An-Najah National University College of Engineering Fluid Mechanics-61341 Chapter [1] Fundamentals 1
The Book (Elementary Fluid Mechanics by Street, Watters and Vennard) Each chapter includes: Concepts Sample problems applications of concepts Problems reinforce understanding Keys to mastering Fluid Mechanics Learning the fundamentals: read and understand the text Working many problems 2
The Book (Street, Watters and Vennard) Only by working many problems can you truly understand the basic principles and how to apply them 3
How to master assigned Material Study material to be covered before it is covered in class Study sample problems until you can solve them closed book Do enough of the practice problems, answer unseen Do the homework Problems you have been assigned 4
Lecture's Objective Students will be able to: 1. Identify what is mechanics / fluid 2. Work with two types of units 3. Understand some fluid properties 4. Apply problem solving strategies 5
What is Mechanics? Mechanics can be defined as that branch of the physical sciences which deals with the effects of forces on objects Branchs of Mechanics 6
What is Fluid Mechanics? Fluid Mechanics is the study of fluids either in motion (fluid dynamics) or at rest (fluid static) and the subsequent effects of the fluid upon the boundaries, which may be either solid surfaces or interfaces with other fluids 7
Definition of a Fluid A fluid is a substance that deforms continuously under the action of shearing forces. If a fluid is at rest, we know that the forces on it are in balance. A simple example is shown in the following figure, in which a timber board floats on a reservoir of water If the force (F) is applied to one end of the board, then the board transmits a tangential shear stress ( ) to the reservoir surface. The board and the water beneath will continue to move as long as F and are nonzero, which means that water satisfies the definition of a fluid 8
States of Fluid A gas is a fluid that is easily compressed. It fills any vessel in which it is contained A liquid is a fluid which is hard to compress. A given mass of liquid will occupy a fixed volume, irrespective of the size of the container A free surface is formed as a boundary between a liquid and a gas above it 9
Why are we Studying Fluid Mechanics on a Civil Engineering Course? The provision of adequate water services such as the supply of potable water, drainage, sewerage are essential for the development of any society. It is these services which civil engineers provide Fluid mechanics is involved in nearly all areas of Civil Engineering either directly or indirectly 10
Why are we Studying Fluid Mechanics on a Civil Engineering Course? Some examples of direct involvement are those where we are concerned with manipulating the fluid: Sea and river (flood) defenses Water distribution / sewerage (sanitation) networks Hydraulic design of water / sewage treatment works Dams Irrigation Pumps and Turbines Water retaining structures 11
Why are we Studying Fluid Mechanics on a Civil Engineering Course? And some examples where the primary object is construction - yet analysis of the fluid mechanics is essential: Flow of air in / around buildings Bridge piers in rivers Groundwater flow Notice how nearly all of these involve water. This course, although introducing general fluid flow characteristics, ideas and principles, will demonstrate many of these principles through examples where the fluid is water 12
System of Units As any quantity can be expressed in whatever way you like it is sometimes easy to become confused as to what exactly or how much is being referred to This is particularly true in the field of fluid mechanics. Over the years many different ways have been used to express the various quantities involved Even today different countries use different terminology as well as different units for the same thing - they even use the same name for different things e.g. an American pint is 0.8 of a British pint! Two system of units are in common use: the International System of Units (SI) and the U.S. Customary System (USCS) It is essential that all quantities be expressed in the same system or the wrong solution will results 13
The SI System of Units The SI system consists of six primary units, from which all quantities may be described For convenience secondary (derived) units are used in general practice which are made from combinations of these primary units Primary Units: the six primary units of the SI system are shown in the table below: 14
The SI System of Units In fluid mechanics we are generally only interested in the top four units from the previous table Notice how the term 'Dimension' of a unit has been introduced in that table This is not a property of the individual units, rather it tells what the unit represents For example a meter is a length which has a dimension L but also, an inch, a mile or a kilometer are all lengths so have dimension of L The above notation uses the MLT system of dimensions 15
Derived Units: there are many derived units all obtained from combination of the above primary units. Those most used are tabulated in the shown table The SI System of Units 16
Conversion Factor Table 17
Conversion Factor Software's Brows the internet for free downloadable conversion factors software s Convert is one 18
Get to Know SI Prefixes When a numerical quantity is either very large or very small, the units used define its size may be modified by using prefix Exponential Form Prefix SI Symbol Multiple 10 9 giga G 10 6 mega M 10 3 kilo k Sub-multiple 10-3 milli m 10-6 micro μ 10-9 nano n 19
Example 1 Solution: 20
Fluid Properties Density Specific weight Specific volume Specific gravity Viscosity Surface tension Capillarity 21
Density The density ( ) of a fluid is defined as its mass per unit volume M (kg) (kg/m 3 ) = V (m3 ) water = 999 kg/m 3 = 1.94 slug/ft 3 air =1.22 kg/m 3 = 0.0023 slug/ft 3 If the density is constant (most liquids), the flow is incompressible If the density varies significantly (e.g. some gas flows), the flow is compressible Although gases are easy to compress, the flow may be treated as incompressible if there are no large pressure fluctuations 22
Specific Weight Specific Weight (γ) is defined as the weight per unit volume or The force exerted by gravity, g, upon a unit volume of the substance Units: Newton's per cubic meter (N/m 3 or Ib/ft 3 ) The relationship between density ( ) and specific weight (γ) can be determined by Newton's 2nd Law as: γ = g γ water = 9804 N/m 3 = 62.41 Ib/ft 3 γ air =12.01 N/m 3 = 0.076 Ib/ft 3 23
Specific Volume and Specific Gravity Specific Volume (α) is defined as volume per unit of mass and has units of (m 3 /kg or ft 3 /slug) Specific Gravity (s.g.) is defined as the ratio of the density of a substance to the density of water at specified temperature and pressure s.g. = fluid / water Specific Weight (γ) of any liquid can be calculated by γ = (s.g.) γ water 24
Ideal Gas Law An ideal gas is a superheated vapor that is at a relatively low p or high T (i.e., not approaching condensation or liquefaction) Ideal gases obey the following equation of state (a combination of Boyle s and Charles' laws), known as the ideal gas law: gp / RT or p / RT where: γ = specific weight R = gas constant p = absolute pressure T = absolute temperature 25
Example 2 5.6 m 3 of oil weighs 46800 N. Find its mass density (ρ) and specific gravity (s.g.) Solution: Weight 46 800 = mg Mass m = 46 800 / 9.81 = 4770.6 kg Mass density ρ = Mass / volume = 4770.6 / 5.6 = 852 kg/m 3 s.g. = fluid / water = 852 / 1000 = 0.852 26
Temperature Temperature (ºC or K, ºF or ºR) measure of a body s hotness or coldness indicative of a body s internal energy Unit conversions: K = ºC + 273.15 ºR = ºF + 459.67 ºF = 1.8 ºC + 32 27
Bulk Modulus of Elasticity For most purposes a liquid may be considered as incompressible, but for situations involving either sudden or great changes in pressure, its compressibility becomes important; also when temperature changes are involved The compressibility of a liquid is expressed by its Bulk Modulus of Elasticity (E). 28
If the pressure of a unit volume of liquid is increased by dp, it will cause a volume decrease dv For any volume V Bulk Modulus of Elasticity dp E dv / V Modulus of Elasticity (E) expressed in units of pressure. For water at 20 o C E = 2.2. GPa. Modulus expressed by of Elasticity (E) of gases can be E dp dp d / d / 29
Example 3 A liquid compressed in a cylinder has a volume of 1 L (1000cm 3 ) at 1 Pa and volume of 995 cm 3 at 2 Pa. What is its bulk modulus of elasticity E? Solution: dp 2 1 E 200MPa dv / V 995 1000 /1000 30
The viscosity of a fluid is a measure of its resistance to flow when acted upon by an external force such as a pressure gradient or gravity Newton s Law of Viscosity Viscosity dv dy where Is the shear stress (Pa or Ib/ft2 ) Is the coefficient of dynamic viscosity (Pa.s or Ib.s/ft2 ) dv dy Is the velocity gradient (rate of the shear strain) The coefficient of kinematic viscosity is defined as the ratio of dynamic viscosity to mass density Units: m 2 /s or ft 2 /s 31
Viscosity Even among fluids which are accepted as fluids there can be wide differences in behavior under stress Fluids obeying Newton's law where the value of µ is constant are known as Newtonian fluids If µ is constant the shear stress is linearly dependent on velocity gradient. This is true for most common fluids Fluids in which the value of µ is not constant are known as non-newtonian fluids 32
Viscosity There are several categories of non-newtonian fluids. These categories are based on the relationship between shear stress and the velocity gradient (rate of shear strain) in the fluid. These relationships can be seen in the graph below for several categories 33
Example 4 The following is a table of measurements for a fluid at constant temperature Determine the dynamic viscosity of the fluid 34
Example 4 (Solution) Using Newton's law of viscosity dv dy Where µ is the viscosity. So viscosity is the gradient of a graph of shear stress against shear strain of the above data, or Plot the data as a graph: 5 dv / dy 4 Shear stress 3 2 1 0 0 0.2 0.4 0.6 0.8 1 Rate of shear strain 35
Example 4 (Solution) Calculate the gradient for each section of the line Thus the mean gradient = dynamic viscosity = 4.98 N.s /m 2 (Pa.s) 36
Example 5 The density of an oil is 850 kg/m 3. Find its specific gravity (s.g.) and Kinematic viscosity if the dynamic viscosity µ is 5 10-3 kg/m.s (N.s /m 2 ) Solution: oil = 850 kg/m 3 water = 1000 kg/m 3 s.g. = fluid / water = 850 / 1000 = 0.85 Dynamic viscosity = µ = 5 10-3 kg/m.s Kinematic viscosity = 3 5 6 2 10 850 5.88 10 m / s 37
Example 6 In a fluid the velocity measured at a distance of 75 mm from the boundary is 1.125 m/s. The fluid has absolute viscosity 0.048 Pa.s and relative density (s.g.) 0.913. What is the velocity gradient and shear stress at the boundary assuming a linear velocity distribution. Solution: m = 0.048 Pa.s Velocity gradient = dv dy 1 1.125 0.075 15s Shear stress = dv dy 0.048 15 0.72 Pa 38
Example 7 39
Example 7 (Solution) 40
Surface Tension and Capillary Surface tension is property used to describe a phenomena observed at the interface between a gas and liquid where intermolecular cohesive forces form an imaginary film capable of resisting tension Capillary action is caused by surface tension between a liquid and solid surface. Water rises in a thin-bore tube 41
Surface Tension and Capillary An important consequence of surface tension (σ) is that it gives rise to a pressure jump across the interface whenever it is curved. Consider a spherical interface having a radius of curvature R. If p i and p o are the pressures on the two sides of the interface, then a force balance gives From which the pressure jump is found to be 42
Surface Tension and Capillary The previous equation holds only if the surface is spherical. The curvature of a general surface can be specified by the radii of curvature along two orthogonal directions, say, R 1 and R 2. A similar analysis shows that the pressure jump across the interface is given by 43
Surface Tension and Capillary For the cylindrical capillary tube (assuming the liquid surface to be a section of a sphere), p o = -γh, p i = 0, p i -p o = γh, and r/r = cos θ Substituting these in yields 44
Capillarity in Circular Glass Tubes 45
Example 8 Solution: 46
Example 9 Solution: 47
Vapor Pressure All liquids possess a tendency to vaporize, that is, to change from the liquid to gaseous phase Such vaporization occurs because molecules are continually projected through the free liquid surface and lost from the body of the liquid as a consequence of their natural thermal vibrations This ejected molecules, being gaseous, then exert their own partial pressure, which is known as the vapor pressure of the liquid Because of the increase of molecular activity with temperature, vapor pressure increases with temperature Boiling: will occur when the pressure above a liquid is equal to or less than the vapor pressure of the liquid When very low pressures are produced at certain locations in the system, pressures may be equal to or less than the vapor pressure the liquid flashes into vapor Cavitation, which is a form of boiling, may occur wherever the local pressure falls to vapor pressure 48
Example 10 Solution: 49
Problem Solving Strategy: IPE, A 3 Step Approach 1. Interpret: Read carefully and determine what is given and what is to be found/delivered. Ask, if not clear. If necessary, make assumptions and indicate them. 2. Plan: Think about major steps (or a road map) that you will take to solve a given problem. Think of alternative/creative solutions and choose the best one. 3. Execute: Carry out your steps. Use appropriate diagrams and equations. Estimate your answers. Avoid simple calculation mistakes. Reflect on/revise your work. 50