Physics 151: Lecture 30, Pg 1

Physics 151: Lecture 30 Physics 151: Lecture 30 Today’s AgendaToday’s Agenda

Today’s Topic:Fluids in MotionBernoulli’s Equation and applications

Physics 151: Lecture 30, Pg 2

Example / Fluid StaticsExample / Fluid Statics

A beaker of mass mbeaker containing oil of mass moil (density = oil) rests on a scale. A block of iron of mass miron is suspended from a spring scale and completely submerged in the oil as in Figure on the right.

Determine the equilibrium readings of

both scales.

Physics 151: Lecture 30, Pg 3

Fluids in MotionFluids in Motion

Up to now we have described fluids in terms of their static properties:

density pressure p

To describe fluid motion, we need something that can describe flow:

velocity v

There are different kinds of fluid flow of varying complexity non-steady / steady compressible / incompressible rotational / irrotational viscous / ideal

See text: 14.5

Physics 151: Lecture 30, Pg 4

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

A1

A 2

v1

v2

streamline

Ideal FluidsIdeal Fluids

Fluid dynamics is very complicated in general (turbulence, vortices, etc.)

Consider the simplest case first: the Ideal Fluidno “viscosity” - no flow resistance (no internal friction) incompressible - density constant in space and time

See text: 14.5

Physics 151: Lecture 30, Pg 5

Flow obeys continuity equation

volume flow rate Q = A·v is constant along flow tube.

A1

A 2

v1

v2

streamline

A1v1 = A2v2

Ideal FluidsIdeal Fluids streamlines do not meet or cross

velocity vector is tangent to streamline

volume of fluid follows a tube of flow bounded by streamlines

See text: 14.6

• follows from mass conservation if flow is incompressible.

Physics 151: Lecture 30, Pg 6

Recall the standard work-energy relationApply the principle to a section of flowing fluid with

volume V and mass m = V (here W is work done on fluid)

V)pp(xApxApW

21222111pressure

)yy(gV)yy(gmW12

12gravity

)vv(VvmvmKW 21

222

1212

1222

1

22

221

212

121

1 gyvpgyvp Bernoulli Equation

y1

y2

v1

v2

p1

p2

V

pressuregravity WWW

KW

Conservation of Energy for Conservation of Energy for Ideal FluidIdeal Fluid

See text: 14.7

Physics 151: Lecture 30, Pg 7

1) 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?

Lecture 30 Lecture 30 Act 1Act 1ContinuityContinuity

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

a) 2 v1 b) 4 v1 c) 1/2 v1 c) 1/4 v1

Physics 151: Lecture 30, Pg 8

Lecture 30 Lecture 30 Act 2Act 2Bernoulli’s PrincipleBernoulli’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.

2) What is the pressure in the 1/2” pipe relative to the 1” pipe?

a) smaller b) same c) larger

v1 v1/2

Physics 151: Lecture 30, Pg 9

DEMO SLIDE The smaller the diameter the lower is the pressure

Physics 151: Lecture 30, Pg 10

ExampleExample A Pitot tube (see Fig. below) can be used to determine

the velocity of air flow by measuring the difference between the total pressure and the static pressure. If the fluid in the tube is mercury, density Hg = 13 600 kg/m3, and h = 5.00 cm, find the speed of air flow. (Assume that the air is stagnant at point A and take air = 1.25 kg/m3.)

Physics 151: Lecture 30, Pg 11

Venturi Meter

v = ?

Can we know what is v from what we can measure ?

h

Hg

air

Physics 151: Lecture 30, Pg 12

ExampleExample

A tank containing a liquid of density r has a hole in its side at a distance h below the surface of the liquid. The hole is open to the atmosphere and its diameter is much smaller than the diameter of the tank.

What is the speed with of the liquid as it leaves the tank.

h

v=?

Physics 151: Lecture 30, Pg 13

ExampleExample

Figure on the right shows a stream of water in steady flow from a kitchen faucet. At the faucet the diameter of the stream is 0.960 cm. The stream fills a 125-cm3 container in 16.3 s. Find the diameter of the stream 13.0 cm below the opening of the faucet.

Physics 151: Lecture 30, Pg 14

ExampleExample

Water is forced out of a fire extinguisher by air pressure, as shown in Figure below. How much gauge air pressure in the tank (above atmospheric) is required for the water jet to have a speed of 30.0 m/s when the water level in the tank is 0.500 m below the nozzle?

Physics 151: Lecture 30, Pg 15

Recap of today’s lectureRecap of today’s lecture

14.4-7 StreamlinesBernoulli’s Equation and applications