© 2002 by CRC Press LLC Two options exist: (1) hydrostatic (externally pressured) thrust bearings, in which the fluid is fed from a high-pressure source to a lubrication film, and (2) hydrodynamic, where the supporting pressure is generated by a viscous pump fabricated on the surface of the thrust bearing itself (see Figure 9.9). Hydrostatic bearings are easy to operate and relatively easy to fabricate. These have been successfully demonstrated in the MIT Microengine program [Fréchette et al., 2000]; the thrust bearing is shown in Figure 9.8, which shows an SEM of the fabricated device cut though the middle to reveal the plenum, restrictor holes and bearing lubrication gap, which is approximately 1 µm wide. Key to the successful operation of hydrostatic FIGURE 9.8 Close-up cutaway view of microthrust bearing showing the pressure plenum (on top), feed holes and bearing gap (faintly visible). (SEM courtesy of C C. Lin.) FIGURE 9.9 Schematic of hydrodynamic thrust bearings and predicted performance (stiffness, in N/m, vs. axial eccentricity) for a typical spiral-groove thrust bearing for use in a high-speed MEMS rotor. High pressure plenum (pumped up by spiral grooves) Inward-pumping spiral grooves Rotor Stator Axis of rotation 1.50E+06 1.30E+06 1.10E+06 9.00E+05 7.00E+05 5.00E+05 Eccentricity -0.45 -0.3 -0.15 0 0.15 0.3 0.45 © 2002 by CRC Press LLC Two options exist: (1) hydrostatic (externally pressured) thrust bearings, in which the fluid is fed from a high-pressure source to a lubrication film, and (2) hydrodynamic, where the supporting pressure is generated by a viscous pump fabricated on the surface of the thrust bearing itself (see Figure 9.9). Hydrostatic bearings are easy to operate and relatively easy to fabricate. These have been successfully demonstrated in the MIT Microengine program [Fréchette et al., 2000]; the thrust bearing is shown in Figure 9.8, which shows an SEM of the fabricated device cut though the middle to reveal the plenum, restrictor holes and bearing lubrication gap, which is approximately 1 µm wide. Key to the successful operation of hydrostatic FIGURE 9.8 Close-up cutaway view of microthrust bearing showing the pressure plenum (on top), feed holes and bearing gap (faintly visible). (SEM courtesy of C C. Lin.) FIGURE 9.9 Schematic of hydrodynamic thrust bearings and predicted performance (stiffness, in N/m, vs. axial eccentricity) for a typical spiral-groove thrust bearing for use in a high-speed MEMS rotor. High pressure plenum (pumped up by spiral grooves) Inward-pumping spiral grooves Rotor Stator Axis of rotation 1.50E+06 1.30E+06 1.10E+06 9.00E+05 7.00E+05 5.00E+05 Eccentricity -0.45 -0.3 -0.15 0 0.15 0.3 0.45 © 2002 by CRC Press LLC 10 Physics of Thin Liquid Films 10.1 Introduction 10.2 The Evolution Equation for a Liquid Film on a Solid Surface 10.3 Isothermal Films Constant Surface Tension and Gravity • van der Waals Forces and Constant Surface Tension • Homogeneous Substrates • Heterogeneous Substrates • Flow on a Rotating Disc 10.4 Thermal Effects Thermocapillarity, Surface Tension and Gravity • Liquid Film on a Thick Substrate 10.5 Change of Phase: Evaporation and Condensation Interfacial Conditions • Evaporation/Condensation Only • Evaporation/Condensation, Vapor Recoil, Capillarity and Thermocapillarity • Flow on a Rotating Disc 10.6 Closing Remarks Acknowledgments 10.1 Introduction Various aspects of fluid mechanics in microelectromechanical systems (MEMS), such as flows in micro- configurations, flow transducers and flow control by microsystems, were reviewed by Ho and Tai (1998). However, the issue of thin liquid films and their dynamics in the context of microelectromechanical systems was left out of the scope of that important work. This chapter is intended to fill this gap. Thin liquid films are encountered in a variety of phenomena and technological applications [Myers, 1998]. On a large scale, they emerge in geophysics as gravity currents under water or as lava flows [Huppert and Simpson, 1980; Huppert, 1982]. On the engineering scale, liquid films serve in heat and mass transfer processes to control fluxes and protect surfaces, and their various applications arise in paints, coatings and adhesives. They also occur in foams [Schramm, 1994; Prud’homme and Khan, 1996], emulsions [Ivanov, 1988; Edwards et al., 1991] and detergents [Adamson, 1990]. In biological applications, they appear as membranes, as linings of mammalian lungs [Grotberg, 1994] or as tear films in the eye [Sharma and Ruckenstein, 1986]. On the microscale in MEMS, thin liquid films are used to produce an insulating coating of solid surfaces, to form stable liquid bridges at specified locations, to create networks of microchannels on patterned microchips [Herminghaus et al., 1999; 2000] and to design fluid microre- actors [Ichimura et al., 2000]. The presence of the deformable interface between the liquid and the ambient (normally gaseous, but possibly also another liquid) phases engenders various kinds of dynamics driven by one or usually several physical factors simultaneously. Liquid films may spontaneously or under the influence of external factors Alexander Oron Technion–Israel Institute of Technology © 2002 by CRC Press LLC 11 Bubble/Drop Transport in Microchannels 11.1 Introduction 11.2 Fundamentals 11.3 The Bretherton Problem for Pressure-Driven Bubble/Drop Transport Corrections to the Bretherton Results for Pressure-Driven Flow 11.4 Bubble Transport by Electrokinetic Flow 11.5 Future Directions Acknowledgments 11.1 Introduction Many microdevices involve fluid flows. Microducts, micronozzles, micropumps, microturbines and microvalves are examples of small devices with gas or liquid flow. It would be extremely desirable to design similar devices for two-phase flows, and many attractive applications can be envisioned if microre- actors and microlaboratories could include immiscible liquid–liquid and gas–liquid systems. Miniature evaporative and distillation units, bubble generators, multiphase extraction/separation units and many other conventional multiphase chemical processes could then be fabricated at microscales. Efficient multiphase heat exchangers could be designed for MEM devices to minimize Joule or frictional heating effects. Even for the current generation of microlaboratories using electrokinetic flow, multiphase flow has many advantages. Drops of organic samples could be transported by flowing electrolytes, thus extending the electrokinetic concept to a broader class of samples. Gas bubbles could be used as spacers for samples in a channel or to act as a piston to produce pressure-driven flow on top of the electrokinetic flow. Flow valves and pumps that employ air bubbles, like those in the ink reservoirs of ink jet printers, are already being tested for microchannels. Drug-delivery and diagnostic devices involving colloids, molecules and biological cells are also active areas of research. Before multiphase flow in microchannels becomes a reality, however, several fundamental problems that arise from the small dimension of the channels must be solved. Most of these problems originate from the large curvature of the interface between two phases in these small channels. As a result, capillary effects and other related phenomena dominate in multiphase microfluidics. Contact-line resistance, for example, is often negligible in macroscopic flows. The contact-line region, defined by intermolecular and capillary forces, is small compared to the macroscopic length scales. However, in microchannels, the contact-line region is comparable in dimension to the channel size. As a result, the large stress in that region (the classical contact-line logarithm stress singularity) can dominate the total viscous dissipation [Kalliadasis and Chang, 1994; Veretennikov et al., 1998; Indeikina and Chang, 1999]; hence, it is inad- visable to have contact lines in microchannels unless one is prepared to apply enormous pressure or electric potential driving forces. One fluid should wet the channel or capillary walls while the other is Hsueh-Chia Chang University of Notre Dame © 2002 by CRC Press LLC 12 Fundamentals of Control Theory 12.1 Introduction 12.2 Classical Linear Control Mathematical Preliminaries • Control System Analysis and Design • Other Topics 12.3 “Modern” Control Pole Placement • The Linear Quadratic Regulator • Basic Robust Control 12.4 Nonlinear Control SISO Feedback Linearization • MIMO Full-State Feedback Linearization • Control Applications of Lyapunov Stability Theory • Hybrid Systems 12.5 Parting Remarks 12.1 Introduction This chapter reviews the fundamentals of linear and nonlinear control. This topic is particularly important in microelectromechanical systems (MEMS) applications for two reasons. First, as electromechanical systems, MEMS devices often must be controlled in order to be utilized in an effective manner. Second, important applications of MEMS technology are controls-related because of the utility of MEMS devices in sensor and actuator technologies. Because the area of control is far too vast to be entirely presented in one self-contained chapter, the approach adopted for this chapter is to outline a variety of techniques used for control system synthesis and analysis, provide at least a brief description of their mathematical foundation, discuss the advantages and disadvantages of each of the techniques and provide sufficient references so that the reader can find a starting point in the literature to fully implement any described techniques. The material varies from the extremely basic (e.g., root locus design) to relatively advanced material (e.g., sliding mode control) to cutting-edge research (hybrid systems). Some examples are provided; additionally, many references to the literature are provided to help the reader find further examples of a particular analysis or synthesis technique. This chapter is divided into three sections, each of which considers both the stability and performance of a control system. The term performance includes both the qualitative nature of any transient response of the system, reference signal tracking properties of the system and the long-term or steady-state perfor- mance of the system. The first section considers “classical control,” which is the study of single-input, single-output (SISO) linear control systems, which relies heavily upon mathematical techniques from complex variable theory. The material in this section outlines what is typically covered in an elementary undergraduate controls course. The second section considers so-called “modern control” which is the study of multi-input, multi-output (MIMO) control systems in state space. Included in this section is Bill Goodwine University of Notre Dame . research. Before multiphase flow in microchannels becomes a reality, however, several fundamental problems that arise from the small dimension of the channels must be solved. Most of these problems originate from. microelectromechanical systems (MEMS) applications for two reasons. First, as electromechanical systems, MEMS devices often must be controlled in order to be utilized in an effective manner. Second, important. properties of the system and the long-term or steady-state perfor- mance of the system. The first section considers “classical control,” which is the study of single-input, single-output (SISO)