Fluid Mechanics Lecture Notes PDF

Chapter 12 © 2016 Pearson Education, Ltd.  Learning Goals for Chapter 12 Looking forward at … the meaning of the density and pressure of a fluid, and how they are measured. how to calculate the buoyant force that a fluid exerts on a body immersed in it. how to use Be...

Chapter 12 © 2016 Pearson Education, Ltd.  Learning Goals for Chapter 12 Looking forward at … the meaning of the density and pressure of a fluid, and how they are measured. how to calculate the buoyant force that a fluid exerts on a body immersed in it. how to use Bernoulli’s equation to relate pressure and flow speed at different points in certain types of flow. the significance of laminar versus turbulent fluid flow, and how the speed of flow in a tube depends on the tube’s size. how viscous flow and turbulent flow differ from ideal flow. © 2016 Pearson Education, Ltd.  Introduction A colorful tail-spot wrasse is about 10 cm long and can float in the ocean with little effort. A manta ray is more than 5 m across and must “flap” its fins continuously to keep from sinking. The differences have to do with fluid mechanics. We begin with fluids at rest and then move on to the more complex field of fluid dynamics. © 2016 Pearson Education, Ltd.  Density An important property of any material, fluid or solid, is its density, defined as its mass per unit volume. A homogeneous material such as ice or iron has the same density throughout. For a homogeneous material, The SI unit of density is the kilogram per cubic meter (1 kg/m3). © 2016 Pearson Education, Ltd.  Density Here are two objects with different masses and different volumes, but the same density. © 2016 Pearson Education, Ltd.  Densities of some common substances Material Density (kg/m3) Air (1 atm, 20°C) 1.20 Ice 0.92 × 103 Water 1.00 × 103 Blood 1.06 × 103 Aluminum 2.7 × 103 Lead 11.3 × 103 Gold 19.3 × 103 © 2016 Pearson Education, Ltd. 7 Fluids States of Matter http://global.britannica.com/media/full/455270/14866 0 PHY1005 - 7 8 Fluids https://exploration.grc.nasa.gov/education/rocket/state.ht ml PHY1005 - 8 © 2016 Pearson Education, Ltd.  Pressure in a fluid A fluid exerts a force perpendicular to any surface in contact with it, such as a container wall or a body immersed in the fluid. Consider a small surface of area dA within a fluid at rest, centered on a point in the fluid. We define the pressure p at that point as: The SI unit of pressure is the pascal, where 1 pascal = 1 Pa = 1 N/m2. © 2016 Pearson Education, Ltd.  Pressure Here are the forces acting on a small surface within a fluid at rest. © 2016 Pearson Education, Ltd.  Pressure is a scalar quantity © 2016 Pearson Education, Ltd.  Pressure at depth in a fluid The pressure at a depth h in a fluid is greater than the pressure p0 at the surface: © 2016 Pearson Education, Ltd.  Pressure at depth in a fluid Each fluid column has the same height, no matter what its shape. © 2016 Pearson Education, Ltd.  Pascal’s law Pressure applied to an enclosed fluid is transmitted undiminished to every portion of the fluid and the walls of the containing vessel: © 2016 Pearson Education, Ltd.  Absolute pressure and gauge pressure When we say that the pressure in a car tire is “32 psi” (2.2 × 105 Pa), we mean that it is greater than atmospheric pressure by this amount. The total pressure in the tire is greater by patm. The excess pressure above atmospheric pressure is usually called gauge pressure. The total pressure is called absolute pressure. If the pressure is less than atmospheric, as in a partial vacuum, the gauge pressure is negative. © 2016 Pearson Education, Ltd.  Pressure gauges This Bourdon-type pressure gauge is connected to a high- pressure gas line. The gauge pressure shown is just over 5 bars (1 bar = 105 Pa). © 2016 Pearson Education, Ltd.  Blood pressure Blood-pressure readings, such as 130/80, give the maximum and minimum gauge pressures in the arteries, measured in “mm Hg” or “torr.” Blood pressure varies with vertical position within the body. The standard reference point is the upper arm, level with the heart. © 2016 Pearson Education, Ltd.  12.4 © 2016 Pearson Education, Ltd.  Archimedes’s principle: Proof step 1 © 2016 Pearson Education, Ltd.  Archimedes’s principle: Proof step 2 https://www.youtube.com/watch?v=0v86Yk https://www.youtube.com/watch?v=0v86Yk14rf8 14rf8  Archimedes’s principle A body immersed in water seems to weigh less than when it is in air. When the body is less dense than the fluid, it floats. The human body usually floats in water, and a helium-filled balloon floats in air. These are examples of buoyancy, a phenomenon described by Archimedes’s principle:  When a body is completely or partially immersed in a fluid, the fluid exerts an upward force on the body equal to the weight of the fluid displaced by the body. © 2016 Pearson Education, Ltd.  12.5 © 2016 Pearson Education, Ltd.  Surface tension The surface of the water acts like a membrane under tension, allowing this water strider to “walk on water.” © 2016 Pearson Education, Ltd.  Surface tension A molecule at the surface of a liquid is attracted into the bulk liquid, which tends to reduce the liquid’s surface area. © 2016 Pearson Education, Ltd.  Surfactant © 2016 Pearson Education, Ltd. © 2016 Pearson Education, Ltd.  Fluid flow The path of an individual particle in a moving fluid is called a flow line. In steady flow, the overall flow pattern does not change with time, so every element passing through a given point follows the same flow line. In steady flow no fluid can cross the side walls of a given flow tube. © 2016 Pearson Education, Ltd.  Fluid flow In laminar flow, adjacent layers of fluid slide smoothly past each other and the flow is steady. At sufficiently high flow rates, the flow can become turbulent. In turbulent flow there is no steady-state pattern; the flow pattern changes continuously. © 2016 Pearson Education, Ltd.  The continuity equation The figure at the right shows a flow tube with changing cross-sectional area. The continuity equation for an incompressible fluid is A1v1 = A2v2. The volume flow rate is dV/dt = Av. © 2016 Pearson Education, Ltd.  The continuity equation The continuity equation helps explain the shape of a stream of honey poured from a spoon. © 2016 Pearson Education, Ltd.  Bernoulli’s equation Bernoulli’s equation is: It is due to the fact that the work done on a unit volume of fluid by the surrounding fluid is equal to the sum of the changes in kinetic and potential energies per unit volume that occur during the flow. © 2016 Pearson Education, Ltd.  Why healthy giraffes have high blood pressure Bernoulli’s equation suggests that as blood flows upward at roughly constant speed v from the heart to the brain, the pressure p will drop as the blood’s height y increases. For blood to reach the brain with the required minimal pressure, the giraffe’s maximum (systolic) blood pressure must be 280 mm Hg! © 2016 Pearson Education, Ltd.  The Venturi meter The pressure is less at point 2 because the fluid flow speed is greater. © 2016 Pearson Education, Ltd.  Lift on an airplane wing Bernoulli’s principle helps to explain how airplanes fly. © 2016 Pearson Education, Ltd.  Lift on an airplane wing Computer simulation of air parcels flowing around a wing, showing that air moves much faster over the top than over the bottom. © 2016 Pearson Education, Ltd.  Viscosity Viscosity is internal friction in a fluid. Due to viscosity, the speed is zero at the pipe walls (to which the fluid clings) and is greatest at the center of the pipe. © 2016 Pearson Education, Ltd.  Viscosity Lava is an example of a viscous fluid. The viscosity decreases with increasing temperature: The hotter the lava, the more easily it can flow. © 2016 Pearson Education, Ltd.  Viscosity © 2016 Pearson Education, Ltd.  Turbulence At low flow speeds, the flow of a fluid of a given viscosity is laminar. When a critical speed is exceeded, however, the flow pattern becomes turbulent. © 2016 Pearson Education, Ltd.  Listening for turbulent flow Normal blood flow in the human aorta is laminar, but a small disturbance such as a heart pathology can cause the flow to become turbulent. Turbulence makes noise, which is why listening to blood flow with a stethoscope is a useful diagnostic technique. © 2016 Pearson Education, Ltd.