Finite element approximation of a two-layered liquid film in the presence of insoluble surfactants

DOI:10.1051/m2an:2008028 www.esaim-m2an.org

FINITE ELEMENT APPROXIMATION OF A TWO-LAYERED LIQUID FILM IN THE PRESENCE OF INSOLUBLE SURFACTANTS

John W. Barrett

and Linda El Alaoui

Abstract. We consider a system of degenerate parabolic equations modelling a thin film, consisting of two layers of immiscible Newtonian liquids, on a solid horizontal substrate. In addition, the model includes the presence of insoluble surfactants on both the free liquid-liquid and liquid-air interfaces, and the presence of both attractive and repulsive van der Waals forces in terms of the heights of the two layers. We show that this system formally satisfies a Lyapunov structure, and a second energy inequality controlling the Laplacian of the liquid heights. We introduce a fully practical finite element approximation of this nonlinear degenerate parabolic system, that satisfies discrete analogues of these energy inequalities. Finally, we prove convergence of this approximation, and hence existence of a solution to this nonlinear degenerate parabolic system.

Mathematics Subject Classification. 65M60, 65M12, 35K55, 35K65, 35K35, 76A20, 76D08.

Received April 24, 2007.

Published online July 30, 2008.

1. Introduction

In [1,2] fully practical finite element approximations were proposed and analysed for a system of nonlinear degenerate parabolic equations modelling a thin film of liquid, laden with insoluble surfactant, on a horizontal substrate in the possible presence of both attractive and repulsive van der Waals forces. In this paper, we extend the approximation and subsequent analysis in [1] to the case when the thin film consists of two layers of immiscible Newtonian liquids with possibly different viscosities. In addition, the model includes the presence of insoluble surfactants on both the free liquid-liquid and liquid-air interfaces, and the presence of both attractive and repulsive van der Waals forces in terms of the heights of the two layers, and possibly the total height of the film.

The model problem, derived using lubrication theory, as it appears in the applied mathematics, physics and engineering literature, seee.g.[4], is the following: Find{u_i(x, t), v_i(x, t), w_i(x, t)}²_i=1 such that

μ^∂u_∂t¹ =∇ ·[¹₃u³₁∇w₁+¹₂u²₁u₂∇w₂−¹₂u²₁∇(σ₁(v₁) +σ₂(v₂))], (1.1a)

Keywords and phrases. Thin film, surfactant, bilayer, fourth order degenerate parabolic system, finite elements, convergence analysis.

∗Supported by EPSRC U.K. grant GR/S35660/01.

1 Department of Mathematics, Imperial College, London, SW7 2AZ, UK.[email protected]

Article published by EDP Sciences c EDP Sciences, SMAI 2008

0

u1(x, t) u2(x, t) v1(x, t)

v2(x, t)

upper, viscosity 1

lower, viscosityμ

x y

Figure 1. Geometry of the two-layered system.

μ^∂u_∂t² =∇ ·[¹₂u²₁u₂∇w₁+u₁u²₂∇w₂−u₁u₂∇(σ₁(v₁) +σ₂(v₂))] +μ∇ ·[¹₃u³₂∇w₂−¹₂u²₂∇σ₂(v₂)], (1.1b)

w₁−w₂=−c1Δu₁+φ₁(u₁)−φ₂(u₂), (1.1c)

w₂=−c2Δ(u₁+u₂) +φ₂(u₂) +φ₃(u₁+u₂), (1.1d)

μ^∂v_∂t¹ =ρ₁μΔv₁+∇ ·₁

2u²₁v₁∇w₁+u₁v₁(u₂∇w₂− ∇[σ₁(v₁) +σ₂(v₂)])

, (1.1e)

μ^∂v_∂t² =ρ₂μΔv₂+∇ ·₁

2u²₁v₂∇w1+u₁v₂(u₂∇w2− ∇[σ1(v₁) +σ₂(v₂)])

+μ∇ ·[¹₂u²₂v₂∇w2−u₂v₂∇σ2(v₂)] (1.1f) in Ω_T, where Ω_T := Ω×(0, T], and Ω is a bounded domain in R^d, d= 1 or 2. Lety be the vertical variable, withy= 0 being the solid horizontal substrate. Thenu₁(x, t) andw₁(x, t) are the height and reduced pressure, respectively, atx∈Ω and timetof the lower liquid having viscosityμ >0, whereasu₂(x, t) andw₂(x, t) are the height and reduced pressure of the upper liquid having unit viscosity. The concentration of insoluble surfactant at the liquid-liquid interface, y = u₁(x, t), is v₁(x, t); and at the liquid-air interface, y = (u₁+u₂)(x, t), is v₂(x, t); see Figure 1. The constants ρ_i, c_i ∈ R>0 are the inverses of the surface Peclet numbers and the modified capillary numbers, respectively, with i = 1 for the y =u₁ interface and i = 2 for the y = u₁+u₂ interface. In addition,σ_i∈C¹(R≥0) withσ_i(s)≥0 andσ_i(s)<0 for alls∈R≥0 is the constitutive equation of state relating the surface tension σ_i to v_i on the ith interface, i.e. surfactant reduces surface tension. An empirical model, proposed in [11], often used in the literature is σ_i(s) := (α_i+ 1) [1 +θ(α_i)s]⁻³−α_i, where θ(α_i) :=_αi+1

αi

¹₃

−1 andα_i∈R>0 relates to the activity of the surfactant. Henceσ_i : [0,1]→[0,1]. We shall assume that the surfactant concentration for each interface is dilute,v_i ∈[0,1], in which case the limitα_i→ ∞ is taken, and the equation of state simplifies to

σ_i(s)≡σ(s) := 1−s i= 1,2. (1.2)

The van der Waals forces,φ_j, j= 1→3 in (1.1c,d) acting simultaneously on the three heights, are given by φ_j(s) =φ⁺_j(s) +φ⁻_j(s), φ⁺_j(s) :=−δjs^−νj, ν_j >3, φ⁻_j (s) :=a_js⁻³, (1.3)

where a_j ∈ R_≥0 is a scaled dimensionless Hamaker constant and δ_j ∈ R_≥0 represents the effect of repulsive van der Waals forces. We shall assume throughout thatδ_i>0,i= 1,2, so that these repulsive forces prevent both films from rupturing,i.e. u_i>0. However, there is noa prioribound below onu_iso (1.1a–f) is a degenerate nonlinear parabolic system, which is fourth order inu_i. This degeneracy makes the analysis/numerical analysis of the system particularly difficult. In addition, as there is no maximum principle for parabolic equations of fourth order, a naive discretization does not guarantee the nonnegativity of the approximation tou_i.

In [1] a finite element approximation to the single-layered surfactant model in presence of van der Waals forces, (1.1a–f) with c₁ = 0, μ= 1, v₁(·,0) = 0 and φ₁ ≡φ₂≡0, was presented. In addition, convergence of this approximation was proved, yielding an existence proof for the degenerate nonlinear parabolic system. It is the aim of this paper to adapt the techniques in [1] to present, and prove convergence of, a finite element approximation to (1.1a–f).

As remarked previously, recall (1.2), the physically relevant values ofv_i lie in the interval [0,1]. Noting this, it is convenient for the analysis is this paper, as it was in [1,2], to replacev_iin non-differentiated terms ofv_iby β¹(v_i); where for a givenM ≥1,β^M :R→(−∞, M] is defined as

β^M(s) = [s−M]₋+M, with [s]₋= min{s,0}. (1.4) This two-layered system introduces new difficulties, and it is also convenient in the cased= 2 to replaceu₁in non-differentiated terms ofu₁, which are not arguments ofφ_i, byβ^M(u₁) for some sufficiently large cut-offM. We will return to the need for these cut-offs later in this section.

Altogether, in this paper we consider the following initial boundary value problem:

(P)Find functions{ui, v_i, w_i}²_i=1: Ω×[0, T]→Rsuch that

μ^∂u_∂t¹ =∇ ·[¹₃[β^M(u₁)]³∇w₁+¹₂[β^M(u₁)]²u₂∇w₂−¹₂[β^M(u₁)]²∇(σ(v₁) +σ(v₂))] in Ω_T, (1.5a) μ^∂u_∂t² =∇ ·[¹₂[β^M(u₁)]²u₂∇w1+β^M(u₁)u²₂∇w2−β^M(u₁)u₂∇(σ(v1) +σ(v₂))]

+μ∇ ·[¹₃u³₂∇w2−¹₂u²₂∇σ(v2)] in Ω_T, (1.5b)

w₁−w₂=−c₁Δu₁+φ₁(u₁)−φ₂(u₂) in Ω_T, (1.5c)

w₂=−c₂Δ(u₁+u₂) +φ₂(u₂) +φ₃(u₁+u₂) in Ω_T, (1.5d) μ^∂v_∂t¹ =ρ₁μΔv₁+∇ ·₁

2[β^M(u₁)]²β¹(v₁)∇w1+β^M(u₁)β¹(v₁) (u₂∇w2− ∇[σ(v1) +σ(v₂)])

in Ω_T, (1.5e) μ^∂v_∂t² =ρ₂μΔv₂+∇ ·₁

2[β^M(u₁)]²β¹(v₂)∇w1+β^M(u₁)β¹(v₂) (u₂∇w2− ∇[σ(v1) +σ(v₂)])

+∇ ·[¹₂u²₂β¹(v₂)∇w2−u₂β¹(v₂)∇σ(v2)] in Ω_T, (1.5f) u₁(x,0) =u⁰₁(x)>0, v₁(x,0) =v⁰₁(x)≥0 ∀x∈Ω,(1.5g) u₂(x,0) =u⁰₂(x)>0, v₂(x,0) =v⁰₂(x)≥0 ∀x∈Ω (1.5h) with no flux boundary conditions on (1.5a,b), (1.5e,f), and homogeneous Neumann boundary conditions on (1.5c,d). The latter can be interpreted as a 90^◦ angle condition on the film surfaces, where they meet the exterior container. In the aboveμ, c_i, ρ_i ∈R>0 are given constants, while σ(·),φ_j(·) and β^M(·) are given by (1.2), (1.3) and (1.4) witha_j, δ₃∈R_≥0,δ₁, δ₂∈R>0 andM ≥1.

The basic ingredients of our approach are two energy bounds combined with a regularization procedure. In particular, for any givenε∈(0,1), we introduce the regularized function

β^M_ε (s) := max{β^M(s), ε}, (1.6)

with yields the regularised system (P_ε); that is, (P) with{ui, v_i, β¹(v_i), w_i}²_i=1replaced by{ui,ε, v_i,ε, β_ε¹(v_i,ε), w_i,ε}²_i=1. On defining the horizontal velocity fieldsVi,ε(x, t, y), whereyis the vertical variable – recall Figure1,

we have from lubrication theory, similarly to [1,2], that μ∂²V1,ε

∂y² =∇w1,ε in Ω_T ×(0, u_1,ε(x, t)), and ∂²V2,ε

∂y² =∇w2,ε in Ω_T ×(u_1,ε(x, t),(u_1,ε+u_2,ε)(x, t)) (1.7a) subject to the boundary conditions

V1,ε(x, t,0) = 0, μ∂V1,ε

∂y (x, t, u_1,ε(x, t)) = ∂V2,ε

∂y (x, t, u_1,ε(x, t)) +∇σ(v_1,ε(x, t)), V1,ε(x, t, u_1,ε(x, t)) =V2,ε(x, t, u_1,ε(x, t)), ∂V2,ε

∂y (x, t,(u_1,ε+u_2,ε)(x, t)) =∇σ(v2,ε(x, t)) ∀(x, t)∈Ω_T, and [V1,ε · ν_∂Ω](x, t, y) = 0 ∀y∈[0, u_1,ε(x, t)],

[V2,ε · ν_∂Ω](x, t, y) = 0 ∀y∈[u_1,ε(x, t),(u_1,ε+u_2,ε)(x, t)]

∀(x, t)∈∂Ω×(0, T); (1.7b) whereν_∂Ωis normal to ∂Ω, and∇, as throughout, is respect to the horizontal variablex, and not the vertical variabley. The above yields for any (x, t)∈Ω_T that

V1,ε(x, t, y) =−y μ

2 i=1

[u_i,ε(x, t)∇wi,ε(x, t)− ∇σ(vi,ε(x, t))] +y²

μ ∇w1,ε(x, t) fory∈[0, u_1,ε(x, t)], (1.8a) V2,ε(x, t, y) =V1,ε(x, t, u_1,ε(x, t)) + (y−u_1,ε(x, t)) y−u_1,ε(x, t)

2 −u_2,ε(x, t)

∇w2,ε(x, t) +∇σ(v2,ε(x, t))

fory∈[u_1,ε(x, t),(u_1,ε+u_2,ε)(x, t)]. (1.8b) We can then recast the corresponding (P_ε) versions of (1.5a,b) and (1.5e,f), withβ^M(u_1,ε) replaced byu_1,ε, as

∂u_1,ε

∂t +∇ ·

_u1,ε

0

V1,ε(·,·, y) dy

= 0, ∂u_2,ε

∂t +∇ · _u2,ε

u1,ε

V2,ε(·,·, y) dy

= 0 in Ω_T, (1.9a)

∂v_1,ε

∂t +∇ ·(V1,ε(·,·, u1,ε)β¹_ε(v_1,ε) ) =ρ₁Δv_1,ε, ∂v_2,ε

∂t +∇ · (V2,ε(·,·, u1,ε+u_2,ε)β_ε¹(v_2,ε) ) =ρ₂Δv_2,ε in Ω_T. (1.9b) In order to derive the crucial energy bounds, we introduce

F_ε(s) = [β¹_ε(s)]⁻¹ and F_ε(1) =F_ε(1) = 0, (1.10) which, on recalling (1.6), implies that

F_ε(s) :=

⎧⎪

⎪⎨

⎪⎪

⎩

s²−ε²

2ε + (lnε−1)s+ 1 s≤ε s(lns−1) + 1 ε≤s≤1

1

2(s−1)² 1≤s.

(1.11)

HenceF_ε∈C^2,1(R), and for later purposes, we note that

F_ε(s)≥ ^s₄² −¹₂ ∀s≥0 and F_ε(s)≥ ^s_2ε² ∀s≤0; (1.12) seee.g.(2.4) in [2].

We will now derive several formal bounds for {ui,ε, v_i,ε, w_i,ε}²_i=1. Testing the u_i,ε equation in (1.9a) with w_i,ε,i= 1,2, combining with the (P_ε) versions of (1.5c,d) and noting (1.7a,b) yields that

d dt

Ω

c1

2 |∇u_1,ε|²+^c₂²|∇(u_1,ε+u_2,ε)|²+ ₂

i=1

Φ_i(u_i,ε)

+ Φ₃(u_1,ε+u_2,ε)

dx+μ

Ω

_u1,ε

0

∂V1,ε

∂y ² dy

dx

+

Ω

_u1,ε+u2,ε

u1,ε

∂V2,ε

∂y ²dy

dx=

Ω

[V1,ε(·,·, u1,ε)∇σ(v1,ε) +V2,ε(·,·, u2,ε)∇σ(v2,ε)] dx, (1.13)

where Φ_j(·) is an antiderivative ofφ_j(·),i.e.Φ_j(·)≡φ_j(·),j = 1→3. Testing thev_i,ε equation in (1.9b) with F_ε(v_i,ε), noting (1.10) and combining yields that

d dt

Ω

2 i=1

F_ε(v_i,ε) dx+ 2 i=1

ρ_i

Ω

F_ε(v_i,ε)|∇vi,ε|²dx=

Ω

[V1,ε(·,·, u1,ε)∇v1,ε+V2,ε(·,·, u2,ε)∇v2,ε] dx. (1.14)

Combining (1.13) and (1.14), and noting (1.2), yields the formal energy identity

d dt

Ω

c1

2 |∇u1,ε|²+^c₂²|∇(u1,ε+u_2,ε)|²+ 2 i=1

[Φ_i(u_i,ε) +F_ε(v_i,ε)] + Φ₃(u_1,ε+u_2,ε)

dx

+μ

Ω

_u1,ε

0

∂V1,ε

∂y ² dy

dx+

Ω

_u1,ε+u2,ε

u1,ε

∂V2,ε

∂y ²dy

dx+

2 i=1

ρ_i

Ω

F_ε(v_i,ε)|∇vi,ε|²dx= 0.

(1.15) Noting (1.8a,b) and Young’s inequality,

|r s| ≤ ^γ₂r²+_2γ¹ s² ∀r, s∈R, γ∈R>0, (1.16) one can derive the following inequalities for any γ^∗∈(¹₂,²₃)

μ _u1,ε

0

∂V1,ε

∂y

²dy=_μ^{1 1}₃u³_1,ε|∇w_1,ε|²+u_1,ε|u_2,ε∇w_2,ε− ∇(σ(v_1,ε) +σ(v_2,ε))|² +u²_1,ε∇w1,ε·[u_2,ε∇w2,ε+∇(σ(v1,ε) +σ(v_2,ε)) ]

≥_μ¹

(¹₃−^γ₂^∗)u³_1,ε|∇w_1,ε|²+ (1−₂¹_γ∗)u_1,ε|u_2,ε∇w_2,ε− ∇(σ(v_1,ε) +σ(v_2,ε))|² , (1.17a) _u1,ε+u2,ε

u1,ε

∂V2,ε

∂y

²dy=¹₃u³_2,ε|∇w2,ε|²+u_2,ε|∇σ(v2,ε)|²−u²_2,ε∇w2,ε· ∇σ(v2,ε)

≥(¹₃−^γ₂^∗)u³_2,ε|∇w2,ε|²+ (1−_2γ¹∗)u_2,ε|∇σ(v2,ε)|². (1.17b) From (1.15), (1.17a,b), (1.10) and (1.4), one can derive uniform bounds on ∇ui,ε in L^∞(0, T;L²(Ω)) and

∇vi,ε inL²(Ω_T). We note the crucial role that the cut-offβ¹(·) onv_i,ε plays in thev_i,εbound, recall (1.10). Of course one could replace β¹(·) with β^M(·), whereM arbitrarily large. However, as it does not appear possible to obtain ana prioriL^∞(Ω_T) bound onv_i,ε, some cut-off above onv_i,ε is required. In addition, the singularity in Φ_i, i = 1,2 at the origin yields the positivity ofu_i,ε. Furthermore, the bound (1.12) together with (1.15) yields that

ΩT[v_i,ε]²₋dxdt≤C ε. As can be seen from the above, it is not necessary to have the cut-offβ^M(·)

onu_1,ε in the coefficients in (P_ε), in order to obtain the formal energy identity (1.15). This cut-off on u_1,ε in these coefficients is required for the second energy bound, see below; and this bound is only required if d= 2.

It is easily deduced, that the effect of this cut-off is just to modify the term (1.17a) in (1.15); that is, u_1,ε is replace byβ^M(u_1,ε).

In order to obtain the second energy bound we define a functionG∈C^∞(R>0) such thatη³∇G(η) =∇η;

that is, fors >0

G(s) =s⁻³ ⇒ G(s) =−¹₂s⁻² ⇒ G(s) = ¹₂s⁻¹, (1.18) where the constants of integration have been chosen to be zero. In addition, we introduceG_M ∈C²(R>0) such that [β^M(η)]³∇G_M(η) =∇η; that is, for allM ≥1 ands >0

G_M(s) = [β^M(s)]⁻³ ⇒ G_M(s) =

G(s) s∈(0, M],

1 2M

(_M^s)²−3 (_M^s) + 3

s≥M. (1.19)

Testing the (P_ε) versions of (1.5a) with G_M(u_1,ε), (1.5b) with G(u_2,ε), (1.5c) with −Δu1,ε and (1.5d) with

−Δu2,ε and combining, formally yields, on noting (1.3), (1.18), (1.19) and the no flux boundary conditions, that

d dt

Ω

[μ G_M(u_1,ε) +G(u_2,ε)] dx+¹₃

Ω

[c₁|Δu1,ε|²+c₂|Δ(u1,ε+u_2,ε)|²] dx +

2 i=1

Ω

(φ⁺_i )(u_i,ε)|∇u_i,ε|²dx+

Ω

(φ⁺₃)(u_1,ε+u_2,ε)|∇(u_1,ε+u_2,ε)|²dx

=−

Ω

(u_2,ε∇w2,ε− ∇[σ(v1,ε) +σ(v_2,ε)])·

1

2[β^M(u_1,ε)]⁻¹∇u1,ε+_μ¹β^M(u_1,ε)u⁻²_2,ε∇u2,ε

dx

−₂¹_μ

Ω

[β^M(u_1,ε)]²∇w1,ε·u⁻²_2,ε∇u2,εdx+¹₂

Ω

u⁻¹_2,ε∇[σ(v2,ε)]· ∇u2,εdx

−²

i=1

Ω

(φ⁻_i )(u_i,ε)|∇ui,ε|²dx−

Ω

(φ⁻₃) ₂

i=1

u_i,ε ∇ ₂

i=1

u_i,ε

2

dx. (1.20)

It follows from (1.20), (1.16), (1.4), (1.3) and the bound

s^−α≤[β^M(s)]^−α≤γ s^−ζ +C(γ, α, ζ, M) ∀s, γ∈R>0, α∈(0, ζ), (1.21) forζ=ν_j+ 1,j= 1,2, and for bothα= 3 and 4, that

d dt

Ω

[μ G_M(u_1,ε) +G(u_2,ε)] dx+¹₃

Ω

[c₁|Δu1,ε|²+c₂|Δ(u1,ε+u_2,ε)|²] dx +

2 i=1

Ω

(φ⁺_i )(u_i,ε)|∇u_i,ε|²dx+

Ω

(φ⁺₃)(u_1,ε+u_2,ε)|∇(u_1,ε+u_2,ε)|²dx

≤C

Ω

β^M(u_1,ε)|u2,ε∇w2,ε− ∇[σ(v1,ε) +σ(v_2,ε)]|²dx+

Ω

u_2,ε|∇[σ(v2,ε)]|²dx +

Ω

[β^M(u_1,ε)]³|∇w_1,ε|²dx+ 2

i=1

Ω

|∇u_i,ε|²dx

. (1.22)

From (1.22), (1.15), and (1.17a,b) withu_1,εreplaced byβ^M(u_1,ε), one obtains thatu_i,εis uniformly bounded in L²(0, T;H²(Ω)). We note that we have used the cut-off onu_1,ε, in order to control the first and second terms on the right hand side of (1.20).

It is the goal of this paper to derive a finite element method that is consistent with the formal energy bounds (1.15) and (1.22).

This paper is organized as follows. In Section2 we formulate a fully practical finite element approximation of the degenerate problem (P) and derive discrete analogues of the energy bounds (1.15), and (1.22) if d= 2 and ν_j ≥ 7, j = 1,2, in (1.3). In Section 3 we prove convergence, and hence existence of a solution to the system (P). In the cased= 1, we prove existence of a solution to (P) withβ^M(u₁) replaced byu₁.

Finally, although there is a vast amount of work in the applied mathematics, physics and engineering litera- ture, there is very little work in the PDE literature on surfactant type problems. To our knowledge, there is no work on the two-layered system (P). For the single-layered system, the only papers that we are aware of are the following. A local existence result without cut-offs is shown in [8] for the pure initial-value problem with very smooth initial data. A global existence result in one space dimension without van der Waals forces and cut-offs can be found in [5], but this result does not allow for σ of the form (1.2). A global existence result, via the convergence of a finite element approximation, in both one and two space dimensions with van der Waals forces and with a cut-off on the surfactant concentration in the coefficients can be found in [1]. The above results are all for the case of an insoluble surfactant. An extension of the existence result in [1] to the case of a soluble sur- factant can be found in [3]. Of course, it is possible to extend the results in this paper to the more complicated two-layered case in the presence of soluble surfactants by combining the ideas here with those in [3].

Notation and auxiliary results

LetD⊂R^d,d= 1 or 2, with a Lipschitz boundary∂Difd= 2. We adopt the standard notation for Sobolev spaces, denoting the norm ofW^m,q(D) (m∈N,q∈[1,∞]) by · m,q,D and the semi-norm by | · |m,q,D. We extend these norms and semi-norms in the natural way to the corresponding spaces of vector and matrix valued functions. For q = 2, W^m,2(D) will be denoted by H^m(D) with the associated norm and semi-norm written as, respectively, · m,Dand| · |m,D. For notational convenience, we drop the domain subscript on the above norms and semi-norms in the case D ≡Ω. Throughout (·,·) denotes the standardL² inner product over Ω, whileq denotes for anyq∈[1,∞] the “dual exponent” such that ¹_q +_q¹ = 1. In addition we define

−η:= _m(Ω)¹

Ω

ηdx ∀η∈L¹(Ω), (1.23)

wherem(D) denotes the measure ofD.

It is convenient to introduce the “inverse Laplacian” operatorG:F →Z such that

(∇Gz,∇η) =z, ηq ∀η∈W^1,q(Ω), (1.24)

where F:=

z∈(W^1,q(Ω)) :z,1q = 0

and Z:={z∈W^1,q(Ω) : (z,1) = 0}. Here and throughout ·,· q

denotes the duality pairing between (W^1,q(Ω))andW^1,q(Ω) for anyq∈(1,2]. The well-posedness ofGfollows from the generalised Lax-Milgram theorem and the Poincar´e inequality

|η|0,r ≤C(|η|1,r+|(η,1)|) ∀η∈W^1,r(Ω) and r∈[1,∞]. (1.25) Throughout C denotes a generic constant independent ofh,τ andε; the mesh and temporal discretization parameters and the regularization parameter. In addition C(a₁, . . ., a_I) denotes a constant depending on the arguments {ai}^I_i=1. Furthermore ·₍₎ denotes an expression with or without the subscript ; similarly for superscripts.

2. finite element approximation

We consider the finite element approximation of (P) under the following assumptions on the mesh:

(A) Let Ω be a convex polygonal domain ifd= 2. Let {T^h}h>0 be a quasi-uniform family of partitionings of Ω into disjoint open simplicesκwithh_κ:= diam(κ) andh:= max_κ∈Thh_κ, so that Ω =∪_κ∈T^hκ. In addition, it is assumed ford= 2 that all simplicesκ∈ T^h are right-angled.

We note that the right-angled simplices assumption is not a severe constraint, as there exist adaptive finite element codes that satisfy this requirement, seee.g.[10].

Associated withT^h is the finite element space

S^h:={χ∈C(Ω) :χ|κ is linear∀κ∈ T^h} ⊂H¹(Ω).

We introduce also

S_≥0^h :={χ∈S^h:χ≥0 in Ω} ⊂H_≥0¹ (Ω) :={η∈H¹(Ω) :η≥0a.e. in Ω},

and similarlyS_>0^h andH_>0¹ (Ω). LetJ be the set of nodes ofT^hand{pj}_j∈J the coordinates of these nodes. Let {χj}j∈J be the standard basis functions forS^h; that isχ_j∈S_≥0^h andχ_j(p_i) =δ_ij for alli, j∈J. We introduce π^h :C(Ω) →S^h, the interpolation operator, such that (π^hη)(p_j) = η(p_j) for allj ∈J. A discrete semi-inner product onC(Ω) is then defined by

(η₁, η₂)^h:=

Ω

π^h(η₁(x)η₂(x)) dx=

j∈J

m_jη₁(p_j)η₂(p_j), (2.1)

where m_j := (1, χ_j) > 0. The induced discrete semi-norm is then |η|h := [ (η, η)^h]¹², where η ∈ C(Ω). We introduce also theL² projectionQ^h:L²(Ω)→S^hdefined by

(Q^hη, χ)^h= (η, χ) ∀χ∈S^h. (2.2)

Similarly to the approach in [7,12] for the thin film equation,i.e.a single-layered system without surfactant, we introduce matrices Λ_ε:S^h→[L^∞(Ω)]^d×d, and Ξ_(M):S_>0^h →[L^∞(Ω)]^d×dsuch that for allz^h∈S^h,χ∈S_>0^h anda.e.in Ω

Λ_ε(z^h),Ξ_(M)(χ) are symmetric and positive semi-definite, (2.3a) Λ_ε(z^h)∇π^h[F_ε(z^h)] =∇z^h, [Ξ_(M)(χ)]³∇π^h[G_(M)(χ)] =∇χ. (2.3b) The construction of Ξ and Λ_εis given in [1]. The construction of Ξ_M is the same as that of Ξ, but withGreplaced by G_M. We note that the right-angle constraint on the partitioning T^h is exploited for these constructions.

Throughout this paper we make use of the fact that the matrices Ξ(χ), Ξ_M(η^h) and Λ_ε(z^h) commute with each other for anyχ, η^h∈S_>0^h andz^h∈S^h.

In addition to T^h, letτ = _N^T be the uniform time step andt_n :=n τ,n= 0→N. For any givenε∈(0,1), we then consider the following fully practical finite element approximation of (P) withσgiven by (1.2), andφ_j given by (1.3):

(P^h,τ_ε ) Forn≥1 find{{U_i,εⁿ, W_i,εⁿ, V_i,εⁿ}²_i=1} ∈[S^h]⁶ such that for allχ∈S^h

μ

U_1,εⁿ −U_1,εⁿ⁻¹

τ , χ

_h +¹₃

[Ξ_M(U_1,εⁿ )]³∇W_1,εⁿ ,∇χ +¹₂

[Ξ_M(U_1,εⁿ )]²Ξ(U_2,εⁿ )∇W_2,εⁿ ,∇χ

=−¹₂

[Ξ_M(U_1,εⁿ )]³²[Ξ_M(U_1,εⁿ⁻¹)]¹²∇[V_1,εⁿ⁻¹+V_2,εⁿ⁻¹],∇χ

, (2.4a)

μ

U_2,εⁿ −U_2,εⁿ⁻¹

τ , χ

_h +

[^μ₃[Ξ(U_2,εⁿ )]³ + Ξ_M(U_1,εⁿ ) [Ξ(U_2,εⁿ )]²]∇W_2,εⁿ ,∇χ +¹₂

[Ξ_M(U_1,εⁿ )]²Ξ(U_2,εⁿ )∇W_1,εⁿ ,∇χ

=−^μ₂

[Ξ(U_2,εⁿ )]³²[Ξ(U_2,εⁿ⁻¹)]¹² ∇V_2,εⁿ⁻¹,∇χ

−

[Ξ_M(U_1,εⁿ )]¹²[Ξ_M(U_1,εⁿ⁻¹)]¹²Ξ(U_2,εⁿ )∇[V_1,εⁿ⁻¹+V_2,εⁿ⁻¹],∇χ

, (2.4b)

c₁

∇U_1,εⁿ ,∇χ

+

φ⁺₁(U_1,εⁿ ) +φ⁻₁(U_1,εⁿ⁻¹), χ _h

−

φ⁺₂(U_2,εⁿ ) +φ⁻₂(U_2,εⁿ⁻¹), χ _h

=

W_1,εⁿ −W_2,εⁿ , χ _h

, (2.4c)

c₂

∇[U_1,εⁿ +U_2,εⁿ ],∇χ +

φ⁺₂(U_2,εⁿ ) +φ⁻₂(U_2,εⁿ⁻¹) +φ⁺_{3 2}

i=1

U_i,εⁿ

+φ⁻_{3 2}

i=1

U_i,εⁿ⁻¹

, χ h

=

W_2,εⁿ , χ h

, (2.4d) μ

V_1,εⁿ−V_1,εⁿ⁻¹

τ , χ

_h +ρ₁μ

∇V_1,εⁿ ,∇χ +

Ξ_M(U_1,εⁿ ) Λ_ε(V_1,εⁿ )∇[V_1,εⁿ +V_2,εⁿ],∇χ +

Ξ_M(U_1,εⁿ ) Ξ(U_2,εⁿ ) Λ_ε(V_1,εⁿ )∇W_2,εⁿ ,∇χ

=−¹₂

[Ξ_M(U_1,εⁿ )]²Λ_ε(V_1,εⁿ )∇W_1,εⁿ ,∇χ

, (2.4e) μ

V_2,εⁿ−V_2,εⁿ⁻¹

τ , χ

_h +ρ₂μ

∇V_2,εⁿ ,∇χ

+

Ξ_M(U_1,εⁿ ) Λ_ε(V_2,εⁿ )∇[V_1,εⁿ +V_2,εⁿ],∇χ +

Ξ_M(U_1,εⁿ ) Ξ(U_2,εⁿ ) Λ_ε(V_2,εⁿ )∇W_2,εⁿ ,∇χ +^μ₂

[Ξ(U_2,εⁿ )]²Λ_ε(V_2,εⁿ)∇W_2,εⁿ ,∇χ +μ

Ξ(U_2,εⁿ )Λ_ε(V_2,εⁿ )∇V_2,εⁿ ,∇χ

=−¹₂

[Ξ_M(U_1,εⁿ )]²Λ_ε(V_2,εⁿ)∇W_1,εⁿ ,∇χ

; (2.4f)

where, for i= 1,2,U_i,ε⁰ ∈S_>0^h andV_i,ε⁰ ∈S^h are approximations ofu⁰_i andv_i⁰, respectively,e.g. U_i,ε⁰ ≡π^hu⁰_i or Q^hu⁰_i and similarly forV_i,ε⁰.

Remark 2.1. We note that the above system decouples into (2.4a–d) and (2.4e,f); that is, one updates the heights and pressures at the new time level, then the surfactant concentrations. (P^h,τ_ε ) is the natural extension of the approximation of the insoluble single-layered surfactant system studied in [1]. In particular, on setting c₁= 0, φ₁≡φ₂≡0 yields thatW_εⁿ ≡W_1,εⁿ ≡W_2,εⁿ andU_εⁿ ≡U_1,εⁿ +U_2,εⁿ ,n= 1→N. Moreover, μ= 1 and v₁⁰ ≡0 yields that V_1,εⁿ ≈ εand V_2,εⁿ ≈V_εⁿ, n= 1→N. Of course, as noted in the introduction, we require Ξ_M(·) for {U_1,εⁿ }^N_n=0, as opposed to Ξ(·), for this two-layered problem in order to obtain our discrete entropy bound, see (2.54) below; which is required only in the case d= 2. In the case d= 1, one can replace Ξ_M(·) by Ξ(·). Finally, as U_i,ε⁰ >0, one can ensure that Ξ_(M)(U_i,εⁿ⁻¹) andφ⁻_j(U_i,εⁿ⁻¹) are well defined for n≥1; see Theorem2.4below.