Capillary Surface Interfaces, Volume 46, Number 7

Total Page:16

File Type:pdf, Size:1020Kb

Capillary Surface Interfaces, Volume 46, Number 7 fea-finn.qxp 7/1/99 9:56 AM Page 770 Capillary Surface Interfaces Robert Finn nyone who has seen or felt a raindrop, under physical conditions, and failure of unique- or who has written with a pen, observed ness under conditions for which solutions exist. a spiderweb, dined by candlelight, or in- The predicted behavior is in some cases in strik- teracted in any of myriad other ways ing variance with predictions that come from lin- with the surrounding world, has en- earizations and formal expansions, sufficiently so Acountered capillarity phenomena. Most such oc- that it led to initial doubts as to the physical va- currences are so familiar as to escape special no- lidity of the theory. In part for that reason, exper- tice; others, such as the rise of liquid in a narrow iments were devised to determine what actually oc- tube, have dramatic impact and became scientific curs. Some of the experiments required challenges. Recorded observations of liquid rise in microgravity conditions and were conducted on thin tubes can be traced at least to medieval times; NASA Space Shuttle flights and in the Russian Mir the phenomenon initially defied explanation and Space Station. In what follows we outline the his- came to be described by the Latin word capillus, tory of the problems and describe some of the meaning hair. current theory and relevant experimental results. It became clearly understood during recent cen- The original attempts to explain liquid rise in a turies that many phenomena share a unifying fea- capillary tube were based on the notion that the ture of being something that happens whenever portion of the tube above the liquid was exerting two materials are situated adjacent to each other a pull on the liquid surface. That, however, cannot and do not mix. We will use the term capillary sur- be what is happening, as one sees simply by ob- face to describe the free interface that occurs when serving that the surface fails to recede (or to change one of the materials is a liquid and the other a liq- in any way) if the tube is cut off just above the in- uid or gas. In physical configurations such as the terface. Further, changing the thickness of the capillary tube, interfaces occur also between these walls has no effect on the surface, thus suggest- materials and rigid solids; these latter interfaces ing that the forces giving rise to the phenomenon yield in many cases the dominant influence for de- can be significant only at extremely small dis- termining the configuration. tances (more precise analysis indicates a large por- In this article we describe a number of such tion of these forces to be at most molecular in phenomena, notably some that were discovered range). Thus, for a vertical tube the net attractive very recently as formal consequences of the highly forces between liquid and wall must by symmetry nonlinear governing equations. These discoveries be horizontal. It is this horizontal attraction that include discontinuous dependence on the bound- causes the vertical rise. Molecules being pulled to- ary data, symmetry breaking, failure of existence ward the walls force other molecules aside in all Robert Finn is professor emeritus of mathematics at directions, resulting in a spread along the walls that Stanford University. His e-mail address is finn@gauss. is only partly compensated by gravity. Liquid is stanford.edu. forced upward along the walls, and cohesive forces 770 NOTICES OF THE AMS VOLUME 46, NUMBER 7 fea-finn.qxp 7/1/99 9:56 AM Page 771 carry the remaining liquid column with it. The low- g ered hydrostatic pressure at the top of the column is compensated by the curvature of the surface, in essentially the same manner as occurs with a soap S bubble. This basic observation as to the nature of the acting forces appears for the first time in the γ writings of John Leslie in 1802. V W An early attempt to explain capillarity phe- nomena was made by Aristotle, who wrote circa 350 B.C. that A broad flat body, even of heavy mater- ial, will float on a water surface, but a long thin one such as a needle will always sink. Any reader with Figure 1. Fluid interface S, support surface W ; γ is the angle access to a needle and a glass of water will have between the two surface normals. little difficulty refuting the statement. On the other hand, Leonardo da Vinci wrote in 1490 on the me- chanics of formation of liquid drops, using ideas very similar to current thinking. But in the ab- γ sence of the calculus, the theory could not be made quantitative, and there was no convincing way to test it against experiments. The achievements of the modern theory de- pend essentially on mathematical methods, and u0 specifically on the calculus, on the calculus of vari- ations, and on differential geometry. When one looks back on how that came about, one is struck by the irony that the initial mathematical insights were introduced by Thomas Young, a medical physician and natural philosopher who made no secret of his contempt for mathematics (and more specifically for particular mathematicians). But it was Young who in 1805 first introduced the math- Figure 2. Capillary tube configuration. ematical concept of mean curvature H of a surface and who showed its importance for capillarity by Theorem 1. The height u(x, y) of a capillary sur- relating it to the pressure change across the sur- face interface lying over a domain Ω in a vertical face: ∆p =2σH, with σ equal to surface tension. gravity field satisfies the differential equation Young also reasoned that if the liquid rests on a support surface W, then the fluid surface S meets (3) div Tu = κu + λ. W in an angle γ (contact angle) that depends only Here λ is a constant to be determined by phys- on the materials and not on the gravity field, the ical conditions (such as fluid volume) and bound- shape of the surface, or the shape or thickness of ary conditions; κ is positive when the denser fluid W; see Figure 1. lies below the interface; in the contrary case, the Young derived with these concepts and from the laws of hydrostatics the first correct approxima- sign of κ reverses. For the problem described tion for the rise height at the center of a circular above, considered by Young, λ =0and (3) becomes capillary tube of small radius a immersed vertically (4) div Tu = κu. in a large liquid bath: For a capillary surface in a cylindrical vertical 2 cos γ ρg (1) u0 ≈ ,κ= ; tube of homogeneous material and general hori- κa σ zontal section Ω, the Young condition on the con- here ρ is the density change across the free sur- tact angle yields the boundary condition face, g the magnitude of gravitational accelera- · tion. See Figure 2. (5) ν Tu = cos γ Young’s chief competitor in these developments on Σ = ∂Ω, with ν the unit exterior normal on Σ. was Laplace, who relied heavily on mathematics. Instead of a tube dipped into an infinite reser- Laplace derived a formal mathematical expression voir as considered by Young, one could imagine a vertical tube closed at the bottom and partially ≡ ≡ q Du (2) 2H div Tu, Tu filled with a prescribed volume of liquid covering 1+|Du|2 the base. In general in this case λ =06 , but addition for the mean curvature H of a surface u(x, y); he of a constant to u converts (3) to (4). From unique- was led to ness properties discussed below, it follows that in AUGUST 1999 NOTICES OF THE AMS 771 fea-finn.qxp 7/1/99 9:56 AM Page 772 sult was obtained again independently by E. Mierse- mann in 1994. In 1830 Gauss used the Principle of Virtual Work, formulated by Johann Bernoulli in 1717, to unify the achievements of Young and of Laplace, and he δ obtained both the differential equation and the Γ boundary condition as consequences of the prin- α ciple. In the Gauss formulation the constant λ in O (3) appears as a Lagrange parameter arising from an eventual volume constraint. Ωα δ Capillarity attracted the attention of many of the leading mathematicians of the nineteenth and early twentieth centuries, and some striking results were obtained; however, the topic then suffered a hia- tus till the latter part of the present century. A great influence toward new discoveries was provided by the “BV theory”, developed originally for min- imal surfaces by E. de Giorgi and his co-workers. In the context of this theory M. Emmer provided Figure 3. Wedge domain. in 1973 the first existence theorem for the capil- lary tube of general section. For further references all cases, and independent of the volume of liquid, to these developments, see [1, 2]. Other directions the surfaces obtained are geometrically the same. were initiated by Almgren, Federer, Fleming, Si- This result holds also when κ =0 and a solution mons, and others and led to results of different exists; however, existence cannot in general then character; see, e.g., [10]. be expected, as we shall observe below. To some extent, these considerations extend to configura- The Wedge Phenomenon tions with κ<0, i.e., with the heavier fluid on top; It is unlikely that anyone reading this article will however, in general both existence and unique- be unfamiliar with the name of Brook Taylor, as ness may fail when κ<0. the Taylor series figures prominently in every cal- If γ is constant, we may normalize it to the culus sequence.
Recommended publications
  • Hydrostatic Shapes
    7 Hydrostatic shapes It is primarily the interplay between gravity and contact forces that shapes the macroscopic world around us. The seas, the air, planets and stars all owe their shape to gravity, and even our own bodies bear witness to the strength of gravity at the surface of our massive planet. What physics principles determine the shape of the surface of the sea? The sea is obviously horizontal at short distances, but bends below the horizon at larger distances following the planet’s curvature. The Earth as a whole is spherical and so is the sea, but that is only the first approximation. The Moon’s gravity tugs at the water in the seas and raises tides, and even the massive Earth itself is flattened by the centrifugal forces of its own rotation. Disregarding surface tension, the simple answer is that in hydrostatic equi- librium with gravity, an interface between two fluids of different densities, for example the sea and the atmosphere, must coincide with a surface of constant ¡A potential, an equipotential surface. Otherwise, if an interface crosses an equipo- ¡ A ¡ A tential surface, there will arise a tangential component of gravity which can only ¡ ©¼AAA be balanced by shear contact forces that a fluid at rest is unable to supply. An ¡ AAA ¡ g AUA iceberg rising out of the sea does not obey this principle because it is solid, not ¡ ? A fluid. But if you try to build a little local “waterberg”, it quickly subsides back into the sea again, conforming to an equipotential surface. Hydrostatic balance in a gravitational field also implies that surfaces of con- A triangular “waterberg” in the sea.
    [Show full text]
  • The Shape of a Liquid Surface in a Uniformly Rotating Cylinder in the Presence of Surface Tension
    Acta Mech DOI 10.1007/s00707-013-0813-6 Vlado A. Lubarda The shape of a liquid surface in a uniformly rotating cylinder in the presence of surface tension Received: 25 September 2012 / Revised: 24 December 2012 © Springer-Verlag Wien 2013 Abstract The free-surface shape of a liquid in a uniformly rotating cylinder in the presence of surface tension is determined before and after the onset of dewetting at the bottom of the cylinder. Two scenarios of liquid withdrawal from the bottom are considered, with and without deposition of thin film behind the liquid. The governing non-linear differential equations for the axisymmetric liquid shapes are solved numerically by an iterative procedure similar to that used to determine the equilibrium shape of a liquid drop deposited on a solid substrate. The numerical results presented are for cylinders with comparable radii to the capillary length of liquid in the gravitational or reduced gravitational fields. The capillary effects are particularly pronounced for hydrophobic surfaces, which oppose the rotation-induced lifting of the liquid and intensify dewetting at the bottom surface of the cylinder. The free-surface shape is then analyzed under zero gravity conditions. A closed-form solution is obtained in the rotation range before the onset of dewetting, while an iterative scheme is applied to determine the liquid shape after the onset of dewetting. A variety of shapes, corresponding to different contact angles and speeds of rotation, are calculated and discussed. 1 Introduction The mechanics of rotating fluids is an important part of the analysis of numerous scientific and engineering problems.
    [Show full text]
  • Artificial Viscosity Model to Mitigate Numerical Artefacts at Fluid Interfaces with Surface Tension
    Computers and Fluids 143 (2017) 59–72 Contents lists available at ScienceDirect Computers and Fluids journal homepage: www.elsevier.com/locate/compfluid Artificial viscosity model to mitigate numerical artefacts at fluid interfaces with surface tension ∗ Fabian Denner a, , Fabien Evrard a, Ricardo Serfaty b, Berend G.M. van Wachem a a Thermofluids Division, Department of Mechanical Engineering, Imperial College London, Exhibition Road, London, SW7 2AZ, United Kingdom b PetroBras, CENPES, Cidade Universitária, Avenida 1, Quadra 7, Sala 2118, Ilha do Fundão, Rio de Janeiro, Brazil a r t i c l e i n f o a b s t r a c t Article history: The numerical onset of parasitic and spurious artefacts in the vicinity of fluid interfaces with surface ten- Received 21 April 2016 sion is an important and well-recognised problem with respect to the accuracy and numerical stability of Revised 21 September 2016 interfacial flow simulations. Issues of particular interest are spurious capillary waves, which are spatially Accepted 14 November 2016 underresolved by the computational mesh yet impose very restrictive time-step requirements, as well as Available online 15 November 2016 parasitic currents, typically the result of a numerically unbalanced curvature evaluation. We present an Keywords: artificial viscosity model to mitigate numerical artefacts at surface-tension-dominated interfaces without Capillary waves adversely affecting the accuracy of the physical solution. The proposed methodology computes an addi- Parasitic currents tional interfacial shear stress term, including an interface viscosity, based on the local flow data and fluid Surface tension properties that reduces the impact of numerical artefacts and dissipates underresolved small scale inter- Curvature face movements.
    [Show full text]
  • Energy Dissipation Due to Viscosity During Deformation of a Capillary Surface Subject to Contact Angle Hysteresis
    Physica B 435 (2014) 28–30 Contents lists available at ScienceDirect Physica B journal homepage: www.elsevier.com/locate/physb Energy dissipation due to viscosity during deformation of a capillary surface subject to contact angle hysteresis Bhagya Athukorallage n, Ram Iyer Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409, USA article info abstract Available online 25 October 2013 A capillary surface is the boundary between two immiscible fluids. When the two fluids are in contact Keywords: with a solid surface, there is a contact line. The physical phenomena that cause dissipation of energy Capillary surfaces during a motion of the contact line are hysteresis in the contact angle dynamics, and viscosity of the Contact angle hysteresis fluids involved. Viscous dissipation In this paper, we consider a simplified problem where a liquid and a gas are bounded between two Calculus of variations parallel plane surfaces with a capillary surface between the liquid–gas interface. The liquid–plane Two-point boundary value problem interface is considered to be non-ideal, which implies that the contact angle of the capillary surface at – Navier Stokes equation the interface is set-valued, and change in the contact angle exhibits hysteresis. We analyze a two-point boundary value problem for the fluid flow described by the Navier–Stokes and continuity equations, wherein a capillary surface with one contact angle is deformed to another with a different contact angle. The main contribution of this paper is that we show the existence of non-unique classical solutions to this problem, and numerically compute the dissipation.
    [Show full text]
  • Arxiv:1808.01401V1 [Math.NA] 4 Aug 2018 Failure in Manufacturing Microelectromechanical Systems Devices, Is Caused by an Interfacial Tension [53]
    A CONTINUATION METHOD FOR COMPUTING CONSTANT MEAN CURVATURE SURFACES WITH BOUNDARY N. D. BRUBAKER∗ Abstract. Defined mathematically as critical points of surface area subject to a volume constraint, con- stant mean curvatures (CMC) surfaces are idealizations of interfaces occurring between two immiscible fluids. Their behavior elucidates phenomena seen in many microscale systems of applied science and engineering; however, explicitly computing the shapes of CMC surfaces is often impossible, especially when the boundary of the interface is fixed and parameters, such as the volume enclosed by the surface, vary. In this work, we propose a novel method for computing discrete versions of CMC surfaces based on solving a quasilinear, elliptic partial differential equation that is derived from writing the unknown surface as a normal graph over another known CMC surface. The partial differential equation is then solved using an arc-length continuation algorithm, and the resulting algorithm produces a continuous family of CMC surfaces for varying volume whose physical stability is known. In addition to providing details of the algorithm, various test examples are presented to highlight the efficacy, accuracy and robustness of the proposed approach. Keywords. constant mean curvature, interface, capillary surface, symmetry-breaking bifurcation, arc- length continuation AMS subject classifications. 49K20, 53A05, 53A10, 76B45 1. Introduction Determining the behavior of an interface between nonmixing phases is crucial for under- standing the onset of phenomena in many microscale systems. For example, interfaces induce capillary action in tubules [23], produce beading in microfluidics [24], and change the wetting properties of patterned substrates [34]. Additionally, stiction, the leading cause of arXiv:1808.01401v1 [math.NA] 4 Aug 2018 failure in manufacturing microelectromechanical systems devices, is caused by an interfacial tension [53].
    [Show full text]
  • Statics and Dynamics of Capillary Bridges
    Colloids and Surfaces A: Physicochem. Eng. Aspects 460 (2014) 18–27 Contents lists available at ScienceDirect Colloids and Surfaces A: Physicochemical and Engineering Aspects j ournal homepage: www.elsevier.com/locate/colsurfa Statics and dynamics of capillary bridges a,∗ b Plamen V. Petkov , Boryan P. Radoev a Sofia University “St. Kliment Ohridski”, Faculty of Chemistry and Pharmacy, Department of Physical Chemistry, 1 James Bourchier Boulevard, 1164 Sofia, Bulgaria b Sofia University “St. Kliment Ohridski”, Faculty of Chemistry and Pharmacy, Department of Chemical Engineering, 1 James Bourchier Boulevard, 1164 Sofia, Bulgaria h i g h l i g h t s g r a p h i c a l a b s t r a c t • The study pertains both static and dynamic CB. • The analysis of static CB emphasis on the ‘definition domain’. • Capillary attraction velocity of CB flattening (thinning) is measured. • The thinning is governed by capillary and viscous forces. a r t i c l e i n f o a b s t r a c t Article history: The present theoretical and experimental investigations concern static and dynamic properties of cap- Received 16 January 2014 illary bridges (CB) without gravity deformations. Central to their theoretical treatment is the capillary Received in revised form 6 March 2014 bridge definition domain, i.e. the determination of the permitted limits of the bridge parameters. Concave Accepted 10 March 2014 and convex bridges exhibit significant differences in these limits. The numerical calculations, presented Available online 22 March 2014 as isogones (lines connecting points, characterizing constant contact angle) reveal some unexpected features in the behavior of the bridges.
    [Show full text]
  • Georgia Institute of Technology in Partial Fulfillment Of
    M INVESTIGATION OF CAPILLARY SPREADING POWER OF EISJLSI01IS J A THESIS Presentad to the Faculty of the Division of Graduate Student: Georgia Institute of Technology In Partial Fulfillment of the Requirements for the Degree Master of Science in Chemical Engineering fey Boiling Gay Braviley September 19/49 0752fl rt 9 AN INVESTIGATION OF CAPILLARY SPREADING POWER OP EMULSIONS Approved: ^i S'-"—y<c\-— /. ./ J Date Approved by Chairman &, \j^e^^cZt^u^ <C~*Si. ^g*i #q t AC XBQRLgDGSWTS The author wishes to express his appreciation to the personnel of Southern Sizing Company for making this study possible. To Dr. J» M« DallaValle, my advisor, I am most deeply indebted for his suggestion of the study and his valuable guidance and interest in the problem. SABLE OF CONTENTS CHAPTER PAGE I. INTRODUCTION AMD SCOPE OF THE PROBLEM 1 II. REVIEW OF THE LITERATURE 3 Capillary Flow 3 Surface Tension 5 Wotting, Spreading, and Penetration 6 III. EQUIPMT 7 Selection 7 Description and Operation 8 IV. SELECTION, STABILITY, AND PHYSICAL PROPERTIES OF EMULSIONS . , 11 V. CAPILLARY SPREADING POWER 19 Development of Equations* •••••••• • 19 Effect of Temperature •• ••••• 26 Effect of Emulsion Concentration • 26 VI. SUT«RY £SB CONCLUSIONS 29 Summary 29 Conclusions •••••••••.••••••••••••• 30 BIBLIOGRAPHY 3J APPMDIX „ 36 * LIST OF TABLES TABLES PAGE I. Physical Properties of Test Emulsion 13 II. Time of Capillary Plow vs. Viscosity-Surface Tension Ratio •••• • • 21 III. Time of Capillary Plow vs. Concentration and Temperature #••.••••»••••••••••••*• ho IV. Variation of Physical Properties of Test Emulsion with Temperature ••••• •• hi V. Variation of Physical Properties of Test Emulsion with Concentration •••••••••••• •• i\Z ' LIST OP FIGURES FIGURES PAGE 1.
    [Show full text]
  • Sharp-Interface Limit for the Navier–Stokes–Korteweg Equations
    Sharp-Interface Limit for the Navier–Stokes–Korteweg Equations Dissertation zur Erlangung des Doktorgrades vorgelegt von Johannes Daube an der Fakult¨at f¨ur Mathematik und Physik der Albert-Ludwigs-Universit¨at Freiburg Februar 2017 Dekan: Prof. Dr. Gregor Herten 1. Gutachter: Prof. Dr. Dietmar Kr¨oner 2. Gutachter: Prof. Dr. Helmut Abels Datum der m¨undlichen Pr¨ufung: 09.11.2016 Contents Abstract v Acknowledgements vii List of Symbols ix 1 Introduction 1 1.1 PhaseTransitions................................ 1 1.2 CapillaryEffects ................................. 2 1.3 Sharp- and Diffuse-Interface Models and the Sharp-Interface Limit . 3 1.4 The Navier–Stokes–Korteweg Model . .... 4 1.5 TheStaticCase.................................. 10 1.6 ExistingResults ................................. 13 1.7 NewContributions ................................ 16 1.8 Outline ...................................... 16 2 Mathematical Background 19 2.1 Notation...................................... 19 2.2 Measures ..................................... 26 2.3 Functions of Bounded Variation . 30 3 The Diffuse-Interface Model 35 3.1 The Double-Well Potential . 35 3.2 TheNotionofWeakSolutions. 37 3.3 APrioriEstimates ................................ 43 3.4 CompactnessofWeakSolutions. 56 3.5 LimitingInterfaces .............................. 60 3.6 Remarks...................................... 63 4 The Sharp-Interface Model 67 4.1 Two-Phase Incompressible Navier–Stokes Equations with Surface Tension . 67 4.2 Hypersurfaces................................... 70
    [Show full text]
  • A Capillary Surface with No Radial Limits
    A CAPILLARY SURFACE WITH NO RADIAL LIMITS A Dissertation by Colm Patric Mitchell Master of Science, Wichita State University, 2009 Bachelor of Science, Wichita State University, 2005 Submitted to the Department of Mathematics and the faculty of the Graduate School of Wichita State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy May 2017 ⃝c Copyright 2017 by Colm Patric Mitchell All Rights Reserved A CAPILLARY SURFACE WITH NO RADIAL LIMITS The following faculty members have examined the final copy of this dissertation for form and content, and recommend that it be accepted in partial fulfillment of the requirement for the degree of Doctor of Philosophy with a major in Applied Mathematics. Thomas DeLillo, Committee Chair James Steck, Committee Member Elizabeth Behrman, Committee Member Ziqi Sun, Committee Member Jason Ferguson, Committee Member Accepted for the College of Liberal Arts and Sciences Ron Matson, Dean Accepted for the Graduate School Dennis Livesay, Dean iii DEDICATION To my wife Jenny, without her gentle shove I may never have gone to college; and to my boys Caleb and Connor, who had many years of dad time taken up in my studies. iv ACKNOWLEDGEMENTS I wish to acknowledge the tireless efforts of Dr. Kirk Lancaster over the last several years. Without his guidance I may never have finished my dissertation. It is unfortunate that he could not be present to chair the committee at the defense. I wish to thank Dr. Thomas DeLillo for stepping in for Kirk to chair the committee at the last minute. v ABSTRACT We begin by discussing the circumstances of capillary surfaces in regions with a corner, both concave and convex.
    [Show full text]
  • Equilibria of Pendant Droplets: Spatial Variation and Anisotropy of Surface Tension
    Equilibria of Pendant Droplets: Spatial Variation and Anisotropy of Surface Tension by Dale G. Karr Department of Naval Architecture and Marine Engineering University of Michigan Ann Arbor, MI 48109-2145 USA Email: [email protected] 1 Abstract An example of capillary phenomena commonly seen and often studied is a droplet of water hanging in air from a horizontal surface. A thin capillary surface interface between the liquid and gas develops tangential surface tension, which provides a balance of the internal and external pressures. The Young-Laplace equation has been historically used to establish the equilibrium geometry of the droplet, relating the pressure difference across the surface to the mean curvature of the surface and the surface tension, which is presumed constant and isotropic. The surface energy per unit area is often referred to as simply surface energy and is commonly considered equal to the surface tension. The relation between the surface energy and the surface tension can be established for axisymmetric droplets in a gravitational field by the application of the calculus of variations, minimizing the total potential energy. Here it is shown analytically and experimentally that, for conditions of constant volume of the droplet, equilibrium states exist with surface tensions less than the surface energy of the water-air interface. The surface tensions of the interface membrane vary with position and are anisotropic. Keywords: capillary forces, droplets, surface energy, surface tension 2 Background For droplets of water in air with negligible inertia forces, the water mass is influenced by gravity and surface tension. These forces dominate at length scales of approximately 2mm, the characteristic capillary length (1-3).
    [Show full text]
  • Viscous Fluids 9
    Viscous Fluids 9 In the previous chapter on fluids, we introduced the basic ideas of pressure, fluid flow, the application of conservation of mass and of energy in the form of the continuity equa- tion and of Bernoulli’s equation, respectively, as well as hydrostatics. Throughout those discussions we restricted ourselves to ideal fluids, those that do not exhibit any frictional properties. Often these can be neglected and the results of the previous chapter applied without any modifications whatsoever. Clearly mass is conserved even in the presence of viscous frictional forces and so the continuity equation is a very general result. Real fluids, however, do not conserve mechanical energy, but over time will lose some of this well-ordered energy to heat through frictional losses. In this chapter we consider such behavior, known as viscosity, first in the case of simple fluids such as water. We study the effects of viscosity on the motion of simple fluids and on the motion of suspended bodies, such as macromolecules, in these fluids, with special attention to flow in a cylinder, the most important geometry of flow in biology. The complex nature of blood as a fluid is studied next leading into a description and physics perspective of the human circulatory system. We conclude the chapter with a discussion of surface tension and capillarity, two important surface phenomena in fluids. In Chapter 13 we return to the general notion of the loss of well-ordered energy to heat in the context of thermodynamics. 1. VISCOSITY OF SIMPLE FLUIDS Real fluids are viscous, having internal attractive forces between the molecules so that any relative motion of molecules results in frictional, or drag, forces.
    [Show full text]
  • A Numerical Study for Liquid Bridge Based Microgripping And
    © 2007 SANTANU CHANDRA ALL RIGHTS RESERVED A NUMERICAL STUDY FOR LIQUID BRIDGE BASED MICROGRIPPING AND CONTACT ANGLE MANIPULATION BY ELECTROWETTING METHOD A Dissertation Presented to The Graduate Faculty of the University of Akron In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy Santanu Chandra December, 2007 A NUMERICAL STUDY FOR LIQUID BRIDGE BASED MICROGRIPPING AND CONTACT ANGLE MANIPULATION BY ELECTROWETTING METHOD Santanu Chandra Dissertation Approved: Accepted: _________________________________ _________________________________ Advisor Department Chair Dr. Celal Batur Dr. Celal Batur _________________________________ _________________________________ Committee Member Dean of the College of Engineering Dr. Minel Braun Dr. George K. Haritos _________________________________ _________________________________ Committee Member Dean of the Graduate School Dr. Jiang “John” Zhe Dr. George R. Newkome _________________________________ _________________________________ Committee Member Date Dr. George G. Chase _________________________________ Committee Member Dr. Gerald Young ii ABSTRACT The last decade had witnessed some very outstanding research on Micro Electro Mechanical Systems (MEMS) that has vast impact on future technologies. However building a complete microsystem requires proper microassembly methods. But microassembly research is facing stiff resistance due to the presence of many dominant forces which appears due to the scaling law. Overcoming these forces has been found to be a major drawback in microgripper research. So the primary the challenge today researches face is the lack of proper manipulation schemes. Understanding the physical forces associated with micro-scale as well as devising techniques to control them is needed in order to design a proper micromanipulation scheme. The motivation of this study is to contribute to the microassembly research by studying the forces in microscale and propose a micromanipulation scheme to control these forces.
    [Show full text]