Physics 106 Lesson 7 Pressure and Fluid Flow

Physics 106 Lesson #7 Pressure and Fluid Flow: Pascal and Bernoulli Dr. Andrew Tomasch 2405 Randall Lab atomasch@umich. edu

Review: Density • Density (Greek letter rho): The mass of uniform object is its density times its volume: m = V water = 1000 kg/m 3 (SI) water = 1 g/cm 3 (CGS) The specific gravity is density expressed in units of the density of water. The specific gravity of x ≡ x / water

Review: Buoyancy • The upward buoyant force helps to balance F B the downward weight, reducing the tension in the string, which is the apparent weight of the block → apparent weight is less than the true weight T + W

Review: Archimedes’ Principle Magnitude of Buoyant Force Weight of Displaced Fluid • The buoyant force acting on a body is equal to the weight of the fluid it displaces

Review: Archimedes’ Principle Applied • An object sinks if ρobject > ρfluid • An object floats if ρobject < ρfluid • An object is neutrally buoyant if ρobject = ρfluid FB ρobject Vobject ρfluid W Completely immerse an object of volume Vobject with density ρobject in a fluid of density ρfluid

Review: Floating Objects • An object floats with some portion of its volume protruding from the liquid • When Floating in Equilibrium: FB Vin W

Review: Surface Tension • Water molecules have an uneven distribution of electrical charge, slightly positive at one end, slightly negative at the other. • The molecules on the water’s surface Demo: soap powered boat stick together because of electrical forces and behave like a membrane. • Small objects more dense than water can be supported on the water’s surface. • Soaps and detergents disrupt this surface tension.

Introduction to Fluids • Atoms in fluids can change relative positions easily, unlike those in solids • Push on a fluid and it will move in any direction to release the pressure, not just away from you • Fluids have a definite volume but no definite shape

Pressure is a scalar. Area is a vector. The direction of an element of area is perpendicular to the surface. • Pressure P is the force perpendicular to a surface divided by the area of the surface: A difference in pressure across a surface or object exerts a net force perpendicular to the surface. Units of pressure: N/m 2 ≡ Pascals (Pa) (also mm or inches of mercury and lbs/in 2)

How Fluids Exert Pressure • Due to their thermal motion, molecules are continuously bombarding the walls of a container even when there is no bulk movement of the air (wind): container • Force = for collisions • Molecules colliding with the container walls produce a pressure (force/area) air molecule

Fluids: Not Just Liquids • We are talking about fluids, not just liquids – air is a fluid – fluids like air (a gas) and water (a liquid) differ primarily in the separation between atoms – in liquids, the atoms are not as far apart as in gasses, making them difficult to compress Fluids Include Liquids & Gases incompressible fluids

Atmospheric Pressure • The weight of the overlying air produces an atmospheric pressure at any depth in the Earth’s atmosphere At sea level atmospheric pressure is: Patm = 1. 013 x 105 Pa ≡ 1 atm gravity atmosphere ground Demo: Magdeburg Sphere Equivalent to the weight of a 10, 000 kg (22 ton) mass distributed over 1 square meter (!)

Hydrostatic Pressure • Hydrostatic Pressure – for a fluid at rest the hydrostatic pressure at a depth h is the weight of the fluid above an area A divided by the area A – the total pressure at depth h is the hydrostatic pressure plus the atmospheric pressure Patm at the top of the fluid: Patm volume h Area = A AKA Pascal’s Law

Pascal’s Principle • We have just seen that the pressure at a depth h in a fluid is the sum of atmospheric pressure applied above the fluid and the hydrostatic pressure: Pascal • This is an example of Pascal’s Principle: “Any change in the pressure applied to a completely enclosed fluid is transmitted undiminished to all parts of the fluid and vessel walls” • Atmospheric pressure is transmitted throughout the fluid and to the vessel walls by Pascal’s Principle

Pascal’s Law Caution Quiz Ahead • At a depth h below the surface of an incompressible fluid: Pascal Gauge Pressure ≡ Difference From Atmospheric The pressure in a static fluid is the same at all points that have the same depth regardless of the container’s Demonstration shape: P A = P B = P C= P D

This is a mercury barometer The pressure at the bottom of the mercury column (A) is the density of mercury x g x h and is the same as at the top of the mercury (B) which is at atmospheric pressure and at the same height. Water has 1/13 the density of mercury, so a water barometer is 13 times taller or 32 feet tall! Concept Test #1 A static fluid in a container is subject to both atmospheric pressure at its surface and Earth’s gravitation. The pressure at A: A. Depends on the cross sectional area of the container B. Depends on the shape of the container C. Is equal to atmospheric pressure 760 mm Hg = 1 Atm = 29. 92 inches Hg

Practical Hydrostatics • Why do your ears hurt when dive into deep water? If you dive to the bottom of the deep end of a pool, 3 m or so, you’ll feel uncomfortable pressure in your ears. This depth corresponds to a gauge pressure of (1000 kg/m 3)(9. 8 m/s 2)(3 m) = 2. 9 x 104 Pa, about a 30% increase over atmospheric pressure. This pressure increase is enough to compress gas inside your eardrum so that is bends inward in a painful way.

Practical Hydrostatics • Why are snorkels always so short? Ever wondered by no one markets 10 ft snorkels? When you are swimming at a depth h, the pressure outside you in the water is Patm+ gh. Inside your lungs, which are directly connected to the air by the snorkel, the pressure is Patm. So your lungs have to breathe against a gauge pressure of gh. You can expand your chest against a pressure only a small fraction above atmospheric pressure. At a depth of 3 m you would injure your lungs. At a depth of 10 m your lungs would implode as the outside water pressure crushed your chest. This is why scuba divers breathe carefully regulated pressurized gas.

Practical Hydrostatics • How does a straw work? Do you really suck a drink up into a straw? Actually, you remove the air from inside the straw thereby lowering the pressure inside to something close to zero. Atmospheric pressure at the surface of the drink acts unopposed to push the drink up into the straw.

Applied Hydrostatics: The Hydraulic Lift • Pascal’s Principle: the pressure induced by pushing down on a fluid is transmitted equally throughout the fluid • This fact is the basis for many useful devices including the hydraulic brakes in your car and the hydraulic jack shown here • A small force applied to the small area A 1 generates a pressure that in turn can apply a large force on a large area A 2 A B

Continuity in Fluid Flow • Mass flowing in = mass flowing out per unit time • For an incompressible fluid mass is equivalent to volume because density is constant What Flows In Must Flow Out Units: m 3/s (volume in)/time = (volume out)/time

Conservation of Mass: The Equation of Continuity Caution Quiz Ahead (mass in) / time Incompressible Fluid (mass out) / time The product of the crosssectional area and flow speed is everywhere the same.

Concept Test #2 A river is 10 m wide and 2 m deep at a certain point where the speed of the flow is 2 m/s. A little later on, it’s 15 m wide and 1 m deep. What is the speed of the flow there? A. B. C. D. 2. 67 m/s 1. 34 m/s 6. 33 m/s 4. 25 m/s

Conservation of Energy: Bernoulli's Equation • Work-Energy Theorem for a drop of fluid: Dividing through by the volume of the drop replaces the mass of the drop with the density of the fluid. nc ≡ nonconservative nc work/volume Gravitational Potential Energy per Volume Kinetic Energy per Volume Bernoulli’s Equation Bernoulli’s equation is the statement of energy conservation in a moving fluid and is shown here for background only. Note that to apply a force to the fluid (and do nonconservative work) requires a pressure difference across the pipe

Bernoulli’s Equation: A Surprising Result • Flow in a horizontal pipe: Continuity Demos: Bernoulli Effects and Venturi Tube Bernoulli As the speed of a fluid increases over a surface, the pressure of the fluid against the surface decreases.

Example: A Damaged Heart Bernoulli’s equation dictates that as the speed of the flowing blood increases through a narrowed area of the vessel to satisfy continuity, the pressure on the inside of the vessel wall decreases, resulting in a tendency to collapse the vessel wall.

Example: Bernoulli in the Shower • The Shower-Curtain Effect: The shower curtain sometimes gets sucked in towards the shower when the water flows • The moving water induces moving currents of air on the shower side of the curtain, which lowers the air pressure slightly on that side

Example: A Leak • Suppose a tank open to the air has a leak a distance h below the surface. • Q: How does the speed v 2 of the exiting water compare with that of a ball dropped from a height h ? • A: It is exactly the same. An exiting drop of water carries away the gravitational potential energy lost from the top of the water as kinetic energy. From a work-energy standpoint, each exiting drop has simply fallen a distance h under the influence of gravity! Patm Point 1: v 1 ≈ 0 h Point 2 PE=0 Here Patm v 2

More Examples If an airplane flies too slowly and/or at too high an angle of attack, turbulent flow occurs over the top of the wing and the wing stalls resulting in a loss of lift and a large increase in drag Most of the lift a wing produces actually comes from deflecting the momentum of the moving air.