Chapter 15, Fluids

Chapter 15, Fluids • This is an actual photo of an iceberg, taken by a rig manager for Global Marine Drilling in St. Johns, Newfoundland. The water was calm and the sun was almost directly overhead so that the diver

Physics 207, Lecture 20, Nov. 10 Goals: • Chapter 15 • Understand pressure in liquids and gases • Use Archimedes’ principle to understand buoyancy • Understand the equation of continuity • Use an ideal-fluid model to study fluid flow. • Investigate the elastic deformation of solids and liquids • Assignment • HW9, Due Wednesday, Nov. 19th • Wednesday: Read all of Chapter 16

Fluids (Ch. 15) • At ordinary temperature, matter exists in one of three states • Solid - has a shape and forms a surface • Liquid - has no shape but forms a surface • Gas - has no shape and forms no surface • What do we mean by “fluids”? • Fluids are “substances that flow”…. “substances that take the shape of the container” • Atoms and molecules are free to move. • No long range correlation between positions.

Fluids • An intrinsic parameter of a fluid • Density units : kg/m3 = 10-3 g/cm3 r(water) = 1.000 x 103 kg/m3 = 1.000 g/cm3 r(ice) = 0.917 x 103 kg/m3 = 0.917 g/cm3 r(air) = 1.29 kg/m3 = 1.29 x 10-3 g/cm3 r(Hg) = 13.6 x103 kg/m3 = 13.6 g/cm3

n A Fluids • Another parameter: Pressure • Any force exerted by a fluid is perpendicular to a surface of contact, and is proportional to the area of that surface. • Force (avector) in a fluid can be expressed in terms of pressure (a scalar) as:

What is the SI unit of pressure? • Pascal • Atmosphere • Bernoulli • Young • p.s.i. Units : 1 N/m2 = 1 Pa (Pascal) 1 bar = 105 Pa 1 mbar = 102 Pa 1 torr = 133.3 Pa 1 atm = 1.013 x105 Pa = 1013 mbar = 760 Torr = 14.7 lb/ in2 (=PSI)

Pressure vs. DepthIncompressible Fluids (liquids) • When the pressure is much less than the bulk modulus of the fluid, we treat the density as constant independent of pressure: incompressible fluid • For an incompressible fluid, the density is the same everywhere, but the pressure is NOT! • p(y) = p0 - y g r = p0 + d g r • Gauge pressure (subtract p0, usually 1 atm) F2 = F1+ m g = F1+ rVg F2 /A = F1/A + rVg/A p2 = p1 - rg y

Pressure vs. Depth • For a uniform fluid in an open container pressure same at a given depthindependentof the container • Fluid level is the same everywhere in a connected container, assuming no surface forces

Pressure Measurements: Barometer • Invented by Torricelli • A long closed tube is filled with mercury and inverted in a dish of mercury • The closed end is nearly a vacuum • Measures atmospheric pressure as One 1 atm = 0.760 m (of Hg)

(A)r1 < r2 (B)r1 = r2 (C)r1>r2 Exercise Pressure • What happens with two fluids?? • Consider a U tube containing liquids of density r1 and r2 as shown: Compare the densities of the liquids: dI r2 r1

(A)r1 < r2 (B)r1 = r2 (C)r1 > r2 Exercise Pressure • What happens with two fluids?? • Consider a U tube containing liquids of density r1 and r2 as shown: • At the red arrow the pressure must be the same on either side. r1 x = r2 (d1+ y) • Compare the densities of the liquids: dI r2 y r1

W2? W1 W1 > W2 W1 < W2 W1 = W2 Archimedes’ Principle • Suppose we weigh an object in air (1) and in water (2). How do these weights compare? • Buoyant force is equal to the weight of the fluid displaced

W2? W1 W1 > W2 W1 < W2 W1 = W2 Archimedes’ Principle • Suppose we weigh an object in air (1) and in water (2). • How do these weights compare? • Why? Since the pressure at the bottom of the object is greater than that at the top of the object, the water exerts a net upward force, the buoyant force, on the object.

y F mg B Sink or Float? • The buoyant force is equalto the weight of the liquid that is displaced. • If the buoyant force is larger than the weight of the object, it will float; otherwise it will sink. • We can calculate how much of a floating object will be submerged in the liquid: • Object is in equilibrium

piece of rock on top of ice Bar Trick What happens to the water level when the ice melts? B. It stays the same C. It drops A. It rises

Exercise V1 = V2 = V3 = V4 = V5 m1 < m2 < m3 < m4 < m5 What is the final position of each block?

Exercise V1 = V2 = V3 = V4 = V5 m1 < m2 < m3 < m4 < m5 What is the final position of each block? But this Not this

Pascal’s Principle • So far we have discovered (using Newton’s Laws): • Pressure depends on depth: Dp = r g Dy • Pascal’s Principle addresses how a change in pressure is transmitted through a fluid. Any change in the pressure applied to an enclosed fluid is transmitted to every portion of the fluid and to the walls of the containing vessel.

Pascal’s Principle in action • Consider the system shown: • A downward force F1 is applied to the piston of area A1. • This force is transmitted through the liquid to create an upward force F2. • Pascal’s Principle says that increased pressure from F1 (F1/A1) is transmitted throughout the liquid. • F2 > F1 with conservation of energy

M dA A1 A10 M dB A2 A10 Exercise Hydraulics: A force amplifierAka…the lever • Consider the systems shown on right. • In each case, a block of mass M is placed on the piston of the large cylinder, resulting in a difference di in the liquid levels. • If A2 = 2A1, compare dA and dB V10 = V1= V2 dA A1= dB A2 dA A1= dB 2A1 dA = dB 2

Fluids in Motion • To describe fluid motion, we need something that describes flow: • Velocity v • There are different kinds of fluid flow of varying complexity • non-steady/ steady • compressible / incompressible • rotational / irrotational • viscous / ideal

Types of Fluid Flow • Laminar flow • Each particle of the fluid follows a smooth path • The paths of the different particles never cross each other • The path taken by the particles is called a streamline • Turbulent flow • An irregular flow characterized by small whirlpool like regions • Turbulent flow occurs when the particles go above some critical speed

Onset of Turbulent Flow The SeaWifS satellite image of a von Karman vortex around Guadalupe Island, August 20, 1999

streamline Ideal Fluids • Fluid dynamics is very complicated in general (turbulence, vortices, etc.) • Consider the simplest case first: the Ideal Fluid • No “viscosity” - no flow resistance (no internal friction) • Incompressible - density constant in space and time • Simplest situation: consider ideal fluid moving with steady flow - velocity at each point in the flow is constant in time • In this case, fluid moves on streamlines

streamline Ideal Fluids • Streamlines do not meet or cross • Velocity vector is tangent to streamline • Volume of fluid follows a tube of flow bounded by streamlines • Streamline density is proportional to velocity • Flow obeys continuity equation Volume flow rate Q = A·vis constant along flow tube. Follows from mass conservation if flow is incompressible. A1v1 = A2v2

(A) 2 v1 (B) 4 v1 (C) 1/2 v1 (D) 1/4 v1 Exercise Continuity • A housing contractor saves some money by reducing the size of a pipe from 1” diameter to 1/2” diameter at some point in your house. v1 v1/2 • Assuming the water moving in the pipe is an ideal fluid, relative to its speed in the 1” diameter pipe, how fast is the water going in the 1/2” pipe?

v1 v1/2 (A) 2 v1 (B) 4 v1 (C) 1/2 v1 (D) 1/4 v1 Exercise Continuity • For equal volumes in equal times then ½ the diameter implies ¼ the area so the water has to flow four times as fast. • But if the water is moving four times as fast then it has 16times as much kinetic energy. • Something must be doing work on the water (the pressure drops at the neck and we recast the work as P DV = (F/A) (ADx) = F Dx )

Lecture 20, Nov. 10 • Question to ponder: Does heavy water (D20) ice sink or float? • Assignment • HW9, Due Wednesday, Nov. 19th • Wednesday: Read all of Chapter 16 Next slides are possible for Wednesday

DV Conservation of Energy for Ideal Fluid This leads to… P1+ ½ r v12= P2+ ½ r v22= const. and with height variations Bernoulli’s Equation P1+ ½ r v12 + r g y1 = constant

(A) smaller (B) same (C) larger Bernoulli’s Principle • A housing contractor saves some money by reducing the size of a pipe from 1” diameter to 1/2” diameter at some point in your house. v1 v1/2 2) What is the pressure in the 1/2” pipe relative to the 1” pipe?

Cavitation Venturi result In the vicinity of high velocity fluids, the pressure can gets so low that the fluid vaporizes.

Applications of Fluid Dynamics • Streamline flow around a moving airplane wing • Lift is the upward force on the wing from the air • Drag is the resistance • The lift depends on the speed of the airplane, the area of the wing, its curvature, and the angle between the wing and the horizontal higher velocity lower pressure lower velocity higher pressure Note: density of flow lines reflects velocity, not density. We are assuming an incompressible fluid.

F L0 L V0 F V0 - V Some definitions • Elastic properties of solids : • Young’s modulus: measures the resistance of a solid to a change in its length. • Bulk modulus: measures the resistance of solids or liquids to changes in their volume. elasticity in length volume elasticity

EXAMPLE 15.11 An irrigation system QUESTION:

EXAMPLE 15.11 An irrigation system

Elasticity

Elasticity F/A is proportional to ΔL/L. We can write the proportionality as • The proportionality constant Y is called Young’s modulus. • The quantity F/A is called the tensile stress. • The quantity ΔL/L, the fractional increase in length, is called strain.With these definitions, we can write

EXAMPLE 15.13 Stretching a wire QUESTIONS:

EXAMPLE 15.13 Stretching a wire

Volume Stress and the Bulk Modulus

Volume Stress and the Bulk Modulus • A volume stress applied to an object compresses its volume slightly. • The volume strain is defined as ΔV/V, and is negative when the volume decreases. • Volume stress is the same as the pressure. where B is called the bulk modulus. The negative sign in the equation ensures that the pressure is a positive number.

The figure shows volume flow rates (in cm3/s) for all but one tube. What is the volume flow rate through the unmarked tube? Is the flow direction in or out? • 1 cm3/s, in • 1 cm3/s, out • 10 cm3/s, in • 10 cm3/s, out • It depends on the relative size of the tubes.

Rank in order, from highest to lowest, the liquid heights h1 to h4 in tubes 1 to 4. The air flow is from left to right. • h1 > h2 = h3 = h4 • h2 > h4 > h3 > h1 • h2 = h3 = h4 > h1 • h3 > h4 > h2 > h1 • h1 > h3 > h4 > h2

Chapter 15, Fluids