On the Two-Phase Navier-Stokes Equations with Boussinesq ... - MMA

ON THE TWO-PHASE NAVIER-STOKES EQUATIONS WITH BOUSSINESQ-SCRIVEN SURFACE FLUID ¨ DIETER BOTHE AND JAN PRUSS Abstract. Two-phase flows with interface modeled as a BoussinesqScriven surface fluid are analysed concerning their fundamental mathematical properties. This extended form of the common sharp-interface model for two-phase flows includes both surface tension and surface viscosity. For this system of partial differential equations with free interface it is shown that the energy serves as a strict Ljapunov functional, where the equilibria of the model without boundary contact consist of zero velocity and spheres for the dispersed phase. The linearizations of the problem are derived formally, showing that equilibria are linearly stable, but nonzero velocities may lead to problems which linearly are not wellposed. This phenomenon does not occur in absence of surface viscosity. The present paper aims at initiating a rigorous mathematical study of two-phase flows with surface viscosity.

Mathematics Subject Classification (2000): Primary: 35R35, Secondary: 35Q30, 76D45, 76T10. Key words: Navier-Stokes equations, surface tension, surface viscosity, equilibria, asymptotic behaviour, Ljapunov functionals, well-posedness, linearization.

1. Introduction In the commonly applied sharp-interface continuum mechanical model of twophase flows, the interfacial momentum transition is usually described by the jump condition −[[S]]ν = σκΓ ν,

which connects the normal stress at the interface with surface tension and curvature. The underlying constitutive law for the surface stress does not account for internal friction within the interface, hence dissipation of kinetic energy is solely due to viscous damping inside the bulk phases. Because of surface mobility, a fluid particle moving in accordance to this model should then experience a smaller drag force compared to a solid particle of otherwise identical properties. Corresponding theoretical predictions by Hadamard [5] and Rybczynski [8] for the settling velocity of a spherical fluid droplet where contradicted by experments of Lebedev [6] and Silvey [10]. At the same time Boussinesq [3], apparently in an attempt to resolve such discrepancies, postulated that a kind of surface viscosity has to be incorporated in the interfacial consitutive law. Much later Levich [7] suggested that the difference in droplets’ settling velocities is due to a Marangoni effect, i.e. due to interfacial stresses induced by surface tension 1

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¨ DIETER BOTHE AND JAN PRUSS

gradients which - in this case - are caused by the presence of surface-active agents, so-called surfactants. Today, it is clear that both effects are strongly related and should be accounted for in the modeling of two-phase flows. Indeed, a coverage of the interface with surfactant will simultaneously lead to a variable surface tension, which depends on the area specific concentration of adsorbed surfactant, and to both shear and dilatational viscosities of the interface. In case of variable surface tension, the constitutive law above only requires the addition of the surface gradient of the surface tension. However, an additional balance equation for the surfactant species mass is of course required. The correspondingly extended model is wellknown in the Engineering Sciences; cf., e.g., [11]. Mathematically, its well-posedness has recently been obtained in [2], locally-in-time. The mathematical formulation for two-phase flow with surface viscosity for the general case of non-planar interfaces is more involved since it requires additional concepts from differential geometry. In fact, it was not until 1960 that Scriven [9] extended the Boussinesq surface stress tensor to material interfaces having arbitrary curvature. This contribution, together with the famous paper of Sternling and Scriven [12] on the Marangoni effect can be seen as the beginning of modern interfacial rheology. Starting from the linear stress-strain relationship of a Newtonian surface fluid - as the Boussinesq-Scriven surface fluid is also named - there currently is an ever-growing interest in understanding even more complex fluids. The reasons are at least threefold: (i) besides classical applications like emulsions and foams, the present developments in Micro-Engineering to create so-called lab-on-a-chip’s requires manipulating liquid quantities in the pico liter range and at such area-tovolume ratios the surface properties are completely dominant; (ii) the increasing use of and reliance on mathematical descriptions of biological processes requires improved models of cells and cell membranes which is closely related to the modeling of liquid interfaces; (iii) on the basis of new measurement techniques, better experiments allow for an increasing quantitative understanding of the interaction forces in twodimensional liquid layers. For more information on both, the modeling of interfacial transport processes, and the field of interfacial rheology, we recommend the monographs [4] and [11]. In the present paper we take the original viewpoint of Boussinesq and Scriven, i.e. we do not consider a separate mass balance but account for surface viscosities, resulting in a momentum balance on the interface. If the interface is material, this results in a Navier-Stokes equation on the free boundary which, however, is of compressible type, since one is not allowed to assume that the mass density on the surface is constant. The model formulated below excludes phase changes and exchange of mass to or from the interface. Therefore it exhibits conservation of total mass of the bulk phases and of interface mass, and the energy equality is valid, for smooth solutions, at least. We show that the equilibria of the system without boundary contact (in absence of external forces like gravity) are zero velocity, and the disperse phase consists of not overlapping spheres with the pressure π being constant in the components of the phases. Moreover, we prove that the energy serves as a strict Ljapunov-functional of the system. We derive the linearization of the problem at such an equilibrium as well as the principal linearization for a general state. It is shown that at an equilibrium the

TWO-PHASE NAVIER-STOKES EQUATIONS WITH SURFACE FLUID

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linearization admits no eigenvalues with positive real part, and that 0 is always a semi-simple eigenvalue with eigenspace of dimension n if conservation of phases is taken into account. This is precisely the dimension of the manifold of equilibria E, and the eigenspace is isomorphic to the tangent space of E. By the generalized principle of linearized stability this indicates that these equilibria are stable and each solution starting near such an equilibrium converges to another one. This will be made precise in a forthcoming paper. We also present an analysis of the principal linearization at an arbitrary state of the system. The relevant boundary symbols are investigated, and it is shown that for small velocities the principal boundary symbol is well-posed in the sense of Hadamard, but it may be illposed for large velocities. The analysis shows that the velocities at the interface are uniquely determined by the geometry of the interface once it is considered as known, and are smooth if the geometry is. Thus illposedness comes from geometry, which indicates that experimentally observed singularities like cusp-shaped bubbles are supported by analysis, without employing more complicated models like non-Newtonian surface fluids. It seems that this paper is the first one to deal with the mathematical analysis of two-phase flows with surface viscosities in the general case of interfaces of arbitrary curvature. However, this paper is just a starting point, since mostly we employ symbolic calculus and formal arguments; in forthcoming papers we intend to set this analysis on rigorous grounds.

2. The Mathematical Model. In this section we introduce the relevant mathematical model which can also be found in [1], [4] and [11]. We consider isothermal flows of two immiscible Newtonian fluids with constant densities. Based on continuum mechanics, balance of mass and momentum inside the phases lead to the Navier-Stokes equations, i.e. ∇ · u = 0,

∂t (ρi u) + ∇ · (ρi u ⊗ u − S) = 0

with stress tensor S = −πI + 2µi D = −πI + µi ∇u + (∇u)T .

Here u = u(t, x) and π = π(t, x) denote the velocity field and the pressure, respectively. These balances hold in Ω1 (t) ∪ Ω2 (t), where Ωi (t) are the domains occupied at time t by two fluids with densities ρi > 0 and viscosities µi > 0, respectively. Hence the material parameters ρi and µi depend on the phase i, but otherwise are assumed to be constant. Whenever a distinction between the different phases is not necessary, the phase index i is omitted. The phases are separated by a phase boundary Γ(t) at which at least one of the material parameters has a jump-discontinuity. At this interface, the additional jump conditions [[ρ(u − uΓ )]] ν

= Φρ ,

[[ρu ⊗ (u − uΓ ) − S]] ν

= Φu ,

appear, where uΓ means the interfacial velocity, and Φρ resp. Φu are sources at the interface for mass and momentum, respectively. Before we discuss these quantities,

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let us introduce the unit normal ν = ν(t, x) at the interface directed into the phase Ω2 (t), and κ = κΓ = −divΓ ν, the sum of the (local) principal curvatures of the interface. More precisely, it should read divΓ(t) , but the dependence on t is dropped whenever this is reasonable. Thus κ(t, x) < 0 at x ∈ Γ(t) iff Ω1 (t) ∩ Br (x) is convex for small r > 0, in particular κ = −(n − 1)/R in case Ω1 is a ball with radius R. The notation [[φ]] stands for the jump of a physical quantity φ across the interface, i.e. [[φ]](t, x) = lim φ(t, x + h ν) − φ(t, x − h ν) , x ∈ Γ(t). h→0+

If the surface is material then balance of interface mass yields ds D ρΓ := ρΓ + ρΓ divΓ uΓ = −Φρ , dt Dt where ρΓ means surface density, uΓ surface velocity, and D/Dt = ∂t + (u|∇) the Lagrangian derivative w.r.t. the velocity field u. Since the partial time-derivative of an interfacial quantity is not defined, in general, we also split the Lagrangian derivative according to D/Dt = ∂t,n + (u|∇Γ ), where ∂t,n stands for the derivative along the purely normal path corresponding to the given velocity field. Similarly, for the surface momentum ρΓ uΓ we obtain the balance law D ds (ρΓ uΓ ) = (ρΓ uΓ ) + (ρΓ uΓ )divΓ uΓ = divΓ SΓ − Φu , dt Dt where SΓ means surface stress, for which a constitutive law has to be added. Below, it is assumed that the two-fluid system under consideration does not endure phase changes like melting or evaporation. Consequently, there are no convective fluxes across the interface, i.e. Φρ = 0, and hence the normal components of the fluid velocities are continuous at the phase boundary, and the interface itself is advected with the flow, i.e. (u1 · ν)|Γ(t) = (u2 · ν)|Γ(t) = V,

where V := uΓ · ν denotes the normal velocity of the interface. This implies also ds D ρΓ = ρΓ + ρΓ divΓ uΓ = 0. dt Dt In the major part of this paper we suppose that the interface is non-material, i.e. ρΓ = 0. Furthermore, we assume no-slip for the velocity u at the interface. Then the jump conditions simplify to [[u]] = 0,

u Γ = u |Γ ,

[[−S]] ν = Φu .

For the surface stress SΓ we use the Boussinesq-Scriven law 1 SΓ = [σ + (λs − µs )divΓ u]PΓ + 2µs DΓ , DΓ = PΓ (∇Γ u + ∇Γ uT )PΓ , 2 where σ > 0 denotes surface tension, λs > µs dilatational, µs > 0 surface shear viscosity, and PΓ = I − ν ⊗ ν means the tangential projection at Γ. In the sequel we assume that σ, λs , and µs are constant. Note that the surface divergence div Γ u does not vanish, in general. The full model of a two-phase flow with surface tension and surface viscosity now reads as


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Balance equations. (2.1)

∇·u=0

∂t (ρi u) + ∇ · (ρi u ⊗ u − S) = 0,

S = −πI + 2µi D,

D = 21 (∇u + ∇uT ),

for t > 0, x ∈ Ωi (t),

for t > 0, x ∈ Ωi (t). for t > 0, x ∈ Ωi (t).

Free boundary conditions. [[u]] = 0, ds dt ρΓ

(2.2)

ds dt (ρΓ u)

V =u·ν

for t > 0, x ∈ Γ(t),

for t > 0, x ∈ Γ(t),

= 0,

= divΓ SΓ + [[S]] ν

SΓ = [σ + (λs − µs )divΓ u]PΓ + 2µs DΓ DΓ = 21 PΓ (∇Γ u + ∇Γ uT )PΓ

for t > 0, x ∈ Γ(t),

for t > 0, x ∈ Γ(t), for t > 0, x ∈ Γ(t).

Initial conditions. (2.3)

u(0, x) = u0 (x), Γ(0) = Γ0 .

for x ∈ Ωi (0),

To complete the model we assume that Ω = Ω1 (t) ∪ Γ(t) ∪ Ω2 (t) is constant, i.e. Ω is a fixed container, with noslip condition u = 0 at ∂Ω, and no boundary intersection, i.e. Γ(t) ∩ ∂Ω = ∅, to avoid the contact angle problem. It is important to observe that in our formulation the total mass Z Z M (t) := ρdx + ρΓ (t)dΓ Ω

Γ(t)

is a conserved quantity. In virtue of div u = 0 in Ω \ Γ, V = (u|ν) and ddts ρΓ = 0, this follows directly from the transport and surface transport theorem. In fact, we have Z Z d ds M (t) = [[ρ]] (u · ν − V )dΓ + ρΓ = 0. dt Γ(t) Γ(t) dt It is of some interest to write the equations explicitly. For u in Ω \ Γ(t) we obtain the usual Navier-Stokes equations. ρ(∂t u + (u · ∇)u) − µ∆u + ∇π = 0,

t > 0, x ∈ Ω \ Γ(t).

For the momentum balance on Γ(t) we have, with the curvature tensor KΓ = −∇Γ νΓ and κΓ = tr KΓ , ρΓ (∂t,n u + u · ∇Γ u) = (2[[µD]]ν − [[π]]ν) + µs PΓ ∆Γ u + λs ∇Γ divΓ u + µs κΓ ∇Γ uν + (σκΓ + (λs − µs )κΓ divΓ u + 2µs tr (KΓ DΓ ))ν.

Note that KΓ is symmetric and observe that for ρΓ = λs = µs = 0 we have the usual free boundary condition which only involves surface tension, namely 2[[µD]]ν − [[π]]ν + σκΓ ν = 0.

If there is surface mass, i.e. if ρΓ > 0, the problem is more complicated than for ρΓ = 0, and conservation of surface mass has to be taken into account: ∂t,n ρΓ + u · ∇Γ ρΓ + ρΓ divΓ u = 0.

Note that it does not make sense physically to assume that ρΓ > 0 is constant, since then total mass will not be conserved, in general. Although in this model there

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is no exchange of mass between the interface and the two bulk phases, one has to take into account change of mass density on the surface, e.g. due to its stretching during motion. For this reason we restrict our attention to the case ρΓ = 0, except for the next section.

3. The Energy Functional and Equilibria One should observe that there are conserved quantities other than total mass, namely the phase volumes, hence also mass of the two phases. This follows from Z Z Z Z d dx = V dΓ = (u|ν)dΓ = div u dx = 0. dt Ω1 (t) Γ(t) Γ(t) Ω1 (t) To construct the energy functional for the system we proceed as follows. First, the transport theorem yields Z Z d ρi 2 |u| dx = ρi [u · ∂t u + div(|u|2 u/2)]dx dt Ωi (t) 2 Ωi (t) Z = ρi [u · ∂t u + (u · ∇)(|u|2 /2)]dx Ωi (t) Z = ρi [∂t u + (u · ∇)u] · u dx Ωi (t) Z = div S · u dx Ωi (t) Z Z = Sνi · u dΓ − S : ∇u dx ∂Ωi (t) Ωi (t) Z Z = Sνi · u dΓ − 2 µi D : ∇u dx. ∂Ωi (t)

Ωi (t)

Hence, adding the equations for i = 1, 2, we obtain the identity (3.1)

1 ∂t ||ρ1/2 u||2Ω + 2||µ1/2 D||2Ω + ([[Sν]]|u)Γ(t) = 0. 2

The next identity follows by integration of the stress condition on Γ(t). Employing the surface transport theorem we get Z Z d ρΓ D ρΓ ρΓ |uΓ |2 dΓ = ( |uΓ |2 ) + ( |uΓ |2 )divΓ u dΓ dt Γ(t) 2 Dt 2 2 Γ(t) Z D D 2 = ( ρΓ + ρΓ divΓ u)|u| /2 + ρΓ u · u dΓ Dt Dt Γ(t) Z D |u|2 D = u· (ρΓ u) + ρΓ udivΓ u − ( ρΓ + ρΓ divΓ u)dΓ Dt 2 Dt Γ(t) Z Z ds = u · (ρΓ u)dΓ = u · ([[S]]ν + divΓ SΓ )dΓ. dt Γ(t) Γ(t)


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Next we have Z Z Z − u · divΓ SΓ dΓ = tr(SΓ ∇Γ uT )dΓ = 2µs tr(DΓ ∇Γ uT )dΓ Γ(t) Γ(t) Γ(t) Z + (σ + (λs − µs )divΓ u)tr(PΓ ∇Γ uT )dΓ Γ(t) Z Z = 2µs tr(DΓ2 ) dΓ + σdivΓ u + (λs − µs )|divΓ u|2 dΓ Γ(t) Γ(t) Z Z 2 = 2µs tr(DΓ )dΓ + − σκV + (λs − µs )|divΓ u|2 dΓ. Γ(t)

Γ(t)

Summing these identities this implies 1 1/2 2 1 1/2 ∂t ||ρ u||Ω + ||ρΓ uΓ ||2Γ 2 2 Z 2 2 = −2 ||µ1/2 D||2Ω + ||µ1/2 D || − (λ − µ )||div u|| + σ Γ s s Γ s Γ Γ

κV dΓ. Γ(t)

Finally, the surface transport theorem yields the relation Z Z d dΓ = − κV dΓ, dt Γ(t) Γ(t) which leads to the energy identity ∂t (3.2)

1 1/2 2 1 1/2 ||ρ u||Ω + ||ρΓ u||2Γ + σ 2 2

Z

dΓ Γ(t)

!

2 2 + 2||µ1/2 D||2Ω + 2||µ1/2 s DΓ ||Γ + (λs − µs )||div Γ u||Γ = 0

This shows that that the energy functional 1 1/2 2 1 1/2 ||ρ u||Ω + ||ρΓ u||2Γ + σ meas Γ(t) 2 2 is a Ljapunov functional for the problem under consideration. Let us look at equilibria of the problem (2.1), (2.2), (2.3), assuming for simplicity that the phases are connected. If (u∗ , Γ∗ ) is a time-independent solution then the energy balance (3.2) yields D ≡ DΓ ≡ divΓ u = 0, as well as ∂t u = V = 0. Hence we obtain ∇u ≡ 0 in Ω by Korn’s inequality, which gives u ≡ 0 from the no-slip condition on ∂Ω. By the equations, this in turn implies ∇π ≡ 0 hence the pressure is constant in each subregion Ωi , and finally from the free boundary condition we get −[[π]] + σκ = 0. In case of no boundary contact Γ∗ ∩ ∂Ω = ∅, boundedness of Ω implies that Γ∗ is a sphere Γ∗ = SR∗ (x0 ) ⊂ Ω and [[π]] = −σ/R∗ . We denote the set of equilibria without boundary contact by E. Note that the constraint meas Ω1 ≡ d0 determines the radius R∗ of such a sphere, we have R∗ = [d0 /|B1 (0)|]1/n , or R∗ = [d0 /|B1 (0)|]1/n /N in case Ω1∗ consists of N balls. We want to prove that Φ is even a strict Ljapunov functional. For this purpose, suppose ∂t Φ(u, Γ) = 0 for some time t0 . This implies D = 0, which once more yields u = 0 by Korn’s inequality and the no-slip condition at ∂Ω. From the equations we then get at time t0 Φ(u, Γ) :=

ρ∂t u + ∇π = 0,

∂t (ρΓ u) = (σκ − [[π]])ν,


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hence ∆π = 0 in Ω \ Γ(t0 ), as well as ∇π = 0 on ∂Ω. By the elliptic continuation principle this yields π = constant in Ω2 , hence also ∂t u = 0 in Ω2 . Therefore we have ∂t u = 0 as well as ∇π = 0 on Γ(t0 ), hence π is constant also on Ω1 , again by the unique continuation property. This finally yields [[π]] = σκ, and so Γ(t0 ) is again a sphere. This shows that states (u, Γ) which are such that Γ does not intersect ∂Ω and dΦ(u, Γ)/dt = 0 are equilibria of the system. Let us have a look at the critical points (u∗ , Γ∗ ) of Φ with the constraint Z dx = d0 . Ψ(Γ) := Ω1

The method of Lagrange multipliers yields a real parameter λ such that Φ0 (u∗ , Γ∗ ) + λΨ0 (Γ∗ ) = 0.

We compute the Gateaux derivatives of the functionals. This yields Z < Ψ0 |h >= hdΓ∗ , Γ∗

and

0

< Φ |(v, h) >=

Z

Ω

ρu · vdx +

Z

Γ

ρΓ u · vdΓ −

Z

1 ( ([[ρ]] + ρs κΓ )|u|2 + σκΓ )hdΓ. 2 Γ

Setting first h = 0 this yields u∗ = 0, then varying h we obtain λ = σκΓ , i.e. κΓ is constant. Thus the critical points of the functional Φ with constraint Ψ = d0 are precisely the equilibiria of the system. Let us summarized what we have proved. Theorem 3.1. Let ρi , µi , σ > 0, λs ≥ µs ≥ 0 be constants. Then the following assertions hold. (a) The energy equality (3.2) is valid for smooth solutions. (b) The equilibria are zero velocities, constant pressures in the components of the phases, the dispersed phase is a union of nonintersecting balls. (c) The energy functional is a strict Ljapunov-functional. (d) The critial points of the energy functional for constant phase volumes are precisely the equilibria. 4. Equilibria and Linearization From now on we assume ρΓ ≡ 0. The full linearization of the problem at an equilibrium (0, π∗ , Γ∗ ) with reference frame Σ := Γ∗ becomes ρ∂t v − 2µdiv D + ∇q = 0 div v = 0, D =

1 2 (∇v

v=0

t > 0, x ∈ Ω \ Σ

T

+ ∇v )

t > 0, x ∈ Ω \ Σ

t > 0, x ∈ ∂Ω

0

(4.1)

2[[µD]]ν − [[q]]ν + σκ (0)hν

+ (λs − µs )∇Σ divΣ v + 2µs divΣ DΣ = 0

[[v]] = 0, DΣ =

1 2 PΣ (∇Σ v

∂t h − (v|νΣ ) = 0

v(0) = v0

h(0) = h0

T

+ ∇Σ v )PΣ ,

t > 0, x ∈ Σ

t > 0, x ∈ Σ

t > 0, x ∈ Σ

x ∈Ω\Σ

x∈Σ


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The eigenvalue problem corresponding to (4.1) reads ρλv − 2µdiv D + ∇q = 0 1 2 (∇v

div v = 0, D = v=0 (4.2)

x∈ Ω\Σ

T

+ ∇v )

x∈ Ω\Σ

x ∈ ∂Ω

0

2[[µD]]ν − [[q]]ν + σκ (0)hν

+ (λs − µs )∇Σ divΣ v + 2µs divΣ DΣ = 0

[[v]] = 0, DΣ =

1 2 PΣ (∇Σ v

λh − (v|νΣ ) = 0

T

+ ∇Σ v )PΣ ,

x∈Σ

x∈Σ

x∈Σ

Suppose λ ∈ C with Re λ ≥ 0 is an eigenvalue with eigenfunction (v, q, h). Then the L2 -inner product of the equation for v with v¯ implies Z 1/2 2 1/2 2 λ||ρ v||2 + 2||µ D||2 + (2[µDν] − [[q]]ν) · v¯dΣ = 0, Σ

and the stress boundary condition yields, integrating by parts as in section 3, Z Z (2[[µDν]]−[[q]]ν)·¯ v dΣ+σ κ0 (0)h(v|ν)dΣ−2µs ||DΣ ||2Σ −(λs −µs )||divΣ v||2Σ = 0. Σ

Σ

Adding these two equations and using λh = (v|ν) we get (4.3) 0 ¯ λ||ρ1/2 v||22 + 2||µ1/2 D||22 − λσ(κ (0)h|h)Σ + 2µs ||DΣ ||2Σ + (λs − µs )||divΣ v||2Σ = 0.

For λ 6= 0 in the right half-plane we have Z Z Z λ hdΣ = (v|νΣ )dΣ = Σ

Σ

div v dx = 0, Ω1

hence h has mean value zero. On functions with mean zero −κ0 (0) = −(n − 1)/R2 − ∆Σ

is selfadjoint and positive semidefinite. Hence in this case, taking real parts, (4.3) implies D = 0, and then v = 0 by Korn’s inequality and the no-slip condition at ∂Ω, hence also h = 0. This implies that no λ 6= 0 in the closed right half-plane can be an eigenvalue. If we consider λ = 0 then again D = 0 by (4.3) hence v = 0, and q is constant in each of the subdomains. The stress boundary condition yields [[q]] = σκ0 (0)h, 2

R a differential equation for h. A special solution is given by h0 = (n−1)σ [[q]], and the solutions of the homogeneous equation κ0 (0)h = 0 are the spherical harmonics of degree one, which form an n-dimensional subspace. Since so far [[q]] is arbitrary we obtain an n + 1-dimensional eigenspace for the eigenvalue λ = 0. On the other hand, conservation of volumes requires that h has mean value zero, which reduces one degree of freedom and gives [[q]] = 0. Summarizing, we have

Theorem 4.1. Let ρi , µi , σ > 0, λs ≥ µs ≥ 0 be constants and assume ρΓ ≡ 0. Then the following assertions hold. (a) The linearization of the problem has no eigenvalues λ with nonnegative real part other than λ = 0. (b) λ = 0 is an eigenvalue with multiplicity n + 1.

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Observe that the equilibrium in question is contained in an n + 1-parameter family of equilibria, where the parameters are the center and the radius of the spheres. The n + 1-dimensional eigenspace for λ = 0 resembles this fact, actually the equilibria form the center manifold of the nonlinear problem at these equilibria. The eigenspace corresponding to the eigenvalue λ = 0 is isomorphic to the tangent space of the manifold E of equilibria of the nonlinear problem. Each equilibrium is stable and solutions starting near such one will converge to another equilibrium. These will assertions be proved in a subsequent paper. 5. The Linearized Problem at Zero Velocities Note that the linearization obtained in the previous section applies to any state of the system with zero velocity. However, in case Σ is not a sphere then, in general, κ0 (0) = tr KΣ2 + ∆Σ will not be negative semidefinite. Thus for arbitrary initial surfaces Σ we do not expect stability, but well-posedness will hold for small initial velocities and arbitrary smooth initial hypersurfaces. To derive the relevant analytic properties of the linearized problem at an arbitrary state (0, Σ) with zero velocity, it is useful to write the linearized equations in explicit form. This leads to ρ∂t u − µ∆u + ∇π = 0,

div u = 0, u=0

(5.1) T

∂t h − (u|νΣ ) = 0,

PΣ [[µ(∇u + ∇u )]]νΣ + µs PΣ ∆Σ u +λs ∇Σ divΣ u + µs κΣ ∇Σ uνΣ = 0, 2[[µ(∂νΣ u|νΣ )]] − [[π]] + σκ0 (0)h

+2µs tr(KΣ ∇Σ u) + (λs − µs )κΣ divΣ u = 0, u(0) = u0 , h(0) = h0 .

t > 0 x ∈ Ω \ Σ,

t > 0, x ∈ Ω \ Σ t > 0, x ∈ ∂Ω t > 0, x ∈ Σ

t > 0, x ∈ Σ t > 0, x ∈ Σ

Here we already decomposed the stress boundary condition into its tangential and its normal part. On the boundary Σ it is convenient to split also the velocity u into tangential and normal velocities. This will give more insight into the structure of the stress boundary condition. We set u = v + wνΣ ,

v = PΣ u,

w = (u|νΣ ).

Then the following relations hold. divΣ u = divΣ v − κΣ w, ∇Σ divΣ u = ∇Σ divΣ v − κΣ ∇Σ w − w∇Σ κΣ ,

(∇Σ u) νΣ = (∇Σ v) νΣ + ∇Σ w = KΣ v + ∇Σ w, tr(KΣ ∇Σ u) = tr(KΣ ∇Σ v) − w tr KΣ2 ,

PΣ ∆Σ u = PΣ ∆Σ v − 2KΣ ∇Σ w + w(PΣ ∆Σ νΣ ).

Similarly, with ∂νΣ νΣ = 0, the decomposition of ∇u leads to ∇u = (∇Σ + νΣ ∂νΣ )(v + wνΣ )

= ∇Σ v + νΣ ⊗ ∂νΣ v + ∇Σ w ⊗ νΣ + (∂νΣ w)νΣ ⊗ νΣ + w∇Σ νΣ .


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Employing the identities 0 = ∂νΣ (v|νΣ ) = (∂νΣ v|νΣ ) + (v|∂νΣ νΣ ) and 0 = ∇Σ (v|νΣ ) = ∇Σ vνΣ + ∇Σ νΣ v, where ∇Σ νΣ = −KΣ , KΣ the curvature tensor of Σ, this yields PΣ (∇u + ∇uT )νΣ = ∂νΣ v + ∇Σ w + KΣ v, (∂νΣ u|νΣ ) = ∂νΣ w.

Thus the stress boundary condition can be written as the following system. (5.2) B0 (v, w)T + B1 (v, w)T + [[(µ[∂νΣ v + ∇Σ w + KΣ w], 2µ∂νΣ w − π)T ]] = (0, −σκ0 (0))T h, where B0 =

"

µs PΣ ∆Σ + λs ∇Σ divΣ

(λs − µs )κΣ divΣ + 2µs tr(KΣ ∇Σ )

−[(λs − µs )κΣ + 2µs KΣ ]∇Σ 0

#

,

and B1 =

"

µ s κΣ KΣ

µs PΣ ∆Σ νΣ + λs ∇Σ divΣ νΣ −(λs − µs )κ2Σ − 2µs tr KΣ2

0

#

.

It is well-known in differential geometry that ∆Σ νΣ = −∇Σ κΣ − (tr KΣ2 )νΣ ; this yields B1 =

"

µ s κΣ KΣ 0

−(µs + λs )∇Σ κΣ

−(λs − µs )κ2Σ − 2µs tr KΣ2

#

.

Note that B1 is of order zero, hence a lower-order perturbation of B0 . It is of interest to observe that −B0 − B1 is selfadjoint and positive semidefinite in L2 (Σ; Rn ). For the case of an equilibrium (0, SR (x0 )), we obtain KΣ = −PΣ /R,

κΣ = −(n − 1)/R,

∇Σ κΣ = 0,

hence in this case " µs PΣ ∆Σ + λs ∇Σ divΣ B0 = −[((λs − µs )(n − 1) + 2µs )/R]divΣ

∆Σ νΣ = (n − 1)νΣ /R2 ,

[((λs − µs )(n − 1) + 2µs )/R]∇Σ 0

#

,

and B1 =

"

µs (n − 1)/R2 0

0 −[(λs − µs )(n − 1)2 + 2µs (n − 1)]/R2

#

.

It is interesting to observe that surface viscosity only regularizes the tangential, but not the normal velocity. The analysis of this problem is by no means easy, we can show that it is well-posed, but postpone this to a forthcoming paper.

12


6. Principal Linearization at Nonzero Velocities For the discussion of well-posedness we need the linearization of the problem at an arbitrary state (u∗ , Σ). Actually, only the principal part of the linearization is of importance. In the bulk phases this will be the same as before, since the equations are linear in u, q, besides the convection term which, however, is of lower order. Thus we have ρ∂t u − µ∆u + ∇π = 0, div u = 0, u=0 u(0) = u0 ,

t > 0 x ∈ Ω \ Σ, t > 0, x ∈ Ω \ Σ t > 0, x ∈ ∂Ω x ∈ Ω \ Σ.

For the dynamics of the free boundary the main part is ∂t h − (u|νΣ ) = 0,

h(0) = h0 .

So, non-trivial is only the linearization of the stress boundary condition. Since it is linear in u and π, as before the linearization w.r.t. (u, π) is given by B0 (v, w)T + B1 (v, w)T + DN (v, w))T , where DN denotes the Dirichlet-to-Neumann operator defined by v DN := [[(µ[∂νΣ v + ∇Σ w + KΣ w], 2µ∂νΣ w − π)T ]]. w It remains to compute the derivative w.r.t. the height function h. For this we observe that we are only looking for terms which contain second order derivatives of h. Since νΓ = νΣ − ∇Σ h + o(|h|1,∞ ), we see that there is no contribution from the term [[SνΓ ]]. Next, we have for a vector field f divΓ f = divΣ f + h tr(KΣ ∇Σ f ) + (∇Σ h|∇Σ f νΣ ) + o(|h|1,∞ ), hence up to first order divΓ = divΣ . Next, we have PΓ = PΣ + νΣ ⊗ ∇Σ h + ∇Σ h ⊗ νΣ + o(|h|1,∞ ), and ∇Γ u∗ = ∇Σ u∗ + h KΣ ∇Σ u∗ + (∇Σ h|∇Σ u∗ )νΣ + o(|h|1,∞ ).

With κ0 (0) = ∆Σ + tr KΣ2 , this implies for the principal linearization C w.r.t. h " # µs (∇Σ u∗ νΣ )∆Σ h + λs ∇2Σ h∇Σ u∗ νΣ Ch = . 2µs tr ∇Σ u∗ ∇2Σ h + (σ + (λs − µs )divΣ u∗ )∆Σ h

Thus the principal part of the linearization of the stress boundary condition reads as follows (6.1)

B0 (v, w)T + DN (v, w)T + Ch = 0.


13

7. The Dirichlet-to-Neumann Operator for the Stokes Equation For the discussion of well-posedness we will need the Dirichlet-to-Neumann operator for the two-pase Stokes equation with flat interface. This means we want to solve the following problem. ρ∂t u − µ∆u + ∇π = 0,

div u = 0, u=f u(0) = 0,

t > 0 x ∈ Rn−1 , y 6= 0,

t > 0, x ∈ Rn−1 , y 6= 0, t > 0, x ∈ Rn−1 , y = 0,

x ∈ Rn−1 , y 6= 0.

Here we have split the spatial variable into (x, y) ∈ Rn−1 × R. We then define the Dirichlet-to-Neumann operator by means of DN f = −[[S]]en = −[[µ(∇u + ∇uT )]]en + [[π]]en .

It is convenient to split u into u = (v, w)T and f = (g, h)T . Then we obtain DN f = (−[[µ]]∇x w − [[µ∂y v]], −2[[µ∂y w]] + [[π]])T ,

by continuity of ∇x w across the boundary y = 0. We employ Laplace transform in t and Fourier transform in the tangential variables x ∈ Rn−1 , to the result ω 2 v − µ∂y2 v + iξπ = 0,

y 6= 0,

(iξ|v) + ∂y w = 0,

y 6= 0,

2

ω w−

µ∂y2 w

+ ∂y π = 0,

v(0) = g, w(0) = h.

y 6= 0,

Here we have set ωj2 = ρj λ + µj |ξ|2 , j = 1, 2. Actually, we should have written something like vˆ, etc. to indicate that these functions have been transformed, but for simplicity we drop the hats. This system of equations is easily solved to the result       a2 −iξ v2 √ √  w2  = e−ω2 y/ µ2  µ2 (iξ|a2 )  + α2 e−|ξ|y  |ξ|  , ω2 ρ2 λ π2 0 for y > 0, and       a1 v1 −iξ √ √ µ  w1  = eω1 y/ µ1  − 1 (iξ|a1 )  + α1 e|ξ|y  −|ξ|  , ω1 π1 ρ1 λ 0

for y < 0. Here ai ∈ Rn−1 and αi have to be determined by the boundary conditions v(0) = g and w(0) = h. We have and

a2 − iξα2 = g = a1 − iξα1 , √ √ µ2 µ1 (iξ|a2 ) + |ξ|α2 = h = − (iξ|a1 ) − |ξ|α1 . ω2 ω1

This yields aj = g + αj iξ, and

j = 1, 2,

√ ω2 + |ξ| µ2 √ α2 = − ( µ2 (g|iξ) − ω2 h), ρ2 λ|ξ|


14

as well as

√ ω1 + |ξ| µ1 √ ( µ1 (g|iξ) + ω1 h). ρ1 λ|ξ| We may now compute the symbol of the Dirichlet-to-Neumann operator to the result " # √ √ ω1 µ1 a1 + ω2 µ2 a2 − (α1 µ1 + α2 µ2 )|ξ|iξ − [[µ]]iξh DN f = . 2(µ2 a2 − µ1 a1 ))iξ + 2(α2 µ2 − α1 µ1 )|ξ|2 + λ(α2 ρ2 − α1 ρ1 ) α1 = −

Simple algebraic manipulations then yield the following symbol for the Dirichletto-Neumann operator. α + βζ ⊗ ζ iγζ (7.1) DN (λ, ξ) = , −iγζ T α+δ

where ζ = ξ/|ξ| and

√ √ α = µ1 ω1 + µ2 ω2 , β = (µ1 + µ2 )|ξ|, √ √ γ = ( µ2 ω2 − µ1 ω1 ) − [[µ]]|ξ|, δ = (ω12 + ω22 )/|ξ|. Observe that for real λ > 0 this matrix is selfadjoint, hence the form v v (DN | ) = α|v|2 + β|(v, ζ)|2 + (α + δ)|w|2 + 2γIm (v|ζ)w¯ w w p is real and bounded from below by c λ + |ξ|2 (|v|2 + |w|2 ) + cλ|w|2 /|ξ| with some constant c > 0. On the other hand, it is bounded from above by p C λ + |ξ|2 (|v|2 + |w|2 ) + C|λ||w|2 /|ξ|, with some larger constant C > 0.

8. Wellposedness of the Boundary Symbol We consider now the case of a flat interface. To study well-posedness in the sense of Hadamard, by means of the Dirichlet-to-Neumann operator from the previous section, we have to consider the following transformed problem for (v, w, h). v v −B0 + DN = Ch w w (8.1) λh = w. The matrices B0 and C are given by µs |ξ|2 + λs ξ ⊗ ξ 0 −B0 = , 0 0 µs b + λs (b|ζ)ζ −C = |ξ|2 . 2µs (Bζ|ζ) + (σ + (λs − µs )tr B Here, as in the previous section, ζ = ξ/|ξ|, and B = ∇x v∗ , b = ∇x w∗ . If (8.1) admits no nontrivial solutions for Re λ ≥ λ0 ≥ 0, ξ ∈ Rn , then we call the problem well-posed in the sense of Hadamard, otherwise illposed. Setting α0 = α + µs |ξ|2 , β 0 = β + λs |ξ|2 , c = µs b + λs (b|ζ)ζ, d = σ + (λs − µs )tr B + 2µs (Bζ|ζ),


15

we first solve

α0 + β 0 ζ ⊗ ζ −iγζ T

iγζ α+δ

v w

=−

c d

|ξ|2 h.

Since we obtain

det := (α0 + β 0 )(α + δ) − γ 2 6= 0 ξ ∈ Rn , Re λ ≥ 0, w = −det−1 (iγ(c|ζ) + (α0 + β 0 )d)|ξ|2 h.

Therefore (8.1) admits nontrivial solutions if and only if (λ, ξ) satisfies the characteristic equation (8.2)

λ + det−1 (iγ(c|ζ) + (α0 + β 0 )d)|ξ|2 = 0.

We keep ζ ∈ Rn , |ζ| = 1 fixed and scale λ = λ(ξ) according to λ = |ξ|τ (1/|ξ|) with some continuous function τ on R+ . Inserting this expression for λ, dividing by |ξ| and passing to the limit |ξ| → ∞ we obtain the limiting equation τ (0) +

d = 0. 2(µ1 + µ2 )

This equation has exactly one positive solution τ (0) = τ0 if and only if d < 0. If this is the case then by the implicit function theorem there is r > 0 such that for each |ξ| > r there is a unique solution λ = |ξ|τ (1/|ξ|) of the characteristic equation with |τ (1/|ξ|) − τ0 | < τ0 /2. Thus d < 0 implies existence of solutions (λ, ξ) such that |ξ| as well as Re λ become arbitrarily large. This is illposedness in the sense of Hadamard. Recall that d = σ + (λs − µs )tr B + 2µs (Bζ|ζ), and herein we may replace B = ∇x v∗ by its symmetric part E∗ . To minimize d we may choose ζ in the direction of the eigenvector corresponding to the smallest eigenvalue of E ∗ , and obtain the quantity d0 := σ + (λs − µs )tr E∗ + 2µs min{(E∗ ζ|ζ) : |ζ| = 1}. We summarize our arguments in Theorem 8.1. Let ρi , µi , σ > 0, as well as λs ≥ µs ≥ 0. Then the linearized localized problem (8.1) is well-posed in the sense of Hadamard if d 0 > 0 and it is illposed if d0 < 0. Observe that only tangential gradients of the tangential velocity enter the quantity d0 . Normal derivatives as well as tangential gradients of the normal velocity are not important. Although this result is far away from the linearization of the full model at a given state (u∗ , Γ∗ ), it gives an indication of what to expect. In a later paper, we intend to prove that the condition of this theorem for well-posedness is the essential one for well-posedness also of the nonlinear problem.

References [1] R. Aris: Vectors, tensors, and the basic equations of fluid mechanics. Dover Publications, 1989. [2] D. Bothe, J. Pr¨ uss, G. Simonett: Well-posedness of a two-phase flow with soluble surfactant, pp. 37-61 in Nonlinear Elliptic and Parabolic Problems (M. Chipot, J. Escher, eds.), Birkh¨ auser 2005.

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[3] M.J. Boussinesq: Sur l’existence d’une viscosit´e superficielle, dans la mice couche de transition separant un liquide d’une autre fluide contigu. Ann. Chim. Phys. 29, 349-357 (1913). [4] D.A. Edwards, H. Brenner, D.T. Wasan: Interfacial Transport Processes and Rheology. Butterworth-Heinemann, Stoneham, 1991. [5] J.S. Hadamard: Mouvement permenent lent d’une sph´ere liquide et visqueuse dans un liquide visqueux. Comp. Rend. Acad. Sci. (Paris) 152, 1735-1738 (1911). [6] A.A. Lebedev: Zhur. Russ. Fiz. Khim. 48, (1916). [7] V.G. Levich: Physicochemical Hydrodynamics. Prentice Hall, Englewood Cliffs, New Jersey, 1962. ¨ [8] W. Rybczynski: Uber die fortschreitende Bewegung einer fl¨ ussigen Kugel in einem z¨ ahen Medium. Bull. Int. Acad. Sci., Cracovic A, 40-46 (1911). [9] L.E. Scriven: Dynamics of a fluid interface. Chem. Eng. Sci. 12, 98-108 (1960). Phys. Rev. 7, 106-111 (1961). [10] A. Silvey: The fall of mercury droplets in a viscous medium. Phys. Rev. 7, 106-111 (1916). [11] J.C. Slattery, L. Sagis, E.-S. Oh: Interfacial Transport Phenomena, 2nd edition. Springer 2007. [12] C.V. Sternling, L.E. Scriven: Interfacial turbulence: Hydrodynamic instability and the Marangoni effect. AIChE J. 5, 514-523 (1959). ¨ lische, Center for Computational Engineering Science, Rheinisch-Westfa Technische Hochschule Aachen, Pauwelsstr. 19, D-52074 Aachen, Germany E-mail address: [email protected] ¨ r Mathematik, Martin-Luther-Universita ¨ t Halle-Wittenberg, Institut fu Theodor-Lieser-Str. 5, D-60120 Halle, Germany E-mail address: [email protected]