Contact angle evolution in the standard model

4.2.1. Preliminaries

Definition 4.15(Regularity). In the following, we consider an open intervalIand the space of functions

V :=C(I×Ω)∩C¹(grΩ⁺)∩C¹(grΩ⁻). (4.21) Note that we assume in particular that the fluid velocitiesv^±aredifferentiableat the contact line and the viscous stress is locallybounded. This is a rather strong assumption in contrast to weak solution concepts which allow an integrable singularityin the viscous stress as long as the corresponding dissipation rate is finite.

46

4.2. Contact angle evolution in the standard model

Remark4.16 (Flow kinematics and the impermeability condition). We note that the impermeability conditions for the free

v^±,n_Σ

=V_Σ on grΣ (4.22)

and the solid surface

v^±,n_∂Ω

=0 on∂Ω, (4.23)

which both hold for the bulk velocities in the standard model, have an important and immediate consequence for the flow kinematics for regular solutions of the standard model. Since (4.22) and (4.23) imply (see Lemma 4.6 above)

v^±,nΓ

=VΓ on grΓ,

it follows that both grΣ and grΓ are invariant subsets for the flow-maps induced by the bulk velocities (see Lemma 4.3). Hence, a fluid particle located in the bulk phases can never reach the solid surface in finite time for a regular solution. Conversely, fluid particles may reach the solid surface in finite time if the impermeability condition on the solid surface is dropped since grΣand grΓare not necessarily invariant with respect to the bulk flow. This kind of flow kinematics is called “rolling motion”. It is discussed in the literature as one of the main qualitative feature of the Interface Formation Model (see [Shi06, Shi08]).

The Continuity Lemma: The following Lemma shows anadditionalcontinuity property for the velocity gradi-ent, whichonlyholds at the contact line. Typically, the gradient of the velocity field has a jump, which is controlled by the interfacial transmission conditions.

Note that we define the gradient of a vectorwin Cartesian coordinates as (∇w)_i,j=∂wi

∂x_j. Lemma 4.17. LetΩ⊂R³,0<θ<π, v∈C(Ω),∇v∈J(Ω,Σ)and

hv,n_∂Ωi=0 on∂Ω, ∇·v=0 inΩ\Σ(t),

where∂Ωis the smooth boundary ofΩ. Then∇v has the following continuity property at the contact line:

Jh∇vα,βiK=0 onΓ, whereα,β are arbitrary vectors in the plane spanned by nΓand n_∂Ω.

Proof. We consider an arbitrary point onΓand showJ∇vτK=0 as well asJh∇v nΓ,n_∂ΩiK=Jh∇v nΓ,nΓiK=0.

This is already sufficient sinceτandnΓare linearly independent. Sincevis assumed to be continuous acrossΣ, the tangential derivatives ofvare continuous

J∇vτK=J∇vt_ΓK=0.

Sincevis tangential to∂Ω, it follows that

h∇v n_Γ,n_∂Ωi=0 ⇒ Jh∇v n_Γ,n_∂ΩiK=0.

It remains to show thatJh∇v n_Γ,n_ΓiK=0. Sincevis solenoidal, we have

0=∇·v=h∇v n_Γ,n_Γi+h∇v n_∂Ω,n_∂Ωi+h∇vt_Γ,t_Γi. Therefore, we can write

Jh∇v nΓ,nΓiK=−(Jh∇v n_∂Ω,n_∂ΩiK+Jh∇vtΓ,t_ΓiK) =−Jh∇v n_∂Ω,n_∂ΩiK.

Chapter 4. Kinematics of moving contact lines

Fromτ=−cosθnΓ−sinθn_∂Ωwe infer (since 0<θ<π) n_∂Ω=− 1

sinθ(cosθn_Γ+τ).

This yields

Jh∇v n_Γ,n_ΓiK= 1

sinθ(cosθJh∇v n_Γ,n_∂ΩiK+Jh∇vτ,n_∂ΩiK)

=Jh∇vτ,n_∂ΩiK

sinθ =hJ∇vτK,n_∂Ωi sinθ =0.

Note that in the 2D case thefullgradient ofvis continuous acrossΓ.

On the Navier boundary condition: We reconsider the Navier condition (3.18). By taking the projection onto n_Γwe have

a^± v^±,n_Γ

+2

D^±n_∂Ω,n_Γ

=0.

Ifvsatisfies the kinematic conditionsv^±·n_Γ=V_Γ, we obtain the jump condition hJDKn_∂Ω,n_Γi_|Γ=−JaKVΓ

2 . (4.24)

Under the assumptions of Lemma 4.17, we havehJDKn_∂Ω,nΓi_|Γ=0 and hence

JaKVΓ=0. (4.25)

Hence, to allow for a regular solution withVΓ6=0, one has to choose the inverse slip lengtha as a continuous function across the contact line, i.e.

λ⁺

η⁺_|Γ=λ⁻

η⁻_|Γ=a_|Γ. In this case, we have the relations

JλK=aJηK (4.26)

and

2hDn_∂Ω,n_Γi_|Γ=h∇v n_∂Ω,n_Γi_|Γ=−aV_Γ. (4.27)

4.2.2. Contact angle evolution

The following Theorem shows that, for sufficiently regular solutions, ˙θhas a quite simple form for a large class of models. Note that the equations (4.28)-(4.32) say nothing about external forces, do not specify the contact angle and the slip length may be a function of space and time. Moreover, we only need thetangential partof the trans-mission condition for the stress. In this sense, the system (4.28)-(4.32) is not closed but describes aclassof models.

The main idea for the proof is the observation that both the Navier and the interfacial transmission condition are valid at the contact line. A regular classical solution has to satisfy both of them. Hence the conditions have to be compatible⁶.

Theorem 4.18. Let Ω⊂R³ (or Ω⊂R²) be a half-space with boundary ∂Ω, σ ≡const, η^±>0, JηK6=0, a∈C(gr∂Ω)and(v,grΣ)with v∈V,grΣaC^1,2-family of moving hypersurfaces with boundary, be a classical

6See Chapter 5 for a detailed discussion on compatibility conditions in mathematical models of dynamic wetting.

48

4.2. Contact angle evolution in the standard model

solution of the PDE-system

∇·v=0 inΩ\Σ(t), (4.28)

JvK=0, PΣJSKnΣ=0 onΣ(t), (4.29) hv,n_∂Ωi=0 on∂Ω\Γ(t), (4.30) aP∂Ωv+2P∂ΩDn_∂Ω=0 on∂Ω\Γ(t), (4.31)

V_Σ=hv,n_Σi onΣ(t) (4.32)

withθ∈(0,π)ongrΓ. Then the evolution of the contact angle is given by Dθ

Dt =aV_Γ 2 =V_Γ

2L. (4.33)

Moreover, in the caseθ=π/2it holds that

(aV_Γ)_|θ=π/2=0, (4.34)

which also means that

Dθ

Dt_|θ=π/2=0.

Proof. Sincevis continuous andv^±∈C¹(grΩ^±), we can choosev_Σ:=v⁺

|grΣ=v⁻

|grΣ∈C¹(grΣ)and apply Theo-rem 4.8 to obtain

Dθ Dt =

(∇v)^±τ,n_Σ

. (4.35)

Recall that the vectorsτandn_Σcan be expressed as

τ=−n_Γcosθ−n_∂Ωsinθ, n_Σ=n_Γsinθ−n_∂Ωcosθ. (4.36) Inserting (4.36) into equation (4.35) yields

Dθ

Dt =cos²θ

(∇v)^±n_Γ,n_∂Ω

−sin²θ

(∇v)^±n_∂Ω,n_Γ

+sinθcosθ

(∇v)^±n_∂Ω,n_∂Ω

−

(∇v)^±n_Γ,n_Γ . (4.37) Notice that the impermeability condition implies that

(∇v)^±n_Γ,n_∂Ω

=0.

Moreover, Lemma 4.17 allows to drop the±-notation. We now exploit that both the jump condition and the Navier condition are active at the contact line. Using the relation (4.27), it follows from the Navier condition

Dθ

Dt =sin(θ)²aV_Γ+sinθcosθ(h∇v n_∂Ω,n_∂Ωi − h∇v n_Γ,n_Γi). (4.38) Sinceτis tangential toΣ, it follows from the continuity of the tangential stress component (4.29) that

hJSKn_Σ,τi=0.

Using Lemma 4.17, we can exploit the continuity property of∇vat the contact line to obtain 0=2JηKhD nΣ,τi ⇔ 0=hD nΣ,τi.

Together with the expansions (4.36) fornΣandτwe obtain

0=sinθcosθ(−hDn_Γ,n_Γi+hDn_∂Ω,n_∂Ωi) + (cos²θ−sin²θ)hDn_∂Ω,n_Γi. (4.39)

Chapter 4. Kinematics of moving contact lines

Using the Navier condition, we can replace the last term to find

0=sinθcosθ(−h∇v n_Γ,n_Γi+h∇v n_∂Ω,n_∂Ωi) + (sin²θ−cos²θ)aV_Γ

2 onΓ. (4.40)

Note that forθ=π/2 this reduces to

aV_Γ=0.

The claim follows by inserting equation (4.40) into the contact angle evolution equation (4.38):

Dθ

Dt =sin²(θ)aV_Γ+sinθcosθ(h∇v n_∂Ω,n_∂Ωi − h∇v n_Γ,n_Γi)

=aV_Γ

sin²θ−1

2(sin²θ−cos²θ)

=aV_Γ 2 .

Note that the incompressibility condition (4.28) can be dropped leading to (see Theorem 4.29) Dθ

Dt =JλK JηK

V_Γ 2 .

Remark4.19 (Free Boundary Problem). Following the proof of Theorem 4.18, it is easy to show that (4.33) also holds for afree boundary formulation, where the Navier-Stokes equations are only solved in the liquid domain;

see (3.24). The outer phase is represented just by a constant pressure field p₀and the jump conditions (4.29) are replaced by

(p₀−p+S)n_Σ=σ κn_Σ on Σ(t).

In particular, the viscous stress componenthSn_Σ,τivanishes and Lemma 4.17 is not required for the proof.

Corollary 4.20. Under the assumptions of Theorem 4.18, a quasi-stationary solution, i.e. a solution with constant contact angle, satisfies

aV_Γ=0 on Γ, (4.41)

which means that either the contact line is at rest ora=0.

Consequently, a regular, non-trivial, quasi-stationary solution only exists ifavanishes at the contact line, i.e.

in the free-slip case. On the other hand free-slip at the contact line implies that the contact angle is fixed for all regular solutions. This result confirms the observation from [Sch01], where it is stated that for a regular solution witha∈(0,∞)andθ≡π/2 “the point of contact does not move”.

Corollary 4.21. Let a≥0 and(v,grΣ)be a regular solution in the setting of Theorem 4.18 that satisfies the thermodynamic condition (3.19). Then (4.33) implies

θ˙≥0 for θ≥θeq and θ˙≤0 for θ≤θeq.

From this result, it follows that the system cannot evolve towards equilibrium with a regular solution in the setting of Theorem 4.18.

Corollary 4.22. Leta≥0 and{v,grΣ} be a regular classical solution of the PDE-system (4.28)-(4.32) in the setting of Theorem 4.18 that satisfies (3.19). Let the initial condition be such that

θ(0,x)>θ_eq ∀x∈Γ(0), whereΓ(0) =∂Σ(0)is assumed to be bounded. Then it follows that

θ(t,x)≥ min

x⁰∈Γ(0)θ(0,x⁰)>θeq

for allt∈I∩[0,∞)andx∈Γ(t). That means that the system cannot relax to the equilibrium contact angle.

50

4.2. Contact angle evolution in the standard model

Proof. Considert∈I∩[0,∞)andxt∈Γ(t)arbitrary. Then there exists anx0∈Γ(0)such that the unique solution x(s)of the initial value problem

x⁰(s) =v(s,x(s)), x(0) =x₀∈Γ(0) (4.42) satisfiesx(t) =x_t. The pointx₀can be found by solving (4.42) backwards in time. By integration of (4.33), we conclude

θ(t,x(t)) =θ(t,x_t) =θ(0,x(0)) + Z t

0

d

dsθ(s,x(s))ds

=θ(0,x₀) + Z _t

0

aV_Γ 2

|{z}

≥0

(s,x(s))ds

≥θ(0,x₀)≥ min

x⁰∈Γ(0)

θ(0,x⁰)>θ_eq.

4.2.3. Empirical contact angle models

As discussed in detail in Chapter 2, the literature contains a large variety of empirical contact angle models that prescribe the dynamic contact angle. For the simplest class of these models, it is assumed thatθcan be described by a relation of the type

θ=f(Ca,θeq), (4.43)

whereθeqis theequilibriumcontact angle given by the Young equation (1.4). Hence for a given system, f is a function of the contact line velocityV_Γ, i.e.

θ=f(V_Γ). (4.44)

If this relation is invertible, one can also write (g:=f⁻¹)

VΓ=g(θ). (4.45)

The following corollary is an immediate consequence of this modeling.

Corollary 4.23. Consider the model described in Theorem 4.18 together with the dynamic contact angle model (4.44). Let f ∈C¹(R). Then, for regular solutions in the sense of Theorem 4.18, the contact line velocity obeys the evolution equation

f⁰(V_Γ) D

DtVΓ=aVΓ

2 . (4.46)

If the model from Theorem 4.18 is equipped with the contact angle model (4.45) withg∈C¹(0,π), the contact angle for regular solutions in the sense of Theorem 4.18 follows the evolution equation

Dθ

Dt =a g(θ)

2 . (4.47)

Remark4.24. From Corollary 4.23 we draw the following conclusions.

(i) By adding on of the empirical models (4.44) or (4.45) to the model from Theorem 4.18 with a fixed slip length, the time evolution ofθandV_Γis, for regular solutions, already completely determined by theordinary differential equation (4.46) or (4.47), respectively. But note that neither the momentum equation nor the normal part of the transmission condition involving the surface tension is used for its derivation. This means that, for regular solutions, neither external forces like gravity nor surface tension forces can influence the motion of the contact line.

Chapter 4. Kinematics of moving contact lines

(ii) If the empirical function satisfies the thermodynamic condition (3.21), i.e.

V_Γ(f(V_Γ)−θeq)≥0 or g(θ)(θ−θeq)≥0,

respectively, there are only constant ormonotonically increasing/decreasingsolutions forθ(t)(in Lagrangian coordinates).

(iii) Moreover, we have the additional requirement thatD_tθ=0 forθ=π/2, which dictatesg(π/2) =0=g(θ_eq) (ora=0 which means thatθis fixed).

Im Dokument Mathematical modeling and Volume-of-Fluid based simulation of dynamic wetting (Seite 68-74)