LECTURES IN ELEMENTARY FLUID DYNAMICS:Physics, Mathematics and ApplicationsJ. McDonoughDepartments of Mechanical Engineering and Mathematics University of Kentucky, Lexington, KY c 1987, 1990, 2002, 2004, 2009Contents1 Introduction 1.1 Importance of Fluids. 1.1.1 Fluids in the pure sciences. 1.1.2 Fluids in technology.
Fluid Dynamics: Physical ideas, the Navier-Stokes equations, and applications to lubrication flows and complex fluids Howard A. Stone Division of Engineering & Applied Sciences. Applied Physics 298r A fluid dynamics tour 17 5 April 2004 Elementary Ideas VIII •Newton’s second law. Fluid Dynamics: Physical ideas, the Navier-Stokes equations,. Applied Physics 298r A fluid dynamics tour 17 5 April 2004 Elementary Ideas VIII.
1.2 The Study of Fluids. 1.2.1 The theoretical approach. 1.2.2 Experimental uid dynamics. 1.2.3 Computational uid dynamics 1.3 Overview of Course.
1 1 2 3 4 5 6 6 8 11 11 13 13 15 15 15 23 23 24 29 29 29 31 32 36 37 37 40 41 41 43 44 45.2 Some Background: Basic Physics of Fluids 2.1 The Continuum Hypothesis. 2.2 Denition of a Fluid.
2.2.1 Shear stress induced deformations. 2.2.2 More on shear stress. 2.3 Fluid Properties.
2.3.1 Viscosity. 2.3.2 Thermal conductivity.
2.3.3 Mass diusivity. 2.3.4 Other uid properties. 2.4 Classication of Flow Phenomena. 2.4.1 Steady and unsteady ows. 2.4.2 Flow dimensionality. 2.4.3 Uniform and non-uniform ows.
2.4.4 Rotational and irrotational ows. 2.4.5 Viscous and inviscid ows. 2.4.6 Incompressible and compressible ows 2.4.7 Laminar and turbulent ows. 2.4.8 Separated and unseparated ows.
2.5 Flow Visualization. 2.5.1 Streamlines.
2.5.2 Pathlines. 2.5.3 Streaklines. 2.6 Summary. I.ii 3 The Equations of Fluid Motion 3.1 Lagrangian & Eulerian Systems; the Substantial Derivative. 3.1.1 The Lagrangian viewpoint.
3.1.2 The Eulerian viewpoint. 3.1.3 The substantial derivative. 3.2 Review of Pertinent Vector Calculus. 3.2.1 Gausss theorem. 3.2.2 Transport theorems. 3.3 Conservation of Massthe continuity equation. 3.3.1 Derivation of the continuity equation.
3.3.2 Other forms of the dierential continuity equation. 3.3.3 Simple application of the continuity equation. 3.3.4 Control volume (integral) analysis of the continuity equation 3.4 Momentum Balancethe NavierStokes Equations. 3.4.1 A basic force balance; Newtons second law of motion. 3.4.2 Treatment of surface forces. 3.4.3 The NavierStokes equations. 3.5 Analysis of the NavierStokes Equations.
3.5.1 Mathematical structure. 3.5.2 Physical interpretation. 3.6 Scaling and Dimensional Analysis. 3.6.1 Geometric and dynamic similarity.
3.6.2 Scaling the governing equations. 3.6.3 Dimensional analysis via the Buckingham theorem. 3.6.4 Physical description of important dimensionless parameters. 3.7 Summary.
4 Applications of the NavierStokes Equations 4.1 Fluid Statics. 4.1.1 Equations of uid statics. 4.1.2 Buoyancy in static uids. 4.2 Bernoullis Equation. 4.2.1 Derivation of Bernoullis equation. 4.2.2 Example applications of Bernoullis equation.
4.3 Control-Volume Momentum Equation. 4.3.1 Derivation of the control-volume momentum equation 4.3.2 Application of control-volume momentum equation. 4.4 Classical Exact Solutions to N.S. 4.4.1 Couette ow. 4.4.2 Plane Poiseuille ow. 4.5 Pipe Flow.
4.5.1 Some terminology and basic physics of pipe ow. 4.5.2 The HagenPoiseuille solution. 4.5.3 Practical Pipe Flow Analysis. 4.6 Summary.CONTENTS 47 47 48 49 49 52 53 56 58 58 60 61 61 69 69 73 78 80 80 81 83 83 85 91 98 99.101. 158List of Figures2.1 Mean Free Path and Requirements for Satisfaction of Continuum Hypothesis; (a) mean free path determined as average of distances between collisions; (b) a volume too small to permit averaging required for satisfaction of continuum hypothesis.
Comparison of deformation of solids and liquids under application of a shear stress; (a) solid, and (b) liquid. Behavior of things that ow; (a) granular sugar, and (b) coee. Flow between two horizontal, parallel plates with upper one moving at velocity U. Physical situation giving rise to the no-slip condition. Structure of water molecule and eect of heating on short-range order in liquids; (a) low temperature, (b) higher temperature.
Eects of temperature on molecular motion of gases; (a) low temperature, (b) higher temperature. Diusion of momentuminitial transient of ow between parallel plates; (a) very early transient, (b) intermediate time showing signicant diusion, (c) nearly steadystate prole. Interaction of high-speed and low-speed uid parcels. Pressure and shear stress. Surface tension in spherical water droplet.
Capillarity for two dierent liquids. Dierent types of time-dependent ows; (a) transient followed by steady state, (b) unsteady, but stationary, (c) unsteady.
Flow dimensionality; (a) 1-D ow between horizontal plates, (b) 2-D ow in a 3-D box, (c) 3-D ow in a 3-D box. Uniform and non-uniform ows; (a) uniform ow, (b) non-uniform, but locally uniform ow, (c) non-uniform ow. 2-D vortex from ow over a step. 3-D vortical ow of uid in a box. Potential Vortex. Laminar and turbulent ow of water from a faucet; (a) steady laminar, (b) periodic, wavy laminar, (c) turbulent. Da Vinci sketch depicting turbulent ow.
Reynolds experiment using water in a pipe to study transition to turbulence; (a) low-speed ow, (b) higher-speed ow. Transition to turbulence in spatially-evolving ow. (a) unseparated ow, (b) separated ow.
Geometry of streamlines. Temporal development of a pathline. 14 14 16 172.2 2.3 2.4 2.5 2.6 2.7 2.8.
212.9 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18 2.19 2.20 2.21 2.22 2.23 2.24 2.25.21 22 26 27 28. 39 39 40 42 44iv 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 4.16 4.17 4.18 4.19 4.20 4.21 4.22 4.23 4.24 4.25LIST OF FIGURES Fluid particles and trajectories in Lagrangian view of uid motion. Eulerian view of uid motion.
Steady accelerating ow in a nozzle. Integration of a vector eld over a surface. Contributions to a control surface: piston, cylinder and valves of internal combustion engine. Simple control volume corresponding to ow through an expanding pipe. Time-dependent control volume for simultaneously lling and draining a tank. Calculation of fuel ow rate for jet aircraft engine. Schematic of pressure and viscous stresses acting on a uid element.
Sources of angular deformation of face of uid element. Comparison of velocity proles in duct ow for cases of (a) high viscosity, and (b) low viscosity. Missile nose cone ogive (a) physical 3-D gure, and (b) cross section indicating linear lengths. 2-D ow in a duct.
Prototype and model airfoils demonstrating dynamic similarity requirements. Wind tunnel measurement of forces on sphere. Dimensionless force on a sphere as function of Re; plotted points are experimental data, lines are theory (laminar) and curve t (turbulent). Hydraulic jack used to lift automobile. Schematic of a simple barometer. Schematic of pressure measurement using a manometer. Application of Archimedes principle to the case of a oating object.
Stagnation point and stagnation streamline. Sketch of pitot tube. Schematic of ow in a syphon. Flow through a rapidly-expanding pipe. Couette ow velocity prole.
Plane Poiseuille ow velocity prole. Steady, fully-developed ow in a pipe of circular cross section. Steady, 2-D boundary-layer ow over a at plate.
Steady, fully-developed pipe ow. Turbulent ow near a solid boundary. Graphical depiction of components of Reynolds decomposition. Empirical turbulent pipe ow velocity proles for dierent exponents in Eq. Comparison of surface roughness height with viscous sublayer thickness for (a) low Re, and (b) high Re.
Moody diagram: friction factor vs. Reynolds number for various dimensionless surface roughnesses. Time series of velocity component undergoing transitional ow behavior.
Simple piping system containing a pump. Flow through sharp-edged inlet. Flow in contracting pipe. Flow in expanding pipe. Flow in pipe with 90 bend. Liquid propellant rocket engine piping system. 48 49 51 54 63 64 66 67 74 76.
103 104 105 109 112 113 114 119 123 124 127 127 130 135 136 137. 139 140 145 149 150 151 152 153Chapter 1IntroductionIt takes little more than a brief look around for us to recognize that uid dynamics is one of the most important of all areas of physicslife as we know it would not exist without uids, and without the behavior that uids exhibit.
The air we breathe and the water we drink (and which makes up most of our body mass) are uids. Motion of air keeps us comfortable in a warm room, and air provides the oxygen we need to sustain life. Similarly, most of our (liquid) body uids are water based. And proper motion of these uids within our bodies, even down to the cellular level, is essential to good health. It is clear that uids are completely necessary for the support of carbon-based life forms. But the study of biological systems is only one (and a very recent one) possible application of a knowledge of uid dynamics.
Fluids occur, and often dominate physical phenomena, on all macroscopic (non-quantum) length scales of the known universefrom the megaparsecs of galactic structure down to the micro and even nanoscales of biological cell activity. In a more practical setting, we easily see that uids greatly inuence our comfort (or lack thereof); they are involved in our transportation systems in many ways; they have an eect on our recreation (e.g., basketballs and footballs are inated with air) and entertainment (the sound from the speakers of a TV would not reach our ears in the absence of air), and even on our sleep (water beds!). From this it is fairly easy to see that engineers must have at least a working knowledge of uid behavior to accurately analyze many, if not most, of the systems they will encounter. It is the goal of these lecture notes to help students in this process of gaining an understanding of, and an appreciation for, uid motionwhat can be done with it, what it might do to you, how to analyze and predict it. In this introductory chapter we will begin by further stressing the importance of uid dynamics by providing specic examples from both the pure sciences and from technology in which knowledge of this eld is essential to an understanding of the physical phenomena (and, hence, the beginnings of a predictive capabilitye.g., the weather) and/or the ability to design and control devices such as internal combustion engines.
We then describe three main approaches to the study of uid dynamics: i) theoretical, ii) experimental and iii) computational; and we note (and justify) that of these theory will be emphasized in the present lectures.1.1Importance of FluidsWe have already emphasized the overall importance of uids in a general way, and here we will augment this with a number of specic examples. We somewhat arbitrarily classify these in two main categories: i) physical and natural science, and ii) technology. Clearly, the second of these is often of more interest to an engineering student, but in the modern era of emphasis on interdis12CHAPTER 1.
INTRODUCTIONciplinary studies, the more scientic and mathematical aspects of uid phenomena are becoming increasingly important.1.1.1Fluids in the pure sciencesThe following list, which is by no means all inclusive, provides some examples of uid phenomena often studied by physicists, astronomers, biologists and others who do not necessarily deal in the design and analysis of devices. The accompanying gures highlight some of these areas.
Atmospheric sciences (a) global circulation: long-range weather prediction; analysis of climate change (global warming) (b) mesoscale weather patterns: short-range weather prediction; tornado and hurricane warnings; pollutant transport 2. Oceanography (a) ocean circulation patterns: causes of El Nio, eects of ocean currents on weather n and climate (b) eects of pollution on living organisms 3. Geophysics (a) convection (thermally-driven uid motion) in the Earths mantle: understanding of plate tectonics, earthquakes, volcanoes (b) convection in Earths molten core: production of the magnetic eld 4.
Astrophysics (a) galactic structure and clustering (b) stellar evolutionfrom formation by gravitational collapse to death as a supernovae, from which the basic elements are distributed throughout the universe, all via uid motion 5. Biological sciences (a) circulatory and respiratory systems in animals (b) cellular processes1.1. IMPORTANCE OF FLUIDS31.1.2Fluids in technologyAs in the previous case, we do not intend this list of technologically important applications of uid dynamics to be exhaustive, but mainly to be representative. It is easily recognized that a complete listing of uid applications would be nearly impossible simply because the presence of uids in technological devices is ubiquitous. The following provide some particularly interesting and important examples from an engineering standpoint. Internal combustion enginesall types of transportation systems 2. Turbojet, scramjet, rocket enginesaerospace propulsion systems 3.
Waste disposal (a) chemical treatment (b) incineration (c) sewage transport and treatment 4. Pollution dispersalin the atmosphere (smog); in rivers and oceans 5. Steam, gas and wind turbines, and hydroelectric facilities for electric power generation 6. Pipelines (a) crude oil and natural gas transferral (b) irrigation facilities (c) oce building and household plumbing 7. Fluid/structure interaction (a) design of tall buildings (b) continental shelf oil-drilling rigs (c) dams, bridges, etc. (d) aircraft and launch vehicle airframes and control systems 8.
Heating, ventilating and air-conditioning (HVAC) systems 9. Cooling systems for high-density electronic devicesdigital computers from PCs to supercomputers 10. Solar heat and geothermal heat utilization 11.
Articial hearts, kidney dialysis machines, insulin pumps4 12. Manufacturing processes (a) spray painting automobiles, trucks, etc.CHAPTER 1. INTRODUCTION(b) lling of containers, e.g., cans of soup, cartons of milk, plastic bottles of soda (c) operation of various hydraulic devices (d) chemical vapor deposition, drawing of synthetic bers, wires, rods, etc. We conclude from the various preceding examples that there is essentially no part of our daily lives that is not inuenced by uids.
As a consequence, it is extremely important that engineers be capable of predicting uid motion. In particular, the majority of engineers who are not uid dynamicists still will need to interact, on a technical basis, with those who are quite frequently; and a basic competence in uid dynamics will make such interactions more productive.1.2The Study of FluidsWe begin by introducing the intuitive notion of what constitutes a uid.
As already indicated we are accustomed to being surrounded by uidsboth gases and liquids are uidsand we deal with these in numerous forms on a daily basis. As a consequence, we have a fairly good intuition regarding what is, and is not, a uid; in short we would probably simply say that a uid is anything that ows. This is actually a good practical view to take, most of the time. But we will later see that it leaves out some things that are uids, and includes things that are not. So if we are to accurately analyze the behavior of uids it will be necessary to have a more precise denition. This will be provided in the next chapter.
It is interesting to note that the formal study of uids began at least 500 hundred years ago with the work of Leonardo da Vinci, but obviously a basic practical understanding of the behavior of uids was available much earlier, at least by the time of the ancient Egyptians; in fact, the homes of well-to-do Romans had ushing toilets not very dierent from those in modern 21st -Century houses, and the Roman aquaducts are still considered a tremendous engineering feat. Thus, already by the time of the Roman Empire enough practical information had been accumulated to permit quite sophisticated applications of uid dynamics. The more modern understanding of uid motion began several centuries ago with the work of L. Euler and the Bernoullis (father and son), and the equation we know as Bernoullis equation (although this equation was probably deduced by someone other than a Bernoulli). The equations we will derive and study in these lectures were introduced by Navier in the 1820s, and the complete system of equations representing essentially all uid motions were given by Stokes in the 1840s. These are now known as the NavierStokes equations, and they are of crucial importance in uid dynamics.
For most of the 19th and 20th Centuries there were two approaches to the study of uid motion: theoretical and experimental. Many contributions to our understanding of uid behavior were made through the years by both of these methods. But today, because of the power of modern digital computers, there is yet a third way to study uid dynamics: computational uid dynamics, or CFD for short. In modern industrial practice CFD is used more for uid ow analyses than either theory or experiment. Most of what can be done theoretically has already been done, and experiments are generally dicult and expensive. As computing costs have continued to decrease, CFD has moved to the forefront in engineering analysis of uid ow, and any student planning to work in the thermal-uid sciences in an industrial setting must have an understanding of the basic practices of CFD if he/she is to be successful. But it is also important to understand that in order to do CFD one must have a fundamental understanding of uid ow itself, from both the theoretical,1.2.
THE STUDY OF FLUIDS5mathematical side and from the practical, sometimes experimental, side. We will provide a brief introduction to each of these ways of studying uid dynamics in the following subsections.1.2.1The theoretical approachTheoretical/analytical studies of uid dynamics generally require considerable simplications of the equations of uid motion mentioned above. We present these equations here as a prelude to topics we will consider in detail as the course proceeds. The version we give is somewhat simplied, but it is sucient for our present purposes.
U =0 and (conservation of mass)DU 1 2 = P + U (balance of momentum). Dt Re These are the NavierStokes (N.S.) equations of incompressible uid ow.
In these equations all quantities are dimensionless, as we will discuss in detail later: U (u, v, w)T is the velocity vector; P is pressure divided by (assumed constant) density, and Re is a dimensionless parameter known as the Reynolds number. We will later see that this is one of the most important parameters in all of uid dynamics; indeed, considerable qualitative information about a ow eld can often be deduced simply by knowing its value.
In particular, one of the main eorts in theoretical analysis of uid ow has always been to learn 1.0 to predict changes in the qualitative nature of a ow 0.8 as Re is increased. In general, this is a very di0.6 cult task far beyond the intended purpose of these 0.4 lectures. But we mention it here to emphasize the 0.2 (a) importance of prociency in applied mathematics in 1.0 theoretical studies of uid ow. From a physical point 0.8 of view, with geometry of the ow situation xed, a 0.6 ow eld generally becomes more complicated as 0.4 Re increases. This is indicated by the accompanying 0.2 time series of a velocity component for three dierent (b) values of Re.
In part (a) of the gure Re is low, and 1.0 the ow ultimately becomes time independent. As 0.8 the Reynolds number is increased to an intermediate 0.6 value, the corresponding time series shown in part 0.4 (b) of the gure is considerably complicated, but still 0.2 with evidence of somewhat regular behavior. Finally, (c) in part (c) is displayed the high-Re case in which the 0.03 0.05 0.07 behavior appears to be random. We comment in passScaled Time (Arbitrary units) ing that it is now known that this behavior is not random, but more appropriately termed chaotic. We also point out that the N.S. Equations are widely studied by mathematicians, and they are said to have been one of two main progenitors of 20th -Century mathematical analysis. (The other was the Schrdinger equation of quantum mechanics.) In the current era it is hoped that such o mathematical analyses will shed some light on the problem of turbulent uid ow, often termed the last unsolved problem of classical mathematical physics.
We will from time to time discuss turbulence in these lectures because most uid ows are turbulent, and some understanding of it1 0.8 0.6 0.4 0.2u000.0020.0040.0060.0080.010.0121Dimensionless Velocity,0.80.0214.80.0214.04514.056CHAPTER 1. INTRODUCTIONis essential for engineering analyses. But we will not attempt a rigorous treatment of this topic. Furthermore, it would not be be possible to employ the level of mathematics used by research mathematicians in their studies of the N.S.
This is generally too dicult, even for graduate students.1.2.2Experimental uid dynamicsIn a sense, experimental studies in uid dynamics must be viewed as beginning when our earliest ancestors began learning to swim, to use logs for transportation on rivers and later to develop a myriad assortment of containers, vessels, pottery, etc., for storing liquids and later pouring and using them. Rather obviously, uid experiments performed today in rstclass uids laboratories are far more sophisticated. Nevertheless, until only very recently the outcome of most uids experiments was mainly a qualitative (and not quantitative) understanding of uid motion.
An indication of this is provided by the adjacent pictures of wind tunnel experiments. In each of these we are able to discern quite detailed qualitative aspects of the ow over dierent prolate spheroids.
Basic ow patterns are evident from colored streaks, even to the point of indications of ow separation and transition to turbulence. However, such diagnostics provide no information on actual ow velocity or pressurethe main quantities appearing in the theoretical equations, and needed for engineering analyses. There have long been methods for measuring pressure in a ow eld, and these could be used simultaneously with the ow visualization of the above gures to gain some quantitative data. On the other hand, it has been possible to accurately measure ow velocity simultaneously over large areas of a ow eld only recently.
If point measurements are sucient, then hot-wire anemometry (HWA) or laser-doppler velocimetry (LDV) can be used; but for eld measurements it is necessary to employ some form of particle image velocimetry (PIV). The following gure shows an example of such a measurement for uid between two co-axial cylinders with the inner one rotating. This corresponds to a two-dimensional slice through a long row of toroidally-shaped (donutlike) ow structures going into and coming out of the plane of the page, i.e., wrapping around the circumference of the inner cylinder.
The arrows indicate ow direction in the plane; the red asterisks show the center of the vortex, and the white pluses are locations at which detailed time series of ow velocity also have been recorded. It is clear that this quantitative detail is far superior to the simple visualizations shown in the previous gures, and as a consequence PIV is rapidly becoming the preferred diagnostic in many ow situations.1.2.3Computational uid dynamicsWe have already noted that CFD is rapidly becoming the dominant ow analysis technique, especially in industrial environments.
The reader need only enter CFD in the search tool of any web browser to discover its prevalence. CFD codes are available from many commercial vendors and as freeware from government laboratories, and many of these codes can be implemented on anything from a PC (often, even a laptop) to modern parallel supercomputers.
In fact, it is not1.2. THE STUDY OF FLUIDSrotating inner cylinder7fluiddicult to nd CFD codes that can be run over the internet from any typical browser. Here we display a few results produced by such codes to indicate the wide range of problems to which CFD has already been applied, and we will briey describe some of the potential future areas for its use.
The gure in the lower left-hand corner provides a direct comparison with experimental results shown in an earlier gure. The computed ow patterns are very similar to those of the experiment, but in contrast to the experimental data the calculation provides not only visualization of qualitative ow features but also detailed quantitative output for all velocity component values and pressure, typically at on the order of 105 to 106 locations in the ow eld. The upper left-hand gure displays predictions of the instantaneous ow eld in the left ventricle of the human heart. Use ofCp1.001.008CHAPTER 1. INTRODUCTIONCFD in biomedical and bioengineering areas is still in its infancy, but there is little doubt that it will ultimately dominate all other analysis techniques in these areas because of its generality and exibility. The center gure depicts the pressure eld over the entire surface of an airliner (probably a Boeing 757) as obtained using CFD. It was the need to make such predictions for aircraft design that led to the development of CFD, initially in the U.
Aerospace industry and NASA laboratories, and CFD was the driving force behind the development of supercomputers. Calculations of the type shown here are routine today, but as recently as a decade ago they would have required months of CPU time.
The upper right-hand gure shows the temperature eld and a portion (close to the fan) of the velocity eld in a (not-so-modern) PC. This is a very important application of CFD simply because of the large number of PCs produced and sold every year worldwide. The basic design tradeo is the following.
For a given PC model it is necessary to employ a fan that can produce sucient air ow to cool the computer by forced convection, maintaining temperatures within the operating limits of the various electronic devices throughout the PC. But eectiveness of forced convection cooling is strongly inuenced by details of shape and arrangement of circuit boards, disk drives, etc.
Moreover, power input to the fan(s), number of fans and their locations all are important design parameters that inuence, among other things, the unwanted noise produced by the PC. Finally, the lower right-hand gure shows pressure distribution and qualitative nature of the velocity eld for ow over a race car, as computed using CFD. In recent years CFD has played an ever-increasing role in many areas of sports and athleticsfrom study and design of Olympic swimware to the design of a new type of golf ball providing signicantly longer ight times, and thus driving distance (and currently banned by the PGA). The example of a race car also reects current heavy use of CFD in numerous areas of automobile production ranging from the design of modern internal combustion engines exhibiting improved eciency and reduced emissions to various aspects of the manufacturing process, per se, including, for example, spray painting of the completed vehicles. It is essential to recognize that a CFD computer code solves the NavierStokes equations, given earlier, and this is not a trivial undertakingoften even for seemingly easy physical problems.
A mathematical model based on continuity and Navier-Stokes equations, considering laminar flow in the gap between the disks, is presented to estimate the drag torque in open multidisks wet clutches. By taking into account the effects of Poiseuille and centrifugal forces, the flow pressure and velocity fields are investigated. The model quantifies the volume fraction of fluids and predicts the evolution of film shape. The drag torque estimated by the model is the sum of drag torque due to shearing of automatic transmission fluid (ATF) and the mist (suspension of ATF in air) film. In order to validate the model, experiments are performed on SAE# 2 test-setup under actual operating conditions of clutches. The model is capable of predicting the drag torque under conditions of variable flow rate and different disks rotational state for higher clutch speed range. Copyright in the material you requested is held by the American Society of Mechanical Engineers (unless otherwise noted).
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