Remarks on more general models

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).

4.3. Remarks on more general models

This situation is not possible for the case ofconstant surface tension, visualized in Figure 4.1(b). In the case θ=π/4 and∂τσ=0 equation (4.49) together with the incompressibility condition

0=h∇v nΓ,n_Γi+h∇v n_∂Ω,n_∂Ωi implies

h∇v nΓ,n_Γi=h∇v n_∂Ω,n_∂Ωi=0.

Therefore, the linear part of the velocity field has a quite simple form. In the reference frame of the solid wall, it is given as

(u,v)(x,y) =VΓ

1+y

L,0 .

Figure 4.1(b) shows the field in a co-moving reference frame. Clearly, the field geometry leads to anincreasein the contact angle (clockwise rotation in this example).

(a) Constant contact angle,∇Σσ6=0. (b) Constant surface tension, ˙θ>0.

Figure 4.1.: Linear velocity fields satisfying Navier slip withL>0 in a co-moving reference frame.

4.3.2. Interfacial slip

Another possible generalization of the model is to allow for slip at the fluid-fluid interface. In this case, one only requires continuity of thenormalcomponent of the fluid velocity, i.e.

hJvK,n_Σi=0 onΣ(t),

which means that there is no mass flux from one phase to the other. To describe the evolution of the interface, one can use both of the fluid velocitiesv^±in the kinematic conditions

VΣ= v^±,nΣ

, VΓ= v^±,nΓ

.

This gives rise to two distinct Lagrangian derivative operators. To formulate the following Theorem, we choose the interfacial velocity field

vΣ:=η⁺v⁺−η⁻v⁻

η⁺−η⁻ =JηvK JηK

. (4.51)

Clearly, v_Σ also satisfies the above mentioned kinematic conditions. Hence we can define a Lagrangian time derivative according tov_Σ.

Lemma 4.28. The Lagrangian derivatives with respect to v⁺, v⁻and v_Σsatisfy the relation (η⁺−η⁻)D^Σ

Dt =η⁺D⁺

Dt −η⁻D⁻

Dt. (4.52)

Chapter 4. Kinematics of moving contact lines

Proof. Each Lagrangian derivative along the contact line may be decomposed as D^±

Dt =∂_t^Γ+v^±_k·∇Γ,

where∂_t^Γis the “contact line Thomas derivative” following the normal motion of the contact line,∇_Γis the gradient alongΓandv^±_k is the component ofv^±tangential to the contact line. Multiplication withη^±yields

η⁺D⁺

Dt −η⁻D⁻

Dt =JηK∂_t^Γ+ (η⁺v⁺_k −η⁻v⁻_k)·∇Γ

=JηK ∂_t^Γ+

η⁺v⁺_k −η⁻v⁻_k η⁺−η⁻ ·∇Γ

!

=JηK D^Σ Dt.

Note that Lemma 4.17 cannot be used for the proof of Theorem 4.29 sincevis no longer assumed to be contin-uous. Moreover, the following statement doesnotrequire the incompressibility of the flow.

Theorem 4.29. LetΩ⊂R³be a half-space with boundary∂Ω,η^±>0,JηK6=0and(v,grΣ)with v∈C¹(grΩ⁺)∩C¹(grΩ⁻)

andgrΣaC^1,2-family of moving hypersurfaces with boundary, be a classical solution of the PDE-system Jhv,n_ΣiK=0, PΣJ−SKn_Σ=∇Σσ onΣ(t),

hv,n_∂Ωi=0,λP∂Ωv+P∂ΩSn_∂Ω=0 on∂Ω\Γ(t), VΣ=

v^±,n_Σ

onΣ(t).

Moreover, let

λ∈C(gr∂Ω⁺)∩C(gr∂Ω⁻) andθ∈(0,π)ongrΓ. Then the evolution of the contact angle is given by

D^Σθ Dt =1

2

V_ΓJλK JηK

−∂_τσ JηK

, (4.53)

where ^D_Dt^Σ is the Lagrangian time-derivative according to the surface velocity field(4.51).

Proof. We start from (4.12) and, by a change of coordinates, obtain D^±θ

Dt =−sin²θ

∇v^±n_∂Ω,n_Γ

+sinθcosθ(

∇v^±n_∂Ω,n_∂Ω

−

∇v^±n_Γ,n_Γ ),

where the two time derivatives may now be different. Using the Navier condition, we can replace the first term according to

D^±θ

Dt =sin²θa^±V_Γ+sinθcosθ(

∇v^±n_∂Ω,n_∂Ω

−

∇v^±n_Γ,n_Γ ).

Note thatamay now be discontinuous atΓ. The next step is to express ˙θby means of the jump in normal stress.

Thanks toη^±>0 we can write

∇v^±nΓ,nΓ

= 1 2η^±

S^±nΓ,nΓ

,

∇v^±n_∂Ω,n_∂Ω

= 1 2η^±

S^±n_∂Ω,n_∂Ω .

Plugging this into the above equation for ˙θ gives D^±θ

Dt =sin²θa^±V_Γ+sinθcosθ 2η^± (

S^±n_∂Ω,n_∂Ω

−

S^±n_Γ,n_Γ

). (4.54)

54

4.3. Remarks on more general models

We introduceJSKby adding a zero term, i.e.

D⁺θ

Dt =sin²θa⁺VΓ+sinθcosθ 2η⁺ (

(S⁺−S⁻)n_∂Ω,n_∂Ω

−

(S⁺−S⁻)n_Γ,nΓ

+

S⁻n_∂Ω,n_∂Ω

−

S⁻nΓ,nΓ

). (4.55)

Using the second version of the equation (4.54), we have η⁻D⁻θ

Dt −sin²θa⁻η⁻V_Γ=sinθcosθ 2

S⁻n_∂Ω,n_∂Ω

−

S⁻n_Γ,n_Γ .

Together with (4.55) we obtain (usingλ^±=a^±η^±) η⁺D⁺θ

Dt −η⁻D⁻θ

Dt =sin²θJλKVΓ+sinθcosθ

2 (hJSKn_∂Ω,n_∂Ωi − hJSKnΓ,nΓi). (4.56) Now we exploit the validity of both the Navier and the jump condition for the stress at the contact line. From PΣJSKnΣ=−∇Σσwe obtain

−∂τσ=hJSKn_Σ,τi= (cos²θ−sin²θ)hJSKn_∂Ω,n_Γi

+sinθcosθ(−hJSKnΓ,nΓi+hJSKn_∂Ω,n_∂Ωi) (4.57) From the Navier condition, i.e.

λ^±VΓ+

S^±n_∂Ω,nΓ

=0, we infer

hJSKn_∂Ω,n_Γi=−JλKV_Γ by taking the trace. Combined with (4.57) we obtain

sinθcosθ(hJSKn_∂Ω,n_∂Ωi − hJSKn_Γ,n_Γi) = (cos²θ−sin²θ)JλKV_Γ−∂τσ.

Plugging in this expression into (4.56), we arrive at η⁺D⁺θ

Dt −η⁻D⁻θ

Dt =JλKV_Γ

sin²θ+cos²θ−sin²θ 2

−∂τσ 2 =1

2(JλKV_Γ−∂τσ).

Now the claim follows from (4.52).

Remark4.30. (i) If the flow is incompressible and the velocity is continuous acrossΣ, equation (4.25) implies that the slip length has to be continuous across the contact line to allow for non-trivial regular solutions. In this case, we haveJλK=aJηKand (4.53) reduces to (4.48).

(ii) If the surface tensionσis constant, we obtain the evolution equation D^Σθ

Dt =V_Γ 2

JλK JηK If both jumps have the same sign, i.e. if

JλK JηK

≥0,

the qualitative behavior of regular solutions is still the same as in Theorem 4.18.

Chapter 4. Kinematics of moving contact lines

4.3.3. Systems with phase change

So far we only discussed the case, when no phase transitions occur. We now generalize the results for non-zero mass transfer across the fluid-fluid interface. Given an interface with interface normal fieldn_Σand normal velocity V_Σ, the one-sidedmass transfer fluxesare defined as

m˙^±=ρ^±(v^±·nΣ−VΣ) on grΣ.

If the interface is not able tostoremass, the mass transfer flux has to be continuous, i.e.

Jm˙K=0 ⇔ JρvK·n_Σ=JρKV_Σ. (4.58) Note that models for dynamic wetting allowing for mass on the fluid-fluid interface have also been considered under the name Interface Formation Model, see [Shi93], [Shi08] and Section 3.5. In the case without interfacial mass, the interfacial normal velocity can be expressed as

VΣ=v^±·nΣ− m˙ ρ^±.

Since the interface is now transported byV_Σ6=v^±·n_Σ, the mass flux influences the evolution of the interface.

From the above relation, it follows that the mass transfer flux is related to the jump in the normal component ofv according to

JvK·n_Σ=J1/ρKm˙ on grΣ. (4.59)

At the contact line, we can expressn_Σvian_Γandn_∂Ω, i.e.

JvK·n_Σ=sinθJvK·n_Γ−cosθJvK·n_∂Ω on grΓ.

For simplicity, we consider in the following the case of two spatial dimensions⁷. If we assumev^±to satisfy the impermeability condition, we find the following relation for the jump ofvat the contact line

sinθJvK_|Γ=J1/ρKm n˙ _Γ. (4.60)

Note that the above relation only holds in two spatial dimensions. A slip tangential to the contact line may be present in three dimensions. In this case, one can only state that

(1− ht_Γ,·it_Γ)sinθJvK_|Γ=J1/ρKm n˙ _Γ.

Theorem 4.31. LetΩ⊂R²be a half-space with boundary∂Ω,η^±>0,JηK6=0and(v,grΣ)with v∈C¹(grΩ⁺)∩C¹(grΩ⁻)

andgrΣaC^1,2-family of moving hypersurfaces with boundary, be a classical solution of the PDE-system m˙PΣJvK+PΣJ−SKnΣ=∇Σσ onΣ(t),

hv,n_∂Ωi=0,λP∂Ωv+P∂ΩSn_∂Ω=0 on∂Ω\Γ(t), V_Σ=

v^±,n_Σ

− m˙

ρ^± onΣ(t) withθ∈(0,π)ongrΓ. Then, the interfacial velocity field v_Σ∈C¹(grΣ)defined as

v_Σ:=JρvK JρK

=ρ⁺v⁺−ρ⁻v⁻

ρ⁺−ρ⁻ ongrΣ (4.61)

7Note that equation (4.58) implies that the “natural” interfacial velocity to be considered here is (4.61), i.e. the bulk velocities should be weighted with the densityρ. Recall that in the case of interfacial slip without mass transfer, the natural interfacial velocity to choose is (4.51), i.e. weighted with the viscosityη. In the present case of interfacial slip with mass transfer, it is not obvious which interfacial velocity to choose. We exclude this problem by restricting the Theorem to the 2D case, where the interfacial velocity following the contact line is unique.

56

4.3. Remarks on more general models

satisfies the consistency conditions

VΣ=hvΣ,nΣi on grΣ, (4.62)

V_Γ=hv_Σ,n_Γi on grΓ (4.63)

and the corresponding evolution of the contact angle is given by JηK

D^Σθ

Dt =JλKV_Γ 2 −∂τσ

2 −Jη/ρK∂τm˙ +m˙

−Jη ∂τ(1/ρ)K−κcotθ sη

ρ {

−cotθ 2

s1 ρ

{

m+˙ 1 2 sinθ

sλ ρ

{ .

(4.64)

In the special case of zero mass flux at the contact line, the above equation simplifies to JηK

D^Σθ

Dt =JλKV_Γ 2 −∂τσ

2 −Jη/ρK∂τm.˙ (4.65)

Proof. We observe that, as a consequence of (4.58), the velocity vΣdefined by (4.61) satisfies the consistency condition (4.62). It also satisfiesv_Σ·n_∂Ω=0 on grΓsincev^±are tangential to∂Ω. Hence, Lemma 4.6 implies that (4.63) also holds. Moreover, it is easy to check thatv_Σcan be expressed in two different ways as

vΣ=v^±− JvK ρ^±J1/ρK

on grΣ. (4.66)

Applying Theorem 4.8 yields, using (4.59), D^Σθ

Dt =

∂τv^±,n_Σ

− ∂

∂ τ JvK ρ^±J1/ρK

,n_Σ

=

∂_τv^±,n_Σ

−∂_τ

JvK ρ^±J1/ρK

,n_Σ

− JvK·τ ρ^±J1/ρK

hτ,∂_τn_Σi

=

∇v^±τ,nΣ

−∂τ

˙ m

ρ^±−κcotθ m˙ ρ^±

=:

∇v^±τ,n_Σ

+R. (4.67)

Here we used the relation

JvK·τ J1/ρK

= m˙

sinθn_Γ·τ=−cotθm˙ on grΓ,

which follows from (4.60). Multiplication of the first term in (4.67) withη^±together with a change of basis vectors yields

η^±

∇v^±τ,nΣ

=−sin²θ η^±

∇v^±n_∂Ω,nΓ

+sinθcosθ η^±(

∇v^±n_∂Ω,n_∂Ω

−

∇v^±nΓ,nΓ

)

=−sin²θ

S^±n_∂Ω,n_Γ

+sinθcosθ

2 (

S^±n_∂Ω,n_∂Ω

−

S^±n_Γ,n_Γ ).

We may now multiply (4.67) byη^±to find JηK

D^Σθ

Dt =JηRK−sin²θhJSKn_∂Ω,n_Γi+sinθcosθ

2 (hJSKn_∂Ω,n_∂Ωi − hJSKn_Γ,n_Γi). (4.68) The tangential stress condition atΓreads

hJSKn_Σ,τi=−∂τσ+m˙JvK·τ=−∂τσ−cotθJ1/ρKm˙². We rewrite the left-hand side using the expansions forn_Σandτ, i.e.

hJSKnΣ,τi= (cos²θ−sin²θ)hJSKn_∂Ω,nΓi +sinθcosθ(hJSKn_∂Ω,n_∂Ωi − hJSKn_Γ,n_Γi)

=−∂_τσ−cotθJ1/ρKm˙².

(4.69)

Chapter 4. Kinematics of moving contact lines

From (4.68) and (4.69) we obtain

JηK D^Σθ

Dt =JηRK−1

2hJSKn_∂Ω,nΓi −∂τσ 2 −cotθ

2 J1/ρKm˙². (4.70)

Using (3.8) we can compute the contact line velocity sinθVΓ=v^±·nΣ− JvK·n_Σ

ρ^±J1/ρK

=sinθv^±·nΓ− m˙ ρ^±.

Hence the contact line velocity reads

V_Γ=v^±·n_Γ− m˙

sinθ ρ^± on grΓ. (4.71)

Equation (4.60) shows that the above expression is indeed well-defined, i.e. the two representations are equal. Note thatΓis no longer a material interface. The mass transfer term can cause a motion of the contact line. Hence the Navier condition atΓreads

S^±n_∂Ω,nΓ

=−λ^±v^±·nΓ=−λ^±

VΓ+ m˙ ρ^±sinθ

.

We, therefore, obtain the jump condition

hJSKn_∂Ω,n_Γi=−JλKV_Γ− m˙ sinθ

sλ

ρ {

(4.72) on grΓ. This finally leads to

JηK D^Σθ

Dt =JλKV_Γ 2 −∂τσ

2 −

s η ∂τ

m˙ ρ

{

− m˙ sinθ

κcosθ

sη ρ

{ +cosθ

2 s1

ρ {

˙ m−1

2 sλ

ρ {

. The claim follows from

s η ∂τ

m˙ ρ

{

=Jη/ρK∂τm˙+m˙Jη ∂τ(1/ρ)K.

4.3.4. Generalized Navier Slip

The idea of the Generalized Navier Boundary Condition (GNBC) described briefly in Section 3.4 is to allow for a deviation of the contact angle from the equilibrium value which is then relaxed by a transient process. Let us consider the standard model in two dimensions in the free surface form (3.24) with the Navier slip condition replaced by an approximation of the formal GNBC (3.26), i.e.

λv_k+ (Sn_∂Ω)_k+σ(cosθ−cosθ_eq)n_Γδ_Γ=0 on ∂Ω atthe contact line. The approximation reads as

λv_k+ (Sn_∂Ω)_k+1

εσ(cosθ−cosθ_eq)nΓ=0 atΓ, (4.73) whereε>0 is a small parameter of dimension length. Leaving aside the issue of the thermodynamical consistency of the formulation, we analyze the contact angle dynamics for a regular solution. By taking the inner product with the contact line normal vector and using the impermeability condition, we obtain from (4.73) the relation

λ

ηV_Γ+hn_Γ,∇v n_∂Ωi+ σ

ε η(cosθ−cosθ_eq) =0. (4.74)

58

4.3. Remarks on more general models

Using the impermeability condition, the rate-of-change of the contact angle reads as (see (4.37))

θ˙=−sin²θh∇v n_∂Ω,n_Γi+sinθcosθ(h∇v n_∂Ω,n_∂Ωi − h∇v n_Γ,n_Γi). (4.75) Thanks to the zero tangential stress conditionhSn_Σ,τi=0, we have

0=sinθcosθ

2 (−h∇v nΓ,n_Γi+h∇v n_∂Ω,n_∂Ωi) + (cos²θ−sin²θ)h∇v n_∂Ω,nΓi. (4.76) Hence, we can express the time derivative of the contact angle as

θ˙=−1

2hn_Γ,∇v n_∂Ωi.

It follows from (4.74) that the contact angle dynamics for a regular solution reads as θ˙=V_Γ

2L+1 ε

σ

2η(cosθ−cosθ_eq)). (4.77)

Remark4.32. (i) The uncompensated Young stress in the GNBC leads to an additional term in the rate-of-change in the contact angle which reads as

1 ε

σ

2η(cosθ−cosθ_eq)).

Obviously, the latter term is negative forθ <θ_eq (and positive forθ>θ_eq) and drives the system towards equilibrium. Therefore, the GNBC may give rise to physically reasonable regular solutions. In particular, we note that the uncompensated Young stress is able to change the direction of the velocity gradient at the contact line according to

−hnΓ,∇v n_∂Ωi=λ

ηVΓ+ σ

ε η(cosθ−cosθeq). (4.78) Hence, the kinematics of a spreading droplet withθ>θeqis changed fundamentally. Now, the fluid particles at the solid boundary may have a larger tangential velocity than the fluid particles slightly above the boundary leading tohnΓ,∇v n_∂Ωi>0. The latter effect looks like anegative“effective slip length”. It is, however, caused by the uncompensated Young stress in the velocity boundary condition.

(ii) Moreover, the relaxation term is proportional to 1/εwhich means that the rate-of-change becomes singular as ε→0. This observation is consistent with the fact that eq. (3.26) (without “smearing out” the delta function) impliesθ=θeq.

(iii) It is important to note that the GNBC gives rise to a functional dependence between dynamic contact angle and contact line speed forquasi-stationarystates. In fact, setting ˙θ=0 leads to the relation

Ca=ηVΓ

σ =L

ε(cosθ_eq−cosθ). (4.79)

Notably, the relationV_Γ=g(θ), withgdefined by (4.79), satisfies the thermodynamic consistency conditions formulated in Section 3.3, i.e.

g(θ_eq) =0 and g(θ)(θ−θ_eq)≥0.

Chapter 4. Kinematics of moving contact lines

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