Chapter IV: Generalized Lubrication Theory on Curved Geometries
4.4 Truncated Minimum Dissipation Principle for Lubrication Flow
Next we apply the lubrication scaling (4.68) to the total free energy. In the height field representation, the free energy of the liquid layer Ωcoating the supporting substrate Γˆ is assumed to be a functional of the height fieldη,
F[ ˜Γ] =F[ξ1, ξ2, η(ξ1, ξ2, t)]. (4.74) Energies which only involve differential operators acting onηare considered local. Local energies are the most common types studied in the literature. For a sufficiently thin liquid layer, an asymptotic series in orders of height fieldη can be derived. For instance, as stated earlier, capillary stress arises from the variation of the total area of the moving free surfaceΓ˜. Recall the expansion of the area differential d ˜Γ from (4.38). Then the surfface energy of a liquid film with homogeneous surface tension coefficientσo can be expanded as
F[ ˜Γ] =σo Z
Γ˜ d ˜Γ =F[ ˆΓ;η] =σo Z
Γˆ1−2Hη+Kη2+1
2gˆαβ ∂η
∂ξα
∂η
∂ξβ +O(η3) d ˆΓ . (4.75) Recall from conservation law (4.66) that the columnar volume density%(ξ1, ξ2, t)instead of the height fieldηserves as the dynamical variable in the governing equation for viscous free surface liquid. By the inversion relations (4.41) and (4.42) between%(η) andη(%) we can conveniently switch back and forth between F[ξ1, ξ2, η] as a functional of the height field or F[ξ1, ξ2, %] of the volume density. The zeroth order term in the free energy usually only concerns the substrate geometry which is treated as a stationary object in the present work. The rate at which free energyFfluctuates with the moving interfaceΓ˜ is given by
dF[ ˜Γ]
dt =δF[%;∂%
∂t] =Zˆ
Γ
δF δ%
∂%
∂td ˆΓ , (4.76)
whereδF/δ%=O(%) is the variational derivative ofF[ ˆΓ;%].
the system is measured by the reference areal energy density[σ](energy per area) which has the same unit (N/m) as surface tension coefficient. With these scaling in mind, the dimensionless variables indicated by the superscript(·)∗ are given by
ˆ
x→[L]ˆx∗, ξ3→[H]z, t→[T]t∗, vα→[V](v∗)α, v3→[V](v∗)3, ˆq→[H][V]ˆq∗,
η→[H]η∗, %→[H]%∗, qα →[H][V](q∗)α F→[σ][H][L]F∗.
(4.77)
The appropriate choices for the reference velocity [V] and the time scale [T] would emerge naturally when we balance viscous dissipation with free energy fluctuation rate.
In what follows, we drop the superscript (·)∗ on all rescaled variables. The relative JacobianJ defined in (4.21) and partial flux coefficientqα from (4.67) are expressed in the rescaled transverse coordinatez,
J(ξ1, ξ2, z) = 1−2Hz+2Kz2, (4.78) qα(z) =Z z
0 vαJdz0. (4.79)
Conservation law (4.66) of the columnar volume density%remains invariant after rescal- ing (4.77), which is expected because mass (volume) conservation is never approximated but enforced exactly in lubrication theory. On the other hand, approximation (4.73) of the viscous dissipation in the rescaled variables leads to the approximate dimensionless dissipation,
1
2D[v] = µ[L]3/([T]2) [σ][L]2/[T]
( Z
Γˆ
"Z η
0
1
2gαβ∂vα
∂z
∂vβ
∂z Jdz
#
| {z }
O(1)+O()+O(2)
d ˆΓ +O(2) )
. (4.80)
Likewise the dimensionless rate of energy variation defined in (4.76) becomes dF[ ˜Γ]
dt =Z
Γˆ
δF
|{z}δ%
O(1)
∂%
∂t d ˆΓ , (4.81)
where the variational derivative δF/δ% is an asymptotic series expended in orders of aspect ratio .
We next consider variation to the virtual power due to external surface traction at the free surface. Let [B]be the charaterstic scale of the external surface traction b. Recall
from (4.38) that after dimensionless virtual power can be approximated as, T[%;v,b] = [B][L]2[V]
[σ][L]2/[T] Z
Γˆ
Av·bz=ηd ˆΓ
= [B][L]
[σ]
Z
Γˆ
Abβvαgαβ+v3b3 z=ηd ˆΓ
= MaZ
Γˆ
J +O(2) bβvαgαβ+O(2) z=ηd ˆΓ
= MaZ
Γˆ
J bβvαgαβ z=ηd ˆΓ +O(2), Ma = [B][L]
[σ] , (4.82) where we assumeξ3-component (recallg3 =n) of tractionˆ bis onlyO(). In such cases, Marangoni numberMa in equation (4.82) is appropriate for characterizing the effect of tangential dominant surface traction such as the thermocapillary effect in Chapter2.
The balance between viscous dissipation (4.80) and free energy fluctuation rate (4.81) immediately suggests the scaling for time and velocity,
µ[L]3
[T]2 =[σ][L]2
[T] ⇐⇒[T] = µ[L]
2[σ] or [V] = 2[σ]
µ . (4.83)
The singular scaling 1/2 in [T] is one of the predominant signatures of lubrication flows: the more slender the characteristic film thickness [H] becomes, the slower the thin liquid film redistributes its mass. As suggested by scaling (4.83), such dilemma can be resolved either by reducing the size of the system [L] ↓ or by increasing the scale of areal energy density [σ] ↑. Although we must be cautious with the rate at which the bulk kinematic energy fluctuates ∼ ρ[V]2[H][L]2. For lubrication theory to be self-consistent, the overall scale of the kinematic energy fluctuation rate must be negligible compared to the leading order terms in the viscous dissipation (4.80) and to the free energy fluctuation rate (4.81) otherwise the conventional lubrication theory doesn’t hold under inertial effect. The magnitude of kinematic energy fluctuation rate can be estimated relative to the free energy fluctuation rate (4.81),
ρ[V]2[H][L]2/[T]
[σ][L]2/[T] = ρ[V][L]
µ
µ[V]
[σ] =2Re =O(2). (lubrication assumption) (4.84) Thanks to the small aspect ratio 1, the applicability of lubrication theory only requires a moderately small Reynolds number from (4.84) which is more general than the true Stokes limitRe = 0. These scaling requirements are quite realistic, for example, when applied to the experiment by Takagi and Huppert (2010) in which 123 cm3 of golden syrup was released on a beach ball of radius [L] = 23 cm. The experimental parameters are kinematic viscosity ν = 0.045 m2/s, surface tension [σ] = 0.078 N/m and density ρ = 1.4 kg/m3. For a liquid film of 1 cm thickness (film thickness was not reported by the authors), the aspect ratio ∼0.04. The lubrication scaling (4.83) leads to a characteristic time [T] ∼ 100 s which agrees with the time scale between
the snapshots of their experiment. In such a case, by equation (4.84) we evaluate the Reynolds numberRe∼0.01∼O() which is more than sufficient for lubrication theory to apply.
In summary, we replace the original viscous dissipation D, free energy F and external traction Tby their truncated versions up toO(2) approximation errors,
1
2D[v] = 1
2D[v] +O(2), (4.85) δF[ ˜Γ;v] =δF[%;∂%
∂t] +O(2), (4.86)
T[ ˜Γ;v,b] =T[%;v,b] +O(2), (4.87) where the truncated functionalsD,FandTare given by,
D[v] =Z
Γˆ
(Z η
0 gαβ∂vα
∂z
∂vβ
∂z Jdz )
d ˆΓ , (4.88)
δF[%;∂%
∂t] =Z
Γˆ
δF δ%
∂%
∂td ˆΓ , (4.89)
T[%;v,b] = MaZ
Γˆ
vαJ gαβbβ|z=ηd ˆΓ . (4.90) Truncated minimum dissipation principle
The governing equation of a thin liquid layer driven by dissipation of its free energy and external tangential traction is given by the solution to the truncated minimum dissipation principle in place of the full version (4.56),
minv
1
2D[v] +δF[%;∂%
∂t]−T[%;v,b], subject to ∂%
∂t + ˆ∇ ·qˆ= 0 on ˆΓ , v=0 on ˆΓ ,
(4.91)
where the incompressiblity condition is enforced through the conservation law of colum- nar volume density %. Since the approximate functionals (4.85)–(4.87) are truncated at O(2), it suffices to look for the necessary conditions for which a unique optimal partial fluxqα defined in (4.67) must satisfy in order to minimize (4.91) up to someO(2)error only.
Let η be a height field (not necessarily optimal) on the boundary-less surface Γˆ andv be the optimal velocity field for the liquid volume bounded by η. We consider a flux variation δqα(ξ1, ξ2, z) be the optimal volumetric flux qα associated with the optimal velocityv. Applying integration by parts in thez-coordinate to the variation of viscous
dissipation yields δ
1
2D[v;δv]=Z
Γˆ
Z η
0 gαβ∂δvα
∂z
∂vβ
∂z Jdzd ˆΓ
=Z
Γˆ
(
gαβδvα∂vβ
∂z Jη
0− Z η
0 δvα ∂
∂z ∂vβ
∂z gαβJdz )
d ˆΓ
=Z
Γˆ
(
gαβδvα∂vβ
∂z J
η−δqα J
∂
∂z ∂vβ
∂z gαβJ
η
+Z η
0 δqα ∂
∂z
"
1 J
∂
∂z ∂vβ
∂z gαβJ
# dz
)
d ˆΓ . (4.92)
Note the boundary terms evaluated atz= 0vanish due to the integral definition (4.67) of partial fluxδqα(ξ1, ξ2, z)and the no-slip conditionδvβ(ξ1, ξ2,0) = 0at the supporting substrate. For an admissible variation δqα such that δqα|z=η = 0, the second term in variation (4.92) drops out. If the free surface were stress-free (zero-traction), the first term should vanish as well. In the presence of tangential interfacial tractions such as Marangoni stresses, the balance between the contribution from viscous dissipation and the external power by tangential traction translates into a boundary condition for qβ(z) atz=η,
∂vβ
∂z J
z=η = Mabβ. (4.93)
After dropping out the first and the second term in the integrand of variation (4.92), we are left with a piece which only involves integration in thez-coordinate. A necessary condition for this term to vanish against all variations is that,
∂
∂z 1
J
∂
∂z
hgαβJ ∂
∂z 1
J
∂qβ
∂z
i= 0. (4.94)
Equation (4.94) is a fourth order ODE subject to four boundary conditions derived earlier,
qβ
z=0= 0, (by definition (4.67))
∂qβ
∂z
z=0= 0, (no-slip condition) qβ
z=η =qβ(η), (choice of variation)
∂
∂z 1
J
∂qβ
∂z
z=η = Mabβ. (traction-free)
(4.95)
Equation (4.94) can be solved by elementary integration, 1
J
∂
∂z
hgαβJ ∂
∂z 1
J dqβ
dz
i=Aα,
gαβJ ∂
∂z 1
J dqβ
dz
=Bα+Aα Z z
0 Jdz0,
∂
∂z 1
J dqβ
dz
= gαβ J
Bα+Aα Z z
0 Jdz0,
∂qβ
∂z =CβJ+J Z z
0
gαβ J
Bα+Aα Z z0
0 Jdz00dz0, qβ =Dβ +Z z
0 JhCβ+Z z0
0
gαβ J
Bα+Aα Z z00
0 Jdz000dz00idz0, (4.96) where coefficients Aα, Bα, Cβ and Dβ are functions of ξ1 and ξ2 only. Applying boundary conditions at z = 0 eliminates Cβ = Dβ = 0. Traction condition at free surface imposes a constraint between Aα and Bα,
Bα+Aα Z η
0 Jdz= MaJgαβbβ|z=η. (4.97) The optimal partial fluxqβ for a domain bounded by the height fieldη (or equivalently the columnar volume density%) is found to be
qβ(ξ1, ξ2, z) =Z z
0Jn Z
z0 0
hMabβ +gαβAα J
Z z00
0 Jdz000− Z η
0 Jdz000idz00odz0. (4.98) Any optimal partial flux qβ must be in the form (4.98) otherwise it would necessar- ily produce O() residues for the truncated minimization problem (4.91) which make δqβ(ξ1, ξ2, z)suboptmal.
The only free coefficient Aα(ξ1, ξ2) left in (4.98) is determined by considering a gen- eral variation δq to the optimal flux qβ such that δqβ(ξ1, ξ2, η) may not be identically zero. Letη(ξ1, ξ2, t)and%(ξ1, ξ2, t) be the optimal interface shape and volume density respectively and δqˆα denote the variation to the total volumetric flux qˆα = qα|z=η. Substituting optimal partial flux (4.98) into viscous dissipation (4.92) yields the two surviving terms for the truncated variation problem (4.91),
δ 1
2D[v] +δF[%;∂%
∂t]−T[%;v,b]
=Z
Γˆ−δqα J
∂
∂z ∂vβ
∂z gαβJ
ηd ˆΓ +Z
Γˆ
δF δ%
%δ
∂%
∂t
d ˆΓ First line of4.96
=Z
Γˆ−δqˆαAαd ˆΓ +Z
Γˆ
δF δ%
%δn−∇ ·ˆ qˆo d ˆΓ
=Z
Γˆ−δqˆαgˆαβAβd ˆΓ − Z
Γˆ
δF δ%
%
∇ ·ˆ δˆqd ˆΓ
=Z
Γˆ
δqˆα
"
−ˆgαβAβ+ ∂
∂ξα δF
δ%
%
#
d ˆΓ . Boundary-less manifold (4.99)
For these terms to vanish, coefficient Aβ must satisfy, Aβ = ˆgαβ ∂
∂ξα δF
δ%
%
. (4.100)
Substituting Aβ into equation (4.98) and evaluating at z = η shall produce the opti- mal flux. The truncated minimum dissipation problem (4.91) is essentially a quadratic programming subject to a linear constrain. Since we pursue an approximation to the optimal solution aiming forO(2) errors, it suffices to further expandqβ with only the O(1)and O() terms retained,
ˆ
qβ(ξ1, ξ2, η) =n−η3
3 ˆgαβ+η4
6 (4Hgˆαβ−IIαβ)o ∂
∂ξα δF
δ%
%
+Manη2
2 ˆgαβ+η3−2
3Hˆgαβ +IIαβoˆbα+O(2), (4.101) where we introduce the tangent vector field
ˆb=bαgˆα (4.102)
on the supporting surface Γˆ as an alternative approximation to the external tangential tractionbon the free surfaceΓ˜. To derive the full evolution equation, we convert height field η to columnar volume density % truncated at O(2) errors,
ˆ
qβ(ξ1, ξ2, %) =− (%3
3 ˆgαβ+%4
6 (2Hgˆαβ +IIαβ) ) ∂
∂ξα δF
δ%
%
+ Ma (%2
2gˆαβ+%31
3Hgˆαβ+IIαβ
)ˆbα+O(2). (4.103)
Substituting the optimal flux (4.103) back into the conservation law (4.66) with un- derscores below the optimal variables dropped yields the governing PDE of the optimal columnar volume density in a coordinate-free form,
∂%
∂t + ˆ∇ ·
"
−DF∇ˆδF
δ% + MaDTˆb
#
= 0 on ˆΓ , (4.104) where the two nonlinear, inhomogeneous mobility tensors, DF and DT, convert the surface projections of interfacial pressure gradient and external tangential traction to volumetric fluxes in the lubrication limit, respectively,
DF = %3
3I +%4
6 2HI +II, DT = %2
2I +%3 1
3HI +II. (4.105) On a flat substrate where the first fundamental formI =I becomes the identity tensor and the second fundamental form II = 0 vanishes everywhere, we deduce η = % and hence recover the general form of classical lubrication equation (2.37) in Section 2.2.
In the absence of external surface traction ˆb(orMa = 0), we arrive at an approximate energy-dissipation law similar to (4.60),
dF[%]
dt =− Z
Γˆ
∇ˆδF δ%
!
DF ∇ˆδF δ%
!
d ˆΓ ≤0. (4.106)
Fluctuations in interfacial energies of local-tpye
In order to put a closure on the evolution PDE (4.104), the free energies must be expanded in the asymptotic orders of slender aspect ratio . Recall from the scaling (4.81) that all energies associated with the thin liquid film are scaled by the reference energy [σ][L]2. An interfacial energy is said to be “local” if its areal energy density at a point of the interface only depends on quantities defined at that point. The notion of localness is made clear in the following two example of interfacial energies.
One of the most common energies in physics is the potential energy due to the gravita- tional force. Letgbe the unit vector opposite to the direction of gravity. For a medium of uniform density, the total gravitational energy can be expressed as a volume integral of the projection of position vector onto the unit vectorg,
G[ ˜Γ] = ρg[L]4 [σ][L]2
Z
Ωx·gdΩ= Bo
Z
Γˆ Z η
0 (ˆx+zn)ˆ ·gJdzd ˆΓ , Bo = ρg[L]2 [σ] ,
(4.107) where Bo is the Bond number. For example in the previous example (Takagi and Huppert, 2010) where a golden syrup film of 1 cm thickness spreads on a beach ball with a radius of23 cm,Bo∼9.5indicates gravity is the dominant driving force in that system. Although the Bond number diminishes rapidly as size of the system [L] goes down. In terms of columnar volume density %, we have
G[ ˆΓ;%] = BoZ
Γˆ(ˆx·g)%+(ˆn·g)1
2%2d ˆΓ +O(2). (4.108) Variation of the truncated gravitational energy is a simple scalar field
δG
δ% = Boxˆ·g+(ˆn·g)%. (4.109) Note for a supporting surface Γˆ almost parallel to the direction of gravity field, it’s necessary to retain terms at least up to the first order in the approximate gravitational energy (4.109) because the zeroth order term xˆ ·g completely vanishes and the first order term(ˆn·g)%becomes the dominant driving mechanism underlying the dynamics of the viscous liquid layer layer.
The surface tension coefficientσoof the liquid/air interface is selected to be the baseline for various types of interfacial energies per area. Whenσo is homogeneous, i.e. constant everywhere on an uncontaminated free surface, the surface energy after nondimension- alization is simply the total area of the (boundary-less) free surface Γ˜,
S[ ˜Γ] =Z
Γ˜ d ˜Γ = 1
Z
Γˆ1−2Hη+2Kη2+21
2gˆαβ ∂η
∂ξα
∂η
∂ξβ +O(3) d ˆΓ . (4.110) In terms of columnar volume density%, the surface energy becomes
S[ ˆΓ;%] = 1
Z
Γˆ d ˆΓ − Z
Γˆ2H%+1 2
h(4H2−2K)%2−( ˆ∇%)·( ˆ∇%)id ˆΓ +O(2).
(4.111)
The first integral in (4.111) represents the area of the fixed supporting substrate Γˆ covered by the viscous liquid layer which should not contribute to energy variation in time. The second integral contains two scalar pieces similar to the case of gravitational energy plus a gradient-type (Dirichlet) energy. To derive its variation, we perform integration by parts on the boundaryless manifold Γˆ which yields
Z
Γˆ−( ˆ∇%)·( ˆ∇%) d ˆΓ =Z
Γˆ% ˆ∆%d ˆΓ , (4.112) where the line integral arising from the manifold boundary vanishes and ˆ∆is the Laplace- Beltrami operator (covariant Laplacian) for surface Γˆ. In the end, we arrive at the expression of the variational derivative of surface energy,
δS
δ% =−2H−(4H2−2K)%−ˆ∆%. (4.113)
Equation (4.113) is essentially the perturbation expansion of the curvature of the free surfaceΓ˜, expressed with the coordinates of the supporting surfaceΓˆ and the columnar volume density%. Identical expressions were obtained by Roy, A. J. Roberts, and Simp- son (2002) and Rumpf and Vantzos (2013). Note the first two terms in the variation (4.113) of the approximate surface energy are entirely due to the curvature of the sub- strate whereas the Laplace-Beltrami term is the leading approximation to the curvature of the free surface. As we already know from Chapter2and Chapter3, for a flat region of the supporting surface Γˆ, substrate curvatures completely disappear and the Lapla- cian of volume density%becomes the sole driving force behind interface deformation of the thin films.
For the classical problem of a thin fluid flowing down an flat plane inclined at an angle α with respect to the(x, y)-plane driven by gravity and capillary stresses, the total free energy F = G +S. The resulting governing equation of film thickness agrees with many previous references (Huppert,1982; Troian et al., 1989; Roy, A. J. Roberts, and Simpson,2002),
∂%
∂t + ˆ∇ (%3
3 ∇ˆ hˆ∆%−Bo (cosα)%i+ Bo (sinα)%3 3
)
= 0, (4.114)
where differential operators are the usual two-dimensional Euclidean ones defined with respect to the Cartesian coordinates of the inclined plane.
4.5 Perfect Dielectric Films Coating Curved Conductors: Dissipation of a Non-