Fluid Mechanics PDF - Chapter 1 & 2

FLUID MECHANICS 1  Problem solving strategy: The first step in learning any science is to master the concepts. The next step is to use the fundamentals by testing this knowledge. This is done by solving significant problems. Solving such problems, especially complicated ones, requires a sy...

FLUID MECHANICS 1  Problem solving strategy: The first step in learning any science is to master the concepts. The next step is to use the fundamentals by testing this knowledge. This is done by solving significant problems. Solving such problems, especially complicated ones, requires a systematic approach. By using a step-by-step approach, a physicist can reduce the length of the solution of a complicated problem into the solution of a series of simple problems. When you are solving a problem, you should seriously use the following steps. Step 1: Problem Statement In simple words, briefly state the problem, the key information given, and the quantities to be found. This is to make sure that you understand the problem and the objectives before you attempt to solve the problem. Step 2: Schematic Draw a realistic sketch of the physical system involved, and list the relevant information on the figure. The sketch should resemble the actual system and show the key features. Indicate any energy and mass interactions with the surroundings. Listing the given information on the sketch helps one to see the entire problem at once. LUID MECHANICS 2 Step 3: Assumptions and Approximations State any appropriate assumptions and approximations made to simplify the problem to make it possible to obtain a solution. Assume reasonable values for missing quantities that are necessary. For example, in the absence of specific data for atmospheric pressure, it can be taken to be 1 atm. Step 4: Physical Laws Apply all the relevant basic physical laws and principles (such as the conservation of mass), and reduce them to their simplest form by utilizing the assumptions made. However, the region to which a physical law is applied must be clearly identified first. For example, the increase in speed of water flowing through a nozzle is analyzed by applying conservation of mass between the inlet and outlet of the nozzle. Step 5: Properties Determine the unknown properties necessary to solve the problem from property relations or tables. Step 6: Calculations Substitute the known quantities into the simplified relations and perform the calculations to determine the unknowns. Pay particular attention to the units and unit cancellations, and remember that a dimensional quantity without a unit is meaningless. Round up your answers to an appropriate number of significant digits. 3 Step 7: Reasoning, Verification, and Discussion Repeat the calculations that resulted in unreasonable values. For example, time is never having a negative value. Also, point out the significance of the results, and discuss their implications. State the conclusions that can be drawn from the results, and any recommendations that can be made from them. Keep in mind that the solutions you present to your instructors, and any physical analysis presented to others, is a form of communication. Therefore neatness, organization, completeness, and visual appearance are of utmost importance for maximum effectiveness. Bearing in mind; Carelessness and skipping steps to save time often end up costing more time and unnecessary anxiety. 4  Contents Problem Solving Strategy Systems of unites Primary unites Derived unites Characterizations of fluid Introduction Chapter 1 1.1- Archimedes principles 1.2- Center of buoyancy Chapter 2 Fluid mechanics and fluid properties 2.1- Newton’s law of viscosity 2.2- Classification of fluids 2.3- Fluids VS. Solids 2.4- Liquids VS. Gasses 2.5- Viscosity in gasses and Liquids 5 Chapter 3 Fluid Kinematics 3.1- Inviscid Flow 3.2-The Navier-Stokes equations. 3.3- Flow through Pipes 3.4- Boundary Conditions in Fluid Mechanics Chapter 4 Fluid dynamics 4.1-Stream Line 4.2-Laminar flow 4.3-Turbulent flow 4.4-Equation of continuity 4.5-Bernoulli’s Equation 4.6-Flow through small orifice 4.7-Discharge over Weirs and notches Chapter 5 Vortex Dynamics 6 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 4/5 of a British pint! To avoid any confusion on this course we will always used the SI (metric) system - which you will already be familiar with. It is essential that all quantities be expressed in the same system or the wrong solution will results. Despite this warning you will still find that that this is the most common mistake when you attempt example questions. The SI System of units The SI system consists of six primary units, from which all quantities may be described. For convenience secondary units are used in general practice which is made from combinations of these primary units. 7 Primary Units The six primary units of the SI system are shown in the table below: Quantity SI Unit Dimension length meter, m L mass kilogram, kg M time second, s T temperature Kelvin, K θ current ampere, A I luminosity candela Cd In fluid mechanics we are generally only interested in the top four units from this table. Notice how the term 'Dimension' of a unit has been introduced in this 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. 8 Derived Units There are many derived units all obtained from combination of the above primary units. Those most used are shown in the table below: Quantity SI Unit Dimension velocity m/s ms-1 LT-1 acceleration m/s2 ms-2 LT-2 force N kg ms-2 M LT-2 kg m/s2 energy (or work) Joule J kg m2s-2 ML2T-2 N m, kg m2/s2 power Watt W Nms-1 ML2T-3 N m/s kg m2s-3 kg m2/s3 pressure ( or stress) Pascal P, Nm-2 ML-1T-2 N/m2, kg m-1s-2 kg/m/s2 density kg/m3 kg m-3 ML-3 specific weight N/m3 kg m-2s-2 ML-2T-2 kg/m2/s2 relative density a ratio 1 no units no dimension viscosity N s/m2 N sm-2 M L-1T-1 kg/m s kg m-1s-1 surface tension N/m Nm-1 MT-2 kg /s2 kg s-2 9 CHARACTRIZATION OF FLUIDS 10 Objectives of the course: -The course will introduce the subject of fluid mechanics and establish its relevance in Physics, Biophysics and Geophysics. - To develop an intuitive understanding of fluid mechanics by emphasizing the physics, and by supplying attractive figures and visual aids to reinforce the physics. -Develop the fundamental principles underlying the subject. -Demonstrate how these are used for the calculation of pressure, flow rates, shear forces and viscosity and their application in different fields of physics. Introduction Fluid mechanics is a branch of applied mechanics concerned with the static and dynamics of fluid - both liquids and gases. The analysis of the behavior of fluids is based on the fundamental laws of mechanics, which relate continuity of mass and conservation of energy with force and momentum. There are two aspects of fluid mechanics, which make it different to solid mechanics namely; the nature of a fluid is much different as compared with solid and in fluids it deals with continuous streams of fluid without a beginning or ending. Fluid is a substance, which deforms continuously, or flows, when subjected to shearing forces. If a fluid is at rest there is no shearing force acting. The analysis of the behavior of fluids is based on the fundamental laws of mechanics which relate continuity of mass and energy with force and momentum together with the familiar solid mechanics properties. 11  Chapter 1 Archimedes’ Principle: Archimedes’ principle states that a system submerged or floating in a fluid has buoyant force Fbuoy acting on it whose magnitude is equal to that of the weight of the fluid displaced by the system. Buoyancy arises from the increase of fluid pressure with depth and the increase pressure exerted in all directions as stated by Pascal's law. Thus, there is an unbalanced upward force on the bottom of a submerged or floating object. An illustration is shown in Figure. Since the "water ball" at left is exactly supported by the difference in pressure and the solid object at right experiences exactly the same pressure environment, it follows that the buoyancy force Fbuoy on the solid object upward is equal to the weight of the water displaced according to Archimedes' principle. Illustration of buoyancy follows Archimedes’ principle Supposing that the volume of the water equivalent is V m3, the buoyancy force Fbuoy is equal to Vρwater g. If the object has mass m, the apparent weight of the solid object shall be (mg - Vρwater g). 12 Let’s consider the general case where an object has density ρ and volume V. Its weight W is equal to ρ Vg. When the object submerged in the fluid of density ρ fluid and displaced a volume V’ then the buoyant force Fbuoy is Fbuoy = ρ fluid V’g. If the volume V is equal to V’ and the object sinks to the bottom, this shall be mean W > Fbuoy and also implies that the density ρ of the object is larger than the density ρ fluid of fluid. Likewise, the object floats. It implies that the density of object is smaller than the density of fluid. If the object is neither sink nor float like a fish swimming in the sea, ρ fluidV’g = ρ Vg. For ship that float on water, the condition ρ fluidV’g = ρ Vg is also satisfies. However, the volume is V’ < V because only a small portion of volume of ship is submerged in water. This implies that the density of ship is less than density of water. Example: A rock is suspended from a spring scale in air and found to be weight of magnitude w. The rock is then submerged completely in water while attached to the scale. The new reading of scale is wsub. Find the expression for the density ρ rock of rock in terms of the scale readings and density of water ρ water. Solution: The weight of rock in air is W = ρ rockVg. The buoyant force is: Fbuoy = ρ waterVg. The submerged weight is: Wsub = ρ rockVg - ρ waterVg. Thus, the ratio of submerged weight and weight in air is: Wsub / W = (ρ rockVg - ρ waterVg) / ρ rockVg 13 This implies that the density of rock: ρrock = (W ρ water ) / ( w - wsub ) Center of Buoyancy: The center of buoyancy for floating and submerged object would determine the stability of the system. For stability or equilibrium, everybody knows that the net force and net torque should be zero, which are Fnet = 0 and Tnet = 0. Therefore, for an equilibrium system, the center of mass COM and center of buoyancy COB should lie in same vertical line. For a total submerged system such as a submarine, the center of mass COM should lie below center of buoyancy COB since the submarine is designed such that it is heavier at the bottom. If the submarine is tilted toward right, the COB is shifted toward right. The torque out of the page with respect to COM is restoring the tilt to equilibrium position. The illustration is shown in Figure. Center of buoyancy for a total submerged system 14 For the floating system such as the aircraft carrier, the COB is below the COM as shown in the following Figure. If there is a tilt, the COB is move toward right align the metacenter. A restoring torque will move the aircraft carrier back to equilibrium. Center of buoyancy for a floating sy 15  Chapter 2 Fluids Mechanics and Fluid Properties Objectives of this section - Define the nature of a fluid. - Show where fluid mechanics concepts are common with those of solid mechanics and indicate some fundamental areas of difference. - Introduce viscosity and show what Newtonian and non-Newtonian fluids are. - Define the appropriate physical properties and show how these allow differentiation between solids and fluids as well as between liquids and gases. 2.1. Fluids: There are two aspects of fluid mechanics which make it different to solid mechanics: 1. The nature of a fluid is much different to that of a solid 2. In fluids we usually deal with continuous streams of fluid without a beginning or end. In solids we only consider individual elements. We normally recognize three states of matter: solid; liquid and gas. However, liquid and gas are both fluids: in contrast to solids they lack the ability to resist deformation. Because a fluid cannot resist the deformation force, it moves, it flows under the action of the force. Its shape will change continuously as long as the force is applied. A solid can resist a deformation force while at rest, this force may cause some displacement but the solid does not continue to move 16  indefinitely. The deformation is caused by shearing forces which act tangentially to a surface. We can then say: A Fluid is a substance which deforms continuously, or flows, when subjected to shearing forces. And conversely this definition implies the very important point that: If a fluid is at rest there are no shearing forces acting. All forces must be perpendicular to the planes which they are acting. When a fluid is in motion shear stresses are developed if the particles of the fluid move relative to one another. When this happens adjacent particles have different velocities. If fluid velocity is the same at every point then there is no shear stress produced: the particles have zero relative velocity. Consider the flow in a pipe in which water is flowing. At the pipe wall the velocity of the water will be zero. The velocity will increase as we move toward the centre of the pipe. This change in velocity across the direction of flow is known as velocity profile and shown graphically in the figure below: Velocity profile in a pipe. 17 Because particles of fluid next to each other are moving with different velocities there are shear forces in the moving fluid i.e. shear forces are normally present in a moving fluid. On the other hand, if a fluid is a long way from the boundary and all the particles are traveling with the same velocity, the velocity profile would look something like this: Velocity profile in uniform flow And there will be no shear forces present as all particles have zero relative velocity. In practice we are concerned with flow past solid boundaries; airplanes, cars, pipe walls, river channels etc. and shear forces will be present. 18 2.2. Newton's Law of Viscosity : How can we make use of these observations? We can start by considering a 3d rectangular element of fluid, like that in the figure below. Fluid element under a shear force The shearing force F acts on the area on the top of the element. This area is given by A = δz δx. We can thus calculate the shear stress which is equal to force per unit area i.e. Shear stress, τ = F/A The deformation which this shear stress causes is measured by the size of the angle Φ and is know as shear strain. In a solid shear strain, Φ is constant for a fixed shear stress τ. In a fluid, Φ increases as long as τ is applied, and the fluid flows. It has been found experimentally that the rate of shear strain (shear strain per unit time, Φ / time) is directly proportional to the shear stress. 19 If the particle at point E (in the above figure) moves under the shear stress to point E' and it takes time t to get there, it has moved the distance x. For small deformations we can write: Shear strain Φ = X / y Rate of shear strain = Φ / t = X / ty = (X/t) (1/y) =v/y Where v is the velocity of the particle at E Using the experimental result that shear stress is proportional to rate of shear strain then, τ = (constant). (v / y) The term (v/y) is the change in velocity with y, or the velocity gradient, and may be written in the differential form dv/dy. The constant of proportionality is known as the dynamic viscosity, μ, of the fluid, giving: τ = μ dv/dy This is known as Newton's law of viscosity. 20  2.3. CLASSIFICATION OF FLUIDS Ideal fluid: A fluid, which is incompressible and having no viscosity, is known as an ideal fluid. Ideal fluid is only an imaginary fluid as all the fluids, which exist, have some viscosity. Real fluid: A fluid, which possesses viscosity, is known as real fluid. All the fluids, in actual practice, are real fluids. Example: Water, Air etc. Newtonian fluid: A real fluid, in which shear stress in directly proportional to the rate of shear strain or velocity gradient, is known as a Newtonian fluid. Example: Water, Benzene etc. 21 Non Newtonian fluid: A real fluid, in which shear stress in not directly proportional to the rate of shear strain or velocity gradient, is known as a Non Newtonian fluid. Exemples : Plaster, Slurries, Pastes etc. Ideal plastic fluid: A fluid, in which shear stress is more than the yield value and shear stress is proportional to the rate of shear strain or velocity gradient, is known as ideal plastic fluid. Incompressible fluid: A fluid, in which the density of fluid does not change which change in external force or pressure, is known as incompressible fluid. All liquid are considered in this category. Compressible fluid: A fluid, in which the density of fluid changes while change in external force or pressure, is known as compressible fluid. All gases are considered in this category. Graphical representation of different fluids 22 2.4.Fluids vs. Solids: The differences between the behavior of solids and fluids under an applied force can be summarized as follow, we have; 1- For a solid the strain is a function of the applied stress (providing that the elastic limit has not been reached). For a fluid, the rate of strain is proportional to the applied stress. 2- The strain in a solid is independent of the time over which the force is applied and (if the elastic limit is not reached) the deformation disappears when the force is removed. A fluid continues to flow for as long as the force is applied and will not recover its original form when the force is removed. 3- As you will have seen when looking at properties of solids, when the elastic limit is reached they seem to flow. They become plastic. They still do not meet the definition of true fluids as they will only flow after a certain minimum shear stress is attained. The following section shows more detailed information concerning the Newtonian and Non-Newtonian fluids, 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. 23  Fluids in which the value of μ is not constant are known as non-Newtonian fluids. There are several categories of these, and they are outlined briefly below. 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 Shear stress vs. Rate of shear strain dv/dy Each of these lines can be represented by the equation; τ = A+B (dv/dy)n Where A, B and n are constants. For Newtonian fluids A = 0, B = μ and n = 1. 24 Here is a brief description for the physical properties of several categories of fluids: - Plastic: Shear stress must reach a certain minimum before flow commences. As an example of plastic material is toothpaste. It is necessary that toothpaste will stay on the brush but it must be easily extrudable. - Bingham plastic: As with the plastic above a minimum shear stress must be achieved. With this classification n = 1. An example is sewage sludge. - Pseudo-plastic: No minimum shear stress necessary and the coefficient of viscosity decreases as the rate of shear increases, e.g. colloidal substances like clay, milk and cement. - Dilatant substances; Viscosity increases as rate of shear increases, e.g. quicksand, Corn flour and vinyl resin pastes. - Thixotropic substances: Viscosity decreases with length of time shear force is applied e.g. jelly, paints and blood. - Rheopectic substances: Viscosity increases with length of time shear force is applied - Visco-elastic materials: Similar to Newtonian but if there is a sudden large change in shear they behave like plastic. There are many visco-elastic materials for example Pitch, wool fibers, nylon, rubber, putty (filler for wood) and egg white. - Blood: Blood is a complex fluid, consisting of a plasma in which are suspended a variety of cells, the predominant ones being the red cells. Measurements show that blood is thixotropic.At high shear rates, normal blood has a viscosity of 25  between 5 and 6 times that of water: but at low shear rates it may be several hundred times of water. Blood’s viscous behavior is partly due to aggregation of the red cells. These aggregates can, and do, form when the shear rate is low, but at higher shear rates they break up, giving a lower viscosity. This however is not the whole story – the viscosity of the interior of the red cells plays a major part. If the red cells were rigid particles, when their concentration reached 65%, blood would have the consistency of concrete. This however does not happen. Blood is still very fluid even at 99% red cell concentration. The reason for this is that the red cells are not rigid but in fact fluid. Thus it is that blood can flow in the small capillaries- the cells deform as they flow. Any condition which leads to more rigid red cells leads to a much greater blood viscosity. An important consequence of the rheological nature of blood is that when it is artificially pumped, as it is in certain type s of heart surgery where the heart is by-passed, special pumps have to be used. These are of a roller type. They pump the blood so that the red cells are not damaged by too high shear rates but yet at a rate sufficiently great so that aggregation does not occur. One final aspect to mention is the effect of drugs oh the blood flow. These in general affect the flow just by dilating or contracting the blood vessels. Thus it is that when a cigarette is smoked, there is a short term reduction in the blood flow due to the nicotine constricting the blood flow in the vessels. This can be seen by measuring the blood flow in the vessels of the ear lobe by passing a light beam through it. It is though, however, that as well, prolonged ingestion of nicotine has a long term effect leading to an increase in aggregation of cells and hence a higher viscosity and reduced blood flow. 26 Transient Ischematic Attack (TIA): A dangerous effect of nicotine is the TIA which is a sudden loss of blood supply to the brain. The right and left vertebral arteries both supply blood to the brain. A constriction in one causes a higher flow speed in it, reducing the pressure and flow in the other. There is also one more - which is not real, it does not exist - known as the ideal fluid. This is a fluid which is assumed to have no viscosity. This is a useful concept when theoretical solutions are being considered - it does help achieve some practically useful solutions. 2.5. Liquids vs. Gasses: Although liquids and gasses behave in much the same way and share many similar characteristics, they also possess distinct characteristics of their own. Specifically - A liquid is difficult to compress and often regarded as being incompressible. A gas is easily to compress and usually treated as such - it changes volume with pressure. - A given mass of liquid occupies a given volume and will occupy the container it is in and form a free surface (if the container is of a larger volume). A gas has no fixed volume; it changes volume to expand to fill the containing vessel. It will completely fill the vessel so no free surface is formed. 27 2.6. Causes of Viscosity in Fluids: 2.6.1. Viscosity in Gasses The molecules of gasses are only weakly kept in position by molecular cohesion (as they are so far apart). As adjacent layers move by each other there is a continuous exchange of molecules. Molecules of a slower layer move to faster layers causing a drag, while molecules moving the other way exert an acceleration force. Mathematical considerations of this momentum exchange can lead to Newton law of viscosity. If temperature of a gas increases the momentum exchange between layers will increase thus increasing viscosity. Viscosity will also change with pressure - but under normal conditions this change is negligible in gasses. 2.6. 2. Viscosity in Liquids There is some molecular interchange between adjacent layers in liquids - but as the molecules are so much closer than in gasses the cohesive forces hold the molecules in place much more rigidly. This cohesion plays an important roll in the viscosity of liquids. Increasing the temperature of a fluid reduces the cohesive forces and increases the molecular interchange. Reducing cohesive forces reduces shear stress, while increasing molecular interchange increases shear stress. 2.7. Viscosity and Temperature: To explain the above mention topic about the difference in viscosity between Liquids and gasses, we can look closer to the molecular inter action mechanism. 28  There is some molecules interchange between adjacent layers in liquids. But the molecules are much closer than in gas that their cohesive forces hold them in place much more rigidly. Thus, it reduces the molecules exchange. This cohesion plays an important role in the viscosity of liquid. As the temperature of a fluid increases, it reduces the cohesive force and increases the molecular interchange. Reducing cohesive forces reduces shear stress, while increasing molecules interchange increases shear stress. Thus, one can see there is a complex relationship between molecules exchange and cohesive force on viscosity. In general the reduction of cohesive force is more than increase of molecules exchange. Thus, the viscosity of liquid is decreased as temperature increases. High pressure can also change the viscosity of a liquid. As pressure increases the relative movement of molecules requires more energy hence viscosity increases. The molecules of gas are only weakly bounded by cohesive force between molecules, as they are far apart. Between adjacent layers, there is a continuous exchange of molecules. Molecules of a slower layer move to faster layers causing a drag, while molecules moving the other way exert an acceleration force. Mathematical considerations of this momentum exchange can lead to Newton law of viscosity. If temperature of a gas increases, the momentum exchange between layers will increase thus increasing viscosity. It can be viewed as the temperature increases, it reduces the cohesive force further increase more molecules exchange that increases the viscosity 29