Chapter 14
PHYSICS 2048C
Fluids
What Is a Fluid?
A
fluid, in contrast to a solid, is a
substance that can flow.
 Fluids conform to the boundaries of any
container in which we put them.
Density and Pressure

Density
To find the density ρ of a fluid at any
point,
we isolate a small volume element ΔV
around that point and measure the
mass Δm of the fluid contained within
that element. The density is then

Some Densities
Pressure

We define the pressure on
the piston from the fluid as
where F is the magnitude of the
normal force on area A.
The SI unit of pressure is the Newton
per square meter, which is given a
special name, the Pascal (Pa).
Some Pressures
Fluids at Rest

Here we want to find an expression
for hydrostatic pressure as a
function of depth or altitude.
m = ρV
Fluids at Rest

The pressure at a point in a fluid in
static equilibrium depends on the
depth of that point but not on any
horizontal dimension of the fluid or
its container.
To find the atmospheric pressure at a
distance d above level 1
Example
 The
U-tube, example 3 page 364, contains
two liquids in static equilibrium: Water of
density ρw (= 998 kg/m3) is in the right arm,
and oil of unknown density ρx is in the left.
Measurement gives l = 135 mm and d = 12.3
mm. What is the density of the oil?
Example
 In
the right arm, the interface is a
distance l below the free surface of the
water
 In the left arm, the interface is a
distance l + d below the free surface of
the oil
Measuring Pressure

The Mercury Barometer a device used to measure the pressure of
the atmosphere
the long glass tube is filled with mercury and inverted with its open
end in a dish of mercury, as the figure shows. The space above the
mercury column contains only mercury vapor, whose pressure is so
small at ordinary temperatures that it can be neglected.
Measuring Pressure


The Open-Tube Manometer
It is a device used to measure the gauge pressure pg of a gas
It consists of a U -tube containing a
liquid, with one end of the tube
connected to the vessel whose gauge
pressure we wish to measure and the
other end open to the atmosphere. We
can find the gauge pressure in terms of
the height h
Pascal's Principle

A change in the pressure applied to an enclosed
incompressible fluid is transmitted undiminished to
every portion of the fluid and to the walls of its
container.

Example for that in your daily live is When you
squeeze one end of a tube to get toothpaste out the
other end
Demonstrating Pascal's
Principle
If you add a little more lead shot to
the container to increase Pext by an
amount ∆pext . The quantities ρ, g, h
are unchanged.
Pascal's Principle and the
Hydraulic Lever

The Equation shows that the
output force Fo on the load
must be greater than the
input force Fi if Ao > Ai, as is
the case in the figure
If we move the input piston
downward a distance di, the output
piston moves upward a distance do,
such that the same volume V of the
incompressible liquid is displaced at
both pistons
Pascal's Principle and the
Hydraulic Lever
we can write the output work as
With a hydraulic lever, a given force
applied over a given distance can be
transformed to a greater force applied over
a smaller distance.
Archimedes' Principle
Plastic sack

This net upward force is a
buoyant force . It exists
because the pressure in the
surrounding water increases
with depth below the surface.
Archimedes' Principle
 When
a body is fully or partially submerged
in a fluid, a buoyant force from the
surrounding fluid acts on the body. The
force is directed upward and has a
magnitude equal to the weight mf g of the
fluid that has been displaced by the body.
Apparent Weight in a Fluid

A spherical, helium-filled balloon has a radius R of
12.0 m. The balloon, support cables, and basket
have a mass m of 196 kg. What maximum load M
can the balloon support while it floats at an altitude
at which the helium density ρHe is 0.160 kg/m3 and
the air density ρair is 1.25 kg/m3? Assume that only
the balloon displaces air.
Solution
 Fg
= mfg
ρ = m/V
The Equation of Continuity

the speed v of the water
depends on the crosssectional area A through
which the water flows
The Equation of Continuity
This relation between speed and crosssectional area is called the equation of
continuity for the flow of an ideal fluid.
It tells us that the flow speed increases
when we decrease the cross-sectional
area through which the fluid flows
The Equation of Continuity
 RV is
the volume flow rate of the fluid
Rm is the mass flow rate
Bernoulli's Equation

If the speed of a fluid element
increases as the element
travels along a horizontal
streamline, the pressure of
the fluid must decrease, and
conversely.
Proof of Bernoulli's Equation
work done on the system is then p1ΔV, and the
work done by the system is –p2 ΔV.
The
Proof of Bernoulli's Equation
 Bernoulli's
Equation
Problem
 block
of wood floats in fresh water with
two-third of its volume V submerged
and in oil with 0.90 V submerged. Find
the density of (a) the wood and (b) the
oil.
Solution
 a-
Let V be the volume of the block. Then,
the submerged volume is Vo= 2/3(V)
 According to Archimedes' principle the
weight of the displaced water is equal to the
weight of the block, so
b- If ρo is the density of the oil, then Archimedes'
principle yields