Chapter 9 Fluid Mechanics

Chapter Objectives Define fluid Density Buoyant force Buoyantly of floating objects Pressure Pascal's principle Pressure and depth Temperature Fluid flow continuity equation Bernoulli's principle Ideal gas law

What is a Fluid? So far we have studied the causes of motion dealing with solids. So far we have studied the causes of motion dealing with solids. That leaves us with gases and liquids. That leaves us with gases and liquids. Liquids and gases are different phases, but have common properties. Liquids and gases are different phases, but have common properties. One common property of gases and liquids is their ability to flow and alter their shape on the process. One common property of gases and liquids is their ability to flow and alter their shape on the process. Materials that exhibit the property to flow are called fluids. Materials that exhibit the property to flow are called fluids.

Density It is a difficult concept to visualize the mass of a fluid because its shape can change. It is a difficult concept to visualize the mass of a fluid because its shape can change. So a more useful measurement is the density of an object. So a more useful measurement is the density of an object. The density of an substance is the mass per unit volume of the substance. The density of an substance is the mass per unit volume of the substance. Because this uses mass, it is called the mass density. Because this uses mass, it is called the mass density. If it uses weight, it is called weight density. If it uses weight, it is called weight density. ρ = SI units = Kg/m 3 rho m v

Densities of Common Substances Substance kg/m 3 Hydrogen Helium0.179 Steam (100 o C) Air1.29 Oxygen1.43 Carbon Dioxide 1.98 Ethanol 8.06 x 10 2 Ice 9.17 x 10 2 Fresh water 1.00 x 10 3 Sea water x 10 3 Iron 7.86 x 10 3 Mercury 13.6 x 10 3 Gold 19.3 x 10 3

Buoyancy The ability of a substance to float in a liquid is based of the densities of the two substances. The ability of a substance to float in a liquid is based of the densities of the two substances. The less dense substance will move to the top, or float. The less dense substance will move to the top, or float. The force pushing on an object while in a liquid or floating is called the buoyant force. The force pushing on an object while in a liquid or floating is called the buoyant force. The buoyant force acts opposite of gravity, and that is why objects seem “lighter” in water The buoyant force acts opposite of gravity, and that is why objects seem “lighter” in water

Archimedes’ Principle When an object is placed in water, the total volume of water is raised the same volume as the Portion of the Object that is submerged. When an object is placed in water, the total volume of water is raised the same volume as the Portion of the Object that is submerged. Archimedes Principle states the Buoyant Force is equal to the weight of water displaced. Archimedes Principle states the Buoyant Force is equal to the weight of water displaced. Use this formula if the object is totally submerged in the fluid. Use this formula if the object is totally submerged in the fluid. F B = F g (displaced fluid) = m f g Buoyant Force Mass Fluid = V f ρ f

Buoyant Force on Floating Objects For an object to float, the Buoyant Force must be equal magnitude to the weight of the object. For an object to float, the Buoyant Force must be equal magnitude to the weight of the object. The density of the object determines the depth of the submersion. The density of the object determines the depth of the submersion. Use the following for an object that is floating on top of the fluid. Use the following for an object that is floating on top of the fluid. The object is not totally submerged. The object is not totally submerged. F B = F g (object) = m o g Mass of Object

Floating versus Submerged A floating object is partially submerged and partially exposed from the fluid. A floating object is partially submerged and partially exposed from the fluid. So large density objects needs to displace a larger volume of water than their own volume in order to stay afloat. So large density objects needs to displace a larger volume of water than their own volume in order to stay afloat. Large ocean going ships are typically very long and wide to increase the surface area pushing on the water. Large ocean going ships are typically very long and wide to increase the surface area pushing on the water. VoVo VfVf ρoρo ρfρf =

Pressure Pressure is a measure of how much force is applied over a given area. Pressure is a measure of how much force is applied over a given area. The SI Unit for pressure is the Pascal (Pa), which is equal to 1 N/m 2. The SI Unit for pressure is the Pascal (Pa), which is equal to 1 N/m 2. The air around us pushes with a pressure. This is called atmospheric pressure, which is about 10 5 Pa. The air around us pushes with a pressure. This is called atmospheric pressure, which is about 10 5 Pa. That amount gives us another unit, the atmosphere (atm). That amount gives us another unit, the atmosphere (atm) Pa = 1 atm = 1 bar = in Hg = 14.7 psi P = F A

Pascal’s Principle When you pump up a bicycle tire, it just doesn’t grow sideways, but also in height. When you pump up a bicycle tire, it just doesn’t grow sideways, but also in height. Pascal’s Principle states that the pressure applied to a fluid in a closed container is transmitted equally to every point of the fluid and to the walls of the container. Pascal’s Principle states that the pressure applied to a fluid in a closed container is transmitted equally to every point of the fluid and to the walls of the container. P inc = F 1 = F 2 A1A1 A2A2 Pressure in a closed container

Pressure and Depth Water pressure increases with depth because the water at a given depth has to support the weight of the water above it. Water pressure increases with depth because the water at a given depth has to support the weight of the water above it. Imagine an object suspended in a fluid. There is an imaginary column that is the same cross-sectional area of the object. Imagine an object suspended in a fluid. There is an imaginary column that is the same cross-sectional area of the object. There is water trapped below pushing up on the object. The water above is pushing down on the object. There is water trapped below pushing up on the object. The water above is pushing down on the object. Since the water is suspended, the two pressures are equal. Since the water is suspended, the two pressures are equal. If one becomes larger, the object will sink or float. If one becomes larger, the object will sink or float.

Fluid Pressure Equation Pressure varies with the depth in a fluid. Pressure varies with the depth in a fluid. That is because there is a larger column of water above the object pushing downward. That is because there is a larger column of water above the object pushing downward. We must also account for atmospheric pressure pushing down on top of the water. We must also account for atmospheric pressure pushing down on top of the water. P = F A === = ρAhgρAhg ρVgρVgmgmg AA A ρhgρhg And taking atmospheric pressure into account, we get the following. P = ρhgρhg Po +Po +

Fluid Flow Fluid flows in one of two ways: Fluid flows in one of two ways: Laminar flow is when every particle of fluid follows the same smooth path. Laminar flow is when every particle of fluid follows the same smooth path. That path is said to be streamline. That path is said to be streamline. Turbulent flow is when there is irregular flow due to objects in the path or sharp turns in the flowing chamber. Turbulent flow is when there is irregular flow due to objects in the path or sharp turns in the flowing chamber. Irregular motions of the fluid are called eddy currents. Irregular motions of the fluid are called eddy currents. Since laminar flow is predictable and easy to model, we will use its characteristics in this book. Since laminar flow is predictable and easy to model, we will use its characteristics in this book.

Continuity Equation Due to the conservation of mass, the amount of fluid as it flows through a chamber is consider to also be conserved. Due to the conservation of mass, the amount of fluid as it flows through a chamber is consider to also be conserved. So m 1 = m 2 But the mass of a gas is hard to find and we do know the density and the space it takes up. ρV 1 = ρV 2 ρA 1 Δx 1 = ρA 2 Δx 2 But what happens when the chamber changes size? ρA 1 v 1 Δt = ρA 2 v 2 Δt It is hard to measure displacement of a gas, but we can measure the time it takes to travel. Density of the gas will be constant and the time will be constant, so… A 1 v 1 = A 2 v 2 Continuity Equation

Bernoulli’s Principle The pressure in a fluid decreases as the fluid’s velocity increases. The pressure in a fluid decreases as the fluid’s velocity increases. This is the principle responsible for lift. This is the principle responsible for lift. As air flows over the top of the wing, the speed must increase because it travels a longer distance. As air flows over the top of the wing, the speed must increase because it travels a longer distance. Because the speed increased, the pressure then decreases. Because the speed increased, the pressure then decreases. Now there is more pressure on the bottom of the wing pushing upward, creating lift! Now there is more pressure on the bottom of the wing pushing upward, creating lift!

Bernoulli’s Equation This equation relates pressure to energy in a moving fluid. This equation relates pressure to energy in a moving fluid. Since energy is conserved, Bernoulli’s Equation is set to be a constant. Since energy is conserved, Bernoulli’s Equation is set to be a constant. For our use, we will then set Bernoulli’s Equation equal to itself under initial and final conditions. For our use, we will then set Bernoulli’s Equation equal to itself under initial and final conditions. P / 2 ρ 1 v ρ 1 gh 1 = P / 2 ρ 2 v ρ 2 gh 2 Pressure Density Velocity Height