Chapter IV: Generalized Lubrication Theory on Curved Geometries
4.5 Perfect Dielectric Films Coating Curved Conductors: Dissipation of a
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-
but also anywhere else on the same surface. “Nonlocalness” naturally emerges when a low-dimensional solution (e.g., two-dimensional surface charge density) is obtained from a high-dimensional problem (e.g., three-dimensional electrostatics). In particular for electrified fluid layers wetting a flat solid wall, the effect of nonlocal terms has been examined through a series of analytic and computational investigations for ideal fluid layers of finite depth (Papageorgiou, Petropoulos, and Vanden-Broeck, 2005) or thin viscous lubrication films (Tseluiko and Papageorgiou, 2007) in the presence of wall-normal electric fields, gravity-driven liquid films down a corrugated inclined wall (Tseluiko, Blyth, Papageorgiou, and Vanden-broeck, 2008) or subject to a periodic electrode of arbitrary shape (Tseluiko, Blyth, Papageorgiou, and Vanden-Broeck,2010) and electrostatically stabilized viscous thin films wetting the underside of a flat wall in the presence of wall-parallel electric fields (Anderson et al., 2017). We refer the interested readers to a recent review article by Papageorgiou (2019) (and references therein), which provides a complete survey on nonlocal terms in the nonlinear theories and models of electrohydrodynamic instabilities in immiscible multi-layer flows for various planar and cylindrical geometries.
On the other hand, previous works on electrohydrodynamic thin films coating substrates of non-planar geometries are very limited and mostly focus on the patterns and waves of dielectric fluids confined in the cylindrical geometries, e.g., annulus between two concentric cylinders (Wray, Papageorgiou, and Matar,2013; L. Wang and J. Liu,2016).
Nonlocalness of electrostatic pressure that appears in most papers cited in the review article (Papageorgiou, 2019) is derived based on the Hilbert transform technique which only applies to planar problems in a periodic rectangular channel. In this section we derive nonlocal terms in the lubrication model of dielectric thin films residing on a curved conductor of an arbitrary (smooth) shape fixed at a constant potential subject to a far- field electrode by computing the first and second variations of the total electrostatic energy in the film-conductor-electrode system. The overall setup is sketched in figure 4.4(a).
Electrostatic energy of a perfect dielectric film
Let Ωext be the exterior region enclosed between the conductor surface Γˆ and the electrode surfaceΓ. It is further divided into the liquid volume Ωand the outer region Ωo=Ωext\Ωoccupied by the gas. We refer toεi=εliq as the inner dielectric constant of the liquid andεo =εair as the outer dielectric constant of the air phase. Let [ψ]be the characteristic potential drop between the conductor and the electrode. After the standard procedure of nondimensionalization, we arrive at the electrostatics equations
∇ ·εo∇ψ(o)= 0 in Ωext\Ω,
∇ ·εi∇ψ(i)= 0 in Ω,
(4.115)
Conductor Conductor
Electrode Electrode
Γˆ Γ˜ Γˆ
Γ Γ
Ω εliq
εair
Air
Ωo Ωext
(a) (b)
ψk(o)
Figure 4.4: (a) Thin dielectric liquid layer (liquid thickness is exaggerated) subject to an electrode with the boundaryΓ (thick solid): Γˆ is the conductor surface (thin solid), Γ˜ is the liquid free surface (dashed),Ω is the thin liquid volume (green) Ωo is the gas region exterior to the conductor and the liquid layer. (b) Domain Ωext for the outer potential correctionψk(o), which overlaps with the inner (liquid) domainΩ.
of the inner and outer potentialsψ(i) and ψ(o) subject to a set of boundary conditions due to continuities of electric potential and displacement field,
ψ= 0 on ˆΓ , ψ(i)=ψ(o) on ˜Γ ,
˜
n·εi∇ψ(i)= ˜n·εo∇ψ(o) on ˜Γ ,
ψ=ψ on Γ ,
(4.116)
where ψ is some potential distribution prescribed on the electrode boundary Γ. The negative gradient of electric potential produces the inner and outer electric fields, E(i) andE(o). Then the total electrostatic energy E of the electrode-film-conductor system is the sum of the volume integrals of energy density both interior and exterior to the liquid layer,
E[Ω] = Ec
(Z
Ωext\Ω
εo
2|E(o)|2dΩ+Z
Ω
εi
2|E(i)|2dΩ )
, Ec = ε0[ψ]2
[σ][L], (4.117) whereEc is the electric-capillary number.
In order to identify the leading order effects on the electrostatic energy due to the presence of a dielectric layer, we introduce a consistent base-state electric potentialψ0 corresponding to the limit of → 0, from which the higher order perturbations are constructed. It’s not difficult to see that base-state potential ψ0 solves the bare-bones
electrostatic problem,
∇2ψ0= 0 in Ωext, ψ0= 0 on ˆΓ , ψ0=ψ on Γ ,
(4.118)
in the absence of a dielectric layer.
Before proceeding with the construction of higher order corrections, we expand differ- ential operators in the close vicinity of the conductor boundary. In such cases, the Laplace operator ∇2 acting on a three-dimensional scalar can be expressed in terms of the surface-adapted coordinate system(ξ1, ξ2, ξ3),
∇2φ(ξ1, ξ2, ξ3) = 1
√g
∂
∂ξ3
√g∂φ
∂ξ3 + 1
√g
∂
∂ξα
√ggαβ ∂φ
∂ξβ
, (4.119)
wheregij andgare the metric coefficients and determinant defined in (4.21) and (4.19) respectively. Inspired by exercise 1.1 of Jackson (1999), we make an important obser- vation here: evaluating the Laplace equation on the surface Γˆ where ξ3 = 0 leads to a constraint relating the surface Laplacian of potential φto its first and second normal derivatives at the surface,
∇2φ(ξ1, ξ2,0) = ∂2φ
∂(ξ3)2
ξ3=0−2H ∂φ
∂(ξ3)
ξ3=0+ ˆ∆φ(ξ1, ξ2,0) (4.120) where ˆ∆ is the Laplace-Beltrami operator of surface Γˆ. Relation (4.120) is exact and is very useful in simplifying the electrostatic problem as we will see later. Recall from the metric inversegij in (4.25) that in the stretched inner coordinate system(ξ1, ξ2, z), the three-dimensional Laplacian and gradient are separated into operators of hierarchical scales,
∇2φ(ξ1, ξ2, z) = 1 2
∂2φ
∂z2 −1 2H∂φ
∂z +O(1), (4.121)
∇φ(ξ1, ξ2, z) = 1
∂φ
∂znˆ+ ˆ∇φ+O(), (4.122) provided conductor surface Γˆ is only gently curved (i.e. O(H) =O(1)) and potential φdoes not exhibit rapid variations alongΓˆ (i.e. O( ˆ∇φ) =O(1)).
Inner and outer perturbation solutions
We solve the electrostatic equation interior and exterior to the thin liquid layer by as- suming two separate—inner and outer—perturbation expansions of the unknown electric field in terms of the small aspect ratio parameter. As stated earlier, perturbation series are constructed around the base-state potentialψ0,
ψ(o)=ψ0+ψ(o)1 +2ψ(o)2 +O(3) in Ωext\Ω, ψ(i)=ψ0+ψ(i)1 +2ψ(i)2 +O(3) in Ω,
(4.123)
where we must emphasize that, the outer corrections ψk(o) are defined on the entire exterior domain Ωext instead ofΩ\Ωext (see figure4.4(b) for clarification).
At the leading order in, we find the governing equations of ψ1(i) and ψ1(o) to be
∂2ψ1(i)
∂z2 = 0 in Ω,
∇2ψ1(o)= 0 in Ωext,
(4.124)
subject to four boundary conditions, three of which are of Dirichlet-type, ψ1(i)= 0 at z= 0,
ψ1(o)=ψ1(i) at z=η, ψ1(o)= 0 on Γ .
(4.125)
The last condition comes from the continuity of displacement flux. The leading order balance in the flux jump condition (4.116) is
εonˆ· ∇ψ0|Γˆ =εih∂ψ1(i)
∂z
z=η+ ˆn· ∇ψ0|Γˆ
i, (4.126)
which then yields the solution to the leading order equations (4.124) ψ1(i)(ξ1, ξ2, z) = εo−εi
εi (ˆn· ∇ψ0|Γˆ)z, ψ1(o)= εo−εi
εi (ˆn· ∇ψ0|Γˆ)η on ˆΓ ,
(4.127)
whereψ1(o) is the harmonic potential in the exterior domain Ωext determined indirectly by boundary condition (4.127) on the conductor surface Γˆ.
The equations governing the inner potential ψ2(i) and outer potential ψ2(o) at the next order are found to be
∂2ψ2(i)
∂z2 −2H∂ψ(i)1
∂z = 0 in Ω,
∇2ψ(o)2 = 0 in Ωext.
(4.128)
The Dirichlet-type boundary conditions at the next order are again straightforward, ψ2(i)= 0 at z= 0,
ψ2(o)|Γˆ+ (ˆn· ∇ψ1(o)|Γˆ)η=ψ2(i) at z=η, ψ2(o)= 0 on Γ .
(4.129)
The continuity of displacement field flux at the free surface Γ˜ deserves careful exam- inations since it involves the expansion of electric field which is a vector instead of a scalar,
∇φ|Γ˜ =∇φ|Γˆ+∇(∇φ)|Γˆηnˆ+O(2). (4.130)
Expansion (4.130) projected to the normal vector n˜ of the free surface Γ˜ gives rise to the normal flux at the free surfaceΓ˜ up to the second order,
˜
n· ∇φ|Γ˜ = ˆn· ∇φ|Γˆ+ˆn1· ∇φ|Γˆ +ηn[∇(∇ˆ φ)|Γˆ]ˆn+O(2), (4.131) wherenˆ1 is the first order correction to n˜ so thatn˜ = ˆn+nˆ1+O(2). However the first order correction nˆ1 must be purely tangential (i.e. nˆ1·nˆ = 0) since both n˜ and
ˆ
n are unit vectors for which |n˜|=|nˆ|+ 2nˆ1·nˆ +O(2) = 1 can be fulfilled only if ˆ
n1·nˆ = 0. If we apply expansion (4.131) to the base state ψ0 which is equipotential on the conductor surfaceΓˆ, then
˜
n· ∇ψ0|Γ˜ = ˆn· ∇ψ0|Γˆ +η ∂2ψ0
∂(ξ3)2
ξ3=0+O(2) (4.132) where we recall in surface-adapted coordinate system(ξ1, ξ2, ξ3), expressionn[ˆ ∇(∇φ)|Γˆ]ˆn simplifies to ∂2φ/∂(ξ3)2 atξ3 = 0. Then by virtue of relation (4.120), it immediately follows from the harmonicity of potentialψ0 that,
∂2ψ0
∂(ξ3)2 = 2H ∂ψ0
∂(ξ3) at ξ3= 0. (4.133)
With identity (4.133) substituted into expansion (4.132), the continuity of displacement flux at next order reads,
εo
hnˆ · ∇ψ1(o)|Γˆ+ 2Hη(ˆn· ∇ψ0|Γˆ)i=εi
h∂ψ2(i)
∂z
z=η+ 2Hη(ˆn· ∇ψ0|Γˆ)i. (4.134) The next order corrections to the inner and outer potentials are found by solving equa- tions (4.128) subject to boundary conditions (4.129) and (4.134),
ψ2(i)= 1
22Hεo−εi
εi (ˆn· ∇ψ0|Γˆ)z2+εo
εi(ˆn· ∇ψ1(o)|Γˆ)z, ψ2(o)= 1
22Hεo−εi
εi (ˆn· ∇ψ0|Γˆ)η2+εo−εi
εi (ˆn· ∇ψ1(o)|Γˆ)η,
(4.135)
where ψ(o)2 is the harmonic potential in the exterior domain Ωext subject to boundary condition (4.135) on conductor surface Γˆ.
Total electrostatic energy in presence of a dielectric layer
The total electrostatic energy E defined in (4.117) has two disjoint contributions, the energy confined within in the dielectric layer and the energy stored in the free space exterior to the film. Recall the outer potential correctionsψk(o)are defined on the entire exterior region Ωext instead of the gas volume Ω\Ωext. Therefore we must subtract off the overlapping part to prevent double-counting,
E[Ω] = Ec
εo
Z
Ωext
1
2|∇ψ(o)|2dΩ−εo Z
Ω
1
2|∇ψ(o)|2dΩ+εi Z
Ω
1
2|∇ψ(i)|2dΩ
. (4.136)
Applying Green’s identity to the first volume integral in (4.136) yields Z
Ωext
1
2|∇ψ(o)|2dΩ
=Z
Ωext
1
2|∇ψ0|2+(∇ψ0)· ∇ψ(o)1 +2
2|∇ψ1(o)|2+2(∇ψ0)· ∇ψ2(o)dΩ+O(3)
=Z
Ωext
1
2|∇ψ0|2dΩ− Z
Γˆ
ψ1(o)(ˆn· ∇ψ0) d ˆΓ
−2 Z
Γˆ
1
2ψ(o)1 (ˆn· ∇ψ1(o)) +ψ2(o)(ˆn· ∇ψ0) d ˆΓ +O(3). (4.137) The second integral in (4.136) can be separated into orders ofas well. The idea here is to interchange Taylor expansion with integration, i.e. Raa+f(x) dx = Raa+f(a) + f0(a)x+f00(a)x2/2 dx+..., for a smooth function f(x). Expanding the integral with index notation leads to,
Z
Ω
1
2|∇ψ(o)|2dΩ
=Z
Ω
1 2
∂ψ(o)
∂ξi
∂ψ(o)
∂ξj gijdΩ
=Z
Γˆ
Z η 0
1 2
"
(ˆgαβ+ξ32IIαβ)∂ψ(o)
∂ξα
∂ψ(o)
∂ξβ + ∂ψ(o)
∂ξ3
∂ψ(o)
∂ξ3 +O((ξ3)2)
#
Jdξ3d ˆΓ
=Z
Γˆ
Z η 0
1 2
"
∂ψ(o)
∂ξ3
∂ψ(o)
∂ξ3 +O((ξ3)2)
#h
1−2Hξ3+ +O((ξ3)2)i dξ3d ˆΓ
= Z
Γˆ
η
2(ˆn· ∇ψ0)2d ˆΓ +2 Z
Γˆη(ˆn· ∇ψ0)(ˆn· ∇ψ1(o)) +H
2 η2(ˆn· ∇ψ0)2d ˆΓ +O(3), (4.138) where we note∂ψ(o)/∂ξαmust beO()since∂ψ0/∂ξα = 0on the equipotential surface ξ3 = 0.
The last volume integral in (4.136) measures the electrostatic energy interior to the dielectric layer due to the interior electric potential ψ(i). In light of the perturbation series from (4.127) and (4.135), we have
Z
Ω
1
2|∇ψ(i)|2dΩ=ε2o ε2i
Z
Γˆ
η
2(ˆn· ∇ψ0)2d ˆΓ +2ε2o
ε2i Z
Γˆη(ˆn· ∇ψ0)(ˆn· ∇ψ1(o)) + H
2η2(ˆn· ∇ψ0)2d ˆΓ +O(3).
(4.139) Rearranging the expanded integrals from (4.137)–(4.139) into orders of leads to the first and the second order corrections to the electrostatic energy of the base-state po-
tential (up to a material constant), Z
Ωext
εo
2|∇ψ(o)|2dΩ− Z
Ω
εo
2|∇ψ(o)|2dΩ+Z
Ω
εi
2|∇ψ(i)|2dΩ
=εo
Z
Ω
1
2|∇ψ0|2dΩ+εi−εo
εi εo
Z
Γˆ
1
2(ˆn· ∇ψ0)2ηd ˆΓ +2εi−εo
εi εo Z
Γˆ
1 2
hH(ˆn· ∇ψ0)2η2+ (ˆn· ∇ψ0)(ˆn· ∇ψ1(o))ηid ˆΓ +O(3).
(4.140) To clean up notations, we define the dielectric contrast parameterκ=εair/εliqas in the EHL model (3.5) and rewrite potential gradient as electric field E = −∇ψ. We shall also drop the(·)(o)superscript on the higher order terms since only the outer correction ψ(o)1 is involved. In terms of columnar volume density %, the truncated final form of total electrostatic energyE reduces to the energy of the base state plus two higher order corrections,
E[ ˆΓ;%] = Ecεair 1
Z
Ωext
1
2|E0|2dΩ + (1−κ)Z
Γˆ
1
2(ˆn·E0)2%d ˆΓ + (1−κ)Z
ΓˆH(ˆn·E0)2%2+1
2(ˆn·E0)(ˆn·E1)%d ˆΓ
. (4.141) The variational derivative immediately follows,
δE
δ% = Ecεair(1−κ)1
2(ˆn·E0)2+2H(ˆn·E0)2%+(ˆn·E0)(ˆn·E1), (4.142) where the outer field correctionE(o)1 is the electric field of the outer potentialψ1(o)which solves the auxiliary electrostatic problem,
∇2ψ1= 0 in Ωext,
ψ1= 0 on Γ ,
ψ1= (1−κ) (ˆn·E0)% on ˆΓ .
(4.143)
In the lubrication limit, the effective pressure (4.142) is weakened due to the presence of a dielectric layer. For a perfectly conductive liquid (i.e. limεliq→∞κ = 0), the familiar electrostatic pressure is recovered at the zeroth order. When liquid material shares the same dielectric property with the gas (i.e. limεliq→εairκ = 1), the entire effective electrostatic pressure (4.142) disappears. This is expected because in such a limit εliq=εair, electric field is no longer discontinuous which means no jump in the Maxwell stresses acting on both sides of the liquid free surface. The first two terms in (4.142) represent the bare-bones electrostatic pressure and a leading order geometric correction due to the dilation of free surface area differential d ˜Γ measured wit respect to d ˆΓ of
the conductor surface. The “nonlocalness” in the effective electrostatic pressure δE/δ%
attributes to the last term in (4.142), which represents a long-range interaction between the base-state electric field E0 and the correction electric field E1. Such interaction is indeed nonlocal (with respect to the conductor surface Γˆ) because E1 =−∇ψ1 is deduced from the potential ψ1 of the auxiliary problem (4.143), the solution of which requires information of the columnar volume density % everywhere on the conductor surfaceΓˆ. The nonlocal term becomes dominant when the surface charge distribution induced by the base-state electric fieldnˆ ·E0 is uniform (e.g., concentric capacitor of two cylinders or spheres of different radius).
Finite element simulations
We illustrate the generalized lubrication model through finite element simulations of the dynamics of a thin dielectric liquid film coating a grounded spherical conductor. In this case, the total free energy of the system has three components, homogeneous surface energy S and gravitational energyG and electrostatic energyE,
F=S+G−E, (4.144)
where the minus sign in front of the electrostatic energy E is due to the fact that for a perfect dielectric film in contact with a conductor at a fixed potential, the external voltage supply is doing work to the liquid film, as opposed to for a charged isolated conductor, surface charges do work to redistribute themselves (Ljepojevic and Forbes, 1995).
Let Vext be the finite element interpolation space of the three-dimensional exterior domain Ωext discretized by quadratic tetrahedral elements and Vˆ be the space of the two-dimensional conductor surface Γˆ discretized by quadratic triangular elements.
ψh0 ∈ Vext and ψ1h ∈ Vext are the finite element projections of the base-state electric potential ψ0 and the correction potential ψ1, respectively. Similarly, we introduce four discretized scalar fields, the columnar volume density ρh ∈ Vˆ, the effective capillary pressure (S/δ%)h ∈ Vˆ, the effective hydrostatic (gravitational) pressure (G/δ%)h ∈ Vˆ and the effective Maxwell (electrostatic) pressure (E/δ%)h ∈ Vˆ. Notations Vi and Xi are reserved for thei-th hat function andi-th global Lagrange nodal position in a finite element space which can be eitherVextorVˆ. The standard finite element inner product h·,··i similar to the one in equation (3.32) is introduced for each discretized domain in terms of volume or surface integrals.
Prior to the time-dependent simulation, we solve the base-state problem only for once,
whose variational formulation reads: find ψ0h∈Vext such that D∇Vi,∇ψh0E= 0 for all Vi ∈Vext, subject to ψ0h(Xi) = 0 for all Xi ∈Γ ,ˆ
ψ0h(Xi) =ψ(Xi) for allXi ∈Γ .
(4.145)
Given an initial distribution of film thickness, at each ensuing time step we solve the following semi-discrete variational problem: find %h∈Vˆ such that
hVi,Dt[%h]i+h∇Vˆ i,DF∇(ˆ δF
δ%)hi= 0, (δF
δ%)h= (δG
δ%)h+ (δS
δ%)h−(δE δ%)h, hVi,(δG
δ%)hi= BohVi,xˆ·g+(ˆn·g)%hi, hVi,(δS
δ%)hi+hVi,2H+(4H2−2K)%hi=h∇Vˆ i,∇ˆ%hi, hVi,(δE/δ%)hi
εair(1−κ)Ec − hVi,(ˆn·Eh0)2
2 +2H(ˆn·Eh0)2%hi=hVi,(ˆn·Eh0)(ˆn·Eh1)i,
(4.146) for all Vi∈Vˆ, andψh1 ∈Vext such that
D∇Vi,∇ψ1hE= 0 for all Vi∈Vext, subject toψh1(Xi) = (1−κ) (ˆn·E0(Xi))%h(Xi) for all Xi∈Γ ,ˆ
ψh1(Xi) = 0 for all Xi∈Γ ,
(4.147)
where Dt is some discrete time-stepping operator (backward Euler, BDF2, etc.). On the surface of a unit sphere, the second fundamental formII has no preferable principle directions since the principle curvatures κ1 = κ2 = −1. The discrete mobility tensor DF is simply the identity tensor multiplied by a nonlinear mobility coefficient,
DF =h1
3(ρh)3−
2(ρh)4iI. (4.148)
The fully discretized finite element system (4.145)–(4.147) is implemented in the com- mercial software COMSOL Multiphysics, Inc. V5.3 (2017).
Plotted in figure 4.5 is a spherical conductor of unity radius concentric with another spherical electrode of radius = 5. A boundary electric potential ψ = 5Y32(ϑ, ϕ) is prescribed on the electrode surface where Ym` is the spherical harmonics of degree ` and order m with polar angle ϑ and azimuthal angle ϕ. The exterior volume Ωext is discretized by245957quadratic tetrahedral elements and the spherical surfaceΓˆ of the conductor by 12550 quadratic triangular elements. Since in equation (4.142) electric- capillary numberEcalways comes with the contrast parameterκand dielectric constant
t= 0 t= 4
t= 0.25 t= 6
t= 0.5 t= 8
t= 0.75 t= 10
t= 1 t= 12
(a) (b)
0.75 1.00 1.25 1.50 1.75 2.00
0.597 2.082
0.5 1.0 1.5 2.0 2.5
0.177 2.511
Figure 4.5: Snapshots of (a) early and (b) late stages of the evolution of liquid columnar volume density% (colored) on a conductor of a unit sphere subject to another spherical electrode of radius 5. Black tubes are field lines of the correction potential ψ1. Tube radius is proportional to field strength|E1|. Parameters used in the simulation: κ= 0, Bo = 0,Ec = 50and= 0.005.
εair, without loss of generality we set κ = 0 and εair = 1 and vary Ec alone. In this example, gravity is turned off (i.e. Bo = 0) and the pattern formation in the dielectric film with aspect ratio = 0.005entirely results from the competition between Maxwell (Ec = 50) and capillary stresses. During the early development 0 ≤ t ≤ 1 shown in figure4.5(a), liquid film immediately responds to the valleys and peaks (eight in total) of the spherical harmonicsY32 in the prescribed potential of the electrode by forming eight charged droplets. The initial growth of triangle-shaped droplets, for example att= 0.5, shows reminiscence of the fine details of the spherical harmonics electrode potential, which however is soon lost to capillary smoothing. Note every charged circular droplet is surrounded by three other droplets of the opposite charge type, as indicated by the connections of electric field lines (black tubes) generated by the correction potentialψ1. In the late stage of the evolution (4≤t≤12), the positions of these droplets are tightly interlocked by the attractive electric forces between each one and its three neighbors.
The surface region between these charged droplets are depleted to about only 18% of the initial film thickness whereas each droplet apex attains more than2.5 times as tall.
In figure4.6is another finite element simulation with the identical setup of the geometry and discretization with the one in figure 4.5except a different electric potential ψ=x is prescribed on the larger spherical electrode of radius 5 in order to mimic a constant electric field parallel to the horizontal plane in the far field. In this simulation, gravity is turned back on (Bo = 1) acting vertically downwards inz-direction on a twice thicker film ( = 0.01) while electrostatic effect is reduced (Ec = 1.25). As shown in figure 4.6(a), a circular pile of liquid four times higher than the background uniform film initially concentrates on the north pole. However, azimuthal symmetry in the system is broken due to the presence of a nearly unidirectional base-state electric field parallel to the horizontal plane. Instead of spreading evenly at every azimuthal angle, the excessive liquid volume is immediately elongated along or against the direction of the electric field E0 depending on the type of charges at the spreading front. In the later stage shown in figure4.6(b), the thinned liquid pile is split into two separate droplets which gradually slide towards the south pole under the vertical pull of gravity. The two droplets absorb significant amount of liquid volume from the relatively uniform liquid reservoir in the background along with their sliding motion and eventually almost recover the maximum thickness of the initial state. In the end at t = 0.3, an equilibrium is about to be reached under the balance between gravitational forces and Maxwell stresses as two circular droplets are suspended at positions between the directions of gravity and base-state electric field. The electric field lines of the correction potentialψ1 plotted in figure 4.6(b) suggest that the two droplets are also attracted to each other nonlocally due to the induced surface charges they carry.
t= 0 t= 0.14
t= 2.5·10−2 t= 0.18
t= 5·10−2 t= 0.22
t= 7.5·10−2 t= 0.26
t= 1·10−1 t= 0.3
(a) (b)
1 2 3 4
0.870 4.000
1.01.52.02.53.03.5
0.783 3.976
Figure 4.6: Snapshots of (a) early and (b) late stages of the evolution of liquid columnar volume density% (colored) on a conductor of a unit sphere subject to another spherical electrode of radius 5. The second column in each panel is the first column rotated by 90◦ about one of the axes. Gravity acts vertically downwards whereas the prescribed electrode potential mimics a constant electric field parallel to the horizontal plane. Black tubes are field lines of the correction potential ψ1. Parameters used in the simulation:
κ= 0,Bo = 1,Ec = 1.25 and= 0.01.