Chapter IV: Generalized Lubrication Theory on Curved Geometries
4.3 Kinematics and Dissipation of Viscous Free Surface Flows
As we shall see later, the columnar volume density % is more suitable as the primary dynamic variable enforcing the conservation law of viscous free surface films than the height field η. However, in many situations physical quantities such as electric field or temperature are usually directly related to the local thickness of the free surface Γ˜ rather than its areal volume density because the height fieldη is a geometric quantity.
While it’s possible to invert the cubic polynomial in (4.41) to obtain an exact solution of η(%), it suffices to construct an asymptotic approximation for the height field η in orders of columnar volume density% for hydrodynamics confined within a thin layer,
η=%+H%2+1
3(6H2−K)%3+O(%4). (4.42) In the limit of slender geometries, the height field η and the volume density % are interchangeable up to higher order corrections depending on the local curvatures.
a non-negative quadratic functional of velocity field. The Helmholtz minimum dissi- pation theorem, named after Helmholtz’s work, states that the Stokes flow (4.43) is a solution to the following minimization problem,
minv
1 2D[v], subject to ∇ ·v = 0 inΩ,
v =v0 on ˆΓ .
(Helmholtz minimum dissipation theorem) (4.47)
In other words, the steady Stokes flow of an incompressible fluid has the smallest rate of viscous dissipation than any other incompressible velocity configurations with the same velocity prescribed on the boundary Γˆ.
The Helmholtz minimum dissipation theorem (4.47) can be proved from a variational argument. Let v be a solution to the Stokes equation (4.43) and δv be an admissible variation to v. By admissible we mean that, δv is divergence-free and δv = 0 on all boundary because the perturbed velocity field v+δv must also be incompressible and agree with the prescribed velocity on the boundaryΓ. Since viscous dissipation (4.46) is a quadric form of strain rate, its variation is exactly composed of a linear and a quadratic part in δv,
1
2D[v+δv] = 1
2D[v] +Z
Ω2µ δe :edΩ+Z
Ω
µ δe :δedΩ, (4.48) whereδe is the strain rate tensor constructed fromδv. The integral linear inδe (hence linear in δv) must vanish for any admissible velocity variationδv,
Z
Ω2µ δe :edΩ=Z
Ω2µ1
2 ∇δv+∇δv>:edΩ
=Z
Ω2µ∇δv:edΩ=Z
Ω2µ(∇iδvj)eijdΩ e is symmetric
=− Z
Ω2µ δvj(∇ieij) dΩ+Zˆ
Γ2µ δvjeijnidΓ
=− Z
Ω
µ δv·(2∇ ·e) dΩ+Zˆ
Γ2µ δvend ˆΓ
=− Z
Ω
δv·(µ∇2v) dΩ+Z
Γˆ2µ δvend ˆΓ Identity (4.45)
=− Z
Ω
δv· ∇pdΩ+Z
Γˆ2µ δvend ˆΓ Stokes equation (4.43)
=Z
Ω
p∇ ·δvdΩ+Z
Γˆ
δv· −pn+ 2µend ˆΓ
= 0, Constraint on velocity variation δv (4.49) where n is understood to be the outward normal vector of liquid domain Ω. Now it’s a straightforward exercise to show that any incompressible flow other than the Stokes solution v with the same velocity prescribed on the boundary can only increase
dissipation rate because 1
2D[e+δe]−1
2D[e] =Z
Ω
µ δe :δedΩ≥0. (4.50)
The Stokes flow (4.43) is indeed an optimal solution to the Helmholtz minimization problem (4.47). It certainly would be desirable if the solution to the Helmholtz mini- mization problem exists and is unique. Under such a condition the Stokes equation and Helmholtz minimization theorem would be two equivalent formulations of viscous flow, Stokes flow (4.43)⇐⇒Helmholtz minimum dissipation (4.47). (4.51) Statement (4.51) is correct although proving the existence and uniqueness of the solution to Helmholtz minimization problem (4.47) is a highly nontrivial task which requires sophisticated tools from functional analysis that are beyond the scope of this work. See Ern and Guermond (2004) for the technical details and the complete proof. We only briefly sketch the idea. The results follows directly from Lax-Milgram theorem which requires the bilinear form of velocity vectorv, in this case the viscous dissipation (4.46), to be coercive, i.e. D[e]&kuk2V, for some suitable normk·kVdefined on the functional space V that velocity field v belongs to. Coercivity of viscous dissipation (4.46) on a Lipschitz domain is guaranteed by the Korn’s first and second inequalities.
Energy-dissipation theorem for free surface flows
Now we consider a slightly different scenario where Ω is the domain enclosed between a fixed supporting surface Γˆ at the bottom and a free surface Γ˜ at the top. In a mixed problem, velocity field (e.g., no slip condition) is prescribed on Γˆ whereas the viscous stress must balance traction forcef on the free surfaceΓ˜. The resulting Stokes equation,
−∇p+µ∇2v =0 in Ω,
∇ ·v = 0 in Ω, v =v0 on ˆΓ , σn˜ =f on ˜Γ ,
(4.52)
is called a “mixed” problem in the sense that it has a mixed set of boundary conditions:
Dirichlet type on Γˆ and Neumann type on Γ˜. If we augment the viscous dissipation with an additional surface dissipation, then we can show that the modified minimum dissipation problem,
minv
1
2D[v]− Z
Γ˜
f·vd ˜Γ , subject to∇ ·v= 0 inΩ,
v=v0 on ˆΓ ,
(4.53)
is an equivalent characterization of the viscous flow described by the Stokes equation (4.52). It’s not difficult to show that any fluid motion satisfying Stokes equation un- der stress boundary condition must be a critical point of the augmented dissipation functional in (4.53) through an expansion procedure similar to (4.48),
δ 1
2D[v]− Z
Γ˜
f ·vd ˜Γ =Z
Ω2µ δe :edΩ− Z
Γ˜
δv·fdΓ
=Z
Ωp∇ ·δvdΩ+Z
Γˆδv·(σn) d ˆΓ +Z
Γ˜δv·(σn˜ −f) d ˜Γ
= 0. δv is admissible andv is traction-free on ˜Γ
(4.54) We again refer to Ern and Guermond (2004) for a detailed proof on the existence and uniqueness of the solution to the augmented minimization (variational) problem (4.58) which eventually yields the equivalence between the Stokes equation (4.52) and the minimum dissipation theorem (4.58) when the fluid is subject to mixed boundary conditions.
In many physical systems, surface traction f arises as a restoring force when the total potential energy F[ ˜Γ] available in the mechanical system undergoes infinitesimal vari- ations in its boundary Γ˜. For example, capillary stress is equivalent to the variation of surface energy which is proportional to the total surface area. Interfacial Maxwell stress in electrostatic systems can be derived by varying electrostatic energy stored in the medium on both sides of the interface. When the liquid interface is covered by fluid lipid membranes, the variation of Helfrich free energy gives rise to the membrane and bending forces (Helfrich, 1973). When surface tension coefficient is inhomogeneous, surface energy variation produces a force tangent to the free surface, known as the Marangoni stress which we have already encountered in Chapter 2. In these cases, the traction integral becomes the shape variation of the energy functional F[ ˜Γ],
Z
Γ˜
f ·vd ˜Γ =−δF[ ˜Γ;v], (4.55) which measures the amount of (signed) power required in order to deform the free surface Γ˜ at a displacement rate v. This idea is very similar to the principle of virtual work in the elastic theory of solids (Gurtin, 1973; Washizu,1982; Marsden and Hughes, 1994).
Except for dissipative systems such as viscous fluids, the name “virtual dissipation” or
“virtual power” is more appropriate since we are dealing with rate of energy instead of work itself.
Then the modified minimum dissipation formulation (4.53) reads, minv
1
2D[v] +δF[ ˜Γ;v]−T[ ˜Γ;v,b], subject to∇ ·v= 0 inΩ,
v=v0 on ˆΓ ,
(4.56)
whereT is the virtual power due to the additional external surface tractions b, T[ ˜Γ;v,b] =Z
Γ˜
b·vd ˜Γ , (4.57)
which do not arise from energy variations, for example, wind shear stress. An important observation due to the energy-dissipation formulation (4.56) is that, the rate at which potential energyF[ ˜Γ]is being depleted by the internal viscous frictions exactly equals to the viscous dissipationD[v]and additional power done by the external surface traction b. To see this, we recall from (4.54) that
Z
Ω2µ δe :edΩ− Z
Γ˜δv·fd ˜Γ = 0 (4.58) for all admissible variation δv. However the velocity field v of the Stokes flow itself qualifies an admissible variation as well. Replacing δv with v in equation (4.58) and substituting the traction forcef defined in (4.56),
Z
Ω2µe :edΩ− Z
Γ˜v·fd ˜Γ =D[v] +δF[ ˜Γ;v]−T[ ˜Γ;v,b] = 0, lead to the identity
δF[ ˜Γ;v] =−D[v] +T[ ˜Γ;v,b]. (4.59) The following is a physical interpretation of the above equation: the difference between external virtual powerT, after being dissipated by the internal viscous frictionsD[v], is equal to the rate at which the total potential energy of the viscous free surface flow varies.
Note when the boundaryΓ˜ is a free surface, it is exactly advected by the instantaneous velocityv evaluated at the free surface, which meansδF[ ˜Γ;v] = dF[ ˜Γ]/dt. Therefore in absence of external surface tractionb, we arrive at the energy-dissipation law for free surface viscous flow,
dF[ ˜Γ]
dt =−D[v]≤0 if b=0. (4.60)
Energy-dissipation law (4.60) states that, if there is no additional power input from external surface traction, the energy of the viscous fluid must always decrease unless it reaches an equilibrium shape.
Kinematic boundary condition
On flat supporting substrates, height field η of the liquid layer is indistinguishable from the columnar volume density % at every pointxˆ ∈Γˆ. The classical lubrication theory which we use to develop the models for thermocapillary and electrohydrodynamic thin film in Chapter 2 and 3 is established around the height field. However, it’s not the case for thin layer flow coating curved substrates. In particular we show that the time- dependent columnar volume density%(ξ1, ξ2, t)is more appropriate as a dynamic variable
than the height fieldη. Let’s consider the following integral evaluated via index notation, Z η
0 (∇ ·v)Jdξ3 =Z η
0
√1g
∂√gvj
∂ξj Jdξ3=Z η
0
√1gˆ
∂√ˆgvαJ
∂ξα + ∂v3J
∂ξ3 dξ3
= 1
√ˆg Z η
0
∂√ˆgvαJ
∂ξα dξ3+v3Jξ
3=η ξ3=0
= 1
√ˆg
∂
∂ξα pˆg
Z η
0 vαJdξ3− ∂η
∂ξα(vαJ)
ξ3=η+v3J
ξ3=η, (4.61) where the last line we have used the Leibniz’s rule for differentiation under the integral sign. Equation (4.61) must be exactly zero due to incompressibility condition∇ ·v= 0.
We immediately recognize the form of surface divergence on manifoldΓˆ in (4.61). This motivates the definition of volumetric flux
ˆ q= ˆgα
Z η
0 vαJdξ3 (4.62)
which is a vector field belong to the tangent bundle of the supporting surface Γˆ. Then equation (4.61) reduces to
∇ ·ˆ qˆ+hv3−vα(ξ1, ξ2, η) ∂η
∂ξα
iJ(ξ1, ξ2, η) = 0 (4.63)
where ∇ˆ stands for the covariant differential operators (e.g., divergence and gradient) on the curved surfaceΓˆ.
On the other hand, the kinematic boundary condition of a moving free surfaceη(ξ1, ξ2, t) states that the material time derivative of the zero contour of level function implicitly defined in (4.32) must remain zero at all times,
h∂
∂t+v(ξ1, ξ2, η)· ∇i(ξ3−η) =−∂η
∂t −vα(ξ1, ξ2, η) ∂η
∂ξα +v3(ξ1, ξ2, η) = 0. (4.64) Substituting kinematic condition (4.64) into the incompressibility identity (4.63) yields
J(ξ1, ξ2, η)∂η
∂t + ˆ∇ ·qˆ= 0. (4.65)
Recall from (4.39) that columnar volume density%is precisely defined as the integral of relative volume JacobianJ along the normal coordinate line ξ3. Hence equation (4.65) is in fact the exact conservation law for the columnar volume density,
∂%
∂t + ˆ∇ ·ˆq= 0, (4.66)
instead of for the height field η. Volume conservation law (4.66) is no surprise: vol- umetric flux q(ξˆ 1, ξ2) has the physical interpretation of total amount velocity vector traversing side walls of the infinitesimal liquid column at x(ξˆ 1, ξ2), which must be bal- anced with the rate at which the volume of the liquid column %(ξ1, ξ2, t) changes in
time. For a boundary-less manifold Γˆ, dynamical equation (4.66) conserves the total volume occupied by the thin liquid layer Ω coating a fixed supporting surface Γˆ at all times,
d dt
Z
Ω dΩ= d dt
Z
Γˆ
%d ˆΓ =Z
Γˆ
∂%
∂t d ˆΓ =− Z
Γˆ
∇ ·ˆ qˆd ˆΓ = 0,
where the last equality is implied by the (covariant) Stokes theorem on a manifold. As it turns out later, it is convenient to introduce the partial volumetric flux coefficient
qα(ξ1, ξ2, ξ3) =Z ξ3
0 vα(ξ1, ξ2, ξ30)J(ξ1, ξ2, ξ30) dξ30 (4.67) in analogy to the “full” flux qˆ = qα(ξ1, ξ2, η)ˆgα. In this sense, the vα-components of the velocity field v are fully specified by the ξ3-derivative of the partial flux vα = J−1∂qα/∂ξ3.
Viscous dissipation in lubrication regime
Let v be the dimensional velocity field within the viscous fluid layer. The essence of Reynolds lubrication theory is encoded in the singuarly scaled ansatz for the liquid velocity field,
v=vα(ξ1, ξ2, ξ3/)gα+v3(ξ1, ξ2, ξ3/)g3, O(ξ3) =O(v3) =O(). (4.68) Here 0 < 1 is some small parameter (e.g., aspect ratio of the liquid layer) which we will identify later. Partial derivative ∂ξ3 of the velocity field v along the transverse direction is amplified by a factor of 1/ while the scaling of the normal component of velocity field v3 = O() ensures mass conservation still applies for the free surface liquid layer by retaining the full incompressibility conditionO(∇αvα) =O(∇3v3). From a mechanical point of view, lubrication scaling (4.68) decouples the “out-of-tangent- plane” componentseα3 of the strain rate tensore from the “in-plane” componentseαβ owning to the scaling O(eα3) > O(eαβ). As a result, the total energy F[ ˜Γ] of the liquid layer (assuming no external surface traction) dissipates mostly through tangential shear against the supporting substrateΓˆ, which is precisely what lubrication theory was invented for.
Such mechanical point of view can be made explicit based on the lubrication ansatz (4.68). Magnitude of the tensor components of the velocity gradient ∇v in orders of the aspect ratio can be estimated from the Christoffel symbols (4.29) and (4.27)
derived earlier,
∇3v3= ∂v3
∂ξ3 + Γ3k3vk= ∂v3
∂ξ3 =O(1),
∇βvα= ∂vα
∂ξβ + Γαkβvk =O(1),
∇3vα= ∂vα
∂ξ3 + Γαk3vk= ∂vα
∂ξ3 −IIβαvβ
| {z }
O(−1)+O(1)
+O(),
∇αv3= ∂v3
∂ξα + Γ3kαvk=IIβαvβ
| {z }
O(1)
+O().
(4.69)
The integrand of the viscous dissipation (4.46) can be expanded with the use of index notation,
2e :e = 2eijeij = (∇jvi)(∇ivj) +glmgij(∇lvi)(∇mvj). (4.70) We look for those leading order terms of the components in (4.70) which couple with
∇3vα,
(∇jvi)(∇ivj) = 2(∇3vα)(∇αv3) +O(1)
= 2∂vα
∂ξ3 −IIβαvβIIγαvγ+O(1)
= 2∂vα
∂ξ3IIγαvγ
| {z }
O(−1)
−2IIβαIIγαvβvγ
| {z }
O(1)
+O(1) (4.71)
glmgij(∇lvi)(∇mvj) =gαβ(∇3vα)(∇3vβ) +O(1)
=∂vα
∂ξ3 −IIναvνgαβ∂vβ
∂ξ3 −IIγβvγ+O(1)
=gαβ∂vα
∂ξ3
∂vβ
∂ξ3
| {z }
O(−2)
−2IIβνvν∂vβ
∂ξ3
| {z }
O(−1)
+IIναvνgαβIIγβvγ
| {z }
O(1)
+O(1), (4.72)
where we have used that fact from (4.19) that gαβ = ˆgαβ +O() +O(2). Summing up (4.71) and (4.72) yields
2e :e = gαβ∂vα
∂ξ3
∂vβ
∂ξ3
| {z }
O(−2)+O(−1)+O(1)
+O(1), (4.73)
where the O(−1) andO(1)term in (4.73) exclusively come from the coupling with the first and second order corrections in metric coefficients gαβ. We remark that, approx- imation (4.73) contains O(1) terms, the same order of magnitude of the residue. We made this choice intentionally: by partially retaining someO(1) error, the approximate dissipation (4.73) is symmetrized while still accurate up to some O(1) error.
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;%].