Chapter 14 Fluids

Fluids at Rest (Fluid = liquid or gas)

Density (  ) unit: kg/m 3 we will consider only  = m/V = constant (incompressible fluid) we will also assume g = constant

Pressure (p) A fluid exerts a force dF normal to any area dA you consider in it. For a fluid at rest, the force is equal and opposite on each side.

Pressure (p) Why is there a force? Microscopically: the fluid particles are in motion and collide with dA Macroscopically: the fluid is at rest

Pressure (p) pressure p is a scalar (no intrinsic direction) reason: it acts normal to any surface dA

Pressure (p) units used: pascal, millibar, atm 1 Pa = 1 N/m 2 1 millibar = 100 Pa 1 atm = 1.013×10 5 Pa

Terminology ‘gauge pressure’ = p – p atm (can be > 0 or < 0) (e.g., read on a car tire pressure gauge) p = absolute pressure (> 0) = atmospheric pressure + gauge pressure = p atm + gauge pressure

Pressure and depth pressure: p coordinate: y pressure decreases with ‘elevation’ y: Derive this result and integrate it

Pressure and depth coordinate: y distance: h > 0 p increases with depth for any shape of vessel Demonstration: depth and shape of container

Exercise The dangers of a long snorkel tube: Find the gauge pressure at the depth shown. Will this cause the snorkeler’s lungs to collapse? Demonstration: atmospheric pressure

Pascal’s Law If any change in pressure  p is applied at one point, it is transmitted to all points in the fluid and to walls enclosing it.

Example: Hydraulic Lift At equilibrium, p = F 1 /A 1 = F 2 /A 2 Demonstration

Two Pressure Gauges Notes on (a) and Exercise 14-9 Notes on (b) first

Homework Hints: Exercise 14-55

Buoyancy and Buoyant Force

A (fully or partially) submerged object feels an upward force equal to the weight of fluid it displaces

(a) fluid element with weight w fluid (b) body of same shape feels buoyant force B = w fluid Demonstration

Surface Tension Molecules of liquid attract each other (else no definite volume) center: net force = 0 surface: net force is directed inward

Surface Tension So the surface acts like a membrane under tension (like a stretched drumhead) The surface resists any change in surface area Strength characterized by ‘surface tension’ Demonstration

Surface Tension   = F/d = cohesive force per unit length surface tension force F =  d We can measure  by just balancing F Do Example Notes on measuring 

cohesion: attraction of like molecules example: liquid-liquid forces (surface tension)

adhesion: attraction between unlike molecules example: liquid-glass forces

(a) adhesion > cohesion: water wets glass (b) adhesion < cohesion: mercury beads up

Capillarity For these two cases, the surface tension force F pushes the column of liquid either up or down: (a) up for water (b) down for mercury Notes on capillary tubes

Homework Announcements Homework Set 5: Correction to hints for (handout at front and on webpage) Recent changes to classweb access (see HW 5 sheet at front and webpage) Homework Sets 1, 2, 3: returned at front (scores to be entered on classweb soon)

Midterm Announcements Friday: review required topics practice problems (from class, HW, new?) Monday: (midterm) you can bring a sheet of notes (both sides) you will be given a list of equations

Fluid Flow (Fluid Dynamics)

Flow Fluid Flow line = path of fluid element Flow tube = bundle of flow lines passing through area A (just a useful construct)

Simplifying Assumptions Steady flow: At any given point in the fluid, its properties (v, , p) don’t change in time

Simplifying Assumptions Steady flow: different flow lines never cross each other fluid entering a flow tube never leaves it

Simplifying Assumptions Incompressible fluid:  = constant No friciton: no ‘viscosity’

Continuity Equation A 1 v 1 = A 2 v 2 the same volume dV of fluid enters and exits tube: dV = volume passing through A in dt = Av dt Notes

Continuity Equation A 1 v 1 = A 2 v 2 along the flow: A = area of flow tube v = speed of fluid if one increases, the other must decrease Notes and Demonstration: water flow

Continuity Equation A 1 v 1 = A 2 v 2 Where the flow lines are crowding together, the fluid speed is increasing

Bernoulli’s Equation only valid for: steady flow, incompressible fluid, no viscosity! Notes

Bernoulli’s Equation if v 1 = v 2 = 0: reduces to previous result for fluid at rest

Bernoulli’s Equation if y 1 = y 2 then for p and v: if one increases, the other must decrease Demonstration

Applications of Bernoulli’s Equation

Venturi Meter (Example 14-10) horizontal flow tube Notes

Note: if viscosity is present, then v decreases with distance from tube center

Venturi Meter: Homework Problem (c) Notes Demonstration

Wing Lift

Can’t predict flow lines but they indicate low pressure above wing, so net force up Demonstration: propellor

Efflux Speed: vertical flow tube Notes

Siphon: flow tube points up, then down First: you must fill the tube There is a limit: H + h < 10 m

Curve Ball: viscosity makes it possible Warm-up demonstrations

Viscosity drags air with spinning ball: low pressure=net force so the ball curves Demonstration

Homework Announcements Homework Set 5: Correction to hints for (handout at front and on webpage) Recent changes to classweb access (see HW 5 sheet at front and webpage) Homework Sets 1, 2, 3: returned at front (scores to be entered on classweb soon)