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. That leaves us with gases and liquids. 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. Materials that exhibit the property to flow are called fluids.

Density m ρ = v SI units = Kg/m3 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. The density of an substance is the mass per unit volume of the substance. Because this uses mass, it is called the mass density. If it uses weight, it is called weight density. m ρ = SI units = Kg/m3 v rho

Densities of Common Substances kg/m3 Hydrogen 0.0899 Helium 0.179 Steam (100 oC) 0.598 Air 1.29 Oxygen 1.43 Carbon Dioxide 1.98 Ethanol 8.06 x 102 Ice 9.17 x 102 Fresh water 1.00 x 103 Sea water 1.025 x 103 Iron 7.86 x 103 Mercury 13.6 x 103 Gold 19.3 x 103

Specific Gravity The specific gravity of a substance is a ratio of a substance’s density to that of water. It gives us an easier scale for comparison of whether objects will float or not. Example: Lead has a specific gravity of 11.4. Which means that lead is 11.4 times more dense than water. Water has a density of 1.00 x 103 kg/m3, so lead has a density of 11.4 x 103 kg/m3. Or water has a density of 1 g/cm3, so lead has a density of 11.4 g/cm3. The object will the larger specific gravity will sink!

Buoyancy 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 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

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. 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. FB = Fg(displaced fluid) = mfg Buoyant Force Mass Fluid = Vf ρf

Buoyant Force on Floating Objects 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. Use the following for an object that is floating on top of the fluid. The object is not totally submerged. FB = Fg (object) = mog Mass of Object

Unknown Densities/Objects When faced with the challenge of identifying an unknown substance, we compare its density to that of water. To do this, we use the Buoyant Force of water to determine the density of the unknown substance. This is done by comparing the weight of the object in air and then again in water. The difference between the two weights would show the Buoyant Force. Use this when finding identifying an unknown substance by its density FB = Fg (in air) - Fg (in water) object Fg (in air) = fluid FB

Pressure F P = A 105 Pa = 1 atm = 1 bar = 29.92 in Hg = 14.7 psi 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/m2. The air around us pushes with a pressure. This is called atmospheric pressure, which is about 105 Pa. That amount gives us another unit, the atmosphere (atm). 105 Pa = 1 atm = 1 bar = 29.92 in Hg = 14.7 psi

Pascal’s Principle 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. Pinc = F1 = F2 Pressure in a closed container A1 A2

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. 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. Since the water is suspended, the two pressures are equal. If one becomes larger, the object will sink or float.

Fluid Pressure Equation Pressure varies with the depth in a fluid. 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. F mg ρVg ρAhg ρhg P = = = = = A A A A And taking atmospheric pressure into account, we get the following. P = Po + ρhg

Temperature Temperature is a measure of the average kinetic energy of the particles in a substance. There are actually two SI units for temperature, kelvin (K) and degrees Celsius (oC). To convert, add 273 to the Celsius measurement. Fahrenheit is the units for temperature in the United States.

Ideal Gas Law PV = nRT PV = NkBT The ideal gas law varies slightly for physics versus chemistry. That is due to Boltzmann’s Constant (kB). kB = 1.38 x 10-23 J/K Chemistry’s version uses the ideal gas constant (R). R = 8.31 J/(mol *K) PV = nRT PV = NkBT n is number of moles P is pressure (N/m2) V is volume (m3) N is number of particles T is temperature (K)

3rd Version of Ideal Gas Law Assuming the amount of gas (N1= N2) remains constant in a closed container, we can derive a 3rd version of the ideal gas law. This version will also help us to see the basis for the 3 gas laws from chemistry (Boyle’s Law, Charles’ Law, and Gay-Lussac’s Law). Notice that And since N1=N2, Leaving us with PV P1V1 P2V2 P1V1 P2V2 N= = = kBT kBT1 kBT2 T1 T2 Because kB is constant, it cancels itself out

Fluid Flow Fluid flows in one of two ways: Laminar flow is when every particle of fluid follows the same smooth path. 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. 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.

Continuity Equation Due to the conservation of mass, the amount of fluid as it flows through a chamber is consider to also be conserved. So m1= m2 But the mass of a gas is hard to find and we do know the density and the space it takes up. ρV1= ρV2 But what happens when the chamber changes size? ρA1Δx1= ρA2Δx2 ρA1v1Δt = ρA2v2Δt 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… A1v1 = A2v2 Continuity Equation

Bernoulli’s Principle The pressure in a fluid decreases as the fluid’s velocity increases. 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. Because the speed increased, the pressure then decreases. Now there is more pressure on the bottom of the wing pushing upward, creating lift!

Bernoulli’s Equation P1 + 1/2ρ1v12 + ρ1gh1 = P2 + 1/2ρ2v22 + ρ2gh2 This equation relates pressure to energy in a moving fluid. 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. P1 + 1/2ρ1v12 + ρ1gh1 = P2 + 1/2ρ2v22 + ρ2gh2 Pressure Velocity Height Density