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ME3560 – Fluid Mechanics Summer 2016 ME 3560 Fluid Mechanics Chapter I. Introduction • Fluid Mechanics is the science that deals with the behavior of fluids at rest or in motion, and the interaction of fluids with solids or other fluids at the boundaries. 1 Chapter I. Introduction Summer 2016 ME3560 – Fluid Mechanics 1.1 Brief History of Fluid Mechanics • One of the first engineering problems was the supply of water to cities for domestic use and for the irrigation of crops. •Roman aqueducts are a good example of water systems constructed at the beginning of civilization. •Roman aqueducts in Segovia, Spain, built around the 1st Century A.D. 2 Chapter I. Introduction ME3560 – Fluid Mechanics Summer 2016 • ►It has been found that from 283 to 133 BC the Hellenistic city of Pergamon (Turkey) built a series of pressurized led and clay pipelines, up to 45 km long that operated at a pressure exceeding 1.7 Mpa (180 m of head, 246.6 psi). • ► The earliest contribution theory to fluid mechanics was made by Archimedes (285–212 BC). He formulated and applied the buoyancy principle. •During the Middle Ages the application of fluid machinery expanded, piston pumps were used for dewatering mines, water and wind mills were perfected to grind grains and forge metal. 3 Chapter I. Introduction ME3560 – Fluid Mechanics Summer 2016 •The figure shows a mine hoist powered by a reversible water wheel (Georgius Agricola 1556). •The development of fluid systems and machines continued during the Renaissance. The scientific method was perfected and adopted throughout Europe. • Simon Stevin (1548–1617), Galileo Galilei (1562–1642), Edme Mariotte (1620–1684), and Evangelista Torricelli (1608–1647) were among the first to apply the scientific method to investigate hydrostatic pressure distributions and vacuums. •Blaise Pascal integrated and refined the work developed by these scientists. 4 Chapter I. Introduction ME3560 – Fluid Mechanics Summer 2016 • The Italian monk Benedetto Castelli (1577–1644) was the first person to publish a statement of the continuity principle for fluids. • ► Sir Isaac Newton (1643–1727) applied his laws to fluids and explored fluid inertia and resistance, free jets, and viscosity. • ► The Swiss engineer Daniel Bernoulli (1700–1782) and his associate Leonard Euler (1707–1789) built upon Newton’s studies and defined the energy and momentum equations. •Bernoulli published in 1738 his treatise Hydrodynamica. This may be considered the first fluid mechanics text. •Jean d’Alembert (1717–1789) developed the idea of velocity and acceleration components, a differential expression of continuity , and his “paradox” of zero resistance to steady uniform motion. 5 Chapter I. Introduction ME3560 – Fluid Mechanics Summer 2016 • ► Fluid mechanics theory up through the end of the 18th century had little impact on engineering since fluid properties and parameters were poorly quantified. •The French school of engineering led by Riche de Prony (1755–1839) along with Ecole Polytechnic and the Ecole Ponts et Chaussees were the first to incorporate calculus and scientific theory to the engineering curriculum. This brought a change to the engineering theory by making it more practical and capable of solving real world problems. •Scientists such as Antonie Chezy (1718–1798), Louis Navier (1785– 1836), Gaspar Coriolis (1792–1843), Henry Darcy (1803–1858) contributed to fluid engineering and theory and were students and/or instructors at these schools 6 Chapter I. Introduction ME3560 – Fluid Mechanics Summer 2016 • ► By the mid 19th century the advances in fluid mechanics were coming from different fronts: •Jean Poiseuille (1799–1869) had accurately measured flow in capillary tubesfor multiple fluids. •Gotthilf Hagen (1797–1884) had differentiated between laminar and turbulent flow in pipes. •Osborn Reynolds (1842–1912) continued Hagen’s work and developed the dimensionless number that bears his name. • ► In parallel work Louis Navier and George Stokes (1819–1903) completed the general equations of fluid motion with friction (Navier– Stokes equations). •James Francis (1815–1892) and Lester Pelton (1829–1908) pioneered work in turbines. •Clemens Herschel (1842–1930) invented the Venturi meter 7 Chapter I. Introduction ME3560 – Fluid Mechanics Summer 2016 • Irish and English engineers such as William Thompson, Lord Kelvin (1824–1907), William Strutt, Lord Rayleigh (1842 –1919), and Sir Horace Lamb (1849–1934) investigated problems such as dimensional analysis, irrotational flow, vortex motion, cavitation, and waves. • ► At the dawn of the 20th century the Wright brothers (Wilbur, 1867– 1912; Orville, 1871–1948) through application of theory and experimentation perfected the airplane. • ► In 1904, Ludwig Prandtl (1875–1953) showed that fluid flows can be derived into a layer near the walls, the boundary layer, where the friction effects are significant and an outer layer where such effects are negligible and the simplified Euler and Bernoulli equations are applicable. 8 Chapter I. Introduction ME3560 – Fluid Mechanics Summer 2016 • ► By the mid 20th century the existing theories were adequate for the tasks at hand and the fluid properties and parameters were well defined, thus supporting a enormous expansion of the aeronautical, chemical, industrial and water resources sector. • ► In the late 20th century, Fluid Mechanics research was dominated by the development of the digital computer. The ability to solve large complex problems, such as global climate modeling or to optimize the design of a turbine blade. •The principles that we will study in this curse apply to flows ranging from very small to extremely large scales. 9 Chapter I. Introduction ME3560 – Fluid Mechanics Summer 2016 1.2 Definition of a Fluid • A fluid is defined as a substance that deforms continuously when acted on by a shearing stress of any magnitude. •When common solids such as steel or other metals are acted on by a shearing stress, they will initially deform (usually a very small deformation), but they will not continuously deform (flow). •Common fluids such as water, oil, and air satisfy the definition of a fluid—that is, they will flow when acted on by a shearing stress. •Some materials, such as slurries, tar, putty, toothpaste are not easily classified since they will behave as a solid if the applied shearing stress is small, but if the stress exceeds some critical value, the substance will flow. The study of such materials is called rheology. 10 Chapter I. Introduction ME3560 – Fluid Mechanics Summer 2016 •To describe the behavior of fluids at rest or in motion, we consider the average, or macroscopic, value of the quantity of interest. •The average is evaluated over a small volume containing a large number of molecules. •The volume is small compared with the physical dimensions of the system of interest, but large compared with the average distance between molecules. •For gases at normal pressures and temperatures, the spacing is on the order of 10−6 mm. For gases, the number of molecules per cubic millimeter is on the order of 1018. •For liquids it is on the order of 10−7 mm. For liquids, the number of molecules per cubic millimeter is on the order of 1021. 11 Chapter I. Introduction ME3560 – Fluid Mechanics Summer 2016 1.3 The Non–Slip Condition • Fluid flow is often confined by solid surfaces, and it is important to understand how the presence of solid surfaces affects fluid flow. •.Consider the flow of a fluid in a stationary pipe or over a solid surface that is nonporous. Experimental observation indicates that a fluid in motion comes to a complete stop at the surface and assumes zero velocity relative to that surface. •That is, a fluid in direct contact with a solid “sticks” to the surface due to viscous effects and there is no slip. •This is known as the non–slip condition. • The fluid property responsible for the non–slip condition is the viscosity. 12 Chapter I. Introduction ME3560 – Fluid Mechanics Summer 2016 • A fluid layer adjacent to a moving surface has the same velocity as the surface •A consequence of the non–slip condition is that all velocity profiles must have zero values with respect to the surface at the points of contact. •Another consequence of the non–slip condition is the surface drag, which is the force a fluid exerts on a surface in the direction of the flow. 13 Chapter I. Introduction ME3560 – Fluid Mechanics Summer 2016 1.4 Classification of Fluid Flows • There is a wide variety of fluid flow problems and it is convenient to classify them based on some common characteristics to group them. Viscous versus Inviscid Regions of Flow •When two fluid layers move relative to each other, a friction force develops between them and the slower layer tries to slow down the faster layer. •This internal resistance to flow is quantified by the viscosity. The viscosity is caused by cohesive forces between the molecules in liquids and by molecular collisions in gases. There is no fluid with zero viscosity. •Flows in which the viscous effects are important are called viscous flows. 14 Chapter I. Introduction ME3560 – Fluid Mechanics Summer 2016 • In many flows of practical interest, there are regions where the viscous forces are negligibly small compared to inertial or pressure forces. • Neglecting the viscous effects in such inviscid flow regions greatly simplifies the analysis without much loss in accuracy. • The development of viscous and inviscid regions of flow as a result of inserting a flat plate parallel to a fluid stream of uniform velocity is shown in the picture. •The fluid sticks to the plate on both sides due to the non – slip condition. • Two zones are present, a viscous flow region (boundary layer) and an inviscid flow region. 15 Chapter I. Introduction ME3560 – Fluid Mechanics Summer 2016 Internal versus External Flow •A fluid flow is internal or external depending on whether the fluid is forced to flow in a confined channel or over a surface. •The flow of an unbounded fluid over a surface such as a plate, a wire, or a pipe is external flow. •The flow in a pipe or a duct is internal flow if the fluid is completely bounded by solid surfaces. •The flow of liquids in a duct which is only partially filled is called open channel flow. Flow of rivers is an example of this type of flows. • Internal flows are dominated by the influence of viscosity throughout the flow field. • In external flows the viscous effects are limited to boundary layers near solid surfaces and to wake regions downstream of bodies. 16 Chapter I. Introduction Summer 2016 ME3560 – Fluid Mechanics Compressible versus Incompressible Flow • A flow is classified as being compressible or incompressible, depending on the level of variation of density during flow. •Incompressibility is an approximation and a flow is said to be incompressible if the density remains constant, that is, the volume of every portion of fluid remains unchanged. •The densities of liquids are essentially constant (incompressible). •Gases on the other hand are highly compressible. However, gas flow can often be considered incompressible if the density changes are under 5 percent, which is usually the case when Ma < 0.3 = = 17 Chapter I. Introduction ME3560 – Fluid Mechanics Summer 2016 • The speed of sound in air (room temperature, sea level) is c = 346 m/s. Therefore, the compressibility effects in air can be neglected at speeds under about 100 m/s (≈ 220 mi/hr). Laminar versus Turbulent Flow • Some flows are smooth and orderly while others are rather chaotic. • The highly ordered fluid motion characterized by smooth layers of fluid is called laminar. • The highly disordered fluid motion that typically occurs at high velocities and is characterized by velocity fluctuations is called turbulent. • Flow that alternates between being laminar and turbulent is called transitional. 18 Chapter I. Introduction ME3560 – Fluid Mechanics Summer 2016 Natural (or Unforced) versus Forced Flow • A forced flow is a flow in which the fluid is forced to flow over a surface or in a pipe by external means such as pump or a fan. • In natural flows any fluid motion is due to natural means such as the buoyancy effect. Steady versus Unsteady Flow • The terms steady and uniform are used frequently in engineering. • The term steady implies no change at a point with time. The opposite of steady is unsteady. •The term uniform implies no change with location over a specified region. 19 Chapter I. Introduction ME3560 – Fluid Mechanics Summer 2016 • The terms unsteady and transient are often used interchangeably, however, in fluid mechanics unsteady applies to any flow that is not steady, and transient applies to developing flows. One–, Two–, and Three–Dimensional Flows • The best way to describe a flow field is through the velocity distribution, thus the flow can be one–, two–, or three –dimensional, depending on the number of coordinate directions required to describe the flow. • In the most general case, a fluid flow is described by three–dimensions [V(x, y, z) or V(r, θ, z)]. • In many instances, the variation of the velocity in certain directions can be small relative to the variation in other directions and can be ignored with negligible error. Thus the flow can be 1–D or 2–D. 20 Chapter I. Introduction ME3560 – Fluid Mechanics Summer 2016 1.5 System and Control Volume •A system is a collection of matter of fixed identity (always the same atoms or fluid particles), which may move, flow, and interact with its surroundings. •A system is a specific, identifiable quantity of matter. It may consist of a relatively large amount of mass or it may be an infinitesimal size. •A system may interact with its surroundings by various means (by the transfer of heat or the exertion of a pressure force, for example). • A system may continually change size and shape, but it always contains the same mass. 21 Chapter I. Introduction ME3560 – Fluid Mechanics Summer 2016 •A control volume, is a volume in space (a geometric entity, independent of mass) through which fluid may flow. •In fluid mechanics, it is difficult to identify and keep track of a specific quantity of matter. • In several cases, the main interest is in determining the forces put on a device rather than in the information obtained by following a given portion of the air (a system) as it flows along. • For these situations it is more adequate to use the control volume approach. •Identify a specific volume in space (a volume associated with the device of interest) and analyze the fluid flow within, through, or around that volume. 22 Chapter I. Introduction ME3560 – Fluid Mechanics Summer 2016 • In general, the control volume can be a moving volume, although for most situations we will use only fixed, non-deformable control volumes. • The matter within a control volume may change with time as the fluid flows through it. • The amount of mass within the volume may change with time. •The control volume itself is a specific geometric entity, independent of the flowing fluid. 23 Chapter I. Introduction ME3560 – Fluid Mechanics Summer 2016 •All of the laws governing the motion of a fluid are stated in their basic form in terms of a system approach. •For example, “the mass of a system remains constant,” or “the time rate of change of momentum of a system is equal to the sum of all the forces acting on the system.” 24 Chapter I. Introduction Summer 2016 ME3560 – Fluid Mechanics 1.6 Dimensions, Dimensional Homogeneity, and Units. •The study of fluid mechanics requires to develop a system for describing the fluid characteristics qualitatively and quantitatively. •The qualitative aspect serves to identify the nature, or type, of the characteristics (such as length, time, stress, and velocity). V =& LT −1 a =& LT −2 F =& MLT −2 •The quantitative aspect provides a numerical measure of the characteristics. The quantitative description requires both a number and a standard (unit) by which various quantities can be compared. m V s m a 2 s F[ N ] 25 Chapter I. Introduction ME3560 – Fluid Mechanics Summer 2016 26 Chapter I. Introduction ME3560 – Fluid Mechanics Summer 2016 27 Chapter I. Introduction Summer 2016 ME3560 – Fluid Mechanics •All theoretically derived equations are dimensionally homogeneous, and all additive separate terms must have the same dimensions. For example, the equation for the velocity, V, of a uniformly accelerated body is V = V0 + at where V0 is the initial velocity, a the acceleration, and t the time interval. In terms of dimensions the equation is LT −1 =& LT −1 + LT −1 and thus this equation is dimensionally homogeneous. 28 Chapter I. Introduction Summer 2016 ME3560 – Fluid Mechanics 1.6.1 Systems of Units •In addition to the qualitative description of the various quantities of interest, it is necessary to have a quantitative measure of any given quantity. •We will consider three systems of units that are commonly used in engineering. - International System (SI) Quantity Unit Length Meter (m) Time Second (s) Mass Kilogram (kg) Temperature Kelvin (K) K = oC + 273.15 Chapter I. Introduction 29 ME3560 – Fluid Mechanics Quantity Unit Force Work Power Newton (N) Joule (J) Watt (W) 1N = 1kg ⋅1m/s 1J = 1N ⋅ m Summer 2016 2 1W = 1J/s = 1N ⋅ m/s W = mg;g = 9.81m/s 2 30 Chapter I. Introduction Summer 2016 ME3560 – Fluid Mechanics -British Gravitational (BG) System. o Quantity Unit Length Time Mass Temperature Force Work Power Foot (ft) Second (s) Slug (slug) Rankine (oR) Pound (lb) R = o F + 459.67 1lb = 1slug ⋅ ft/s 2 W = mg;g = 32.2ft/s 2 lb⋅ft lb⋅ft/s 31 Chapter I. Introduction Summer 2016 ME3560 – Fluid Mechanics -English Engineering (EE) System. •In the EE system, units for force and mass are defined independently. o Quantity Length Time Mass Temperature Force Work Power Unit Foot (ft) Second (s) Pound mass (lbm) Rankine (oR) Pound (lb) R = o F + 459.67 1lb = 1lbm ⋅ 32.2ft/s 2 1slug = 32.2lbm W = mg;g = 32.2ft/s 2 lb⋅ft lb⋅ft/s 32 Chapter I. Introduction ME3560 – Fluid Mechanics Summer 2016 1.7 Modeling in Engineering •An engineering device can be studied either experimentally or analytically. •The experimental approach is advantageous because it deals with the actual physical system and the desired quantity is determined by measurement. •The experimental approach is expensive, time consuming and often impractical. Additionally the system to be studied might not exist. •On the other hand, the analytical approach (including numerical approach) has the advantage of being fast and inexpensive. •The results obtained are subject to the accuracy of the assumptions, approximations, and idealizations made in the analysis. 33 Chapter I. Introduction ME3560 – Fluid Mechanics Summer 2016 • The description of most scientific problems involve equations that relate the changes in some key variables to each other. • Usually, the smaller the increment chosen in the changing variables, the more general and accurate the description. •Therefore, differential equations are used to investigate a wide variety of problems in engineering and sciences. • However, may problems can be studied without the need of using differential equations. • In this class we will learn different tools, additional to the solution of differential equations to study problems in Fluid Mechanics 34 Chapter I. Introduction ME3560 – Fluid Mechanics Summer 2016 1.8 Continuum • Matter is made up of atoms that are widely spaced in the gas phase. •However, it is very convenient to disregard the atomic nature of a substance and treat it as a continuous, homogeneous matter with no holes, that is, a continuum. • The continuum idealization allows the treatment of properties as point functions and to assume that properties vary continuously in space without discontinuities. •This assumption is valid as long as the size of the system considered is large relative to the space between the molecules. This will be the case in all problems we analyze in this course of Fluid Mechanics 35 Chapter I. Introduction ME3560 – Fluid Mechanics Summer 2016 •To describe the behavior of fluids at rest or in motion, we consider the average, or macroscopic, value of the quantity of interest. •The average is evaluated over a small volume containing a large number of molecules. •The volume is small compared with the physical dimensions of the system of interest, but large compared with the average distance between molecules. •For gases at normal pressures and temperatures, the spacing is on the order of 10−6 mm. For gases, the number of molecules per cubic millimeter is on the order of 1018. •For liquids it is on the order of 10−7 mm. For liquids, the number of molecules per cubic millimeter is on the order of 1021. 36 Chapter I. Introduction Summer 2016 ME3560 – Fluid Mechanics 1.9 Measures of Fluid Mass and Weight 1.9.1 Density •The density of a fluid is defined as its mass per unit volume. m kg slug ρ= , 3 3 V m ft •The specific volume of a fluid is defined as the ratio of the volume occupied by the volume to its mass. V m 3 ft 3 ft 3 v = = , , ρ m kg slug lbm 1 •The density of liquids is assumed to be constant –incompressible fluids •The density of gases depends on the temperature and pressure of the system. For example, for ideal gases: p = ρRT 37 Chapter I. Introduction Summer 2016 ME3560 – Fluid Mechanics 1.9.2 Specific Weight •Specific weight (γ) is weight per unit volume. γ = ρg 1.9.3 Specific Gravity •Specific Gravity (SG) is defined as the ratio of the density of the fluid to the density of water at 4 °C (39.2 °F): ρ=1.94 slugs/ft3=1000 kg/m3. SG = ρ ρH [email protected] 2 = o C γ γ H [email protected] 2 o C 38 Chapter I. Introduction ME3560 – Fluid Mechanics Summer 2016 1.10 Viscosity •If P is applied to the upper plate, it will move continuously with a velocity, U. •The fluid in contact with the upper plate moves with a velocity, U. •The fluid in contact with the bottom fixed plate has a zero velocity. •The fluid between the two plates moves with velocity u=u(y)=Uy/b •A velocity gradient du/dy=U/b, develops in the fluid between the plates. •The experimental observation that the fluid “sticks” to the solid boundaries is referred to as the no-slip condition. 39 Chapter I. Introduction Summer 2016 ME3560 – Fluid Mechanics •It can be experimentally determined that P U ∂u α → τα A b ∂y •τ shear stress in a fluid in motion •∂ u/ ∂ y. Rate of shearing strain (Velocity gradient) • For a large number of fluids the relation between shear stress and velocity gradient is linear: ∂u •µ Absolute (dynamic) viscosity τ =µ ∂y Dynamic viscosity is property that relates shearing stress and fluid motion M ⋅L 1 ⋅ 2 Shear Stress M T L Dimensions : = = L 1 Velocity Gradient LT / T L N ⋅s kg = Pa ⋅ s •1poise = 0.1 N⋅s/m2 Units : 2 = m m ⋅s 2 Chapter I. Introduction 40 ME3560 – Fluid Mechanics Summer 2016 •Fluids for which the shearing stress is linearly related to the rate of shearing strain (also referred to as rate of angular deformation) are designated as Newtonian fluids. •µ = µ (T) •For gases µ increases as T does. •For liquids µ decreases as T does. 41 Chapter I. Introduction ME3560 – Fluid Mechanics Summer 2016 •Fluids for which the shearing stress is not linearly related to the rate of shearing strain are designated as non-Newtonian fluids. •Quite often viscosity appears in fluid flow problems combined with the density in µ the form υ= ρ •ν kinematic viscosity •The dimensions of ν are L2/T • BG units are ft2/s • SI units are m2/s. •CGS units are cm2/s = St (stoke) 42 Chapter I. Introduction ME3560 – Fluid Mechanics Summer 2016 1.11 Vapor Pressure (pv) • Vapor pressure (saturation pressure) is a thermodynamic property and it is the pressure at which phase change from liquid to gas (boiling) occurs. •Under certain circumstances in flowing fluids low pressures can be generated such that cavitation may occur. http://www.youtube.com/watch?v=GpklBS3s7iU&feature=related 43 Chapter I. Introduction Summer 2016 ME3560 – Fluid Mechanics 1.12 Compressibility of Fluids 1.12.1 Bulk Modulus •A property that is commonly used to characterize compressibility is the bulk modulus, Eν, defined as dp dp Ev = − = dV / V dρ / ρ •The bulk modulus has dimensions of pressure, FL−2. •The units for Ev are lb/in.2 (psi) and N/m2 (Pa). 44 Chapter I. Introduction Summer 2016 ME3560 – Fluid Mechanics 1.12.2 Compression and Expansion of Gases •When gases are compressed (or expanded), the relationship between pressure and density depends on the nature of the process. •Isothermal process p ρ = cons; Ev = p •Isentropic Process (frictionless compression (expansion), no heat is exchanged with the surroundings) p ρ k = cons; Ev = kp k = c p / cv ; R = c p − cv 45 Chapter I. Introduction Summer 2016 ME3560 – Fluid Mechanics 1.12.3 Speed of Sound •The velocity at which small disturbances propagate in a fluid is called the acoustic velocity or the speed of sound, c dp c= = dρ Ev ρ •For gases (isentropic process) c= kp ρ = kRT 46 Chapter I. Introduction ME3560 – Fluid Mechanics Summer 2016 1.13 Ideal Gas Law •The equation for ideal or perfect gases known as equation of state for an ideal gas is: •p is the absolute pressure p ρ= •ρ the density RT •T the absolute temperature •R the gas constant pabs = p gage + patm patm lb = 101.33 kPa = 14.7 2 (psi) in • This equation closely approximates the behavior of gases under normal conditions when the gases are not approaching liquefaction. 47 Chapter I. Introduction ME3560 – Fluid Mechanics Summer 2016 1.14 Surface Tension •At the interface between a liquid and a gas, or between two immiscible liquids, forces develop in the liquid surface which cause the surface to behave as if it were a “membrane” stretched over the fluid mass. •These types of surface phenomena are due to the unbalanced cohesive forces acting on the liquid molecules at the fluid surface. • Molecules in the interior of the fluid mass are surrounded by molecules that are attracted to each other equally. However, molecules along the surface are subjected to a net force toward the interior. 48 Chapter I. Introduction ME3560 – Fluid Mechanics Summer 2016 •As a result of this unbalanced force a hypothetical membrane is created at the interface. •A tensile force may be considered to be acting in the plane of the surface along any line in the surface. •The intensity of the molecular attraction per unit length along any line in the surface is called the surface tension (σ). σ = F/l. •For a given liquid the surface tension depends on temperature and the other fluid it is in contact with at the interface. •The dimensions of surface tension are FL−1. With units of lb/ft and N/m. • The value of the surface tension decreases as the temperature increases. 49 Chapter I. Introduction ME3560 – Fluid Mechanics Summer 2016 Determination of the Pressure inside a Drop •If the spherical drop is cut in half (as shown), the force developed around the edge due to surface tension is 2πRσ. •This force must be balanced by the pressure difference, ∆p, between the internal pressure, pi, and the external pressure, pe, acting over the circular area, πR2. 2πRσ = ∆pπR 2 2σ ∆p = pi − pe = R 50 Chapter I. Introduction ME3560 – Fluid Mechanics Summer 2016 •Another phenomena associated with surface tension is the rise (or fall) of a liquid in a capillary tube. •If a small open tube is inserted into water, the water level in the tube will rise above the water level outside the tube. In this situation we have a liquid–gas–solid interface. •In this case, there is an attraction (adhesion) between the wall of the tube and liquid molecules which is strong enough to overcome the mutual attraction (cohesion) of the molecules and pull them up the wall. • Hence, the liquid is said to wet the solid surface 51 Chapter I. Introduction Summer 2016 ME3560 – Fluid Mechanics •The height, h, is a function of σ, R, γ, and the angle of contact, θ, between the fluid and tube. •An equilibrium analysis yields the following relations 2π Rσ cos θ = γπ R 2 h 2σ cos θ h= γR •The angle of contact is a function of both the liquid and the surface. •For water in contact with clean glass θ ≈ 0°. •h is inversely proportional to R. 52 Chapter I. Introduction ME3560 – Fluid Mechanics Summer 2016 •If adhesion of molecules to the solid surface is weak compared to the cohesion between molecules, the liquid will not wet the surface and the level in a tube placed in a nonwetting liquid will actually be depressed. •Mercury is a good example of a nonwetting liquid when it is in contact with a glass tube. •For nonwetting liquids the angle of contact is greater than 90°, and for mercury in contact with clean glass θ ≈ 130°. 53 Chapter I. Introduction Summer 2016 ME3560 – Fluid Mechanics Pe 54 Chapter I. Introduction Summer 2016 ME3560 – Fluid Mechanics Read Sections: 1.7 Compressibility of Fluids 1.7.1 Bulk Modulus 1.7.2 Compression and Expansion of Gases 1.7.3 Speed of Sound 1.9 Surface Tension 55 Chapter I. Introduction