Contents 1. Introduction 2. Mass balance and diæusion 3. Momentum balance and Korteweg Stresses 4. One dimensional mixing layer problems 5. Dynamic an

Transcription

1 Non-solenoidal Velocity Eæects and Korteweg Stresses in Simple Mixtures of Incompressible Liquids Daniel D. Joseph z, Adam Huang, and Howard Hu Department of Aerospace Engineering and Mechanics University of Minnesota, Minneapolis, MN Abstract We study some basic problems of æuid dynamics of two incompressible miscible liquids modeled as a simple mixture in which the volume of the mixture does not change on mixing. In general, the expansion æ = divu in these problems does not vanish. The velocity in such a mixture can be decomposed into a solenoidal and an expansion part. The expansion velocity is induced by diæusion which is proportional to the gradient of the volume fraction in a simple mixture. The expansion can be large at certain times and places. We have carried out an analysis of transient or dynamic interfacial tension for the problem of smoothing of an initial discontinuity of composition across a plane and spherical surface. The dynamic tension at the spherical interface decays as t,1=2 ; it has two terms, one term arises from the Korteweg stress and it gives rise to a stress opposing the internal pressure as in the case of equilibrium tension if the Korteweg coeæcient has the appropriate sign. The other term arises from the expansion velocity and is proportional to the rate of change of viscosity with volume fraction. This term has the wrong sign for interfacial tension in the case of glycerin and water solutions but has the right sign when the light æuid is more viscous. In the context of the new theory, we derive a new and elementary solution which describes diæusion of binary species along a pipe. An analysis of this solution to small disturbances leads to a nonseparable problem which is solved by a ænite element method. The numerical study indicates stability under all circumstances. z At a time 30 years ago when I was trying to prove theorems Fritz showed me by example how applied math ought to apply. He had an enormous impact on my development as an empirical scientist. I admire him greatly and treasure our years of friendship. 1

2 Contents 1. Introduction 2. Mass balance and diæusion 3. Momentum balance and Korteweg Stresses 4. One dimensional mixing layer problems 5. Dynamic and instantaneous interfacial tension 6. Jump of the normal stress across a plane mixing layer 7. Spreading of a spherical diæusion front and Korteweg stresses 8. The eæects of convection and diæusion 9. Two- and three-dimensional problems 10. Basic solution for diæusion in a pipe 11. Stability of the basic solution for diæusion in a pipe 12. Conclusions and discussion Acknowledgment References 2

3 1 Introduction In this paper we study some basic problems of æuid dynamics of two incompressible miscible liquids modeled as a simple mixture in which the density of the mixture is connected to the densities of the two constituents by a linear relation in the volume fraction. Since the density of such a mixture of incompressible liquids is changed by diæusion, the expansion æ = divu does not vanish in general. The velocity in such a mixture can be decomposed into a solenoidal and an expansion part. The expansion velocity is induced by diæusion which is proportional to the gradient of the volume fraction in a simple mixture. In all the previous studies of incompressible mixing liquids known to us èmiscible displacements, binary convection, Taylor dispersion, reaction and diæusion, transport of diæusing dyes, Marangoni convection, diæusion controlled solidiæcation, etcè, it is universally and incorrectly assumed that divu =0. One aim of our study is to ænd the situations in which divu =0 is a good approximation. In fact divu = 0 is exact when the diæerence of density of the two constituents is zero and when there is no diæusion. It is a good approximation in some situations when the density diæerence is small, as in a Boussinesq approximation and when the diæusion is slow. However, even in the Boussinesq or slow diæusion approximation there are problems in which divu is large at certain times and places. We ænd that the expansion velocity isalways important where the gradients of the volume fraction are suæciently great, most especially at early times in the mixing layer arising from the smoothing of a initial plane or spherical discontinuity of composition. Another case and place where divu = 0 is a bad approximation are in driven problems of mixing layer where gradients of composition èor volume fractionè can be maintained in a competition between convection 3

4 and diæusion. Dynamical eæects can arise in thin mixing layers where the gradients of composition are large. This possibility was already recognized in discussions given by Korteweg ë1901ë in which he proposes a constitutive equation which includes the stresses induced by gradients of composition which could give rise to eæects which mimic surface tension in regions where the gradients are large. The small but interesting history of thought about ersatz interfacial tension in diæusing liquids is given in a paper by Joseph ë1990ë. The presence of sharp interface in slow diæusion in rising bubbles of water in glycerin is reported there. The shape of such interfaces resemble familiar shapes which can be seen in immiscible liquids. A similar parallel description of drops of miscible and immiscible liquids occurs in the evolution of the falling drops into a vortex ring èbaumann, Joseph, Mohr, Renardy ë1991ëè. Of particular interest is a membrane which spans the ring and must rupture before a free ring is formed. Such a membrane on a 9è10 glycerin-water ring falling in a 3è2 glycerin-water solution appears in panel e of Fig 1 in the paper by Arecchi, Buah-Bassuah, Francini, Perez-Garcia and Quercioli ë1989ë. It is hard to explain this membrane without acknowledging some type of interfacial tension. Atypical formulation for ænding the motion and shapes of miscible drops like the one used by Kojima, Hinch and Acrivos ë1984ë which uses divu = 0 in each æuid and classical interface conditions misses out on slow diæusion on the one hand and gradient stresses on the other. More recently it has been suggested by Barkey and Laporte ë1990ë that morphological instabilities observed in electrochemical deposition could have their origins in the æelds and interfacial dynamics that drive growth with diæusion controlled structure observed on a scale of microns, corresponding 4

5 to the mass-transfer boundary layer thickness. In another recent study Garik, Hetrick, Orr, Barkey and Ben-Jacob ë1991ë :::reported on the stability of the interface between to miscible æuids of closely matching viscosities when one is driven into the other. For the case where the æuids diæer only in solute concentration, we ænd that spontaneous cellular convective mixing can develop. We suggest that this interfacial patterning is a surface tension eæect distinct from viscous ængering; the latter can occur simultaneously ::: On the basis of the above experimental results, we hypothesize that the global morphology of depositional growth, i.e., the number of branches, the stability ofthe branch tips, and the way it ælls space èits ëdimension"è is determined by the hydrodynamic stability of the interface between the depleted æuid near the growth and the bulk æuid provided the gradient is suæciently sharp to provide an eæective liquid-liquid interface. Since leading edges grow fastest, hydrodynamic modulation of the liquid-liquid interface ça la Hele-Shaw would determine branch position, just as cellular mixing will. In electrodeposition the existence of a sharp gradient sustained by the growing deposit is experimentally supported. Wehave carried out an analysis of transient or dynamic interfacial tension for the problem of smoothing of an initial discontinuity of composition across a plane and spherical surface. The idea is to evaluate the jump in the normal stress across the mixing layer which in this problem reduces to a jump in the mean normal stress èthe pressureè. We ænd no such jump across a plane layer but there is a jump proportional to the curvature across the spherical surface. The dynamic tension at the spherical interface is proportional to q D=t where D = Oè10,6 cm 2 =secè and t is the time. There are two terms in the expression è7.10è for the interfacial tension; one term arises from the Korteweg stress and it gives rise to a stress opposing the internal pressure as in the case of equilibrium pressure if the Korteweg coeæcient has the appropriate sign. A second term arises from the expansion velocity and is proportional to the rate of change of viscosity with volume fraction. This term has the wrong sign for interfacial tension in the case of glycerin and 5

6 water solutions but has the right sign when the light æuid is more viscous. 2 Mass balance and diæusion Recently Joseph ë1990ë has reconsidered the equations of æuid dynamics of two incompressible miscible liquids with gradient stresses. The density of incompressible æuids can vary with concentration ç and temperature, but not with pressure. The velocity æeld u of such incompressible æuids is not in general solenoidal, divu 6= 0. A conservation form of the left-hand side of the diæusion equation which diæers from the usual substantial derivative of ç by the term ç divu, is implied by requiring that the mass per unit total volume of one of the liquids in a material is conserved in the absence of diæusion. Suppose that æ is the density of one liquid per unit total material volume V, æ = m æ =V where m æ is the mass of æ. Then d dt Z V ètè ædv =, q æ æ nds; è2.1è says that the mass of æ in V can change only by diæusion across the boundary of V. In the usual way we ænd that dæ dt + æ divu =,divq æ è2.2è where q æ is the æux of æ. Of course, the substantial time derivative of the density çèæè is not zero when dç dt = ç0 èæè dæ dt è2.3è 6

7 dæ dt +èuærèæ 6= è2.4è that is, in the usual case. Hence, the continuity equation gives divu =, 1 dç ç dt 6=0: è2.5è In general æ = ç æ ç where ç æ = m æ =V æ is the density of the æuid æ and ç = Væ=V is the volume fraction. Under isothermal conditions ç æ is a constant and we may work with çèçè and ç satisæes è2.2è. Suppose ç is the density of the other liquid per total unit volume. Then ç = ç + æ and dç dt + ç divu =,divq ç: The continuity equation may be written as dç dt + dæ +èç + æè divu =0: dt è2.7è Hence, using è2.2è and è2.6è in è2.7è, we ænd that è2.6è divèq ç + q æ è=0: è2.8è The sum of the æuxes of the mass of each constituent across the boundary of any material volume V must vanish èq ç + q æ è æ nds =0: è2.9è to conserve the total mass. If the volume V of a mixture of two liquids does not change on mixing, then V = V æ + V ç and the density can be expressed in terms of the volume fraction ç = V æ =V of one of the constituents by the form 7

8 çèçè =ç æ ç + ç ç è1, çè è2.10è where ç æ and ç ç are the densities of æ and ç, handbook values. Mixtures satisfying è2.10è will be called simple mixtures. Equation è2.10è is correct to within 1è for glycerin and water mixtures èsee Joseph ë1990ëè. The volume fraction is the natural variable connecting density and diæusion in simple mixtures. Since æ = ç æ ç and ç = ç ç è1, çè and conserved in the absence of diæusion, it is natural to express the constitutive equation for the æuxes q æ and q ç as a nonlinear Ficks' law for each constituent in terms of the volume fraction of one of them q æ =,D æ èçèrèç æ çè; q ç =,D ç èçèr ëç ç è1, çèë è2.11è with diæerent diæusion functions and assume that ç æ and ç ç are constants, as in the isothermal case, then Z n æ èq ç + q æ Z èd ç ç ç, D æ ç æ èn ærçds =0 in each and every material volume V, so that the ratio of diæusion functions D ç D æ = ç æ ç ç è2.12è is a constant. Since the density of a simple mixture of incompressible liquids changes by virtue of diæusion of the volume fraction, the velocity æeld cannot be solenoidal ècf. è2.5èè. However, Galdi, Joseph, Preziosi and Rionero ë1991ë have shown that if è2.10è holds, then è2.2è and è2.5è imply that divw =0; è2.13è 8

9 where From è2.8è, we also get w = u, èç æ, ç ç è ç æ ç ç q æ : è2.14è div bw =0; where bw = u, èç æ, ç ç è ç æ ç ç q ç : In this case we may introduce a streamfunction. Landau and Lifshitz ë1959ë have considered diæusion without explicitly taking up the case of incompressible liquids. They write what might at ærst glance be thought to be the usual diæusion equation ètheir è57.3èè where ç dc dt =,divi c = m æ =m = æ=ç è2.15è è2.16è is the mass fraction, m is the total mass and i is said to be the diæusion æux density, which we shall specify presently, according to our understanding. Substituting è2.16è into è2.15è using è2.5è we get ç dæ=ç dt = dæ dt + æ divu: è2.17è This shows that è2.15è is perfectly consistent with mass conservation argument è2.1è provided that divi = divq æ è2.18è is the divergence of the æux of æ, say the æux of solute. Landau and Lifshitz develop a coupled thermodynamic theory for i and the heat æux under the condition that the concentration gradients are small 9

10 èwhich is not the main case of interest hereè. When temperature and pressure gradients vanish, they ænd that i =,æ grad^ç =,æ ^ç gradc =,çd p;t è2.19è where ^ç is the chemical potential and D is the diæusion coeæcient. simple mixtures c = æ=ç = ç æ ç=çèçè and For ç gradc = ç æç ç çèçè gradç è2.20è After combining è2.17è and è2.18è with è2.15è, with constant ç æ and ç ç, we ænd that + divèçuè = div ëd æèçèrçë D æ èçè = ç ç Dèçè ç æ ç + ç ç è1, çè : è2.21è è2.22è Chemical engineers learn how to deal with diæusion from Bird, Stewart and Lightfoot ë1958ë who use a mixture theory and based on a number averaged velocity u i of species i, a mass averaged velocity u = ç i NX i=1 ç i u i =ç where = N i m i =V is the density of the species i, m i is the the mass of one of them, N i is the number of them in V and ç = NX i=1 ç i. Diæusion equations are deæned from mass conservation consideration by introducing the relative æux j i def = ç ièu i, uè; Then, the mass balance equations NX i=1 j i =0: + divçu =0; + divç iu i =0; è2.25è 10

11 may be combined into diæusion equations + divç iu =,divj i è2.26è Mixture theories assume interpenetrating continua where molecules of diæerent species, say glycerin and water, occupy the same point but move with diæerent velocities. Short range forces which glue liquids of diæerent type together and force cooperative motions are ignored Bird, et al. do not explicitly consider simple mixture of incompressible liquids where the volume fraction is the nature variable of composition. However, John Brady has shown us how this kind of theory can be extracted from the mixture theory of Bird, et al. The volume fraction ç of the ith species is given by ç i = çv i =M i ; NX i=1 ç i =1 è2.27è where M i is the mass per mole and V i is the volume of the ith species in one mole. In binary mixtures N =2,ç 1 = ç, ç 2 =1, ç and ç = ç 1 + ç 2 = M 1 V 1 ç + M 2 V 2 è1, çè è2.28è which is always true whether or not there is a volume change of mixing. When there is no volume change V 1 and V 2 are constants and wemay compare è2.10è and è2.28è to show that èæ;ç;ç æ ;ç ç è = èç 1 ;ç 2 ;M 1 =V 1 ;M 2 =V 2 è. when V 1 is constant, we may write Moreover, ç æ + divçu! =,divj 1 : è2.29è The relative æux can be identiæed with q æ when q æ is expressed as in è2.11è.the relation divèq æ + q ç è = 0 which holds in general is satisæed identically for simple mixture satisfying è2.11è and è2.12è because in this case q æ + q ç = j 1 + j 2 = 0. Bird, et al. model diæusion, their j i, in the same manner as 11

12 Landau and Lifshitz, j 1 replaces i in è2.19è, so that all authors will arrive at the same results when the consequences of incompressibility are modeled by simple mixtures è2.10è. We have investigated the consequences for mass balance and diæusion of the assumption that the volumes of incompressible constituent do not change on mixing. This is expressed by è2.10è which shows that the density of such a mixture may change by dilution and that the volume fraction is the material variable connecting mass and diæusion. It follows from our theory that the velocity u is not solenoidal but that a linear combination w èor bwè of the velocity and a species æux are solenoidal. These eæects could also be obtained from equations contained in the theories presented by Landau and Lifshitz ë1959ë and by Bird, et al. ë1958ë, but they seem not to have been explored. Many pairs of liquids will give rise to small volume changes upon mixing. These liquids are only approximate simple mixtures. It is probable that nearly all the interesting cases which are not already well described by the theory of perfect incompressible mixtures could be treated as a to-bedeveloped perturbation of the perfect case. In areas of applications, problems of mixing liquids èmiscible displacements, binary convection, Taylor dispersion, reaction and diæusion, transport of diæusing ëpassive" scalars like dyes, Marangoni convection, solidiæcation problems, etcè, it is universally and incorrectly assumed that divu = 0. Presumably the practitioners of these arts know what they are doing and recognize that they are making an approximation, like the Boussinesq approximation. In fact, though there are surely many situations in which the assumption that divu = 0 is a good one, there are others in which divu = ç æ, ç ç ç ç ç æ divq æ = ç ç, ç æ ç ç div ëd æ èçèrçë è2.30è is large when rç is large, as is true when mixing liquids are placed into 12

13 sudden contact. It is clear already from è2.30è that if gradients are moderate divu will be small if the prefactor or the diæusion coeæcient D æ èçè is small. For glycerin ç ç =1:26g=cm 3 and water ç æ =1g=cm 3, the prefactor 0.26è1.26 is not negligible, but the diæusion coeæcient D æ = Oè10,6 cm 2 =secè is. It follows then that the assumption that divu = 0 is a slow diæusion rather that Boussinesq approximation. 3 Momentum balance and Korteweg stresses To extract the consequences of the balance of momentum it is desirable to frame the theory in term of a material particle. A natural method for this is to apply balance laws to a material volume which in the continuous limit is a particle of æuid mass. The same perception is behind the use of a mass averaged velocity in mixture theories. In both cases we defer to the statement that the laws of dynamics are framed relative to the velocity u of a volume of æxed mass. Hence it is the u in w which will enter into the balance of momentum. The possibility that stresses are induced by gradients of concentration and density in diæusing incompressible miscible liquids, as in the theory of Korteweg ë1901ë, can be considered. Such stresses could be important in regions of high gradients giving rise to eæects which can mimic surface tension. We have already seen, in è2.30è, that it is just the same region of high gradients where the volume changes due to dilution cause the strongest departures from the classical approximation divu =0. We are going to study the superposition of non-classical eæects of volume changes divu 6= 0due to diæusion and Korteweg stresses. In the isothermal case, ç varies with ç alone, as in è2.10è and in the notation of Joseph ë1990ë the Korteweg's expression for the stress due to the 13

14 combined eæects of gradients of ç and çèçè are T è2è ij = ^æ ç ; i j i j è3.1è where The governing equations are ^æ =èç æ, ç ç è 2 æ 1 + æ 2 +2çèç æ, ç ç è ; ^æ = æ 1 èç æ, ç ç è 2 + æ 2 : and èç æ, ç ç è dç dt + ç divu =0; è3.2è dç dt + ç divu = rèd ærçè ç du dt =,ræ + divtd + çg; where T D is the stress deviator deæned by T D ij = 2çD ij, 2 3 æ ijç divu + ç ij ; ç ij = ^æ ç, 1 o i j i j 3 æ ij n^æjrçj2 +^ær 2 ç è3.3è è3.4è è3.5è and æ is the mean normal stress. where The continuity equation è3.2è may be replaced with divw =0 and w = u, çd æ rç è3.6è ç = èç ç = ç æ è ç ç : è3.7è We should be thinking of glycerin ç and water æ, then çé0. Then, using è3.6è, we may eliminate divu from è3.5è T D ij =2çD ij, 2 3 æ ijççdiv èd æ rçè+ç ij è3.8è 14

15 and rewrite è3.4è as where ç du dt =,r ëæ + Qèçèë + div n 2çD ëuë+^ærç ærç o + çg; Qèçè = 1 3 ^æjrçj ççdiv èd ærçè, 2 3 ^ær2 ç: è3.9è è3.10è In writing è3.9è we have assumed that ^æ is constant. It will be convenient now to also assume that ^æ is a constant. We now adopt è3.3è, è3.6è and è3.9è as our system of equations governing the evolution of simple mixtures of incompressible liquids. These are æve equations for the components of u, æ and ç. In this preliminary study we shall restrict our attention to some one-dimensional problems for which there is a strong decoupling of equations, but some basic issues can be addressed. To keep our discussion of basic issues concrete we will use estimates of material parameters for Glycerin-Water systems. One reference for this is the article by Segur ë1953ë. In Figure 1 we have reproduced Segur's experimental data ë1953ë for the viscosity ç. This can be excellently correlated by the expression ç = ç G exp ç æç + æç 2 + æç 3 ç ; è3.11è where the coeæcients æ, æ and æ depend on temperature T in the way shown in the ægure, ç G is the viscosity of pure glycerin and ç G = 14:99 poise at 20 æ C. The density of glycerin ègè and water èw è mixtures is given to within 1è by è2.10è with èç æ ;ç ç è = èç W ;ç G è ç è1; 1:26èg=cm 3 at 20 æ C. Unfortunately we do not have the global dependence of the diæusion coeæcient D æ èçè. Small gradient theories of diæusion are inadequate for mixing layers in which ç takes on all allowed values from zero to one. A representative value Dèçè over diæerent concentration, taken from Segur èp. 328è is D =5æ10,6 cm 2 =sec. We will use this representative value in our estimates. 15

16 ln( µ ) (centipoise) T=0 ( α= 19.8, β=17.6, γ= 6.66) T=20 ( α= 16.2, β=14.9, γ= 5.92) T=40 ( α= 13.3, β=12.2, γ= 4.99) T=60 ( α= 10.8, β=9.47, γ= 3.83) T=80 ( α= 9.14, β=7.93, γ= 3.26) T=100 ( α= 7.78, β=6.63, γ= 2.79) viscosity of glycerol solutions volume concentration of water in glycerol solutionsφ 1.0 Figure 1: èafter Segur ë1953ëè Viscosity of glycerol solutions at temperature 0, 100 æ C. The expression ç = ç G expèæç + æç 2 + æç 3 è æts the experimental data. 16

17 4 One dimensional mixing layer problems We shall suppose that u = uèx; tèe x, where x increases upward against gravity. In this case Hence where 0 = divw " è u, çd æ : u = Aètè+u e èx; tè u e def çd = æ is the expansion velocity which arises from mixing. è4.1è è4.2è In theories in which divu = 0 is assumed, u e = 0 and, of course, u e = 0 when the æuids are density matched. Using è4.1è and è4.2è we ænd that " + Aètè+çD æ or =è1, è! D æ + Aètè " è è1, ççèd æ : è4.3è The momentum equation in one dimension is given by " è + u@u é : 4 3 ç@u + 2 è! ^æ ç= 2 ; + çg æ e x è4.4è where u is given in terms of Aètè and ç by è4.1è. We need æ to satisfy è4.4è when, say, uèx; tè and çèx; tè are prescribed at the boundary. The problem of diæusion is decoupled from è4.4è when Aètè = 0. And Aètè = 0 if there is a value x such that for all t, u and the diæusion æux = =0. This is the case at an impermeable wall across which the velocity and the æux of water must vanish. It is also true for mixing problems on 17

18 unbounded domains for which u and = vanish at x = æ1. These problems are canonical for the development of mixing layers from initially discontinuous data which are considered below. When Aètè = 0, u = u e = çd æ è4.5è and è4.3è reduces " è è1, ççèd æ : If we switch to use the classical diæusion coeæcient Dèçè given by è2.22è, the above diæusion equation then " Dèçè è : è4.6è In the simplest case, we assume that D is independent of ç, è4.6è is the classical diæusion equation. With appropriate boundary conditions è4.6è can be solved easily. Then u is given by è4.5è, without any considerations from dynamics and the momentum equation è4.4è determine æèx; tè directly. Various issues which arise in the dynamical theory of simple mixtures can be framed in terms of the one-dimensional problems considered below. 5 Dynamic and instantaneous interfacial tension H. Freundlich in his ë1926ë treatise on colloid and capillary chemistry in discussing the methods of measuring interfacial tension between immiscible liquids and the theory of the phenomenon, notes that :::there is little new to be said :::We have only to remember here we are in the end always dealing with solutions. For the 18

19 one liquid will always be soluble in the other to some degree, however small. Hence the dynamic tension of liquids, when ærst brought into contact, is to be distinguished from the static tension, when the two liquids are mutually saturated. Not only do liquids which are not miscible in all proportions have amutual surface tension; even two completely miscible liquids, before they have united to form one phase, exhibit a dynamic interfacial tension. For we get by careful overlaying of any two liquids a deænite meniscus, a jet of one liquid may be generated in another, and so on. The tension decreases rapidly during the process of solution, and becomes zero as soon as the two liquids have mixed completely. Freundlich ë1926ë cites the measurements of the dynamic tension by Quinke ë1902ë of ethyl alcohol in contact with aqueous salt solutions èsulfates of zinc, copper, etc.è. These two liquids are miscible in all proportions. Quinke used the method of drop weight to make his measurements. In these liquids the drop, as it emerges, does not pass into streaks, but keeps at ærst its shape. He found values between 0.8 and 3 dyne=cm. Smith, Van den Ven and Mason ë1981ë have reported a maximum value of 1 dyne=cm for the force corresponding to a ëtransient interfacial tension" between a 2000 cs and a 1 cs silicone oil. According to the authors, these are two mutually soluble liquids whose interdiæusion is suæciently slow to enable this measurement to be made. They note that In principle there exists between any two separated æuid phases which have a chemical potential diæerence, an instantaneous interfacial tension which may or may not persist with time. We are unaware of reports in the literature of measurements of interfacial tension between two miscible liquids. It is clear that in the case of two liquids miscible in all proportions we are not dealing with an equilibrium situation; there is no equilibrium tension. Rather, we are looking at stress eæects due to diæerences in density and composition and possibly even temperature which inæuence the positions occupied by interdiæusing æuids. One could imagine that when the gradients 19

20 of composition are large, as in the boundary layer between two regions of diæerent composition suddenly put into contact, that these stresses give rise to an eæect which might be called ëtransient interfacial tension." Smith, Van den Ven and Mason ë1981ë present an expression for the chemical potential based on expressions for the free energy in a nonuniform system given by van der Waals ë1893ë and Cahn and Hillard ë1954ë writing Zx 0 è! 2 S æ è dx è5.1è,x 0 where S æ is the interfacial tension, ç is the local composition èthe mole fraction of component 1è and x 0 is the ëinterfacial region." The composition is assumed to satisfy a diæusion equation ç t = Dç xx with diæusion constant D. If at t = 0 +, ç = ç + for x é 0 and ç, for x é 0 and thereafter ç is continuous at x = 0, then çèx; tè =ëç +, ç, ë fèçè; fèçè = erfcèçè; ç = x=2 p Dt and S æ is proportional to ëç +, ç, ë 2 Z x 0,x 0 1 ç expè,2ç2 è è! 2 dx = ëç +, ç, ë 2 2ç p Dt Z ç 0,ç 0 expè,2ç 2 èdç At small times the breadth of the diæusion layer scales with p Dt. Then the gradient theory leads to a square root singularity for the dynamic tension. Their experiments indicate that F = S æ cosç decays exponentially and does not follow the t,1=2 decay that would be required if ç were constant, where ç is the contact angle. It is noteworthy that though the rate of decay of F with time varies between 0.6 to 1.4, the extrapolated value of F to zero time does not vary and leads reproducibly to a force of 1 dyne=cm. They conclude that ë... present experiments do indeed conærm that an instantaneous interfacial tension exists between mutually miscible liquids." 20

21 H. Ted Davis ë1988ë has written an interesting paper, ëa theory of tension at a miscible displacement front," in which he supplies a constant of proportionality for the expression è5.1è, which he develops independently starting from the Irving-Kirkwood pressure tensor and some simplifying assumptions. He then uses some estimate of constants in his theory to construct a table of values of S æ èæ in his notationè, given in his table 3.1, varying from about 6:3 æ 10,2 dyne=cm for t = 1s and D = 10,9 cm 2 =s to 10,5 dyne=cm for t = 4000s and D =10,5 cm 2 =s. He notes that ëfrom the entries in this table it follows that the tension of a diæusive mixing zone between miscible æuids, while small, is nevertheless not zero." The theory used by Smith, et al. ë1981ë and by Davis ë1988ë evidently requires that one assume wrongly that the density of a mixture of incompressible æuids is constant. Davis restricts his analysis to a two-component regular solution in which the densities of the components 1 and 2 are n 1 = ~ çn and n 2 =è1, ~ çèn. ~ ç is the mole fraction of component 1 and n is the total density, which he says is constant in a regular solution. We shall reinterpret the Davis work for simple mixtures by replacing the mole fraction ~ ç with the mass fraction ^ç = m æ =m of an incompressible liquid èsay, waterè in a mixture èsay, water and glycerinè of total m = m æ + m ç where m ç is the mass fraction of glycerin. Then èn; n 1 ;n 2 è should be replaced by èm=v; m æ =V æ ;m ç =V ç è=èç; ç æ ;ç ç è, where V is the total material volume and ç æ and ç ç are the ordinary èconstantè densities èof water and glycerinè listed in the handbooks. Moreover, if our regular solution keeps its volume after mixing, then n = ç ~ ç + çè1, ~ çè= m V M æ m + m V ç 1, m æ m ç = æ + ç = ç æ ç + ç ç è1, çè =çèçè and the regular solution of Davis is a simple mixture. Obviously, a mixture of incompressible liquids does not have a constant density even though the 21

22 density of each of its constituents is constant at a æxed temperature. Davis ë1988ë expresses well the notion that gradients of composition can lead to anisotropic forces which mimic the eæects of interfacial tension: When two miscible æuids are placed in contact they will immediately begin o mix diæusively èand convectively if their densities are such as to drive convectionè across the concentration front formed at the zone of contact. Although no interface will form at the concentration front, the composition inhomogeneities can give rise to pressure anisotropies and therefore to tension at the mixing zone between the contacted æuids. Diæusive mixing will continuously broaden the mixing zone and reduce the pressure anisotropy and the associated tension. The purpose of this short paper is to examine with the aid of a molecular theory of inhomogeneous æuid the magnitude and rate of reduction of the tension by diæusive mixing of the zone of contact of miscible æuids. The results found here suggest that instabilities in miscible frontal displacement may be similar to those in ultralow tension immiscible frontal displacement, with the added caveat that in the miscible process the tension decreases continuously in time. The type of calculation of dynamic tension given above, as well as the calculation to be carried out in section 6 given rise to a pressure diæerence across a spreading plane layer. This is not a good analogy to interfacial tension which gives rise to a pressure diæerence proportional to curvature and vanishes across plane layers. The calculation of forces over a spherical layer advanced in section 7 does contain curvature terms, but the analogy is not far reaching, even in the spherically symmetric case. 6 Jump of the normal stress across a plane mixing layer We shall now examine the problem considered in section 5 without assuming that density is constant and using the one dimensional problem deæned by è4.4è, è4.5è and è4.6è. This is the canonical initial value problem for mixing 22

23 layers, the smoothing-out of a discontinuity inç at a plane. At t =0,water lies above glycerin ç =1for xé0; ç =0for xé0 è6.1è where ç is the water fraction. We are on an inænite domain and ç =1for x!1; ç =0for x!,1 è6.2è for all té0. In this situation è4.5è holds and the velocity is proportional to the volume fraction gradient which is inænite at t =0 +. For simplicity we take the diæusion coeæcient D to be independent of ç and for glycerin-water mixture D is of order 10,6 cm 2 =sec. Then classical diæusion equation è4.6è has a similarity solution with ç = p 1 ç Zç,1 Using è6.3è we may express è4.5è as e,ç2 dç è6.3è x ç = q 2 èdtè : è6.4è q u = u e = ç D 2è1,ççè = ç 2 p çè1,ççè q D t ç ç t ç0 px 2 Dt exp ç, ç x 2 p Dt ç 2 ç : è6.5è The diæusion layer can be deæned from the place,x o where è = ç,1=2 =,0:495 to the place x o where è =0:495, or by,m éçémwith m about 2. The thickness of the diæusion layer is æx = x 0, è,x 0 è=4m 23 q èdtè è6.6è

24 and it tends to 1 with t. Equation è6.5è shows that the expansion velocity which enters into dynamics can be considerable at early times inside the diæusion layer. The gradient of ç is the machine which drives the velocity. The velocity decays as q D=t. It is of interest to calculate the jump of the stress across the mixing layer. To ænd the jump in the stress we integrate è4.4è over the diæusion layer. Outside of this layer the derivatives of ç vanish and Zx + u@u dx =, ëæë, g,x 0 çèçè Zx 0,x 0 çèçèdx è6.7è where ëæë =æèx 0 ;tè, æè,x 0 ;tè: Equation è6.7è shows that the Korteweg stresses do not enter into the stress jump across the plane mixing layer. This is unlike the calculations of section 5, but like true interfacial tension in which curvature supports a jump in stress. Using the continuity equation it can be easily seen that the contribution due to inertia is always zero Zx 0 Zx 0 + u@u u dx dx = Dç ç,x 0,x 0 Zx è! dx =0; which is also true for general diæusion coeæcient Dèçè. Therefore ëæë =,g Zx 0 è6.8è,x 0 çèçèdx: è6.9è The jump in normal stress is simply the static pressure diæerence across the mixing layer. 24

25 7 Spreading of a spherical diæusion front and Korteweg stresses The problem of the spreading of a spherical front with gravity neglected is good for bringing out how Korteweg stresses may enter the normal stress balance when the curvature is not zero. In fact this kind of calculation was carried out for an equilibrium phase change cavitation bubble in the absence of diæusion or motion by Korteweg ë1901ë. A critical discussion of the Kortewegs equilibrium calculation can be found in Joseph ë1990ë. At t = 0 a spherical mass of radius r 0 of one liquid is inserted into an inænite reservoir of a second liquid. The two liquids are miscible in all proportions. We can imagine a sphere of glycerin in a reservoir of water. The governing equations are è3.3è, è3.6è and è3.9è written for spherically symmetric solutions with one radial component of velocity uèr;tè, which vanishes at r = 0 and r = 1. Under these conditions divw = 0 implies that èr;tè uèr;tè=u e èr;tè=çd æ : The diæusion equation è3.3è may then be written = è! è7.2è where the water fraction çèr;tè = 1 when rér o, t = 0 and çèr;tè = 0 when r é r 0, t=0. When the thickness of the mixing layer at r = r 0 is small, it is locally like a plane and the 2nd term on the right side of è7.2è may be neglected. This reduces our problem to the one considered in section 6 centered on r = r 0 and it has the same self similar solution with when 2ç p Dt ç r 0. ç = r, r 0 2 p Dt ; è7.3è 25

26 The momentum equation è3.9è may be written as " + @r ëæ u r ç^æ ç + ç 2 ç + 2^æ r 2 è7.4è where çèçè is given by è2.10è and çèçè by è3.11è. After integrating over the mixing layer from r 1 = r 0, 2m p Dt to r 2 = r 0 +2m p Dt, we ænd that Z r 2 r 1 ç ç çèçè @r, r ç ç 2 ç dr ç,æ, + ^æ ç ç 2 r 1 è7.5è Outside the mixing layer èr 1 ètè;r 2 ètèè, ç is essentially constant and u is essentially zero. The contribution due to the inertia at the left-hand side of è7.5è is again found to be zero as in the case of the plane layer. After writing Z r2 r 1 dr = u ç r 2 r Z r2, r 1 r 1 ç 0 èçè u dr and putting terms outside the mixing layer to zero we get Z r2 r 1 8 é : 4ç0 èçè u 2^æ r è! 2 9 ; dr =,ëæër2 r 1 : è7.6è è7.7è Now we evaluate è7.7è at very early times, when the mixing layer is very thin, r 2, r 1 = 4m p Dt and r ç r 0. Using the same approximations with r ç r 0 in the two terms of the integral of è7.7è, we ænd that 2 Z r2 r 0 r 1 8 é : 2ç0 ^æ è! 2 9 ; dr = 2 r 0 26 Z r2 r 1 è èè! 2 2 ç0 èçèçd 1, çç, ^æ dr

27 Finally, 1 r 0 s D t Z m,m æèr 2 ;tè, æèr 1 ;tè= 2 r 0 s D t è 2 ç0 èçèç 1, çç, ^æ è ç 02 dç: D Z m,m è è ç 0 èçèç ^æ, 0:5 ç 02 dç: 1, çç D è7.8è è7.9è For glycerin and water solutions at 20 æ C we may evaluate è7.9è using values for ç 0 and ç near to è3.11è as æèr 1 ;tè, æèr 2 ;tè= 2 r 0 s D t " 164:5,^æ D, 428:7 è è7.10è with D about 7:5 æ 10,6 cm 2 =sec, but we do not have any knowledge about the value of Korteweg stress coeæcient ^æ. Equation è7.10è reminds one of interfacial tension with a time dependent tension T ètè whose values are given by comparing the right-hand side of è7.10è with 2T ètè=r 0. There are two terms in the expression for the dynamic interfacial tension; one term arises from the Korteweg stress and it gives rise to a stress opposing the internal pressure as in the case of equilibrium pressure if the Korteweg coeæcient ^æ has a negative sign. A second term arises from the expansion velocity and is proportional to the rate of change of viscosity with volume fraction. This term has the wrong sign for interfacial tension in the case of glycerin and water solutions but has the right sign when the light æuid is more viscous. 8 The eæects of convection on diæusion In section 5 and 6 we studied problems in which velocity and stresses are induced by gradients of the volume fraction in simple mixtures. In these problems the gradients of ç are the engine which drives motion and the 27

28 motion is important only at early times. In other cases the motion is driven externally and the distribution of ç is driven by a balance of diæusion and convection. In this case, as in the other, the assumption that divu = 0 can lead to large errors. The eæects of expansion due to mixing on the distribution of velocity can be assumed by elementary analysis of steady æow. In the steady case, equations è4.1è to è4.4è reduce to u = A + u e ; è8.1è u e = çèçè du 2 2 dx =, d d ëæ + Qèçèë + dx dx From è8.3è we ænd that çd dç 1, ççdx ; A dç dx = D d2 ç dx ; ç du dx + ^æ è8.2è è8.3è è! 2 3 dç 5 + çèçèg æ ex : è8.4è dx ç = C 1 + C 2 expèax=dè è8.5è and from è8.2è and è8.1è we get and where C 1, C 2 u e = çc ç 2 Ax 1çç A exp D " u = A 1+ çc ç 2 Ax çè 1, çç exp D ç è8.6è è8.7è and A are to be determined from the boundary conditions. The variation of eæective pressure æèxè is determined by C 1, C 2 and A, after putting è8.5è and è8.6è into è8.4è. If, for example, we set Dirichlet conditions by prescribing çè0è = ç 0 and çèlè =ç L, then çè0è = ç 0 +èç L, ç 0 è expèax=dè, 1 expèal=dè, 1 : è8.8è 28

29 The distribution of ç between 0 and L depends on the balance between diæusion and convection. We may deæne a diæusion length ` = D=A; è8.9è or an eæective diæusion parameter S = D molecular diæusion diæusion velocity = = LA convective diæusion convective velocity : Then the distribution of concentration is ç = ç 0 +èç L, ç 0 è expè ç x è, 1 SLç ; 1 exp S, 1 and the expansion velocity in è8.2è can be evaluated u e èxè = exp ç 1 S ç, 1, ç h ç0 èexp ç 1 S Açèç L, ç 0 è exp ç x ç SL ç ç, x 1è+èçL, ç 0 èèexp SL è8.10è è8.11è ç, 1è i: è8.12è If S is very small, then ç = ç 0 for most values of x, with a narrow mixing layer of thickness of order of ` near x = L. And the expansion velocity u e, neglected in analysis which assume divu = 0, will be important inside this layer. We may estimate that u e è0è ç 0; u e èlè ç A çèç L, ç 0 è è8.13è 1, çç L when S is small. The eæects of the expansion velocity are conæned to the narrow mixing layer and the assumption that divu = 0 is valid outside the mixing layer. Equation è8.13è also indicates that the expansion velocity inside the mixing layer is of the same order as the constant convection velocity A if the density ratio ç is not too small. If S is not small, and this is a realizable possibility in many situations, then the expansion velocity will not be small and will not be conæned to boundary layer. The velocity A can be determined from considerations involving momentum in the Hele Shaw example to be considered in the next section. 29

30 9 Two- and three-dimensional problems In this section we will formulate initial-boundary value problems for simple incompressible binary mixtures which are miscible in all proportions. We will leave away the Korteweg stresses and emphasize non-classical eæects arising from the observation that divu 6= 0 in general for such mixtures. We have found that u e = u = w + u e ; è9.1è çd rç = rhèçè; è9.2è 1, çç curlu = curlw; è9.3è divu = divu e = r 2 h: è9.4è The governing equations expressing the balance of mass, diæusion of species and momentum + u æru =,r divw = divèu, u e è=0; + ræçw = divèd gradçè è9.6è çp + 2 ç 3 ç divu è9.5è + div h çèru + ru T è i è9.7è where ç = 1, çç and ç = çèçè depends on ç; usually çèçè is a rapidly varying function. We should think of ç as the water fraction of the glycerinwater mixture. The diæusion è9.6è show that the water fraction is advected with the solenoidal velocity w. When written in terms of the mass average velocity, è9.6è + u ærç + ç divu = div è çd 1, çç rç! è9.8è 30

31 The weight diæerence ç = èç G, ç W è=ç G between the two species of our binary mixture, e.g., glycerin and water is a primary parameter which measures the extent of non-solenoidality. The expansion velocity is proportional to ç and is zero for two species with the same density. Turning next to boundary conditions at a solid wall, we note that the diffusive æux of any species across an impermeable, bounding surface vanishes. If n is the outward normal at such a surface èfrom the æuid to solidè, we have n ærç =0 è9.9è This implies that u e æ n =0 è9.10è at an impermeable boundary. There is no reason to suppress the slip velocity of each of the two species of our binary, but for several reasons, it seems appropriate to enforce a no-slip boundary condition for the mass-averaged velocity u. If the velocity of a solid wall is prescribed as U, then at the wall u = U: è9.11è Of course, u vanishes at a stationary wall. The paper by Mo and Rosenburger ë1991ë on molecular-dynamic simulations of æow with binary diæusion inatwo-dimensional channel with atomically-rough walls establishes the noslip condition via the mutual cancellation of the nonvanishing, opposing slip velocities of the components. Suppose that the velocity of U of the solid wall is prescribed, n is the normal and t is any tangent vector on the wall, n æ t = 0. u e æ n = n ærç = 0 and u = U on the wall, we get w æ n = U æ n; Then since è9.12è 31

32 w æ t = U æ t, u e æ t: è9.13è We can think that ç is determined by the diæusion equation and that nærç = 0. Then u e æ t can be thought as given and è9.12è and è9.13è look like prescribed, non-classical conditions for w. Of course, in practice everything is coupled, but the description just given is good for sorting it out. 10 Basic solution for diæusion in a pipe The problem being taken up here is in a three-dimensional domain with walls, but the solution is one dimensional. We don't think that the problem or its solution has been given before. Consider a pipe of arbitrary but constant cross section and length. The entrance to the pipe is at x =0and the exit at x = L. The pressure is taken to be the same at x = 0 and x = L so pressure does not drive æow, and ç 0 6= ç L are prescribed water fractions at the entrance and exit. We satisfy the condition that u = 0 on the wall by taking u = u e +w =0everywhere. The diæusion equation è9.8è than reduces to è d D dx 1, çç dç dx! =0: è10.1è The solution ç = æèxè of è10.1è, which satisæes the prescribed conditions æè0è = ç 0 ; æèlè =ç L è10.2è is given by æèxè = 1 ç n 1, è1, çç0 è 1,x=L è1, çç L è x=lo è10.3è It is easy to verify from è10.3è and follows directly from è10.1è that çd dæ 1, çæ dx = u e = const: è10.4è 32

33 where the expansion velocity is given by u e = e x u e. This velocity is balanced by a volume-averaged velocity w = e x w ox where w ox =,u e : è10.5è 11 Stability of the basic solution for diæusion in a pipe Let us consider the stability of the basic solution ç =æèxè; u = u e + w ox e x =0; è11.1è where w ox = D ln 1,çç 0 1,çç 1 and æèxè is given by è10.3è, u e by è9.2è. Following the standard procedure of linear stability analysis, we introduce small perturbations to above solution, linearize the equations, and use L and ç w L 2 =ç w as length and time scale to normalize the equations. We then introduce the temporal mode e çt. The resulting dimensionless linearized equations governing the perturbations are ç 0 çu + x y çç + w ox + w dæ x dx " 2ç 0 ç 0 @y =0 è " ç @y " ç u = w x + çç 1, çæ, w çç ox 1, çæ 2ç è è11.2è è11.3è è11.4è è11.5è è11.6è 33

34 v = w y + çç 1, è11.7è where w x, w y are x; y components of w, respectively, è = p ç 0divu, and ç = ç GD ç G, ç 0 = ç W ç G e æèxè. Using equations è11.6è and è11.7è we can eliminate w from the continuity and diæusion çç = çæç, dæ dx è çaèxè, aèxèw oxç! è u + w ox aèxèç, çaèxè, ç =0 è11.8è + w ox è11.9è where æ is the two dimensional Laplacian, and aèxè = ç=è1, çæèxèè. Equation è11.4è, è11.5è, è11.8è and è11.9è are four equations for u; v; ç and è. The boundary conditions we take for this system are v = p = ç = 0 at x = 0 and x =1,u = v = =0aty = 0 and y = r, with r being the width of the channel. Admittedly these conditions are rather artiæcial, but it is very diæcult to specify simple, realistic boundary conditions for the physical process we are trying to model here. We could imagine that the two ends of the channel are porous plates which help maintain the concentration of water, while the no-slip condition forces the tangential velocity v to vanish on these plates. The system of partial diæerential equations we just derived have variable coeæcients depending on x. We can reduce it into a system of ordinary diæerential equations if we leave away the boundaries in the y direction. We shall ærst consider this case. As we said before when ç is zero, the problem degenerates to the classical diæusion problem. The basic concentration distribution becomes a linear function of x, and the stability problem is reduced to that of the usual one- 34

35 dimensional diæusion equation, while the velocity æeld is not involved. The eigenvalues are then given by ç 0 =,çènçè 2 ;n=1; 2; :::: è11.10è When ç is small but not zero, we can perturb this problem by expanding the unknowns as well as the eigenvalues into series of ç, for example, ç = ç 0 + ç 1 ç + ç 2 ç 2 + Oèç 3 è; è11.11è with the zeroth order solution given by the conventional diæusion problem. Carrying out this procedure to ærst order, we found that ç 1 is zero. However, it is not possible to carry on this perturbation beyond ærst order analytically. For general cases like glycerin and water, of which ç is about 0.21, which is not too small, we solved the system of diæerential equations numerically by the ænite element method. More about the numerical method is said later. Results of this case are shown in ægure 2, in which the diæerence between the ærst eigenvalue and ç 0 is presented as functions of ç for diæerent values of ç 1 and ç 0. In this one-dimensional case, we found ç, ç 0 is positive. This means that non-solenoidality is a destabilizing factor, although the inæuence is rather small. We also note that for the cases we calculated ç is nearly proportional to ç and the viscosity ratio m = ç W ç G is not a parameter. In two dimensions, the problem is not separable in general. However, we can get an explicit solution if we look at waves which are so short that the variable coeæcients in the linearized equations are essentially constant over the length of one wave. We may then freeze these coeæcients at each and every point èx 0 ;y 0 è and seek stability locally to disturbances of the form e çt e iæèx,x 0è e iæèy,y 0è è11.12è where æ 2 + æ 2!1for short waves. We ænd that æu + æv + çèæ 2 + æ 2 èaç + iæaw ox ç =0; è11.13è 35

36 0:0020 0:0015 ç, ç 0 0:0010 èç L ;ç 0 è=è1:0; 0:0è è0:9; 0:2è 0:0005 0:0000 è0:8; 0:4è,0:4,0:2 0 ç 0:2 0:4 Figure 2: ç, ç 0 as functions of ç, in one-dimensional case, ç = 0:01, ç 0 =,0: :0100 0:0050 m =0:01 0:10 ç, ç 0 0:0000 1:0,0:0050,0: :0 10:0,0:4,0:2 0 ç 0:2 0:4 Figure 3: ç, ç 0 as functions of ç, wall eæect included. ç =0:01 and r =0:1. m = ç W ç G is the viscosity ratio. ç 0 is as in ægure 2. 36

37 h ç + iæwox + çèæ 2 + æ 2 è i ç +æ 0 èu + aw ox ç, iæççè =0; ç 0 çu =,iæè, 2æ 2 ç 0 u +2iæç 0 0u, æç 0 èæu + ævè; ç 0 çv =,iæè +ëiç 0 0, æç 0 ëèæu + ævè, 2æ 2 ç 0 v: è11.14è è11.15è è11.16è After eliminating è, we ænd that h ç0 ç + ç 0 èæ 2 + æ 2 è+iç 0 0æ i èæu, ævè =0: è11.17è Equation è11.17è determines two eigenvalues, ç 1 =, ç 0 ç 0 èæ 2 + æ 2 è+i ç0 0 ç 0 æ è11.18è and ç 2 =,çèæ 2 + æ 2 è, iæw ox + çæ2 a 2 æ 2 + æ 2 æ02 : è11.19è The ærst term of ç 2 is negative, which arises from the conventional diæusion equation. The last term of è11.19è is destabilizing. However, this term is dominated by èæ 2 + æ 2 è in the limit of large æ 2 + æ 2. Hence the eæect of non-solenoidality of the velocity æeld on the growth rate is minimal for very short waves. We next consider the wall eæects by restricting the mixture to a rectangular domain. We take a slim channel with a width of 0.1 length for our calculation. In this case the conventional diæusion solution still exists because in which the velocity can be taken zero everywhere. The perturbation of ç works only when the two constituents have the same viscosity. In more general cases, the system has to be solved numerically. We have developed a general code to solve eigenvalue problems of systems of PDE's. The ænite element method is used to approximate the equations. However, this leads to very large matrices, which deæes the use of common direct eigenroutines. We use a modiæed Arnoldi iterative method developed by Saad ë1989ë to ænd 37