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1 FLUID MECHANICS Technical English - I 1 th week Fluid Mechanics FLUID STATICS FLUID DYNAMICS Fluid Statics or Hydrostatics is the study of fluids at rest. The main equation required for this is Newton's second law for non-accelerating bodies, i.e. ΣF=0 Fluid Dynamics is the study of fluids in motion. The main equation required for this is Newton's second law for accelerating bodies, i.e. ΣF=ma fluid / mechanics / statics / hydrostatics / rest / motion / acceleration Density (): mass per unit volume (kgsn /m 4 ) Specific weight (): weight per unit volume or specific gravity (kg/m 3 ) Viscosity: The resistance of a fluid to flowing and movement A thick fluid has high viscosity (very viscous) A thin fluid has low viscosity = g Specific volume (v= 1 /): Relative density (s): volume per unit mass the ratio of the density of the fluid to the density of water at +4 o C (From mass / weight / density / specific gravity / specific volume / relative density viscosity / flowability / fluidity / resist / resistance / high flow / thick / thin / viscous The variation of viscosity with temperature Most liquids become less viscous as the temperature is raised. H Due to interaction between fluid molecules, the fluid flow will resist a shearing motion. The viscosity is a measure of this resistance Stationary Plate Moving Plate constant force F constant speed U (From However, the viscosity of a gas will increase with temperature. du, where is shear stress, is dynamic viscosity dy kinematic viscosity temperature / raise / increase / cold / hot / high flow / liquid / gas resist / shearing motion / moving plate / stationary / constant / shear stress 1

2 Surface Tension: Surface tension is the tendency of the surface of a liquid to behave like a stretched elastic membrane. Capillarity: Capillarity, or capillary motion, is the ability of a liquid to flow in narrow spaces without the assistance of, or even in opposition to, external forces like gravity. 96/fluid-properties-density-viscositysurface-tension-capillarity There is a natural tendency for liquids to minimize their surface area. For this reason, drops of liquid tend to take a spherical shape in order to minimize surface area. surface tension / tendency / stretched / elastic / membrane / spherical shape IMPORTANCE OF CAPILLARITY IN CIVIL ENGINEERING. Moisture increases due to capillary rise. Water is a key factor in most cases of deterioration of structural elements, such as walls and columns. If the exterior and interior faces of a wall are connected by capillary passages, severe wetting at the interior of building may occur because of capillarity. capillarity / narrow space / wall / deterioration / exterior / interior / moisture / wetting A static fluid can have no shearing force acting on it, and that Any force between the fluid and the boundary must be acting at right angles to the boundary. Pressure at a point For an element of fluid at rest, the element will be in equilibrium - the sum of the components of forces in any direction will be zero. The sum of the moments of forces on the element about any point must also be zero. Pascal s Law: The pressure at a point in a fluid at rest is independent of direction as long as there are no shearing stresses present. static / boundary / shearing force / equilibrium / component / moment and force pressure / point / rest / independent / direction / shearing stress Pressure variation in a fluid at rest (Hydrostatic distribution) Variation of pressure vertically in a fluid under gravity Equality of pressure at the same level in a static fluid P P 1 = - g A (z z 1 ) Thus in a fluid under gravity, pressure decreases with increase in height z = (z z 1 ) In other words, in an incompressible fluid at rest the pressure varies linearly with depth. p = h = g h 1 h P 1 P P 1 = P Pressure in the horizontal direction is constant (h 1 =h ). In other words, the pressure is the same for any two points at the same elevation in a continuous mass of fluid. pressure variation / distribution / gravity / incompressible / linearly / depth horizontal / level / direction / constant / continuous

3 Absolute pressure (P absolute ): Measured relative to a perfect vacuum (absolute zero pressure) it s always positive. So it is equal to gauge pressure plus atmospheric pressure. Gage pressure (P gage ): Measured relative to the local atmospheric pressure. So it is equal to absolute pressure minus atmospheric pressure. It can be negative or positive. Head (h): The vertical height of any fluid of density ρ which is equal to this pressure. Atmospheric pressure can be measured using a barometer: Vacuum p=0 P atm =10.33 t/m p=0 L p=p atm = g P absolute = P gage + P atm P gage = h h = P gage / Force balance; P atm = ρgl ρ is the density of the fluid, g is the gravitational constant head / absolute pressure / gage pressure / vacuum atmospheric pressure / barometer / vacuum / glass pipe / piezometric tube / mercury Measurement of Pressure Manometers Mechanical Pressure Measurement Instruments Piozemeter Tube U-tube Manometer Inclined-tube Manometer Electronic Pressure Measurement Instruments If the surface is a plane the force can be represented by one single resultant force, acting at right-angles to the plane through the center of pressure. For a horizontal submerged plane the pressure, p, will be equal at all points of the surface. Thus the resultant force will be given by F = pressure x area of plane = p A = ( h) A p h manometer / piezometer tube / u-tube / instrument / strain gauge / oscillation submerged plane surface / resultant force / center of pressure / hydrostatic paradox For a vertical or inclined surface The net pressure force (Resultant acts through the center of pressure, C P ) F r = γ h c A h c : the vertical distance from the fluid surface to the centroid (center of gravity) of the area. A: the area of the surface x c and y c : coordinates of the center of gravity of surface I xc : the moment of inertia of the surface about the x axis I xyc : the product of inertia of the surface about the x and y axes passing through the center of gravity of surface I xc y p = y c + y A x p c I xyc = x c + y A inclined surface / resultant / centroid / center of gravity / moment of inertia c For a vertical or inclined surface (The other alternative approach) b F y F L x /3 H Weight of water above sloping surface W= V b inclined surface / resultant / centroid / center of gravity / pressure prism F x H H/3 PRESSURE PRISM 3

4 Resultant force and the angle the resultant force makes to the horizontal F F x F y F x F y tan 1 F F y x Horizontal forces The resultant horizontal force of a fluid above a curved surface is: F x = Resultant force on the projection of the curved surface onto a vertical plane. F x acts horizontally through the center of pressure of the projection of the curved surface onto an vertical plane. Vertical forces The resultant vertical force of a fluid above a curved surface is: F y = Weight of fluid directly above the curved surface. and it will act vertically downward through the center of gravity of the mass of fluid. The principle of Archimedes: The buoyancy acting on a submerged object is equal to the weight of the displaced fluid due to the presence of the object. This law is valid for all fluid and regardless of the shape of the body. It can also be applied to both fully and partially submerged bodies. Archimedes screw pump curved surface / resultant force / horizontal forces / vertical forces / projection Archimedes Principle / buoyancy / submerged / displaced fluid Linear motion: Rigid-body rotation: a x If a body returns to its equilibrium position when displaced it is said to be in a stable equilibrium position. Conversely it is in an unstable equilibrium position if, when displaced (even slightly), it moves to a new equilibrium position. For the completely submerged body, as long as the center of gravity falls below the center of buoyancy, the body is in a stable equilibrium position with respect to small rotations. stable / unstable / equilibrium / center of gravity / metacentre point tan a a x g dy a dx r P Po g A general class of problems involving fluid motion in which there are no shearing stresses occurs when a mass of fluid undergoes rigid-body motion. pressure variation / fluid motion / rigid-body motion / shearing stresses / rotation Classification of flow types Compressible or Incompressible flows Uniform or non-uniform flows Steady or unsteady flows Laminar or turbulent flows One-, two-, and three-dimensional flows Compressible or Incompressible: All fluids are compressible - even water - their density will change as pressure changes. Under steady conditions, and provided that the changes in pressure are small, it is usually possible to simplify analysis of the flow by assuming it is incompressible and has constant density. classification / compressible / incompressible / uniform / non-uniform / steady unsteady / laminar / turbulent Uniform flow: If the flow velocity is the same magnitude and direction at every point in the fluid it is said to be uniform. Non-uniform: If at a given instant, the velocity is not the same at every point the flow is non-uniform. (From Steady: A steady flow is one in which the conditions (velocity, pressure and cross-section) may differ from point to point but DO NOT change with time. Unsteady: If at any point in the fluid, the conditions change with time, the flow is described as unsteady. magnitude / uniform / non-uniform / steady / unsteady / condition / describe 4

5 Steady uniform flow: Conditions do not change with position in the stream or with time. An example is the flow of water in a pipe of constant diameter at constant velocity. Steady non-uniform flow: Conditions change from point to point in the stream but do not change with time. An example is flow in a tapering pipe with constant velocity at the inlet - velocity will change as you move along the length of the pipe toward the exit. Unsteady uniform flow: At a given instant in time the conditions at every point are the same, but will change with time. An example is a pipe of constant diameter connected to a pump pumping at a constant rate which is then switched off. Unsteady non-uniform flow: Every condition of the flow may change from point to point and with time at every point. For example waves in a channel. Laminar and turbulent flow: In laminar flow the motion of the particles of fluid is very orderly with all particles moving in straight lines parallel to the pipe walls. Laminar flow Re < 000 low velocity Dye does not mix with water Fluid particles move in straight lines Simple mathematical analysis possible Rare in practice in water systems. Transitional flow Turbulent flow 000 > Re < 4000 Re > 4000 medium velocity high velocity Dye stream wavers Dye mixes rapidly and completely in water - mixes Particle paths completely irregular slightly. Average motion is in the direction of the flow Cannot be seen by the naked eye Changes/fluctuations are very difficult to detect. Must use laser. Mathematical analysis very difficult - so experimental measures are used Most common type of flow. stream / tapering / pipe / pump / pumping / wave / channel laminar / turbulent / transitional / filament dye / particle / path / fluctuation / detect Although in general all fluids flow three-dimensionally (3D), with pressures and velocities and other flow properties varying in all directions, in many cases the greatest changes only occur in two directions or even only in one. In these cases changes in the other direction can be effectively ignored making analysis much more simple. One-dimensional flow: The flow parameters (such as velocity, pressure, depth etc.) at a given instant in time only vary in the direction of flow and not across the cross-section. Example: the flow in a pipe. Two-dimensional flow: The flow parameters vary in the direction of flow and in one direction at right angles to this direction. Example: flow over a weir toe. Streamlines: The lines that are tangent to the velocity vectors throughout the flow field. Streamtube: The imaginary tubular surface formed by streamlines along which the fluid flows. Pathline: The line traced out by a given particle as it flows from one point to another. one-dimensional / D / 3D / velocity / pressure / depth / weir / toe streamlines / streamtube / pathline / imaginary / tabular / immersed object Lagrangian Method: This method involves following individual fluid particles as they move about and determining how the fluid properties associated with these particles change as a function of time. That is, the fluid particles are identified and their properties determined as they move. Eulerian Method: The fluid motion is given by completely prescribing the necessary properties (pressure, density, velocity, etc.) as functions of space and time. From this method we obtain information about the flow in terms of what happens at fixed points in space as the fluid flows past those points. The rate of mass stored = the rate of mass in - the rate of mass out min Conservation of mass the continuity equation dm m in m out dt m V mout V : volume Lagrangian and Eulerian Approach / prescribing / properties governing equations / conservation of mass / continuity equation / rate 5

6 Flow rate The Bernouilli Equation dl dz p Mass flow rate: The mass of the fluid flow through the control surface per unit time Volume flow rate Discharge: The discharge is the volume of fluid flowing per unit time. Multiplying this by the density of the fluid gives us the mass flow rate. flow rate / mass flow rate / volume flow rate / discharge Integration of Euler s equation Bernoulli s equation 1 p1 V p V gz1 gz Flow work + kinetic energy + potential energy = constant Bernoulli s equation has some restrictions in its applicability, they are: Flow is steady; Density is constant (which also means the fluid is incompressible); Friction losses are negligible; The equation relates the states at two points along a single streamline, (not conditions on two different streamlines). steady flow / incompressible flow / negligible / friction losses p 1 dx The Bernouilli Equation Conservation of energy Pressure Kinetic Potential Total energy per + energy per + energy per = energy per unit weight unit weight unit weight unit weight p V g z H Pressure Velocity Potential Total head + head + head = head pressure energy / kinetic energy / potential energy / total head Pitot tube / Venturi meter / tank / sharp edge orifice Conservation of momentum (Newton s second law) Newton s nd Law can be written: The rate of change of momentum of a body is equal to the resultant force acting on the body, and takes place in the direction of the force. Force = rate of change of momentum d F (mv ) dt F Q ( u u 1 ) Conservation of momentum (Newton s second law) The force in the x-direction F x Q u x u 1x and the force in the y-direction F y Q u y u 1y We then find the resultant force by combining these vectorially: Fresultant Fx Fy and the angle which this force acts at is given by F 1 y tan Fx momentum / resultant force momentum / resultant force 6

7 Application of the Momentum Equation 1. Force due to the flow of fluid round a pipe bend.. Force on a nozzle at the outlet of a pipe. 3. Impact of a jet on a plane surface. 4. Force due to flow round a curved vane. The force exerted by the fluid on the solid body touching the control volume is opposite to F R. So the reaction force, R, is given by R = - F R Dimensionless Number Symbol Formula Numerator Denominator Importance Reynolds number NRe Dvr/m Inertial force Viscous force Fluid flow involving viscous and inertial forces Froude number N Fr u /gd Inertial force Gravitational force Fluid flow with free surface Weber number N We u rd/s Inertial force Surface force Fluid flow with interfacial forces Mach number N Ma u/c Local velocity Sonic velocity Gas flow at high velocity Drag coefficient C D F D /(ru /) Total drag force Inertial force Flow around solid bodies Friction factor f t w /(ru /) Shear force Inertial force Flow though closed conduits Pressure coefficient C P Dp/(ru /) Pressure force Inertial force Flow though closed conduits. Pressure drop estimation pipe bend / nozzle / water jet / curved vane dimensionless number / Reynolds, Froude, Weber, Mach numbers / coefficient Search about the hydrostatic paradox; and decide on which one of the following is correct? P A = P B = P C = P D = P E or P A P B P C P D P E 7

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