Chengzhe Zhou

Introduction: Patterning Liquid Flow by Interfacial Forces

Contributions of this thesis

The evaluation of the discrete objective J(H,D) is restricted to a finite-dimensional (since it is discretized) hypersurface, implicitly defined by F(H,D) = [0>,. Let Γ (or γ) be the boundary of the material (or spatial) domain Ω (or ω) and N (orn) be the unit normal vector of the boundary.

Cuspidal Formation in Thermocapillary Thin Liquid Films

Cusp Formation in Physical Systems

In Section 2.2 we outline the derivation of the fourth-order nonlinear diffusion equation governing the evolution of the liquid film interface driven by interfacial traction forces in general. In Section 2.6 we present an asymptotic analysis of the conical region that reveals the presence of a stable fundamental mode that acts as an attractor condition.

Figure 2.1: Liquid (a) line type and (b) point type cusp formation in a thin film subject to a negative disjoining pressure from van der Waals forces that promote dewetting of the film from the bottom substrate

Thin Film Equation on a Flat Substrate

The Navier-Stokes equation (2.1) is known to be a special form of the Cauchy momentum equation (Leal, 2007). Under the assumption of vanishing Reynolds number, we arrive at the Reynolds lubrication equation in the bulk of the thin liquid film.

Thermocapillary Growth of Ultrathin Viscous Film

The dimensionless numbers CaandMa differ from the standard definitions by factors of the minor aspect ratio parameter. This is one of the distinguishing features of the smear theory: interfacial forces scale differently in the slender limit.

Figure 2.3: Sketch of a linearly unstable thin molten film overlay by a gas layer. The gas/liquid bilayer is subject to a very large vertical temperature gradient enforced by the uniform temperature difference ∆T = T hot − T cold maintained across the very

Stability Considerations by Analogy to Gradient Flows

Switching the order of the operators∇kand∂/∂τ followed by applying the first Green's identity to the first integral in equation (2.72) gives. However, the free energy form (2.68) of the thin film system depends on the total volume of the liquid.

Figure 2.4: Plots of Φ(H), 0.2×dΦ/dH and 0.005×d 4 Φ/dH 4 for the thermocapillary equation

Numerical Solution of Nonlinear Thermocapillary Equation

Introducing the singular time τs where the local peak of the Hapex film reaches H = 1—the singular point of equation (2.64)—gives the scaling relations governing the conical tip region, namely (1−Hapex/(τs−τ)∼(1− Hapex)-3and(1-Hapex)/X2 ∼(1-Hapex)-1.The top panel shown in Figure 2.6 shows 3D views of a developing corner with a conical tip at four defined times, as obtained from the simulation of the finite elements of the complete nonlinear equation (2.64).

Figure 2.5: Self-similar formation of conical cusp from numerical solution of equation (2.64) for rectilinear (X ) and axisymmetric (R) geometry

Asymptotic Analysis of Self-Similar Cusp Formation

The fact that dS0/dξ belongs to the four roots (two real and two complex) of the fourth degree polynomials ξ(ξ3+ξ/4) = 0 leads to four possible modes of far-field behavior. However, combined with the two symmetry conditions (2.105) at η = 0 that W1 must satisfy, this gives a total of five constraints on the solution of the fourth-order differential equation—problem (2.98) with the boundary conditions (2.105) and ( 2.106) is effectively too specific.

Figure 2.8: Phase portrait (arrow) of dynamical system (2.113): a hyperbolic fixed point at (0, 0), a saddle point (0, 1) and the two unique trajectories (solid line) that are continuous at all orders.

Conclusion

In the end, we arrive at the expression of the variational drain of surface energy. The notion of the first variation of the form of the capacitive functional C[Γ] (formally known as the Gateaux derivative) is defined as the infinitesimal change in capacitance when the vacuum domain Ω is deformed into ω under a certain choice of the configuration mapping x=χ(X, ) (Kasumba and Kunisch , 2014).

Figure 3.1: Atomic force microscopy images reveal formation of unintended structures when replicating design patterns in TiO 2 films

Computational Electrohydrodynamic Lithography of Dielectric Films 44

Thin Film Model of Electrohydrodynamic Lithography (EHL)

In this section we briefly review the derivation of the interfacial pressure P and tangential traction (if any) Fk present in the thin film equation (2.37). This is also reflected in free energy functional F[H, D] of the electrohydrodynamic thin film equation (3.17). Due to the vast literature, it is not our intention here to provide a comprehensive overview of the finite element method.

Figure 3.3: Lagrange polynomial basis {N 1 (ξ 1 , ξ 2 ), ... , N 9 (ξ 1 , ξ 2 ) } and nine correspond- correspond-ing nodes for bi-quadratic interpolation in the canonical domain [ − 1, 1] × [ − 1, 1].

EHL as a Constrained Inverse Problem and Its Optimal Control

ΛLD(H, D)δDdΩdτ, (3.55) where δJ[H, D;δH] and δJ[H, D;δD] are the unrestricted (free) variations of the objective functionalJ with respect to its arguments. The two operators LH and LD in (3.54) and (3.55) are the linearizations of the nonlinear operator N(H, D) over the optimal solutions H and D, respectively. The condition that the first variation δL in (3.53) vanishes for any spatio-temporal variation δΛ of the optimal multiplier Λ simply restores the non-linear constraint, i.e.

Adjoint-Based Discrete Optimal Control of EHL

The computational complexity of the primal (3.68) and dual (3.70) is closely related to the dimension of the matrix and s. An even bigger problem is the error of the imprecise gradient approximated by the finite difference approach. According to definition (3.36), the ij-entry of the global stiffness matrix K can be expressed as

Figure 3.10: Flow chart: discrete state equation (blue) for H k , discrete adjoint equation (red) for Λ k and discrete control equation (green) for C k .

Nonlinear Optimization and Results

The deviation of the analytical solution from the optimal flat electrode D = 1 decreases towards zero as the objective J is minimized. At iteration step 40, shown in Figure 3-11, the maximum spatial error of the optimal topography is already brought below 0.01. Small-scale features of the optimal electrode shown in Figure 3.15(c), e.g. ridges and horns, as usual immediately cause localized growth resembling the boundary of the heart pattern.

Figure 3.11: Sequence of electrode topography profiles 1 − D on a periodic square domain [0, 4.5] × [0, 4.5] at iteration step 0, 5, ..

Conclusion and Outlook

In the height field representation, the free energy of the liquid layer Ω covering the supporting substrate. On the other hand, the approximation (4.73) of the viscous distribution in the rescaled variables leads to the approximate dimensionless distribution,. We also calculate the necessary components needed in the integral formula of the first and second derivatives.

Generalized Lubrication Theory on Curved Geometries

Development of Lubrication Theory on Curved Geometries

An alternative perspective on the formulation of thin film flow was introduced by Rumpf and Vantzos (2013), who reformulated the evolution of thin film as a unique flow state resulting from the interaction between the potential energies (e.g. the surface energy of liquid/gas interface and gravitational energy) and viscous dissipation in the liquid due to tangential frictions against the curved substrate in the thin film boundary. The evolution of a viscous thin film on a curved geometry was then approximated based on the underlying gradient flow structure that is faithful to the energy dissipation law in the liquid system, as opposed to the equations of motion. Finally, in Section 4.5, we present a new non-local model resulting from the total electrostatic energy of a curved conductor in the presence of a thin dielectric liquid coating.

Differential Geometry of Thin Layer Domain

Given knowledge of the surface parameterization (4.1), we are now in a position to construct a curvilinear coordinate system (ξ1, ξ2, ξ3). The effect of curvature should of course be apparent from the first and second fundamental shape of the surface Γˆ. Then, by Lagrange's identity, (a×b) (c×d) = (a c)(b d)−(b c)(a d), the metric determinant of the free surface can be expanded in orders of the local height fieldη,.

Figure 4.1: Riemannian 2-manifold of an embedded surface Γ ˆ in the ambient Euclidean space R 3 through coordinate chart ϕ(ξ 1 , ξ 2 ), the local tangent plane of which is spanned by the induced basis vectors g ˆ α .

Kinematics and Dissipation of Viscous Free Surface Flows

However, the velocity field v of the Stokes current itself also qualifies as an allowable variation. On flat supporting substrates, the height field η of the fluid layer at any point xˆ ∈Γˆ is indistinguishable from the columnar volume density %. For example, as mentioned before, capillary tension arises from the variation of the total area of ​​the moving free surface Γ˜.

Truncated Minimum Dissipation Principle for Lubrication Flow

One of the most common energies in physics is the potential energy due to gravity. In terms of columnar volume density %, we have that. 4.108) Variation of the truncated gravitational energy is a simple scalar field. Equation (4.113) is essentially the perturbation expansion of the curvature of the free surface Γ˜, expressed with the coordinates of the supporting surface Γˆ and the columnar volume density%.

Perfect Dielectric Films Coating Curved Conductors: Dissipation of a

Then, the total electrostatic energy E of the electrode-film-conductor system is the sum of the volume integrals of the energy density both inside and outside the liquid layer. Recall the external potential corrections ψk(o) are defined over the entire external region Ωext instead of the gas volume Ω\Ωext. In terms of % columnar volume density, the final truncated form of the total electrostatic energyE reduces to the ground state energy plus two higher order corrections.

Figure 4.4: (a) Thin dielectric liquid layer (liquid thickness is exaggerated) subject to an electrode with the boundary Γ (thick solid): Γ ˆ is the conductor surface (thin solid), Γ ˜ is the liquid free surface (dashed), Ω is the thin liquid volume (green

Conclusion

In a similar way, we expand the integrand of the fourth integral in the second common derivative (5.85). The first derivative of the total volume can be expressed as a surface integral using the divergence theorem. We will show later that the integrals in the first two lines of the second variation of the form (5.158) lead directly to the famous Rayleigh charge limit QRa.

Shape Analysis and Energy Stability of Conductive Liquids

Energy Stability of Electrically Charged Conductive Liquid

In 1882, Lord Rayleigh (1882) proposed a concise derivation of the theoretical estimate of the maximum amount of charge that a nearly spherical liquid drop can carry while still remaining stable, which is now known as the Rayleigh charge limit. Thinking along the same lines alone, K. first developed a prototype variational method from a consideration of the variation of free energy. It is worth noting that Rayleigh's method is Eulerian: coordinates and potentials do not vary with deformation of the fluid body.

Lagrangian Specification in Continuum Mechanics

The deformation gradient F is a two-point tensor which maps the tangent space of the material configuration at position X ∈ Ω to the tangent space of the spatial configuration at position x∈ω. Similarly, we define the velocity gradient as a measure of the rate at which a material is deformed. Similarly, the volume element of the spatial configuration is given by the determinant of the spatial metric tensor gij,.

Figure 5.1: Triad of parameter, material and spatial frames: material coordinates X send (ξ 1 , ξ 2 , ξ 3 ) from the parameter space E 3 into the material frame Ω in the ambient space R 3 with boundary Γ ; Spatial frame ω and its boundary γ in R 3 are conv

Electrostatic Energy and Shape Variations of a Perfect Conductor

The second derivative of the capacity functionalC with respect to is re-evaluated in the material frame. Using the symmetries of the quadratic forms in expression (5.68), we rewrite the second total derivative of the capacitance functionally as the sum of four volume integrals. Finally, if we evaluate the second total derivative (5.92) at = 0 (that is, material frame), then the second shape variation of capacitance is functional in the direction of the.

Figure 5.2: Material shape Γ (solid black line) is deformed into the spatial shape γ (dashed black line) under material velocity V (black arrow) with normal (red) compo-nents V N N and tangential (blue) ones V Γ .

Geometric Variations of Surface Area and Volume

The second derivative of spatial volume difference dω is given directly by ∂2J/∂2 in To convert the second derivative of the total volume. Since v derived from χ(X, ) is tangent everywhere to the surface γ of the sphere, the first derivative of Rωdω is zero by formula (5.120). The second fundamental form in this case is equivalent to the negative of the identity tensor everywhere on the surface of a unit sphere, yielding the second derivative of the total volume.

Energy Variation and Stability of Charged Conductive Liquids

The second variation of total potential energy with respect to at = 0 has two contributions from material velocity V and acceleration A, d2F[γ]. Finally, we arrive at the constrained (i.e. volume-saving) second shape variation of the potential energy F[γ] evaluated at a critical shape Γ provided that the material velocity field V of a configuration map χ( , ) satisfies the constraint ( 5.143), . Normal field Θ on the unit sphere can be expanded in terms of the spherical harmonics Y`m, .

Conclusion

This is due to the translational symmetry in the electrostatic energy and in the surface energy. The widespread spherically symmetric sink current moving tangentially to the conic surface predicted by Zubarev (2001) is one of the many possibilities in the self-similar framework for inviscid conic formation. Under these assumptions, Euler's equation of inviscid fluids simplifies to the unsteady Bernoulli's equation everywhere in the fluid volume (Batchelor, 2000).

Dynamic Cone Formation in Conductive Liquids: Inviscid Theory

Interfacial Cone Formation in Electrically Conductive Liquids

The novelty of his work lies in the similar scaling of an unsteady flow field, which results in non-trivial inertial forces contributing to the stress balance on the conical surface. Second, in the inviscid regime, conical cusp formation is a dynamically self-similar process for perfectly conducting fluids, during which capillary, Maxwell, and inertial forces all dissipate in the vicinity of the cone apex. New hydrodynamic patterns in the vicinity of the cone apex such as counterflows, stagnation point and oscillatory pressure field during dynamic cone formation are also revealed for the first time.

Previous Theoretical Developments of Dynamic Cone Formation

They hypothesized that the pulsation of a liquid meniscus "attached" to a static Taylor cone must be caused by small vertical displacements in the precise position of the Taylor electrode. Unlike Chung, Cutler and Miskovsky (1989), Zubarev included a spherically symmetric sink flow in the far field as illustrated in Figure 6.1(b) and predicted that the fluid should move at the sinking point Xc along the tangent to the surface. In the inviscid simulation of a charged drop, Burton and Taborek (2011), although unaware of Zubarev's work, accurately recovered the universal self-similar scale.

Inviscid Theory of Electrohydrodynamics Free Surface Flows

Not surprisingly, scaling (6.1) will completely eliminate all material parameters in the invisible model as shown later. 2∇Φ· ∇Φ+P = 0 in Ωliq, (6.4) where the excess constant of integration in the unstable Bernoulli's equation (6.4) is set to zero without loss of generality. In the level function representation (6.2) this simply means that the level function F(X, T) is advected by the velocity field U.

Figure 6.2: Dynamic cone formation of a conducting liquid (yellow) between two plate electrodes viewed in (a) laboratory frame with coordinates X = (X, Y, Z) and (b) axisymmetric self-similar frame χ = (r, z) or (r s , θ)

Family of Asymptotic Solutions to Self-Similar Dynamic Cone Formation 185

Numerical Solutions via Patched Boundary Integral Equation

Discussion and Conclusion

Dynamic Cone Formation in Conductive Liquids: Boundary Integral

Boundary Integral Method for Interfacial Inviscid Flow

Spline Interpolation, Gaussian Quadrature and Elliptic Integral

Discretization of Axisymmetric Boundary Integral Equation

Solutions to Self-Similar Cone Formation via Newton-Raphson Iteration . 228