as
X ( D +)
(5.10) and
and
(5.11) Parameter A is the distance measured along the normal to the dividing surface; X ( D + ) and X ( D - ) are the values of X at S ( D + ) and S ( D - ) , respectively. The principal curvatures of the dividing surface are denoted by K I and K Z ( 7 7, p. 21 1). 112
l&EC FUNDAMENTALS
Equations 5.9, 5.10, 5.11, 5.19, and 5.20 may now be combined to obtain Li,){[P$
d
where we define
+ [+(€)I - [P$l‘D’
V,,)
+ p,al
=0
(5.21)
where in rectangular Cartesian coordinates
[p$
v*a](*)
+
dX} ,a
. ..
K:--
[+*a](*)
dX
+ [+(Dl - [PU4 V(E) = - ea,,
hE) - [P+I V(,) - eaja
= I:p$VD)
(5.23)
Some writers take the interfacial region to be continuous (3; 9; 70, p. 339). In this case Equation 5.23 simplifies to [P$
V(E)l
+ [+(E)]
=
- ea,a
(5.24)
Over-all Mass Balanctt at Dividing Surface of a Continuous Interf’acial Region
T o obtain the over-all interfacial mass balance from Equation 5.23, we identify $=1,
+=o
(=O,
dX
+
[p
vi
-
v * ~ ] ( * ) dX}
{l(D-)
,a
A@+)
Xl,@(D)a(mapI(*)
[tri
dX}
(7.3) ,a
Equation 7.2 bears certain similarities to a previously proposed interfacial momentum balance (73, Equation 2.8). T o approach a usable result, we extend the approximation which we made to obtain Equation 6.6: We neglect the effect of differences in density and velocity between the exact and approximate solutions in the interfacial region as well as the effect of derivatives of these differences. In addition, we state that the same external body force vector acts on all species and that it is not a function of A. Under these restrictions we have
and
~ ( 01 [ P ] ( ~ V(c) ) =
(6.2)
-e(msaa)a,a
where
e(maaa)a,a=
- K I h)(l - K Z A)
(6.1)
This leaves us with [P
(1
},a
Since each term in the integrand is assumed to be continuous, and since the portion of the dividing surface considered is arbitrary, we obtain as our principal result [P$ W I
fi](*)
[p
(5.22)
(1
- K i X)(1
-
Kz
A) dX
[p
4 ( D )
= [ p vi]
(7.5)
From this point of view the Newtonian surface fluid model (72; 7, p. 232 ; 73) states further that
9
e(mom.)ia,a=
(7.6)
tap),a
where the surface stress tensor, tap, is taken as the most general linear function of the surface rate-of-deformation tensor, Sap: This appears to be as far as we can go without making any approximations. Unfortunately, Equation 6.2 is not usable as it stands. Its form is much the same as that of the mass balance suggested by Scxiven (72, Equation 20). The simplest approximation is to neglect the effect of the derivative of the difference in density and of the difference in velocity between the exact and approximate solutions in the interfacial region to obtain
With Equations 6.4 and 6.5, Equation 6.2 becomes [P
wl - [PI
V ( € )= 0
(6.6)
If one regards the interfacial region as continuous, it follows from Equations 5.24, 6.1, and 6.4 that [P V ( E ) I = 0
(6.7)
Over-all Momentum Balance and Dividing Surface of Continuous Interfacial Region
To obtain the over-all interfacial momentum balance we identify Q with the ve:locity vector, 3 with the external body force vector, and 4 with the negative of the stress tensor. In tensor form these identifications are
tap =
+
[U
(K
- e) s],’
aap
1 sap =
[X’;a
uj;p
+2
e
&a
(7.7)
+x’;~
(7.8)
uj;al
By u’ we mean the velocity a t the dividing surface in the approximate solution. Parameters u, K , and e are defined as surface tension, the surface dilational viscosity, and the surface shear viscosity, respectively ; these quantities should in turn be thought of as functions of the thermodynamic state variables. In the static situation u may be identified with the equilibrium surface tension ; identification with the equilibrium surface tension in the dynamic situation would assume that u is a function of (say) concentration alone. With Equations 7.5 and 7.6, Equation 7.2 becomes [ p vi V ( E ) ]
-
[p
vi]
V(E) -
[tij
E$]
=
(Xiia
(7.9)
@),p
If one regards the interfacial region as continuous, it follows from Equations 5.24, 7.1, and 7.6 that [p
vi V(E)]
-
[t’j
5,]
=
(xi;a
tap),@
(7.10)
Future Work
At present in the analysis of multiphase dynamic problems, Equation 5.23 (or Equation 5.24) is the general balance equation usually applied a t the phase interface. Unfortunately, the application of this equation usuVOL. 6
NO. 1
FEBRUARY 1 9 6 7
113
ally requires an assumption, as illustrated by the two empirical statements, Equations 7.6 and 7.7, which were made in the discussion of a balance equation for momentum. While in the absence of a better approach it is probably worth elaborating on empiricisms such as Equation 7.7, serious consideration must be given to means by which the behavior of materials in the interfacial region may be studied. With this information we could approach a problem more rigorously through Equation 4.13.
c(D)ug
The author is grateful for financial support from the National Science Foundation (GK-450) during a portion of this work.
a(D)"
b(D)+a
(A. 8)
b(D)fi,4
A useful expression for the third ground form tensor in terms of the mean curvature, H ( D ) (77, p. 205), and the Gaussian curvature (or total curvature), K ( D ) (77, p. 183), is (77, p. 205). c(D)og =
H ( D ) b(D)ug
- K(D) a(D)up
(A. 9 )
From Equations A. 7 and A. 9 we have a,g = (1
Acknowledgment
=
- X2 K ( D ) ) a ( ~ ) a+ g (2 H ( D ) X2 - 2 X) b ( ~ ) " g
(A. 10)
Equations A. 2 and A. 10 may be used to obtain an expression for the ratio of the determinant of the metric tensor corresponding to S(') to that corresponding to W ) ,
Appendix
-!= 1-4H(D)
The derivation given here of Equation A. 19 is an alternative to those given by Buff (3) and Eliassen (6). Let the y i ( i = 1, 2, 3) represent a rectangular Cartesian coordinate system fixed in space. Let the u"(a = 1, 2) represent a system of surface coordinates on surface S in space. The metric tensor, a,@ (77, p. 167), is defined as:
a(D)
+ (4
H(D)'
f 2 K ( D ) ) X2 -
H(D) K(D)
X3
+
K(D)'
X4
(A. 11)
In obtaining this expression we use the definition for the double contravariant form for the metric tensor (77, p. 167)
(A. 12) the definition of the mean curvature (77, p. 203)
We may express the differential element of surface area in terms of the determinant of the metric tensor, a =
1_
eYs aoy
2
(A. 2)
aga
and the differential elements of the surface coordinates as (77, P. 171),
dS
=
4;du'
du2
(A. 3)
A surface SA), whose points are at a constant distance X along the normal to a reference surface S ( D ) (the singular or dividing surface of the text) is said to be parallel to S @ ) . (See definition of parallel surfaces following Equation 4.2.) Let the u" be an arbitrary coordinate system defined on S ( D ) and imposed on all surfaces which are parallel to S ( D ) by projecting along the normal 6. In this way the coordinates, y', of a point on S"' with the same surface coordinates as a point Y ( D ) on S ( D ) are given by
yi = Y(D)'
+
(A. 4)
f(D)i
In what follows we use this expression to obtain a relationship between the determinant, u ( D ) , of the metric tensor for the surface, S ( D ) , and the determinant a of the metric tensor for a parallel surface, s@). From Equations A. 1 and A. 4 the metric tensor for S(') is given by
(A. 13) and an expression for the determinant, b ( D ) of the second ground form associated with S ( D ) (77, p. 205) b(D)
=
(A. 14)
K(D) a(D)
Elimination of the mean and Gaussian curvatures in terms of the principal curvatures of S ( D ) ( 7 7, p. 21l),
2 H(D)
Ki
f
(A. 15)
Kz
and K(D)
=
(A. 16)
K1 K 2
results in a particularly simple expression
(A. 17) In Figure 3 denote X ( D + ) as the distance between S ( D ) and measured along the normal to S ( D ) . If we choose as a particular set of orthogonal (possibly curvilinear) coordinates X and the surface coordinates u"(a = 1, 2) [which asexplained above are defined on all surfaces parallel with respect to S ( D ) ] , an integral over R ( D + ) may be written as
S(D+)
A dR =
LC,,
A dSdX
(A. 18)
This in turn may be simplified by means of Equations A. 3 and A. 17 to give This expression may be simplified in terms of the second ground form tensor of S ( D ) by noting that (77, p. 202) (since we are working in rectangular Cartesian coordinates)
(1
-
K2
X) dX dS
(A. 19)
A similar expression may be obtained for an integral over R(D-). From Equations A. 5 and A. 6 we have a,@ =
a(Dhg
-2
b(D)ag
+ X2
C(D)U@
(A. 7 )
where the third ground form tensor of the surface S ( D ) has been introduced, 114
l&EC FUNDAMENTALS
Nomenclature
ass, au@
=
covariant and contravariant forms of surface metric tensor ( 7 7, p. 167). See Equation A. 1
portions of a material region identified in Figures 1 and 2 = portions of interfacial region identified in Figures 1 and 3 = material boundaries identified in Figures v 1 and 2 = boundaries in Figures 1 and 3 between interfacial region and bulk phases, such that bulk constitutive equations are applicable outside interfacial region and in Approach of Buff such that approximate solutions [based upon Equation 5.3 as ] with the balance equation a t S D )agree the exact solution (based upon Equation 4.13) outside the interfacial region. These boundaries are parallel to ScD)and move with the same displacement velocity =
-
vo,
= singular or dividing surface in Figures 1, 2,
arid 3 = material boundaries within interfacial region and identified in Figure 1 = surfaces defined to be normal to all surfaces parallel with respect to S ( D ) . See Figure 3 = time = pair of surface parameters or surface coordinates defined on SCD). This parametrization is imposed on all surfaces which are parallel to S ( D )by projecting along normal t = velocity vector = ( v , n ) ,( v . t ) = surface velocity tensor defined as u,xl;paap = displacement velocity of ScD) defined by -hquation . 5.6 = a coordinate system fixed in space ^
I
GREEKLETTERS = source of quantity being balanced per unit
mass = surface flux tensor of quantity being bal-
=
= = = =
ar,ced. -oa,a may be thought of as an (artificial) source per unit area of the quantity being balanced; see Simplest Approach defined by Equation 5.22 principal curvatures ( I 7, p. 21 1) of singular or dividing surface S ( D ) distance measured along normal to singular or dividing surface S(O) values of A at S ( D + ) and S(D-), respectively, in Figure 3 unit normal vector (tensor components of unit normal vector) to surface S ( D ) de, fined such that vectors (dxi/dul, dxi/bu2, ti:) have same orientation as tangent vectors to x-coordinate curves,-Le., orientation is positive ( 7 7 , page 197). I t is assumed here that t points from into RtD+)in Figures 1 and 3 and from R(-) into R(+)in Figure 2
P
= density
4449 h,@cc
= = = =
*
+a
flux of quantity being balanced (+.n),( + . E ) surface flux tensor defined as ξpaaO quantity being balanced per unit mass
SUPERSCRIPTS
( D f ) , (D-)
+, -
I n General Balance at Singular or Dividing Surface of Phase Interface, denote limiting values of superscripted variable as S ( D )is approached through N D + )and RcD-),respectively I n Approach of Buff these denote quantities in the exact solution [based upon actual constitutive equations and balances at $ ( D l of the form of Equation 4.131 for regions and R ( D - 1 , respectively = I n Simplest Approach these denote quantities in approximate solution [based upon the bulk constitutive equations and balance a t S(O) of the form of Equation 5.31 for regions R(+) and R ( - ) , respectively. I n Approach of Buff meaning is similar for regions and respectively =
OTHERS =
[ I [ [
I(+),
I
illustrated by Equation 4.9
= illustrated by Equation 5.4 I(-)
l(*)
= =
illustrated by Equations 5.6 and 5.8 used in Equation 5.22 et seq. to denote [ I(+) in R ( D + ) and [ I(-) in R(D-1
Literature Cited
(1) Aris, R., “Vectors, Tensors, and the Basic Equations of Fluid Mechanics,” Prentice-Hall, Englewood Cliffs, N. J., 1962. (2) Buff, F. P., “Handbuch der Physik,” S. Flugge, ed., Vol. 10, Springer-Verlag, Berlin, 1960. (3) Buff, F. P., J . Chem. Phys. 25, 146 (1956). (4) Buff, F. P., Saltsburg, H., Zbid., 26, 23 (1957). (5) zbid.,P. 1526* (6) Eliassen, J. D., Ph.D. thesis, University of Minnesota, 1963. (7) Eliassen, J. D., Scriven, L. E., “Dynamics of Interfacial Regions with Particular Regard for Bending,” Chemical EnEineering Monograph, University of Minnesota, - Department . JUG 1963. (8) Ericksen, J. L., “Handbuch der Physik,” Vol. 3/1, S. Flugge, ed., Springer-Verlag, Berlin, 1960. (9) Gibbs, J. W., “Collected Works of J. W. Gibbs,” Vol. 1, p. 219, Yale University Press, New Haven, 1948. (10) Hirschfelder, J. O., Curtiss, C. F., Bird, R. B., “Molecular The3ry of Gases and Liquids,” Wiley, New York, 1954. (11) McConnell, A. J., “Applications of Tensor Analysis,” Dover, New York, 1957. (12) Scriven, L. E,, Chem. Eng. Sci. 12, 98 (1960). (13) Slattery, J. C., Zbid., 19, 379 (1964). (14) Zbid., p. 453. (15) Truesdell, C., Toupin, R. A., “Handbuch der Physik,” Vol. 3/1, S. Flugge, ed., Springer-Verlag, Berlin, 1960. (16) Willmore, T. J., “Introduction to Differential Geometry,” Oxford, London, 1959. RECEIVED for review September 7 , 1965 ACCEPTED August 31, 1966
VOL. 6
NO. 1
FEBRUARY 1967
115
