The last two decades have brought a revolution in miniaturization of space technology. Thanks to improved microelectronic sensors and MEMS devices, nanosatellites can perform communication and scientific studies previously limited to large satellites, significantly reducing the financial barriers to space access. But development of a reliable, long-running, small-scale propulsion system for orbital maneuvers remains a key challenge. One solution is the microfluidic electrospray propulsion (MEP) thruster under development at NASA’s Jet Propulsion Laboratory (JPL).

This thesis analytically addresses aspects of the MEP system’s propellant management, specifically, capillary flow in the groove network delivering fluid propellant from the reservoir to the emitters. Building upon the reduced-order model of viscous capillary flow in straight V-shaped channels (“V-grooves”) of Weislogel (1996) and Romero and Yost (1996), we prove stability of steady-state and self-similar flows. Because the MEP design requires an electric field above the grooves, and further calls for grooves which curve and bend in three dimensions, we extend earlier V-groove models to include these effects, and also perform stability analyses of the new models. The results not only validate the use of V-grooves as a robust propellant delivery system, but also provide a theoretical basis for the design of future microfluidic devices with compact, three-dimensional designs and electric fields.

In order to lay the groundwork for future studies of early-time behavior of propellant on emitter tips before the Taylor cone necessary for ion emission is formed, we develop the technique of generalized linear stability analysis (Farrell and Ioannou, 1996) of capillary flow of thin viscous films coating curved surfaces (governed by the equation first developed by Roy and Schwartz, 1997). This methodology was first applied to films coating cylinders and spheres by Balestra et al. (2016, 2018); we instead apply the technique and analyze for the first time a viscous-capillary instability arising on a torus coated with a uniform thin film.

Besides the capillary fluid dynamics results, two additional pieces of work are included in the thesis. First, in an unorthodox application of Noether’s Theorem to non-Lagrangian gradient flow equations, we show that each variational symmetry of the governing functional induces a constraint on the evolution of the system. Second, to support JPL’s efforts to directly detect a “fifth force,” we introduce and implement numerical methods for computation of the scalar Cubic Galileon Gravity (CGG) field at solar system scales.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Troian, Sandra M.}, } @phdthesis{10.7907/PEJ5-1626, author = {Zhou, Chengzhe}, title = {Collection of Solved Nonlinear Problems for Remote Shaping and Patterning of Liquid Structures on Flat and Curved Substrates by Electric and Thermal Fields}, school = {California Institute of Technology}, year = {2020}, doi = {10.7907/PEJ5-1626}, url = {https://resolver.caltech.edu/CaltechTHESIS:12092019-191651654}, abstract = {There has been significant interest during the past decade in developing methods for remote manipulation and shaping of soft matter such as polymer melts or liquid metals to pattern films at the micro- and nanoscale. The appeal of low-cost fabrication of micro-optical devices for beam shaping or metallic films to produce high order cuspidal arrays for antireflective or self-cleaning coatings has driven considerable interest in the fundamentals associated with film shaping and liquid curvature. Physicists and applied mathematicians have uncovered new rich ground in examining the complex behavior of high order, nonlinear partial differential equations describing the motion and response of liquid structures driven to redistribute and reorganize by externally applied thermal and electric fields. For the problems relevant to this thesis, which focuses on liquid structures at small scales, the applied fields induce surface forces which act only at the moving interface. Because the surface-to-volume ratios tend to be very large however, the corresponding forces are considerable in magnitude and dominate the formation and growth processes described. In all cases examined, once the driving forces are removed and the operating temperatures dropped below the melting point, the patterned films and liquid shapes rapidly solidify in place, leaving behind structures with molecularly smooth surfaces, an especially advantageous feature for micro-optical applications.

The first part of this thesis examines the nonlinear dynamics of free surface films in the long wavelength limit coating either a flat or curved substrate. We examine the long wavelength limit in which inertial forces are suppressed in comparison to viscous forces such that the system reacts instantaneously to interfacial forces acting in the direction normal to the moving interface, such as capillary and Maxwell forces, or in the direction parallel to the moving interface, such as thermocapillary forces. In the first example, we demonstrate by analytic and numerical means how a system designed to incur large runaway thermocapillary forces can pattern films with conic cusps whose tips undergo self-focused sharpening through a novel self-similar process. This finding expands the known categories of flows that can generate cusp-like shapes and introduces a new lithographic method for remote, one-step fabrication of cuspidal microarrays. We next examine a lithographic technique known as Electrohydrodynamic Lithography in which remote patterned electric field distributions projected onto the surface of a dielectric film generate Maxwell stresses which cause growth and accumulation toward regions of highest field gradients. Here we solve the inverse problem associated with the governing fourth-order nonlinear interface equation by appealing to a control-theoretical approach. This approach reveals the optimal electrode topography required to generate precise complex liquid patterns within a given time interval. Numerical implementation of this algorithm yields high fidelity pattern replication by essentially incorporating proximity corrections which quench undesirable interference effects of material waves. We then extend the long wavelength analysis to a liquid layer coating a curved manifold and demonstrate how a desired film shape can be obtained by novel application of the Helmholtz minimum dissipation principle. We illustrate this solution method by deriving the nonlocal tensorial partial differential equation for the evolution of a slender, perfectly conducting or insulating liquid film supported on a curved electrode. Finite element simulations demonstrate the complex shapes which can result, including formation of liquid accumulation sites and flow instabilities not accessible to films supported on a planar substrate.

The second part of this thesis focuses exclusively on geometric singularities which result from nonlinear effects caused by the coupling of capillary and Maxwell forces in perfectly conducting liquids. Here, we do not restrict ourselves to the long wavelength approximation but instead examine systems with comparable lateral and transverse dimensions. We probe the energy stability of such systems using a convective Lagrangian approach. The exact variational characterization of equilibrium shapes and their stability is derived in the most general form without restriction to coordinate system or shape deformations. This formulation unmasks several terms, typically not evident in calculations restricted to normal deformations of an electrified spherical drop. Our result provides new insights into the energy stability of equilibrium shapes that do not necessarily have constant interface curvature or uniform surface charge distribution. We then turn attention to the classical problem of a conical meniscus produced in an electrified liquid body. The analysis by G. I. Taylor (1963) first determined that the hydrostatic equilibrium shape for a liquid body subject only to capillary and Maxwell forces is given by a cone with an opening angle of 98.6°. However, the fact that such a cone represents an unsteady configuration is often ignored. We revisit the inviscid analysis by Zubarev (2001) who proposed that conic cusps in perfectly conductive liquid evolve through a time-dependent self-similar process. Using the unsteady Bernoulli’s equation, he analyzed the force balance at the moving interface and obtained an asymptotically correct self-similar solution dominated by a sink flow far from the evolving apex whose streamlines orient nearly parallel to the moving surface. In addition to the sink flow our analysis, supported by accurate, high resolution numerical solutions of the boundary integral equations, independently reveals a two-parameter family of non-spherically symmetric self-similar solutions whose velocity streamlines intercept the conic surface at an angle. This new family of solutions not only properly account for the interplay between capillary, Maxwell and inertial forces but generate advancing and recoiling type interface shapes, which substantially alter current understanding of the formation and acceleration of dynamic cones.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Troian, Sandra M.}, } @phdthesis{10.7907/thhf-5478, author = {Albertson, Theodore Glenn}, title = {Simulations of Conic Cusp Formation, Growth, and Instability in Electrified Viscous Liquid Metals on Flat and Curved Surfaces}, school = {California Institute of Technology}, year = {2020}, doi = {10.7907/thhf-5478}, url = {https://resolver.caltech.edu/CaltechTHESIS:05182020-133853408}, abstract = {It is well known that above a critical field strength sufficiently large to overcome damping by capillary forces, the free surface of a perfectly conducting liquid will spontaneously deform into one or more sharp protrusions known as conic cusps. Such cusps undergo tip sharpening while rapidly accelerating toward regions of highest electric field strength, eventually giving rise to beams of ions and/or charged droplets . These charged beams form the basis for liquid metal ion sources (LMIS) commonly used in focused ion beam systems, scanning ion microscopy, micromilling, ion mass spectrometry, implantation, and lithography. During the past few decades, there has been growing interest in optimizing the formation, growth, and stability of conic cusps in liquid metals for a new class of efficient and highly miniaturizable satellite micropropulsion devices consisting of microarrays of externally wetted solid needles coated with a film of liquid metal propellant. The thrust levels generated by such microarrays is suitable for propulsion of small satellites and precision pointing maneuvers for larger satellites.

This thesis addresses the formation, growth, and instability of conic cusp formations in perfectly conducting, electrified viscous liquids on flat and curved surfaces. We use finite element simulations based on the arbitrary Lagrangian-Eulerian (ALE) method for coupling the vacuum and liquid domains across the accelerating interface. The simulations in Chapters 2–4 describe the evolution of liquid flow subject to electric field distributions generated by opposing flat parallel and solid electrodes. In particular, we examine in Chapter 2 the growth of a small liquid protuberance on an otherwise flat viscous liquid layer of perfectly conducting fluid subject to an initial uniform electric field. Previous studies in the literature have established that tip sharpening proceeds via a self-similar process in two distinct limits: the Stokes regime at Re = 0 and the inviscid regime Re → ∞. These simulations, conducted at fixed capillary number Ca and for 0.1 ≤ Re ≤ 50,000, which span the viscous to inviscid regimes, demonstrate that the conic tip *always* undergoes self-similar growth irrespective of Reynolds number. Field self-enhancement due to conic cusp tip sharpening is shown to generate divergent power law growth in finite time (so-called blowup behavior) of the interfacial and volumetric forces acting at the advancing tip. The computed blow up exponents at the tip surface associated with the various terms in the Navier-Stokes equation and interface normal stress condition reveal the different forces at play as Re increases. Rescaling of the tip shape by the capillary stress exponent yields excellent collapse onto a universal conic tip shape with interior half-angle dependent on the magnitude of the Maxwell stress. The simulations clearly show that the interior cone angle adopts values both above and below the Taylor cone angle value of 49.3°. Additional details of the modeled flow dispel prevailing misconceptions that dynamic cones resemble conventional Taylor cones or that viscous stresses at finite Re can be neglected. In Chapter 3, we demonstrate how the rapid acceleration of the curved liquid interface also generates a thin surface boundary layer with very high local strain rate in the vicinity of the conic tip. The value of the surface vorticity along the moving interface is shown to be in excellent agreement with theoretical predictions. More importantly, the results in Chapters 2 and 3 demonstrate that the velocity streamlines are always at an oblique angle to the moving interface, contrary to commonly held belief that the streamlines always lie tangent to the moving boundary. In Chapter 4, we extend the simulations to include variation of the capillary number and find that for sufficiently high Re and Ca, the advancing interface develops significant oscillations. Fourier analysis of these interface oscillations indicates that the extracted instability wavelength characteristic of flows at smaller values of Re tends to exceed the simplified theoretical prediction based on inviscid flow. By contrast, the extracted instability wavelength for the largest values of Re examined tends to fall below the inviscid prediction.

In Chapter 5, we explore the effect of substrate curvature on the flow and stability of electrified films by examining the behavior of a thin viscous film of perfectly conducting liquid on two types of curved surfaces. These shapes, which include a solid conical needle with a spherical cap tip and a solid parabolic needle, are intended to mimic substrates used in some externally wetted microemitter arrays in LMIS systems. For the simulations in Chapter 5, the needle is situated below a counter electrode perforated with a circular aperture. The films are shown to develop both on-axis and off-axis cusp-like protrusions depending on the parameter range examined. In particular, the formation of off-axis protrusions are directly traced to substrate shapes which manifest an abrupt change in curvature, as present in a solid conical needle with a spherical cap tip. The simulations reported here are anticipated to help optimize fabrication of externally wetted needle shapes for use in a variety of LMIS systems.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Troian, Sandra M.}, } @phdthesis{10.7907/Z92J68X1, author = {Fiedler, Kevin Robert}, title = {Fundamentals of Thermocapillary Sculpting of Liquid Nanofilms and Applications to Thin Film Micro-Optics}, school = {California Institute of Technology}, year = {2017}, doi = {10.7907/Z92J68X1}, url = {https://resolver.caltech.edu/CaltechTHESIS:03062017-120100812}, abstract = {This doctoral thesis describes experimental work conducted as part of ongoing efforts to identify and understand the source of linear instability in ultrathin liquid films subject to large variations in surface temperature along the air/liquid interface. Previous theoretical efforts by various groups have identified three possible physical mechanisms for instability, including an induced surface charge model, an acoustic phonon model, and a thermocapillary model. The observed instability manifests as the spontaneous formation of arrays of nano/microscale liquid protrusions arising from an initially flat nanofilm, whose organization is characterized by a distinct in-plane wavelength and associated out-of-plane growth rate. Although long range order is somewhat difficult to achieve due to thin film defects incurred during preparation, the instability tends toward hexagonal symmetry within periodic domains achieved for a geometry in which the nanofilm is held in close proximity to a cooled, proximate, parallel, and featureless substrate.

In this work, data obtained from a previous experimental setup is analyzed and it is shown how key improvements in image processing and analysis, coupled with more accurate finite element simulations of thermal profiles, lead to more accurate identification of the fastest growing unstable mode at early times. This fastest growing mode is governed by linear instability and exponential growth. This work was followed by re-examination of real time interference fringes using differential colorimetry to quantify the actual rate of growth of the fastest growing peaks within the protrusion arrays. These initial studies and lingering questions led to the introduction of a new and improved experimental setup, which was redesigned to yield larger and more reproducible data sets. Corresponding improvements to the image analysis process allowed for the measurement of both the wavelength and growth rate of the fastest growing mode simultaneously. These combined efforts establish that the dominant source of instability is attributable to large thermocapillary stresses. For the geometry in which the nanofilm surface is held in close proximity to a cooled and parallel substrate, the instability leads to a runaway process, characterized by exponential growth, in which the film is attracted to the cooled target until contact is achieved.

The second part of this thesis describes fabrication and characterization of microlens arrays and linear waveguide structures using a similar experimental setup. However, instead of relying on the native instability observed, formation and growth of liquid shapes and protrusions is triggered by pre-patterning the cooled substrate with a desired mask for replication. These preformed cooled patterns, held in close proximity to an initially flat liquid nanofilm, induce a strong non-linear response via consequent patterned thermocapillary stresses imposed along the air/liquid interface. Once the desired film shapes are achieved, the transverse thermal gradient is removed and the micro-optical components are affixed in place naturally by the resultant rapid solidification. The use of polymer nanofilms with low glass transition temperatures, such as polystyrene, facilitated rapid solidification, while providing good optical response. Surface characterization of the resulting micro-optical components was accomplished by scanning white light interferometry, which evidences formation of ultrasmooth surfaces ideal for optical applications. Finally, linear waveguides were created by this thermocapillary sculpting technique and their optical performance characterized. In conclusion, these measurements highlight the true source of instability in this geometry, and the fabrication demonstrations pave the way for harnessing this knowledge for the design and creation of novel micro-optical devices.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Troian, Sandra M.}, } @phdthesis{10.7907/Z9R20ZBS, author = {Nicolaou, Zachary George}, title = {Symmetry and Variational Analyses of Fluid Interface Equations in the Thin Film Limit}, school = {California Institute of Technology}, year = {2017}, doi = {10.7907/Z9R20ZBS}, url = {https://resolver.caltech.edu/CaltechTHESIS:09082016-145215432}, abstract = {This thesis concerns a class of nonlinear partial differential equations up to fourth order in spatial derivatives that models thin viscous films. In Chapter 1, we review the derivations of thin film equations from the fundamental transport equations. Section 1.1 contains the derivation for a thermocapillary driven film to familiarize the reader with the key long-wavelength approximation that has been successful in modeling a myriad of thin viscous films. In Section 1.2, we consider the coupling between a thin viscous layer and a much thicker fluid layer with much larger viscosity and conductivity and show how a novel, non-local thermocapillary thin film equation can be derived to model such a system. We then review the wider class of thin film equations in Section 1.3, note the important Cahn-Hilliard variational form of these equations, and demonstrate that classic mathematical results concerning the inverse problem of the calculus of variations permit an algorithmic procedure for discovering Lyapunov functionals. In Chapter 2, we review applications of symmetry methods to partial differential equations. Section 2.1 contains an original geometrical motivation for the study of self-similar reductions which draws an analogy with the fixed points of dynamical systems. In Section 2.2, we derive for the first time the full set of symmetries of the fully two-dimensional thin film equations. We then enumerate the possible symmetry reductions of the thin film equations, and discover several which have not been previously recognized. In Chapter 3, we consider rotationally invariant, steady droplet solutions and their stability. In Section 3.1, we derive stability criteria for thermocapillary-driven droplets, and show a novel correspondence between droplet stability, droplet volume, and droplet Lyapunov energy. We consider thin films under other forces in Section 3.2 and make new predictions about conditions under which such films develop into droplets, columns, or jets of fluid. In Chapter 4, we consider the scale invariant symmetry reductions of thin film equations. In Section 4.1 we describe the extraordinarily rich variety of such solutions in the spreading of a insoluble surfactant on a thin viscous film, identify previously unrecognized scale invariant solutions which are well-behaved at the origin, and demonstrate their relevance with finite element simulations. Lastly, in Section 4.2, we illustrate for capillary driven films some numerical solutions to the novel reductions we uncovered in Chapter 2. Each chapter concludes with a Notes section which summarizes the new results contained therein and relates them to the wider literature.}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Troian, Sandra M.}, }