# zbMATH — the first resource for mathematics

Microflows and nanoflows. Fundamentals and simulation. Foreword by Chih-Ming Ho. (English) Zbl 1115.76003
Interdisciplinary Applied Mathematics 29. New York, NY: Springer (ISBN 0-387-22197-2/hbk). xxi, 817 p. (2005).
The main differences between fluid mechanics at microscales and in the macrodomain can be broadly classified into four areas: $$1^\circ$$ noncontinuum effects, $$2^\circ$$ surface-dominated effects, $$3^\circ$$ low Reynolds number effects, $$4^\circ$$ multiscale and multiphysics effects. The monograph under review presents a systematical presentation of all questions connected with fundamentals and simulation of microflows and nanoflows. This material is divided into three main categories: a) gas flows (chapters 2–6), b) liquid flows (chapters 7–13), c) simulation techniques (chapters 14–18).
In Ch. 1 many concepts and devices are given which are discussed in detail in the monograph. Following historical reasons, Ch. 1 begins with some prototype of Micro-Electro-Mechanical-systems (MEMS) devices and discusses such fundamental concepts as breakdown of constitutive laws, new flow regimes, and modeling issues encountered in microfluid and nanofluid systems. Fluid-surface interactions for liquids are discussed such as electrokinetic effects and wetting, important at very small scales. The question of full-system simulation of MEMS is stated, and the concept of macromodeling is introduced.
Ch. 2 presents the basic equations of fluid dynamics for both incompressible and compressible flows and discusses appropriate nondimensionalizations for low-speed and high-speed flows. Most of the flows encountered in microsystems are, in general, of low speed, however micropropulsion applications may involve high-speed supersonic flows. Compressible Navier-Stokes equations are considered with general boundary condition for velocity slip, which are applied to a regime corresponding to second-order correction in Knudsen number.
In Ch. 3 shear-driven gas microflows are considered with the objective of modeling a certain class of flows arising in microsystems. In particular, shear-driven microflows are the flows between the rotor and base plate of a micromotor, and the flows between stationary and movable arms of a comb-drive mechanism. The authors concentrate on prototype flows such as linear Couette flow, and flow in shear-driven microcavities and grooved microchannels, in order to overcome the difficulties of flow physics for complex engineering geometries. At first, analytical and numerical results are presented for steady Couette flow in the slip flow regime. Then the development and validation of an empirical model for steady Couette flow are presented in the transition and free-molecular flow regimes. Simulation results and analysis for oscillatory shear-driven flows in the entire Knudsen regime are given. Flows in prototype complex geometries, such as the microcavity and grooved microchannel flows, are included.
In Ch. 4 models for pressure-driven gas flows in the slip, transition and free molecular flow regimes are presented. The authors are interested in microchannel, pipe, and duct flows as having primary engineering importance with analytical solutions caused by their simple geometry. For transition and free-molecular flow regimes a unified flow model is developed which can accurately product the volumetric flowrate, velocity profile, and pressure distribution in the whole Knudsen regime for pipes and ducts, and also for the minimal Knudsen number.
In Ch. 5 heat transfer in gas microflows is considered. Thermal creep (transpiration) effects is analyzed important for channels with tangential temperature gradients on their surfaces, in particular, a microchannel surface with a prescribed heat flux subjected to temperature variations along its surface together with results on thermal creep flows. Then other temperature-induced flows are studied, and the validity of the heat conduction equation is investigated in various limiting cases. Combined effects of thermal creep, heat conduction, and convection in pressure-, force-, and shear-driven channel flows are also investigated.
Ch. 6 is devoted to rarefied gas flows encountered in applications other than simple microchannels. At first, the lubrication theory is considered with special attention on the slider bearing and squeezed film problems. Then follow the separated flows in internal and external geometries in the ship flow regime. Further, the theoretical and numerical results for Stokes flow past a sphere are presented. The classical Stokes drag for external flows, including rarefaction effects in the ship flow regime, is reviewed with the presentation of drag formulae for the pressure-driven flows past a stationary sphere confined in a pipe. Their verifications are given in connection with numerical simulations in the slip flow regime, that shows drastic variations in the drag coefficient as a function of Knudsen number and the cylinder/sphere blocking ratio. The limiting results are considered applicable to liquid flows past solid electrically neutral spheres. Recent findings on gas flows through microfilters are summarized together with the investigation of high-speed rarefied flows in micronozzles, with are used for controlling the motion of microsatellites and nanosatellites.
With Ch. 7 starts the second part “Liquid flows” of the monograph. Here the authors review and explore ideas of microflow control elements at the usage of electrokinetic flow control schemes, with do not require any moving components. Electroosmotic and electrophoretic transport is covered in detail both for steady and time-periodic flows, and electrophoresis is presented allowing separation and detection of similar size particles based on their polarizability.
Ch. 8 is devoted to surface tension-driven flows and capillary phenomena involving wetting and spreading of liquid thin films and droplets for modeling of classical engineering applications such as coating and lubrication. For microscopic delivery on open surfaces, electrowetting and thermocapillary along with dielectrophoresis have been employed to move continuously on discrete streams of fluid, for example droplets along specified paths on glass surfaces. A new method of actuation exploits optical beams and photoconductor materials in conjunction with electrowetting. Such electrically or lithographically defined paths can be reconfigured dynamically using electronically addressable arrays that respond to electric potential, temperature, or laser beams and control the direction, timing, and speed of fluid droplets. Here microfluidic transport mechanisms are studied based on capillary phenomena taking advantage of the relative importance and sensitivity of surface tension in microscales. Particularly, the authors study how temperature, electric potential, and light can affect the value and possibly the sign of surface tension.
In Ch. 9 the basic ideas of micromixers and chaotic advection are presented and analytic solutions for prototypical problems are given. In microchannels the flow is laminar and steady, so diffusion is controlled solely by the diffusivity coefficient of the medium, thus requiring excessive amounts of time for complete mixing. Examples of passive and active mixers are discussed which have been used in microfluidic applications. Some quantitative measures of characterizing mixing are provided, based on the concept of Lyapunov exponent from chaos theory as well as some convenient ways for their computation.
Ch. 10 is devoted to simple liquids is nanochannels described by standard Lennard-Jones potentials. A key difference between the simulation of the fluid transport in confined nanochannels, where the critical channel dimension can be a few molecular diameters, and at macroscopic scales is that the well-established continuum theories based on Navier-Stokes equations may not be valid in confined nanochannels. Therefore atomistic scale simulations, in which the fluid atoms are modeled explicitly or semi-explicitly and the motion of the fluid atoms is calculated directly, shed fundamental insights on fluid transport. Here density profiles, diffusion transport and Navier-Stokes equations validity are discussed for simple fluids in confined nanochannels. Finally, the slip conditions at solid-fluid interfaces are discussed and experimental and computational results together with conceptual models of slip are presented. Also the lubrication problem first discussed in Ch. 7 is revisited, and the Reynolds-Vinogradova theory for hydrophobic surfaces is presented.
Water and its properties in various forms is one of the most actively investigated areas because of its importance in nature. After introducing some definitions and atomistic models for water, the authors present in Ch. 10 the static and dynamic behavior of water in confined nanochannels. In Ch. 11 the fundamentals and simulation of electroosmotic flow in nanochannels are discussed. The significance of the finite size of ions and the discrete nature of solvent molecules are highlighted. A slip boundary condition which can be used in the hydrodynamic theory for nanochannels electroosmotic flows is presented. The physical mechanisms that lead to charge inversion and corresponding flow reversal phenomena in nanochannel electroosmotic flows are discussed.
The last Ch. 13 of the second part focuses on functional fluids and on functionalized devices, specifically on nanotubes. Here details of the physical mechanisms involved in self-assembly are presented, and examples of patterns are given which are formed using magnetic fields for magneto-rheological fluids and electrophoretic deposition for electro-rheological fluids. The authors give a brief introduction to carbon nanotubes and ion channels in biological membranes, and present results on electrolyte transport through carbon nanotubes together with concepts showing that the transport of electrolytes can be augmented by using functionalized nanotubes and electric fields.
Ch. 14 of the last part “Simulation techniques” contains three main numerical methodologies to analyze flows in microdomains: $$1^\circ$$ High-order finite element (spectral element) methods for Navier-Stokes equations; formulations for both incompressible and compressible flows in stationary and moving domains are presented. $$2^\circ$$ Meshless methods with random point distribution. $$3^\circ$$ The force coupling method for particulate microflows. These are three different classes of discretization. In Ch. 15 the theory and numerical methodologies are discussed for simulating gas flows at mesoscopic and atomistic levels. Here an overview of the Boltzmann equation is given, describing in some detail gas-surface interactions with benchmark solutions for validation of numerical codes and macromodels. The main result relevant for bridging microdynamics and macrodynamics is the Boltzmann equation, which is discussed by using lattice Boltzmann methods as the $$H$$-theorem. In Ch. 16 the theory and numerical methodologies for simulating liquid flows are discussed at atomistic and mesoscopic levels. In Ch. 17 the authors turn to simulating full systems across heterogeneous domains, i.e. fluid, thermal, electrical, structural, chemical etc. Several reduced-order modeling techniques for the analysis of microsystems are introduced, such as generalized Kirchhoff networks, black box models, Galerkin method. The advantages and limitations of various techniques are discussed. Ch. 18 considers some applications of these techniques to several examples in microflows. Here reduced-order modeling of squeezed film damping is investigated by applying equivalent circuit, Galerkin, mixed-level and black box models. A compact model for electrowetting is discussed. Some of the software packages are summarized that are available for reduced-order simulation.
The reviewed monograph is the first systematic fundamental presentation of the subject. It is suitable for graduate students and researches in fluid mechanics, physics and in electrical, mechanical and chemical engineering.

##### MSC:
 76-02 Research exposition (monographs, survey articles) pertaining to fluid mechanics 76Dxx Incompressible viscous fluids 76N15 Gas dynamics (general theory) 76Mxx Basic methods in fluid mechanics 76A02 Foundations of fluid mechanics
Full Text: