Lee Kong Chian Faculty of Engineering and Science Department of Mechanical & Material Engineering UEME 2123 Fluid Mechanics 1 UEME 2123 Fluid Mechanics 1 Course Outcomes 1. Determine the pressure forces in static fluids. 2. Apply continuity and momentum equations to solve fluid flow problems. 3. Analyze fluid problems by using dimensional analysis. 4. Apply fluid mechanics principles to solve fluid flow applications. 5. Conduct fluid mechanics experiments with data analysis. Lee Kong Chian Faculty of Engineering and Science 2 UEME 2123 Fluid Mechanics 1 Course Outline o Course delivery • Lectures, Week 1-14 (2+1 hours lectures per week) • Tutorials, Week 2-13 (1 hour session per alternate week: Check Time Table) • Lab Practical, Week 2-13 (3 hours session: Check Time Table) o Assessment • Final Exam (60%) • Test (10%) • Assignment (20%) • Lab Practical (10%) o Course Textbook Munson, Young, and Okiishi, Fundamental of Fluid Mechanics, 7th Edition, John Wiley and Sons o Extra Help • WBLE • Consultation Hours Lee Kong Chian Faculty of Engineering and Science 3 UEME 2123 Fluid Mechanics 1 Lecture Topics Topic Book Chapter Topic 1: Basic Fluid Mechanics Chapter 1: Introduction Chapter 2: Fluid Statics Topic 2: Analysis of Flow Chapter 3: The Bernoulli Equation Topic 3: Conservation Principles Chapter 5: Finite Control Volume Analysis Topic 4: Dimensional Analysis and Chapter 7: Dimensional Similitude Analysis Topic 5: Introduction to Boundary Chapter 8: Flow over Immersed Layer Flows Bodies Topic 6 : Flow in conduits Chapter 12: Turbomachines Topic 7 : Open Channel Flows Lecturer in Charge Mr. Prakas (prakassp@utar.edu.my) Dr. Bee (beest@utar.edu.my) Chapter 10 : Open-Channel Flow **Note: Please consult the respective lecturer in charge of the topics Lee Kong Chian Faculty of Engineering and Science 4 UEME 2123 Fluid Mechanics 1 Lectures o Week 1-14 (2+1 hours lectures per week) o Attendance is compulsory! Tutorials o Week 2-13 (1 hour session per alternate week: Check Time Table) o Attendance is compulsory! Lab Practical o Complete 2 Lab Practical Session o 2 Session x 3 hours (refer to Time Table) o Submit Lab Report in one week time Lee Kong Chian Faculty of Engineering and Science 5 UEME 2123 Fluid Mechanics 1 Prerequisite Knowledge o Ordinary and Partial Differentiation o Indefinite and Definite Integrals o Surface and Volume Integrals o Rectangular and Cylindrical-Polar Coordinate Systems o Vector Calculus, Gradient, Divergence, and Curl Operations o Definition of Scalars, Vectors, and Tensors Lee Kong Chian Faculty of Engineering and Science 6 Lee Kong Chian Faculty of Engineering and Science UEME 2123 Fluid Mechanics 1 Chapter 1 Introduction 7 Chapter 1: Introduction Main Topics 1. 2. 3. 4. 5. 6. 7. 8. Characteristics of Fluids Dimensions, Dimensional Homogeneity, and Units Measures of Fluid Mass and Weight Ideal Gas Law Viscosity Compressibility of Fluids Vapor Pressure Surface Tension Lee Kong Chian Faculty of Engineering and Science 8 1.1 Characteristics of Fluids Fluid Mechanics o Study of the behaviour of fluids when subject to applies forces o Two subcategories • • Fluid statics: Behaviour of fluids at rest Fluid dynamics: Behaviour of fluids in motion o Why study fluid mechanics? • Fluids everywhere o o o o o Everyday phenomenon Environmental flows Biological flows Medical devices Aerodynamics Lee Kong Chian Faculty of Engineering and Science 9 1.1 Characteristics of Fluids Fluid Mechanics o What is a fluid? • • Substance which continuously deform (strained) when subject to a shear stress Solids, although deforming initially, do not do so continuously o Generally consists of liquids and gases •A liquid takes the shape of the container it is in and forms a free surface in the presence of gravity •Liquid is difficult to compress •A gas expands until it encounters the walls of the container and fills the entire available space •Gases cannot form a free surface Lee Kong Chian Faculty of Engineering and Science 11 1.1 Characteristics of Fluids The Continuum Assumption While a body of fluid is comprised of molecules, most characteristics of fluids are due to average molecular behaviour. o What does it mean? • It means that a fluid regardless of its molecular nature, can be treated as a continuous medium. o What is the result or benefit of this assumption? • Fluid parameters such as density and velocity can be considered continuous functions of position with a value at each point in space. Each individual matters Only their average matters 11 1.1 Characteristics of Fluids Fluid Properties o Different fluids flow differently • This is because different fluids have different characteristics (for example water, oil, honey, tar, air) o Quantification of these fluids therefore requires the definition of fluid properties • Density, specific volume, specific gravity • Bulk modulus of compression • Vapour pressure • Surface tension 12 1.2 Dimensions, Dimensional Homogeneity, and Units Primary and Secondary Quantities Primary quantities or known as Basic dimensions: Length, L, time, T, mass, M, and temperature, θ. Secondary quantities: Area = L2, Velocity = LT-1 , Density = ML-3 System of Dimensions MLT system: Mass[M], Length[L], time[T], and Temperature[θ] FLT system: Force[F], Length[L], time[T], and Temperature[θ] Lee Kong Chian Faculty of Engineering and Science 13 1.2 Dimensions, Dimensional Homogeneity, and Units Dimensions Associated with Common Physical Quantities Lee Kong Chian Faculty of Engineering and Science 14 1.2 Dimensions, Dimensional Homogeneity, and Units Dimensionally Homogeneous All theoretically derived equations are dimensionally homogeneous; that is, the dimensions of the left side of the equation must be the same as those on the right side, and all terms have the same dimensions. Example: πΏπ−1 = πΏπ−1 + πΏπ−2π General homogeneous equation: valid in any system of units. Restricted homogeneous equation : restricted to a particular system of units. Example: Restricted homogeneous equation 2 ππ‘ π = 16.1π‘2 → π = 2 General homogeneous equation π = 16.1π‘2 is only valid for the system unit using feet and seconds, where π = 32.2ππ‘/π 2 Lee Kong Chian Faculty of Engineering and Science 15 1.2 Dimensions, Dimensional Homogeneity, and Units Example 1: Restricted and General Homogeneous Equations During a study of a certain flow system the following equation relating the pressures π1 and π2 at two points were developed: πππ π2 = π1 + π·π In this equation π is a velocity, π is the distance between the two points, π· a diameter, π the acceleration of gravity, and π a dimensionless coefficient. Is the equation dimensionally consistent? Solution: 16 1.2 Dimensions, Dimensional Homogeneity, and Units International System (SI) Length Time Mass Temperature Force/Weight Work Power Gravity Acceleration : meter (m) : second (s) : kilogram (kg) : Kelvin (K); KοΌ°C+273.15 : Newton (N); 1 NοΌ1 kg·m/s2 : Joule (J) ; J οΌ1 N·m : Watt (W) ; WοΌJ/sοΌN·m/s : g = 9.81 m/sec2 British Gravitational System (BG) Length Time Force/Weight Temperature Mass Gravity Acceleration : feet (ft) : second (s) : pound-force (lbf) : Fahrenheit (°F) or Rankine (°R); °R = °F+459.67 : slug : g οΌ 32.174 ft/s2 Lee Kong Chian Faculty of Engineering and Science 17 1.2 Dimensions, Dimensional Homogeneity, and Units Conversion Factor Lee Kong Chian Faculty of Engineering and Science 18 1.3 Measures of Fluid Mass and Weight Density o The density of a fluid, ρ is defined as mass per unit volume, π= π V o Density is used to characterize the mass of a fluid system. o In SI the units are kg/m3. o The density of water is 1000 kg/m3. Specific Volume The specific volume, ν, is the reciprocal of density or volume per unit mass, v= 1 π = V π Lee Kong Chian Faculty of Engineering and Science 19 1.3 Measures of Fluid Mass and Weight Specific Weight o The specific weight of a fluid, γ (gamma), is defined as its weight per unit volume or density times gravitational acceleration, πΎ = ππ o Under conditions of standard gravity (g= 9.81m/s2), water at 15°C has a specific weight of 9800 N/m3. Specific Gravity The specific gravity of a fluid, SG, is defined as the ratio of the density of the fluid at some specified temperature to the density of water at 4°C (1000 ππ/π3), SG = π ππ»2π@4β π = 1000 ππ/π3 Lee Kong Chian Faculty of Engineering and Science To measure SG 20 1.4 Ideal Gas Law Ideal Gas Law o Gases are highly compressible with changes in gas density directly related to changes in pressure and temperature through the equation, Note: π R is 286.9 J/kg.K π= π π not 8.31 J/mol.K!! o The pressure in ideal gas law must be expressed in absolute pressure (abs), in which the pressure is measured relative to absolute zero pressure (perfect vacuum). o However, in engineering it is common practice to measure pressure relative to the local atmospheric pressure called gage pressure (gage). o The standard sea-level atmospheric pressure is 14.6996 psi (abs) or 101.33kPa (abs). Lee Kong Chian Faculty of Engineering and Science 21 1.4 Ideal Gas Law Example 2: Atmospheric Conditions in Earth and Mars The temperature and pressure at the surface of Mars during Martian spring day were determined to be -50°C and 900 Pa, respectively. (a) Determine the density of the Martian atmosphere for these conditions if the gas constant for the Martian atmosphere is assumed to be equivalent to that of carbon dioxide (188.9 J/kg-K). (b) Compare the answer from part (a) with the density of earth’s atmosphere during a spring day when the temperature is 18°C and the pressure 101.6 kPa (abs). Solution: **To be discussed in Lecture… Lee Kong Chian Faculty of Engineering and Science 22 1.4 Ideal Gas Law Example 2: Atmospheric Conditions in Earth and Mars Solution: Lee Kong Chian Faculty of Engineering and Science 23 1.5 Viscosity How to describe the “fluidity” of the fluid? o Substance which continuously deforms when subject to a shear (tangential stress). o Introduce concept of viscosity to describe the ‘fluidity’ of a fluid, i.e., how easily it flows. o If a solid material is placed between the two plates and the top plate acted by force P (see Figs. below), shear force, τ will act tangentially on the surface area of top plate, A, π= π π΄ → π = ππ΄ Lee Kong Chian Faculty of Engineering and Science 24 1.5 Viscosity How to describe the “fluidity” of the fluid? What happens if the solid is replaced with a fluid? o When the force P is applied to the upper plate, it will move continuously with a velocity, U (velocity becomes important!). o The fluid “sticks” at the fixed plate and is referred to as the no-slip condition (y=0, u=0). o The fluid between the two plates moves with velocity u=u(y) that would vary linearly, π¦ π’=π π o And the velocity gradient is, ππ’ π = ππ¦ π Lee Kong Chian Faculty of Engineering and Science 25 1.5 Viscosity How to describe the “fluidity” of the fluid? In a small time increment, πΏπ‘ the top plate would also be displaced a small distance, πΏπ and the vertical line AB would rotate to a new position, AB’ forming a small angle, πΏπ½, πΏπ tan πΏπ½ ≈ πΏπ½ = π Since πΏπ = ππΏπ‘, it follows that, πΏπ½ = ππΏπ‘ π We note that in this case, πΏπ½ is a function not only of the force P (which governs U) but also of time, πΏπ‘. Thus, it is not reasonable to attempt to relate the shearing stress, π to πΏπ½ as is done for solids. Rather, we consider the rate at which is changing and define the rate of shearing strain, πΎ as, πΏπ½ π ππ’ πΎ = lim = = πΏπ‘ →0 πΏπ‘ π ππ¦ Lee Kong Chian Faculty of Engineering and Science 26 1.5 Viscosity How to describe the “fluidity” of the fluid? A continuation of this experiment would reveal that as the shearing stress π, is increased by increasing P (recall that π = π/π΄), the rate of shearing strain is increased in direct proportion—that is, ππ’ π ∝ π ∝ πΎ ππ¦ This the shearing stress and rate of shearing strain (velocity gradient) can be related with a relationship of the form, ππ’ π=π ππ¦ where the constant of proportionality, π is called the absolute viscosity or dynamic viscosity of the fluid. The unit of π in SI; kg/m-s or Ns/m2 or Pa-s This equation indicates plots should be linear with the slope equal to the viscosity as illustrated in the Figure. 27 1.5 Viscosity Newtonian and Non-Newtonian Fluid o Fluids for which the shearing stress is linearly related to the rate of shearing strain are designated as Newtonian fluids. o Most common fluids such as water, air, and gasoline are Newtonian fluid under normal conditions. o Fluids for which the shearing stress is not linearly related to the rate of shearing strain are non-Newtonian fluids. Lee Kong Chian Faculty of Engineering and Science 28 1.5 Viscosity Newtonian and Non-Newtonian Fluid o Shear thinning fluids (Pseudoplastic): The viscosity decreases with increasing shear strain rate – the harder the fluid is sheared, the less viscous it becomes. Many colloidal suspensions and polymer solutions are shear thinning. Latex paint is an example. o Shear thickening fluids (Dilatant): The viscosity increases with increasing shear rate – the harder the fluid is sheared, the more viscous it becomes. Water-corn starch mixture and water-sand mixture are examples. o Bingham plastic: neither a fluid nor a solid. Such material can withstand a finite shear stress without motion, but once the yield stress is exceeded it flows like a fluid. Toothpaste and mayonnaise are common examples. 29 1.5 Viscosity Kinematic and Dynamic Viscosities o Measure of a fluid’s resistance to deformation and hence flow o Acts like friction between layers of fluid when they are forced to move relative to each other. o Kinematic viscosity, π and Dynamic viscosity, π are related through, π= π π o The dimensions of kinematic viscosity are L2/T. o The units of kinematic viscosity in SI system is m2/s or Stoke, abbreviated St. Lee Kong Chian Faculty of Engineering and Science 30 1.5 Viscosity Example 3: Newtonian Fluid Shear Stress Solution: **Refer to Next Slide… Lee Kong Chian Faculty of Engineering and Science 31 1.5 Viscosity 32 1.5 Viscosity Viscosity and Temperature o For liquids, the viscosity decreases with an increase in temperature. o For gases, an increase in temperature causes an increase in viscosity. o WHY? Lee Kong Chian Faculty of Engineering and Science 33 1.5 Viscosity Viscosity and Temperature o In liquid, the molecules are closely spaced, with strong cohesive forces between molecules, and the resistance to relative motion between adjacent layers is related to these intermolecular force. o As the temperature increases, these cohesive force are reduced with a corresponding reduction in resistance to motion. Thus, viscosity is reduced by an increase in temperature. o The Andrade’s equation, π = π·ππ΅/Τ where D and B are constants. Lee Kong Chian Faculty of Engineering and Science 34 1.5 Viscosity Viscosity and Temperature o In gases, the molecules are widely spaced and intermolecular force negligible. o The resistance to relative motion mainly arises due to the exchange of momentum of gas molecules between adjacent layers. o As the temperature increases, the random molecular activity increases with a corresponding increase in viscosity. o The Sutherland equation, π = πΆπ 3/2 π+π where C and S are constants. Lee Kong Chian Faculty of Engineering and Science 35 1.6 Compressibility of Fluids Bulk Modulus o Liquids are usually considered to be incompressible, whereas gases are generally considered compressible. o How do we measure the Compressibility of a fluid? o Bulk modulus, πΈπ£, is used to characterize compressibility of fluid given by, ππ ππ πΈπ£ = − = π∀/ ∀ ππ/π o The bulk modulus of a substance is a measure of how incompressible/resistant to compressibility that substance is. o The bulk modulus has dimensions of FL-2. Lee Kong Chian Faculty of Engineering and Science 36 1.6 Compressibility of Fluids Compression and Expansion of Gases o When gases are compressed or expanded, the relationship between pressure and density depends on the nature of the process (isothermal or isentropic). o For isothermal process (constant temperature) πΈπ£ = π or π π =constant o For isentropic process (constant entropy) πΈπ£ = ππ or π ππ =constant o Where k is the ratio of the specific heat at constant pressure, ππ, to the specific heat at constant volume, ππ£ and is related to the gas constant, R as, π = π π − ππ£ Lee Kong Chian Faculty of Engineering and Science 37 1.6 Compressibility of Fluids Speed of Sound o The velocity at which small disturbances propagate in a fluid is called the speed of sound. o The speed of sound is related to change in pressure and density of the fluid medium through, π= ππ ππ = πΈπ£ π o For gases undergoing an isentropic process, with πΈπ£ = ππ, so that, π= ππ π o and making use of the ideal gas law, it follows that, π = ππ π Lee Kong Chian Faculty of Engineering and Science 38 1.6 Compressibility of Fluids Example 4: Speed of Sound and Mach Number 39 1.7 Vapor Pressure Vapor Pressure and Boiling o If liquids are simply placed in a container open to the atmosphere, some liquid molecules will overcome the intermolecular cohesive forces and escape into the atmosphere. o If the container is closed with small air space left above the surface, and this space evacuated to form a vacuum, a pressure will develop in the space as a result of the vapor that is formed by the escaping molecules. o When an equilibrium condition is reached, the vapor is said to be saturated and the pressure that the vapor exerts on the liquid surface is termed the VAPOR PRESSURE, ππ£. o Vapor pressure decreases with increasing height, which lowers the fluid boiling point. Lee Kong Chian Faculty of Engineering and Science 40 1.7 Vapor Pressure Vapor Pressure and Boiling o Boiling, which is the formation of vapor bubbles within a fluid mass, is initiated when the absolute pressure in the fluid reaches the vapor pressure. o An important reason for our interest in vapor pressure and boiling lies in the common observation that in flowing fluids it is possible to develop very low pressure due to the fluid motion, and if the pressure is lowered to the vapor pressure, boiling will occur. o For example, this phenomenon may occur in flow through the irregular, narrowed passages of a valve or pump. When vapor bubbles are formed in a flowing fluid, they are swept along into regions of higher pressure where they suddenly collapse with sufficient intensity (known as shockwave) to actually cause structural damage. Lee Kong Chian Faculty of Engineering and Science 41 1.8 Surface Tension Surface Tension o 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 “skin” or “membrane” stretched over the fluid mass. o Although such a skin is not actually present, this conceptual analogy allows us to explain several commonly observed phenomena. o Surface tension, π (unit is in force per unit length, not per unit area!!) is the intensity of the molecular attraction per unit length along any line in the surface. The force due to = surface tension The force due to pressure difference 2ο°Rο³ ο½ οpο°R2 2ο³ οp ο½ pi ο pe ο½ R Where pi is the internal pressure and pe is the external pressure 42 1.8 Surface Tension Capillary Action in Small Tube o For Water (see Fig. a), 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. o However, for Mercury (see Fig. c), the 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. o In Fig. (b), the downward weight is balanced by the vertical force due to surface tension, ο§ο° R h ο½ 2ο°Rο³ cos ο± 2 hο½ 2ο³ cosο± ο§R 43 1.8 Surface Tension Role of Surface Tension o Surface tension effects play a role in many fluid mechanics problems including the movement of liquids through soil and other porous media, flow of thin film, formation of drops and bubbles, and the breakup of liquid jets. o Surface phenomena associated with liquid-gas, liquid-liquid or liquid-gas-solid interfaces are exceedingly complex. CA=40° CA=130° 44 1.8 Surface Tension Example 5: Single or Double Edge Razors? As shown in the previous video, surface tension forces can be strong enough to allow a double-edge steel razor blade to “float” on water, but a single blade will sink. Assume that the surface tension forces act at an angle θ relative to the water surface as shown in Fig. P1.84. (a) The mass of the double-edge blade is 0.64 × 10−3 kg, and the total length of its sides is 206 mm. Determine the value of θ required to maintain equilibrium between the blade weight and the resultant surface tension force. (b) The mass of the single-edge blade is 2.61 × 10−3 kg, and the total length of its sides is 154 mm. Explain why this blade sinks. Support your answer with the necessary calculations. Solution: **To be discussed in Lecture… 45 1.8 Surface Tension 46
