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### Chapter 15: Fluid Motion

- Characteristics of fluids

- Microscopically molecules of a fluid do not have long-range
- order. But liquids do have short-range order unlike gases

- Fluids can flow and conform to the boundaries of a container

Fluids

- Fluids cannot sustain a shearing stress

- Density for

mass

,volume

for uniform solid or liquid

unit kg/m3 (at STP for a gas)

in SI unit

Fluids (cont’d)

- Pressure for area

and normal force

in SI unit

unit pascal

other useful units:

height of a column of Hg corresponding to this pressure (mm)

Pressure of a liquid at rest (uniform density)

fluid level

A

Pressure

weight of the imaginary box

imaginary

box

A

Pressure of a fluid at rest : gauge pressure

atmospheric

pressure

gauge pressure:

fluid level

A

Pressure (cont’d)

imaginary

box

atmospheric pressure

A

Pressure measurement using a liquid:

(measures absolute pressure)

From

as

atmospheric

pressure

Barometer

If the liquid is mercury, for 1 atm :

Pressure measurement using a liquid :

(measures gauge pressure)

level 1

Manometer

gauge pressure

level 2

liquid

tank

manometer

- Pressure applied to an enclosed fluid is transmitted

undiminished to every portion of the fluid and walls

of the containing vessel

weight

piston

(area A)

pressure at P

Pascal’s law

pressure due

to weight w:

w/A

atmospheric

pressure

w

liquid

pressure due to

liquid above P

P

Pascal’s law (cont’d)

- Consider a submerged massless object filled with the same fluid
- as the fluid that surrounds the object

The object is at rest

mass of fluid in the object

buoyancy

opposite dir.

Buoyancy

buoyant force

- Now fill the object with another material

mfg

- Consider a portion of fluid at rest in a container surrounded by an
- imaginary boundary represented in dashed line.
- Since the portion of the fluid defined by the surface in dashed line
- is at rest, the net force on this portion due to pressure must be
- equal to that of the weight of the fluid inside the surface, and
- opposite in direction.

Buoyancy

Fnet

buoyant force

X

mg

- The same argument can be applied when the
- imaginary portion of the fluid is replaced by an
- object that occupies the same space.

- The buoyant force on a partly or completely submerged object
- is equal to the weight of the displaced fluid:

Buoyancy

Object sinks

- If

apparent weight

- If

Object floats

The object will rise until a part of it comes out above the fluid

Surface when the average density increases to

What fraction of an iceberg is submerged in the sea water?

Let’s assume that the total volume of the iceberg is Vi. Then the weight of the iceberg Fgi is

Let’s then assume that the volume of the iceberg submerged in the sea water is Vw. The buoyant force B caused by the displaced water becomes

Since the whole system is at its static equilibrium, we obtain

Therefore the fraction of the volume of the iceberg submerged under the surface of the sea water is

About 90% of the entire iceberg is submerged in the water!!!

- A fake or pure gold crown?

Is the crown made of pure

gold?

Tair =7.84 N

Twater =6.86 N

rgold=19.3x103 kg/m3

- Floating down the river

What depth h is the bottom of

the raft submerged?

A=5.70 m2

rwood=6.00x102 kg/m3

- Incompressible ( density is constant at any position)
- No internal friction (no viscosity)
- Steady (non-turbulent) flow- the velocity at a point is
- constant in time.

flow tube

flow line : The path of an individual particle in a

moving fluid

steady flow: A flow whose pattern does not change

with time. Every element passing

through a given point follows the same

flow line

streamline : A curve whose tangent at any point is

in the direction of the fluid velocity at

that point

flow tube : The flow lines passing through the edge

of an imaginary area such as A

Ideal fluid flow

A

flow lines

Continuity equation I (incompressible fluid)

The mass of a moving fluid does not change as it flows.

The volume of the fluid that passes through

area A during a small time interval dt :

flow tube

Continuity equation

- In an ideal fluid the density is constant.
- In a time interval dt the mass that flows
- into Area 1 is the same as the mass that
- flows out of Area 2.

Bernoulli’s equation

The velocity of the fluid coming out of a hole in a tank as

shown in the figure can be calculated using Bernoulli’s

equation.

At the top surface the velocity

of the fluid is zero. The pressures

at the top surface and at the hole

are the same, namely, the

atmospheric pressure.

Example

Suppose a U-shaped piece of pipe is completely submerged in

water, filled with water, and then turned upside down under water.

As you slowly pull the top of the U-shaped piece of pipe out of

water, the water does not run out of the pipe. WHY?

Example

Suppose a U-shaped piece of pipe is completely submerged in

water, filled with water, and then turned upside down under water.

As you slowly pull the top of the U-shaped piece of pipe out of

water, the water does not run out of the pipe. WHY?

Example

Air cannot enter the pipe. As the water starts running out of the pipe,

a near vacuum is created in the topmost region of the inverted U. The

pressure here drops to near zero. The atmospheric pressure on the

surface of the water in the bucket pushes the water into the U-shaped

pipe.

If a U-shaped hose or pipe connects a liquid-filled container at a

higher altitude to a container at a lower altitude over a barrier, the

liquid can be siphoned into the container at the lower altitude.

Atmospheric pressure helps to push the liquid over the barrier.

Example

A water hose 2.50 cm in diameter is used by a gardener to fill a

30.0-liter bucket. The gardener notices that it takes 1.00 min to fill

the bucket. A nozzle with an opening of cross-sectional area 0.500

cm2 is then attached to the hose. The nozzle is held so that water

is projected horizontally from a point 1.00 m above the ground. Over

what horizontal distance can the water be projected?

Example

y1 =3.00 m

- Example : A water tank

- Consider a water tank with a hole.
- Find the speed of the water
- leaving through the hole.

y

x

(b) Find where the stream hits the ground.

Viscosity is internal friction in a fluid, and viscous forces oppose

the motion of one portion of a fluid relative to another.

Viscosity and turbulence

If a fluid in laminar flow flows around an obstacle, it exerts a viscous

drag on obstacle. Frictional forces accelerate the fluid backward

against the direction of flow and the obstacle forward in the direction

of flow.

adjacent layers of fluid slide

smoothly past each other and

flow is steady

laminar flow

Viscosity and turbulence

When the speed of a flowing

fluid exceeds a certain critical

value the flow is no longer laminar.

The flow patter becomes extremely

irregular and complex, and it changes

continuously in time. There is no

steady flow pattern. This chaotic flow

Is called turbulence.

Viscosity and turbulence

The upper edge of a gate in a dam runs

the water surface. The gate is 2.00 m high

and 4.00 m wide and is hinged along the

horizontal line through its center. Calculate

the torque about the hinge arising from

the force due to the water.

2.00 m

4.00 m

Solution

Problems

Denote the width and depth at the bottom of the gate by w and H.

The force on a strip of vertical thickness dh at a depth h is:

and the torque about the hinge is

After integrating from h=0 to h=H, you get the torque:

- An object with height h, mass M, and a uniform cross-sectional area A
- floats upright in a liquid with density r.
- Calculate the vertical distance from the surface of the liquid to the
- bottom of the floating object in equilibrium.
- (b) A downward force with magnitude F is applied to the top of the object.
- At the new equilibrium position, how much farther below the surface of
- the liquid is the bottom of the object than it was in part (a)?
- Calculate the period of the oscillation when the force F is suddenly
- removed.

Solution

(a) From Archimedes’s principle so

(b) The buoyant force is: With the result of part (a) solving

for x gives:

(c) The force is always in the direction toward the equilibrium, namely, a

restoring force Therefore the “spring constant” is

and the period of the oscillation is

You cast some metal of density rm in a mold, but you are worried that there

might be cavities within the casting. You measure the weight of the casting

to be w, and the buoyant force when it is completely surrounded by water

to be B. (a) Show that is the total volume

of any enclosed cavities. (b) If your metal is copper, the casting’s weight

is 156 N, and the buoyant force is 20 N, what is the total volume of any

enclosed cavities in your casting? What fraction is this of the total volume

of the casting?

Solution

(a) Denote the total volume V. If the density of air is neglected, the buoyant

force in terms of the weight is:

Therefore

The total volume of the casting is

(b)

The cavities are 12.4% of the total volume.

A

A

A U-shaped tube with a horizontal portion of

length contains a liquid. What is the difference

in height between the liquid columns in the

vertical arms (a) if the tube has an acceleration

toward the right? (b) if the tube is mounted on

a horizontal turntable rotating with an angular

speed with one of the vertical arms on the

axis of rotation?

Solutions

- Consider the fluid in the horizontal part of the tube. This fluid with mass
- , is subject to a net force due to the pressure difference between
- the ends of the tube, which is the difference between the gauge pressures
- at the bottoms of the ends of the tubes. Now this difference is
- and the net force on the horizontal part of the fluid is
- or
- (b) Similarly to (a) consider the fluid in the horizontal part of the tube. As in (a)
- the fluid is accelerating. The center of mass has a radial acceleration of
- magnitude so the difference in heights between the columns
- is

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