Surface tension in a reactive binary mixture of incompressible fluids

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1 Surfae tension in a reative binary mixture of inompressible fluids Henning Struhtrup Institute for Mathematis and its Appliations, 400 Lind Hall, 07 Churh Street S.E, Minneapolis MN55455 John Dold Mathematis Department, UMIST, Manhester M60 QD, Britain Abstrat Based on the Cahn-Hilliard free energy, a thermodynami model for a reative binary mixture of inompressible and misible fluids is derived with a distributed form of surfae tension. The model desribes hemistry, diffusion, visosity and heat transfer as well as stresses produed by and at right angles to onentration gradients. It allows for a rih spetrum of proesses and some of these are disussed briefly. Saling arguments based on experimental data of Abid et al. [], lead to onsiderable simplifiations. Saffmann-Taylor stability analysis here extended to interfaes of finite thikness agrees with the experimental findings of Abid et al. who argued that the buoyant instability of a reation diffusion front is ontrolled by a form of surfae tension at the front. Keywords: Binary mixtures, Surfae tension, Irreversible thermodynamis, Hele-Shaw ells Introdution Reent experiments [] on misible, buoyantly unstable reation diffusion fronts in Hele-Shaw ells give an indiation that the observed instabilities are governed by a form of surfae tension, a property that is normally assoiated with interfaes between immisible fluids. Sine an interfae between misible fluids is not vanishingly thin the stress that orresponds to surfae tension within the medium must be distributed over the region where the onentrations of different omponents hange. The goal of this artile to provide a sound theoretial foundation for this observation by means of a thermodynami derivation and analysis of the governing equations for the proess. In order to emphasize the basi physis, we shall onsider a binary reating mixture, instead of a more realisti reation with several speies. The fluids used in the experiment are almost inompressible and we shall restrit the model to inompressible fluids. The onstant mass densities of the two onstituents differ and therefore a hange in the onentration, by diffusion and/or reation, leads to a hange in the total density. The mixture therefore appears to be ompressible it is quasi-inompressible in the sense that density hanges an arise through hanges in omposition but not through hanges in pressure alone [9, ]. As struhtr@ima.umn.edu John.Dold@umist.a.uk

2 is usually the ase for inompressible fluids [3], the pressure does not appear in the density s equation of state, but must be onsidered as an additional variable [9, ]. Our thermodynami onsiderations are based on the free energy of the mixture whih ontains terms for the energy as well as for the entropy of mixing. For these ontributions we adopt the elebrated Cahn-Hilliard model [5] aording to whih the energy of mixing depends on the onentration and its gradient. The latter is known to lead to an effetive surfae tension whih dereases as the gradient dereases [5, ]. As a onsequene, the surfae tension of a typial hemial reation wave is very small due to its omparatively large thikness. Surfae tension ould be measured in [] only beause the differene between the densities of reatants and produts was small enough to ensure that a ompeting buoyany-driven instability was also very small. The balane laws for masses of the onstituents, momentum, and energy do not form a losed set of field equations a priori, but require onstitutive equations for the so-alled thermodynami fluxes. In this paper, the onstitutive equations are derived by means of the lassial thermodynamis of irreversible proesses TIP [7]. The basi assumption is the loal equilibrium hypothesis whih in our ase gives rise to the Cahn-Hilliard expressions for energy and entropy in non-equilibrium. In partiular, the evaluation of the seond law of thermodynamis with the Cahn-Hilliard free energy, aording to TIP, leads to expressions for the stress tensor and the hemial potential of the mixture, whih govern both diffusion and hemistry in the mixture. Moreover, as a diret onsequene of the quasi-inompressibility, the pressure enters the theory as an arbitrary parameter whih must be hosen as additional variable. As well as the pressure, the stress tensor ontains the usual visous ontributions plus a term whih relates to apillary fores. Sine the pressure is somewhat arbitrary, it an be redefined or normalised so that the apillary term preisely orresponds to surfae tension, i.e. it represents fores perpendiular to the onentration gradient. TIP also relates the thermodynami fluxes to the orresponding thermodynami fores through linear relations phenomenologial equations. Linear onstitutive laws are in aordane with experimental observations onerning diffusion, heat transfer and visous stresses, but not hemial reations. Therefore, the latter are modelled by the Arrhenius law, whih is set up properly so that the positivity of the entropy prodution due to hemial reation is guaranteed. In partiular, it will be seen that an inrease in the pressure shifts the hemial equilibrium towards the denser omponent Le Chatelier s priniple, although it does not diretly inrease the density by itself. In general, our approah is a ontribution to what is known as Diffusive-Interfae Methods in Fluid Mehanis see the extensive review by Anderson et al. [] for a list of referenes. Most work in this area is related to immisible fluids, phase boundaries, et., whih are typial of systems where surfae tension an be expeted to play a signifiant role. Our results and methods are losest to the work by Lowengrub and Truskinovsky [], but differ in the treatment of the pressure as well as in the emphasis on misible fluids in our onsiderations. The paper by Antanovskii [3] served us for inspiration, but our treatment differs onsiderably from it, sine we exploit the seond law more thoroughly and thus avoid heuristi arguments. To our knowledge the present work is also the first to ombine diffusive-interfae methods with flame-like reations. Joseph [9] investigated surfae tension in misible fluids experimentally and theoretially. He added surfae tension to the momentum balane but onsidered the lassial diffusion equation Fik s law. As will be seen in the ourse of our paper, the diffusion may be affeted by pressure

3 gradients and intermoleular fores. In misible fluids, however, the influene of the latter will hardly be seen expliitly, but merely affet the measurement of the diffusion oeffiient. Pressure gradients may ause partial demixing of the onstituents, if the density differene is large; with Joseph s very small density differenes, Fik s law is indeed appropriate. Altogether, our model, desribing surfae tension, hemistry and diffusion as well as visosity and heat transfer, allows for a rih spetrum of proesses and some of these are briefly onsidered, for example the aforementioned influene of the pressure on diffusion flux and hemial equilibrium. Most of this however, must be referred to later investigations. The importane of the various proesses depends on physial parameters, suh as diffusion oeffiient, reation rate, density differene, intermoleular fores, et. A proper saling, based on the experimental data of [] allows for a onsiderable simplifiation of the equations. The stability results that were used in [] are based on the well-known Saffmann-Taylor stability analysis [6] whih is originally appliable at interfaes of vanishing thikness. We shall show that the same analysis an be used as well in the ase of a finite, but small thikness of the interfae. Binary mixture of two ideal fluids. Ideal fluids In an inompressible ideal fluid the state variables are independent of pressure. Aordingly mass density ϱ and speifi internal energy u are funtions of temperature T alone. However, beause of the well known thermodynami relation u p T = T ϱ ϱ T p + p ϱ ϱ p, the density T must be independent of temperature as well [3] and is therefore a onstant. In an ideal fluid the speifi heat v is assumed to be onstant and its speifi internal energy ut is given by ut = v T T 0 +u 0 where u 0 is the speifi energy at temperature T 0. Due to the inompressibility, the Gibbs equation redues to T ds =du so that the speifi entropy s of the ideal fluid is given by where s 0 is the entropy at temperature T 0. st = v ln T T 0 + s 0. Binary mixture We onsider the mixture of two inompressible ideal fluids with the onstant mass densities ϱ, ϱ, partile masses m, m, and speifi heats v, v. Under the assumption that there are no volume hanges due to mixing exess volume we an write for the density of the mixture ϱ = ϱ + ϱ 3 where is a onentration parameter varying between in fluid and 0 in fluid. Obviously, the partial mass densities of the onstituents are given by ϱ, ϱ. The energy density of 3

4 the mixture is given by the sum of the partial energies plus an extra term whih aounts for the interation between the onstituents, ϱu = ϱ u + ϱ u + U mix 4 = ϱ v T T 0 +u 0 + ϱ v T T 0 +u 0 + Umix. Similarly the entropy of the mixture is not simply the sum of the two partial entropy densities, but ontains an additional term, the entropy of mixing, ϱs = ϱ s + ϱ s + S mix 5 = ϱ v ln T + s 0 + ϱ v ln T + s 0 + S mix. T 0 T 0 The entropy of mixing is onsidered later in setion.4..3 Energy of mixing The energy of mixing U mix results from the different intermoleular fores between partiles of different speies [5]. We denote the potential between two partiles of types α, β by V αβ ri α r β i ; V αβ is a funtion of the distane ξ = ri α r β i between the partiles. The total potential energy of the n α dr α partiles of type α in the volume element dr α and the n β dr β partiles of type β in dr β is given by V αβ r α i r β i n α r α i nβ r β i dr α dr β. We now introdue the vetors x i, ξ i by x i = r α i + r β i, ξi = ri α r β i so that r α i = x i + ξ i, r β i = x i ξ i and dr α dr β =dx dξ. Thus, the potential energy of partile pairs α, β with middle point x i in the volume element dx and distane vetor ξ i in the volume element dξ is given by V αβ ξ n α xi + ξ i nβ xi ξ i dx dξ. Integration over all values of the distane vetor and division by dx gives the potential energy density at x i stemming from all pairs with middle point in dx, namely { U mix = γ αβ V αβ ξ n α xi + ξ i nβ xi ξ if α = β i dξ with γαβ = if α β. The oeffiient γ αβ guarantees that pairs of the same speies do not ontribute twie to the energy. The potential V αβ deays rapidly with inreasing ξ so that it is reasonable to approximate n α using the expansion n α xi ± ξ i nα x i ± ξ i nα x i. Moreover, making use of the onentration by means of the relations n = ϱ m, n = ϱ m and summing over all possible pairs of speies, we find U mix = a + b + a + a 6 x i x i 4

5 with oeffiients a, b given by relevant moments of the interation potentials, ϱ ϱ a = V ξ ϱ V m m ξ ϱ V m ξ dξ, m b = ϱ ϱ 6 V ξ ϱ V m m ξ ϱ V m ξ ξ dξ. m 7 The final two terms in 6, involving the oeffiients a αα = ϱα m Vαα α ξdξ, an be absorbed into the energies u α 0 of the single onstituents and need not be onsidered any further. The oeffiient a has the dimension of an energy and the ratio b/a = l ap defines a fundamental length sale l ap for apillary fores in the mixture. It should only be of the order of a few moleular diameters. In partiular, between immisible fluids one would expet to find only a slightly diffuse interfae of approximate mean thikness l ap. The Taylor expansion used above for n α is likely to prove most aurate, however, when the interfae thikness is large ompared with l ap, as it would be at an interfae between misible fluids. It is worth noting that, aording to our simple model, the energy of mixing is independent of temperature. A similar derivation an be found in Cahn & Hilliard s elebrated paper [5]. These authors, however, ompute the potential energy assoiated with a single partile, and not, as we have done, the energy of a pair of partiles. In order to arrive at the above result 6, Cahn and Hilliard had to perform an integration by parts, whih appears to be somewhat artifiial and is not needed in our approah. More refined models for the energy of mixing an be found in [5]..4 Entropy of mixing In order to alulate the entropy of mixing from a simple argument we onsider a volume element dx where we have n moleules of fluid and n moleules of fluid. Under the assumption that there are as many plaes in dx as partiles, namely n + n, we alulate the entropy of mixing from the well known Boltzmann formula as S mix = k ln n + n! n! n! = k n ln n n + n + n ln n n + n where k denotes Boltzmann s onstant and we have used the Stirling formula. Again, the number densities an be related to the onentration by n = ϱ m, n = ϱ m to obtain [ ϱ S mix = k ln + ϱ m + ϱ ln + ϱ m ]. 8 m ϱ m m ϱ m.5 Free energy The speifi free energy of the mixture is given by f = u Ts = ϱ ϱ f + ϱ f + U mix TS mix 9 with the free energies of the single onstituents f, f given by f α T =u α Ts α = α v T T 0 T ln T + u α 0 Ts α 0 T 0 and with U mix, S mix given by 6, 8. 5

6 0 F â Figure : The free energy 0 with f f /kt/m =0. for â =0,,.5, 3. Depending on the strength of the intermoleular fores, the free energy 9 is either onvex or has the form of a double well potential. To illustrate this, we an onsider a ase having equal moleular masses and mass densities of the onstituents, m = m = m, ϱ = ϱ = ϱ, and ignore the onentration gradients in U mix. We an then write F = f f kt/m = f f + â +ln + ln 0 kt/m with the abbreviation â = a ϱkt/m. In this simplified ase, F eases to be onvex F 0, if â. Figure shows the free energy 0 for some values of â. A non-onvex free energy allows for the separation of two phases with different onentrations, i.e. a gap of misibility []. For very large values of â, when the interation energy exeeds the thermal energy, the equilibrium onentrations are given by 0, and the mixture appears to be immisible. In the ontext of this paper we are interested mainly in fully misible binary mixtures and shall onsider ases where the interation energy is dominated by the thermal energy, for whih â<. 3 Field equations The thermodynamis of inompressible binary mixtures provides a means of determining the onentration, veloity v i and temperature T fields as funtion of spae and time. For this purpose we have the onservation laws of mass, momentum and energy whih must be furnished with onstitutive relations for diffusion flux, mass prodution rate, heat flux and stress tensor. 3. Balane laws The balanes of mass for the onstituents, read t ϱ + x ϱ k v k = τ, t ϱ + x ϱ k v k = τ where τ is the mass prodution rate and vk α denotes the entre-of-mass veloity of onstituent α. The mass balane of the mixture is the sum of the partial mass balanes and assumes the usual form t ϱ + x ϱvk k =0 6

7 with the entre-of-mass veloity v k of the mixture given by ϱv k = ϱ v k + ϱ v k. If the diffusion flux J k is defined as J k = ϱ v k v k= ϱ v k v k then the equations an be ombined to give ċ = ϱ Jk τ, ϱ ϱ v k = ϱ ϱ ċ ϱ in whih = t +v k denotes the material time derivative. The first equation is the evolution equation for the onentration while the seond relates the hange of volume to the hange in onentration. Thus, as long as the onstant densities of the onstituents are different, ϱ ϱ, the mixture appears to be ompressible it is quasi-inompressible as desribed earlier [9, ]. The balanes for momentum and energy of the mixture are assumed to have the same form as for a single fluid, see [3], ϱ v i t ik = ϱg i, ϱ u + q k = t ij v i x j. 3 Here, t ij denotes the stress tensor, q k is the heat flux vetor and g i is an external body fore, usually gravity. With, 3 we have six equations for the unknowns {T,, v k }, so that the system appears to be overdetermined. However, it will be seen that the pressure p enters as an additional variable. This is normally the ase in the theory of inompressible fluids, see [, 3]. 3. The seond law of thermodynamis The equations, 3 ontain quantities whih are not a priori related to the fields {T,, v k }, namely the diffusion flux J k, the mass prodution rate τ, the stress tensor t ij and the heat flux q k. In this setion we shall exploit the seond law of thermodynamis in order to find onstitutive equations for these additional quantities. We shall follow the guidelines of lassial thermodynamis of irreversible proesses TIP, see [7] for a lassial survey. Roughly speaking, we onstrut a balane equation for the entropy and ensure the positiveness of the entropy prodution by an appropriate hoie of the onstitutive equations. The starting point is the speifi entropy 5,8 whih is a funtion of {T,} or, by eliminating the temperature T with the energy 4,6, a funtion of {u,, χ}, where χ is defined as χ = x i x i. Note, that the dependene on χ leads beyond standard lassial TIP, where the entropy depends on equilibrium variables only, and not on gradients. The total differential of the speifi entropy reads Gibbs equation ds = s s T du + d + dχ 4 u,χ χ u, aording to whih the temperature T an be defined in the usual way, suh that s T = u.,χ 7

8 Considering entropy s and energy u to be funtions of {T,, χ} we find from 4 s s = u,χ u T,χ T = f T,χ T s s χ = u, χ u T, T χ = f T, T χ. Note that by 9 the free energy f is a funtion of {T,, χ}. With these relations the Gibbs equation leads to Noting that χ = ϱṡ = T ϱ u ϱ T f ċ ϱ f T χ χ. ċ = v k x i x i x i x i x i we an eliminate the time derivatives of u and by means of the balane laws, 3 to get ϱṡ = q k + ϱ f + Πδ ij + t ij + ϱ f vi + 5 T χċ T χ x i x j x j + ϱ f T ϱ ϱ ϱ ϱ f ϱ ϱ Jk Π τ ϱ ϱ x i χ x i ϱ ϱ where Π denotes the generalized pressure, whih is an arbitrary salar funtion sine the two terms ontaining Π anel eah other by means of. The generalized pressure will be disussed below after we have finished disussing the onstitutive theory. It is onvenient to introdue the irreversible heat flux Q k, the irreversible stress tensor P ij and the hemial potential µ as q k = Q k ϱ f, χċ t ij = Πδ ij ϱ f + P ij, 6 χ x i x j µ = ϱ f ϱ ϱ ϱ ϱ ϱ Π. ϱ ϱ χ ϱ ϱ By referring to Q k and P ij as irreversible we mean that only these parts of heat flux and stress tensor ontribute to the prodution of entropy. After some rearrangement equation 5 assumes the form of a balane law, ϱ f x i x i involving the entropy flux vetor ϱṡ + φ k =Σ and the entropy prodution rate Σ=τ µ T φ k = Q k T µ T J k + P jj v k /T + Q k + J k µ 3T T + P ij v i. 7 T x j The angle brakets on the indies i, j denote the symmetri trae-free part of a tensor, suh as P ij = P ij 3 P kkδ ij. Aording to the seond law of thermodynamis, the entropy prodution rate Σ must be positive or zero. 8

9 3.3 Phenomenologial equations It is onvenient to write the entropy prodution rate Σ as a sum of thermodynami fluxes F a and thermodynami fores X a, Σ= a F a X a 0, in whih the fluxes and fores are, from 7 { F a = τ, { X a = µ T, 3T P jj, Q k, J k, v k, /T, µ T } T P ij, } v i,. In linear irreversible thermodynamis the positiveness of the entropy prodution is ensured by the so-alled phenomenologial equations [7], by whih x j F a = b L ab X b with L ab positive definite for eah value of a. The phenomenologial oeffiients L ab must be taken to be funtions of the same variables as entropy and energy, namely temperature T, onentration, and onentration gradient x i. In lassial irreversible thermodynamis the phenomenologial equations are somewhat simplified by the so-alled Curie priniple, whih states, in its simplest form, that thermodynami fluxes of tensorial rank n salars, vetors, tensors,... are oupled only to fores of the same tensorial rank [7]. This statement follows from arguments on isotropy of the body under onsideration, and is not stritly true in our ase, where the phenomenologial oeffiients L ab an depend on the onentration gradient whih is inherently not isotropi [4, ]. However, being interested in the simplest model obtainable from the seond law, we do not onsider any ross effets but assume that the fluxes are oupled solely to their orresponding fores. Alternatively, we an argue that the model applies to relatively weak gradients in onentration, with hanges ourring over very many moleular diameters. In this ase the off-diagonal elements of the matrix L ab should, effetively, be zero and the phenomenologial equations then read J k = D µ, Q k = κ T, T P jj =3ν v k, P ij =η v i x j. 8 Here, D is the diffusion oeffiient, κ the thermal ondutivity, and η and ν are the shear and bulk visosity, respetively; all oeffiients must be positive. We did not set up a similar equation for the hemial reation rate τ sine a linear law, τ = γ µ T, is physially orret only lose to hemial equilibrium and annot be used for the desription of flame-like reations. We shall onsider the hemistry in Setion 3.5. To lose this setion, we examine the hemial potential for our binary mixture of inompressible fluids with the free energy 9. We an write 6 3 as µ = µ µ 9 9

10 in whih we define the separate hemial potentials for the onstituents, as µ = f T kt ln + ϱ m m ϱ m + Π+ b [ ] + a b ϱ µ = f T kt ln + ϱ m + ϱ Π+ b [ ] + a + b. 0 m ϱ m Note that it is not imperative to assign the individual terms to the hemial potentials of the onstituents in preisely this way, whih, all the same, seems to be the most natural hoie. In partiular, µ α will redue to the free enthalpy f α + Π ϱ α for a single onstituent. 3.4 Pressure and apillary fores Due to 6,8 the stress tensor reads t ij = Πδ ij ϱ f + ν v k δ ij +η v i. χ x i x j x j It is evident that Π, introdued in 5 as an arbitrary salar funtion, indeed plays the role of the pressure. Moreover Π an be added as the sixth variable in the set {T,, v i, Π} and will be determined by the field equations and boundary onditions for the stress tensor. However, beause of the way that Π was introdued in the theory, we have some freedom over its preise definition and an deompose Π into the usual interpretation of the pressure p and any arbitrary salar funtion of the other variables, Π = p + ΠT,, χ. One partiular hoie for ΠT,, χ leads to a transparent and sensible representation of the stress tensor. We set Π=p ϱ f χ and introdue the tensor of apillary stresses as S ij = ϱ f δ ij. χ x i x j Sine this defintion results in the identity, S ij x j = 0, this part of the stress tensor ontributes only fores perpendiular to the gradient of onentration. Thus, S ij desribes a distributed form of the surfae tension of any diffusive layer between regions with different onentration. The omplete stress tensor is a sum of three terms, the pressure p, the visous stresses P ij and the tensor of apillary stresses S ij t ij = pδ ij + S ij + P ij. Other redefinitions of Π are possible, see [] for example, but will not be disussed here. Our hoie for the definition of pressure was taken from [3]. However, although that artile starts off with the exploitation of the seond law, it does not make full use of its possibilities. Instead it derives its expression for the tensor S ij based on heuristi arguments from the energy balane and results in the fator ϱ u χ in plae of ϱ f χ the two results for our simple model where we have ϱ f χ = ϱ u χ. In fat, there is no essential differene between 0 = b, see equations 4,9.

11 3.5 Chemistry In this setion we study the hemistry of our model for a binary mixture with the sole reation γ A + γ B γ A + γ B. γ α and γ α denote the stoihiometri oeffiients for the reatant and the produts, respetively; A and B stand for the omponents,. Due to onservation of mass we have m γ γ = m γ γ. The mass prodution rate τ of onstituent an be written in terms of the reation rate density Λas τ = m γ γ Λ. In order to have a positive prodution of entropy due to hemial reations, τ µ/t 0 must hold or, with the reation rate Λ and 9,, where A and A are the hemial affinities, defined as ΛA A 0 3 A = m γ µ + m γ µ, A = m γ µ + m γ µ. As we have said before, a linear law for the reation rate is appropriate only lose to hemial equilibrium. For flame-like reations there are good reasons to believe that the reation rate density should be of the form Λ=K exp A kt A exp kt with a positive onstant K so that the positivity of 3 is ensured. We make the entropi terms of the hemial potentials 0 expliit by setting µ = µ kt ln + ϱ m, µ = µ kt ln + ϱ m 4 m ϱ m m ϱ m and obtain the well-known Arrhenius law [7] γ n Λ=K nγ n γ +γ exp m γ µ + m γ µ kt nγ nγ n γ +γ exp m γ µ + m γ µ kt written with the number densities n = ϱ m, n = ϱ m, n = n + n. From now on, we shall restrit our attention to the autoatalyti reation A + B B with the stoihiometri oeffiients γ = γ =,γ =0,γ =. This implies that the moleular masses of the partiles are the same, m = m = m, so that mn = ϱ. The Arrhenius law for this partiular reation reads Λ=K ϱ ϱ ϱ exp µ + µ kt/m ϱ exp µ. 5 ϱ kt/m

12 4 Γ 0 â Figure : The funtion Γ = ln + â for â =0,,,3,5. Let us onsider the hemial equilibrium for the mixture so that Λ = 0, or exp { G} = ϱ exp {F + P }. 6 ϱ The latter equation is the law of mass ation for the reation in question where we have introdued the abbreviations F = f T f T = v v T 0 k/mt k/m T ln T + u 0 u 0 T 0 kt/m s 0 s 0 k/m P = p 7 kt/m ϱ ϱ [ ] G = a b + b ϱ kt/m [ ] a b b. ϱ kt/m The ourrene of the onentration gradients in G ompliates the interpretation of the law of mass ation 6 but we an state some prinipal properties nevertheless: A higher value of F shifts the equilibrium towards larger values of the onentration, whih stabilizes onstituent. The value of F is mainly determined by the energy and entropy onstants u α 0, sα 0. An inrease of the pressure p shifts the hemial equilibrium towards the denser omponent this is a refletion of Le Chatelier s priniple. The influene of the intermoleular fores on the hemial potential is ontained in the funtion G. Negleting the onentration gradient and the differene in the densities we have G = â. Figure shows the funtion Γ = ln + â for several values of the parameter â = a ϱkt/m. In equilibrium Γ = F + P, and we an read offfrom the figure that a larger value of â stabilizes the single onstituents. For â, where the free energy is non-onvex, there are three possible equilibrium onentrations for a ertain range of F + P. Only the outer onentrations, however, orrespond to stable equilibria, while the intermediate value for the onentration belongs to an unstable ase.

13 3.6 Summary of results With the results of the last setion we have a losed system of field equations for the variables T,, p, v i. The equations read ċ = ϱ Jk τ, ϱ ϱ v k = ϱ ϱ ċ ϱ ϱ v i + p S ik P ik = ϱg i 8 x i ϱ v T + Q k v i = P ij + ϱ ϱ x j ϱ h h ċ. Here, we have written the energy balane 8 4 in terms of the variables; v is the heat apaity of the mixture, given by ϱ v = ϱ v + ϱ v. The heat release due to hanges of the onentration is proportional to the differene of generalized enthalpies h, h of the onstituents, whih are given below in equation 30. It is interesting that only the irreversible parts of heat flux and pressure tensor appear in this equation. The onstitutive equations for mass prodution rate τ, diffusion flux J k, irreversible heat flux Q k, tensor of apillary stresses S ij and tensor of visous stresses P ij read τ = K exp µ + µ J k = D µ µ T S ij = b kt/m ϱ exp µ ϱ kt/m, Q k = κ T, 9 δ ij x i x j, P ij = ν v k δ ij +η v i x j. The positive oeffiients K, D, κ, ν, and η must be determined from experiments, the theory is not able to say anything about their values or their dependene on the variables {T,, α}. The same is true for the oeffiients of apillary fores a and b whih follow from intermoleular potentials by 7. Finally we give the generalized enthalpies of the two onstituents whih read h = u + p b [ ] + a b, 30 ϱ h = u + p b [ ] + a + b. ϱ The hemial potentials µ α, µ α are given by 0,4. 4 A simplified model for the Hele-Shaw ell In this setion we redue our model for the desription of proesses in Hele-Shaw ells. In partiular we shall use the experimental data of [] in order to weigh the relative importane of the several ontributions to the equations. 3

14 4. Isothermal proesses and simplified hemistry Our main interest in this paper lies in the interplay between hemial reation and surfae tension. In order to fous attention on this behaviour, we introdue some assumptions whih will lead to onsiderable simplifiations. Isothermal proesses: When the heat release of the reation is small, or the speifi heat v is large, or the heat ondutivity is large, one an expet very small variations in temperature. The temperature an then be assumed to be onstant, T = T 0, and the energy balane 8 4 an be disarded. Indeed, this ondition is met very well in the experimental Hele-Shaw ells of [], where the heat of reation is almost immediately absorbed by the walls. Full reation: We assume that the energy and entropy onstants u 0 α, s 0 α are large ompared to the ontributions of pressure and apillary fores in µ α 4, so that µ T 0 u 0 T 0s 0, µ T 0 u 0 T 0s 0. Furthermore we presume that µ T 0 µ T 0 so that the first term in the mass prodution rate 9 dominates and τ redues to τ = K where K = K exp u 0 + u 0 T 0 s 0 s 0 3 kt 0 /m Now, the reation rate K and the hemial equilibrium are independent of pressure and apillary fores. Equation 3 desribes an autoatalyti reation from onstituent = to onstituent = 0. Constituent alone is stable, but as soon as a bit of onstituent is added, a reation will start and proeed until onstituent is fully onsumed. 4. Diffusion flux and misibility In this setion we disuss the diffusion flux. In partiular we shall define a misible system and introdue an assumption on the dependene of the diffusion oeffiient D on the onentration. To simplify notation we set p = p ϱ kt 0 /m, â = a ϱ kt 0 /m, b = b ϱ kt 0 /m 3 and obtain the diffusion flux from the derivative of the hemial potentials as J k = D k [ â m x b k x i x i + ϱ p +â [ + ϱ x b 3 ]] +. k x i x i x i x i x i x i Insertion of this flux into the balane of the onentration 8 gives the speifi form of the Cahn-Hilliard equation for our model. The first term,, stems from the entropy of mixing, and therefore refers to the entropi part of the diffusion flux. This term forbids a total demixing of the onstituents =or = 0 and, if it dominates, leads to a perfetly mixed fluid as long as there are no hemial reations to be sure. Aording to the first term in the seond line a pressure gradient may produe a diffusive flux if the densities of the two onstituents differ. This gives rise to some demixing along the pressure gradient. Total demixing, of ourse, is prohibited by the entropi diffusion. 4

15 d â Figure 3: The funtion d for â =0,0.5,,.5,.99. All other terms are related to the apillary fores and also lead to demixing to some degree. Summarizing we an say that there are entropi fores the first term and energeti fores pressure gradient and apillary fores ating against eah other. Total demixing is forbidden by the speifi form of the entropi ontribution to the diffusive flux. Furthermore it beomes lear from 3 that the relative weight of the energeti fores dereases with inreasing temperature. A misible system an be defined as a mixture where the entropi fores dominate. The diffusion oeffiient D is allowed to be a funtion of the onentration. Sine we want to onsider a reative proess where the onentration goes from =to = 0, it is unpratial to hoose D = onstant. sine we then had to fae divisions by Zero in the entropi ontribution. To overome this problem we set from now on D k m = D â with D = onst. 34 This hoie produes the usual form of Fik s law, J k = D, in the absene of apillary fores and density differenes. Moreover, with 34, we have only one single oeffiient, D, whih relates diffusion flux J k and onentration gradient, and it is this oeffiient whih is measured in experiments just beause an experimentalist will assume Fik s law in this form. To illustrate the influene of the interation energy â on D figure 3 shows the normalized diffusion oeffiient d = 0.5â â 0.5 for several values of â. It is only for values of â> that this funtion differs onsiderably from the ase without interation energy â = 0. In a misible system, where diffusion is dominated by the entropi effets, we have â and the influene of the interation energy manifests itself in a smaller diffusion oeffiient D but not in its dependene on onentration. 4.3 The flame We now introdue sales and dimensionless variables whih will enable us to weight the different terms in the equations against eah other. We assume the flame proess to be the most 5

16 dominating and this gives rise to the following sales: t = tt 0, t 0 = ϱ K, x i = x i L, L = D / K, v k = v k L t 0 The time sale t 0 is the inverse of the reation frequeny, and the length sale L is in the order of magnitude of the thikness of a flame. The onentration is dimensionless of order and must not be saled; aordingly the onentration gradient is of order /L. We shall be interested in ases where the densities of the onstituents differ only slightly. Therefore we introdue a small parameter ε ϱ by ε ϱ = ϱ. ϱ We adopt the values for our sales and oeffiients from [] where we find hene L = 0 4 m t s, L 0 t 0 = D 9 m =.4 0 ϱ s, ε ϱ =6 0 4 L =7 0 6 m, t 0 =0.035s. Moreover, we find in [] an approximate value for the surfae tension, σ = b x x dx b L =5 N 0 6 m. 35 As we have said in Setion.3, the length sale of the apillary fores is given by l ap = b/a. This length must be expeted to be of the order of some moleular diameters, l ap m. k Using the density and moleular weight of water we have ϱ m T J T m 3 0 = 300 K and therefore b L = b ϱ L kt 0 /m 5 0 9, â = b lap. 36 With â, interation energy and thermal energy are of the same order of magnitude. And indeed, a value â = O is what we expet for misible fluids in whih the moleular fores play a signifiant role. With the above dimensionless parameters, the equations for volume hange and onentration read v k = ε ϱ x k +ε ϱ t + v k, x k t + v Ĵ k k = + ε ϱ + x k x k the dimensionless diffusion flux reads Ĵ k = + b x k L â [ + ε ϱ â 3 + x i x i x k p +â + b x k x k L 3 + x k x i x i x i x i x k ]. x k x i x i 6

17 b Our above approximations show that both, and ε L ϱ, are so small that the influene of the orresponding terms in the diffusion flux on the onentration profile an be ignored. Keeping only the leading order terms we obtain drastially simplified equations, v k x k =0, t + v k =. 37 x k x k x k With no parameters in 37 referring expliitly to the intermoleular fores, it seems that the latter affet neither flame struture nor speed. This is not true, however, sine, as we have pointed out in Setion 4., the intermoleular fores affet the measurement of the diffusion oeffiient and are thus impliit in the saling. The density differene is assumed to be so small that the quasi-inompressibility an be ignored, and the mixture appears to be inompressible. Aordingly, if the reatant is at rest initially, and no external fores at on it, the mixture will remain resting while the flame travels through. If one aounted for the differene in the densities, one finds that the flame will push or drag the unburned reatant with the veloity ε ϱ v F where v F is the flame speed. This would lead to an Oε ϱ orretion, and these are ignored in the onentration equation. It follows that the flame is desribed by Fisher s equation [6, 0, 4] t = x k x k. 38 It is well known, that this equation, also known as KPP equation [0], allows traveling wave solutions with dimensionless speed v F = whih onnet the onstant states =0,=. 4.4 The Hele-Shaw ell Next, we onsider the momentum equation for the geometry of the Hele-Shaw ell. In a Hele- Shaw ell the flow is onfined between two parallel plates. The distane l between the plates is rather small, and one an assume that the flow is purely two-dimensional. With the oordinates hosen suh that the plane of the Hele-Shaw ell is spanned by x α = {x,x } = {x, y} we have for the veloity and the momentum equation reads v i = {v α x α,z, 0} ϱ v α + p x α S αi x i η v α x β x β η v α z z = ϱg α. Under the assumption that the typial length sale λ in the plane is muh larger than the distane of the plates l, wehave η v α + η v α x β x β z z = η v α λ + η v α x β x β l z z η v α l z z = η v α z z where x α and z are suitable dimensionless oordinates. Thus, the frition with the walls is dominating while the frition inside the fluid plays no role. It is well known that a flow between 7

18 τ n dv L τ Fluid Fluid R = dφ x y Figure 4: Volume element dv for determining the jump ondition. parallel plates will develop a paraboli profile Poisseuille flow whih an be written with the mean veloity v α = l l 0 v αdz as [ z z ] v α = 6v α. l l Introduing this expression into the momentum equation and averaging along the plate separation l yields [6, 8] vα ϱ t v β + p x α S αβ x β + η l v α = ϱg α ; 39 v α x β in whih the bars indiate mean values. Here, we have assumed that the onentration is independent of the oordinate z, so that z S αz =0. As we have said before, the flame travels through the fluid without induing any signifiant flow veloity. Moreover, the fluid is not pushed by a pressure gradient and the only external fore on the fluid is related to buoyany. Sine the density differene is very small, we an expet only small veloities and heneforth ignore the non-linear term in 39. Surfae tension will only play a role in the reation zone of thikness L. Thus we an onsider two regions of single fluids, onneted by jump onditions whih we shall determine now. To this end we onsider a small volume element around the reation zone of size dv =llr dφ where R dφ is the ar length of the urve =0.5 with the radius of urvature R, and L is the thikness of the reation wave. L must be hosen so that the onentration gradient at R ± L/ is negligible. In fat L should stritly be reinterpreted as an intermediate asymptoti parameter that is large ompared to the thikness of the reation wave but small ompared with the radius of urvature R and the plate separation l. Although the results below an be written more formally in these terms, we shall simply take onentration gradients to be zero at R ± L/ and ontinue to think of L as measuring the thikness of the reation wave, albeit as a slight overestimate, with L R. The value of L does not appear in the final result. Figure 4 shows the projetion into the {x, y} plane; n denotes the normal to the urves of onstant onentration whih is also the diretion of propagation, and τ denotes the tangent vetors. In a frame where n α = {0, } one finds τ α = {± os dφ, sin dφ} {±, dφ}. We write for the surfae tension S αβ = b x γ x γ δ αβ x α = b x β 8 δ αβ n α n β =Σδ αβ n α n β n

19 The jump ondition follows by integration of the momentum balane over the volume element and subsequent salar produt with n α. Under the assumption that all quantities do not hange onsiderably along the ar length we find after division by lr dφ p p σ R = n α where σ denotes the total surfae tension 35, [ ϱ v ] α t +η l v p + p α ϱg α dl + σ = ΣdL. L L pdl R The Hele-Shaw approximation is only valid for urvatures l/r and the flame thikness L is muh smaller than the distane l, so that L/R and the last term on the right hand side of 40 an be ignored. In immisible fluids the thikness of the interfae is so small a few moleular diameters that also the first term on the right is negligible. In our ase of a reation zone, however, this term may beome important and must be onsidered. It is worth omparing the gravitational ontribution ϱgl with the surfae tension term σ R. stritly be negleted for radii of urvature R than the the thikness of the reation wave. 40 Indeed, the gravitational fore an only σ ϱgl = m. This is still muh larger For further simplifiation we assume the same visosity for the two fluids [] and no variation of veloity inside the reation zone. Being interested in a stationary flame proess, we an ignore the time derivative and find finally the jump ondition aross the reation zone of thikness L as p p σ R =ηl l n αv α n α 4.5 Saffmann-Taylor stability analysis ϱg α dl. With the potential of gravity U = g α x α = gx the momentum equation for the single fluid in the stationary ase an be written as p x α + η l v α = ϱ U x α with v α x α =0. Following the lassial work of Saffmann & Taylor [6], we introdue the veloity potential φ by v α = φ x α with φ x α x α =0. 4 Sine ϱ, η = onstant in the single fluid the momentum equation is easily integrated, p = ϱu η l φ. We are interested in small disturbanes of a plane flame travelling through a fluid at rest in diretion n α = ±{, 0}, with the disturbane of the interfae given by ξ = a exp [iky + ωt], 9

20 ξ refers again to the line =0.5. Therefore, the loation of the boundaries of the reation zone is given by x, = ξ ± L and the interfae ondition reads η l [φ φ ]+ϱ ϱ gξ + σ d ξ dy ηl dξ l dt = ϱ + ϱ gl + ϱgdl. 4 Note, that urvature is given by R = ξ / + ξ 3/ ξ, where the hoie of sign follows from figure 4, and that the veloity of the fluid at the disturbed interfae is given by dξ dt. The integral on the r.h.s. gives the weight per unit area of the reation zone and does not hange when the reation zone is displaed. For the potentials one finds φ = φ 0 aω k exp [iky kx + ωt], φ = φ 0 + aω k exp [iky + kx + ωt] as the solutions of 4 whih give the veloity dφ dx = ξ = ωa exp [iky + ωt] atx = 0 and for whih the disturbanes vanish at infinity [6]. The onstants of integration φ 0 α are set equal to zero in the work of Saffmann & Taylor without disussion, but are important here in order to ensure the validity of the jump ondition for the undisturbed interfae ξ = 0. One finds η [ φ 0 l φ 0 ] ϱ + ϱ = gl + ϱgdl, so that the weight of the reation zone is absorbed in the onstants φ 0 α. The jump ondition 4 gives for the disturbane η ω l k Lk σk + gϱ ϱ =0. Being interested in waves with wavelengths λ = π/k large ompared to the distane l, we an ignore the frition term Lk and arrive at the final result for the dispersion relation ω =ϱ ϱ gl 4η k σl 4η k3. 43 This, indeed, is the elebrated Saffmann-Taylor result, whih therefore holds also in ase of a finite but thin interfae between two fluids. Instabilities will develop when ω is positive, i.e. if ϱ >ϱ, orresponding to lower density of the produt, as is the ase in the experiments of Abid et al. []. The experiments showed a wavelength of approximately m and assuming that the instability is governed by the fastest mode, ω = σl η k 3, k = ϱ ϱ g 3σ, 44 it was possible to measure the surfae tension 35. By hanging the angle α of the Hele-Shaw ell against the vertial, so that the effetive gravitational aeleration is g os α, Abid et al. found experimentally that the wavelength λ =π/k is proportional to / g os α. And it is this proportionality whih allows the onlusion that the instability is governed by surfae tension. If for instane a urvature dependent flame speed were to be of ontrolling importane [8], one would have expeted a proportionality to /g os α. 0

21 5 Conlusion In this paper we have developed a onsistent thermodynami model for a reative binary mixture of inompressible fluids with distributed surfae tension effets. The model allows for the desription of a wide range of proesses, and this artile has only provided a glimpse of this variety. In partiular, we have used our model for the desription of experiments in Hele-Shaw ells [] with buoyany driven instabilities. Our theoretial findings reinfore the interpretation by Abid et al of these experimental results and provide them with a sound theoretial framework. The measured values of the parameters a and b, whih together aount for the intermoleular fores, math expetations very well. In fat, a notieable aspet of the intermoleular fores requires that thermal energy v T and the intermoleular potential energy U mix 4 should be of omparable importane. That is, the dimensionless parameter â = a/ϱkt/m is of the order O. We have shown that the value â arises naturally from the measurement of b and the presumed range of the interation fores, see equation 36. As long as â<, the two fluids are misible for all onentrations i.e. without a gap of misibility, see setion.5, and Fik s law of diffusion prevails in most ases. The influene of the intermoleular fores on the diffusion is hidden in the measurement of the diffusion oeffiient, and ould only be more fully qualified by detailed measurements of onentration profiles. However, if the density differene is larger, hemial equilibrium and diffusion will be strongly influened by hanges in the pressure and its gradient. Investigations of other problems based on the model provided in this artile are planned for the future. Aknowledgements: We would like to thank Paul Ronney University of Southern California for bringing the phenomenon to our attention, for useful disussions and for making a preprint of the paper by Abis et al. available to us. This study was arried out at the Institute for Mathematis and its Appliations IMA during its program on Reative Flows and Transport Phenomena and we wish to thank the IMA for their hospitality and their support. Referenes [] M. Abid, J.-B. Liu & P.D. Ronney Surfae tension and fingering of misible interfaes. Submitted to Phys. Rev. Lett, 999 [] D.M. Anderson, G.B. MFadden & A.A. Wheeler Diffusive-interfae methods in fluid mehanis. Annu. Rev. Fluid Meh. 30, [3] L.K. Antanovskii A phase field model of apillarity. Phys. Fluids 7, [4] M.R. Booty, R. Haberman & A.A. Minzoni The aomodation of travelling waves of Fisher s type to the dynamis of the leading tail. SIAM J. Appl. Math. 53, [5] J.W. Cahn & J.E. Hilliard Free Energy of a Nonuniform System. I. Interfaial Free Energy. J. Chem. Phys. 8, [6] R.A. Fisher The wave of advane of advantageous genes. Ann. Eugenis 7, [7] S.R. de Groot & P. Mazur Non-equilibrium thermodynamis. Dover, New York 984

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