Fluids - DynamicsLevel 1 Physics

Fluid FlowSo far, our discussion about fluids has been when they are at rest. We willNow talk about fluids that are in MOTION.

An IDEAL FLUID is non-viscous No internal friction is incompressible Density R.T.S. is when its motion is steady

A fluid's motion can be said to be STREAMLINE, or LAMINAR. The path itself is called the streamline. By Laminar, we mean that every particle moves exactly along the smooth path as every particle that follows it. If the fluid DOES NOT have Laminar Flow it has TURBULENT FLOW in which the paths are irregular and called EDDY CURRENTS.

Fluid Flow Image

Diagram at right showsa streamlined bodytraveling through a wind tunnel at differenttime intervals.

Mass Flow RateMass Flow Rate

Mass of fluid per second that flows through a tube.

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V = Ax

x = vΔt∴V = AvΔt

ρ =m

V⇒ V =

m

ρ

V =m

ρ= AvΔt

m

Δt= ρAvMass Flow Rate

Equation of Continuity

Mass is neither CREATED or DESTROYED. It is ALWAYS CONSERVED!

As fluid flows through pipe, the amount of mass flowing through the pipe remains the same

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m

Δt= Avρ

Δm1 = Δm2A1v1ρ1Δt = A2v2ρ 2Δt

A1v1 = A2v2

The density of the fluid does not change (incompressiblefluid)

Equation of Continuity

Example In the condition known

as atherosclerosis, a deposit forms on the arterial wall and reduces the opening through which blood can flow. In the carotid artery in the neck, blood flows three times faster through a partially blocked region than it does through an unobstructed region. Determine the ratio of the effective radii of the artery at the two places

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A1v1 = A2v2

A = πr2

πru2vu = πro

2vo

ruro=vovu= 3 =1.7

Subscript u unobstructed regionSubscript o obstructed region

Bernoulli’s PrincipleDaniel Bernoulli, was curious about how the velocity changes as the fluid moves through a pipe of different area. He also wanted to incorporate pressure into his idea.

Changes in pressure reults ina Net Force directed towardsthe low pressure region

Bernoulli's Equation

Let’s look at this principle mathematically.

Work is done by a section of water applying a force on a second section in front of it over a displacement. According to Newton’s 3rd law, the second section of water applies an equal and opposite force back on the first. Thus is does negative work as the water still moves FORWARD. Pressure*Area is substituted for Force.

X = L

F1 on 2

-F2 on 1

Bernoulli's Equation

A1

A2

v1

v2

L1=v1t

L2=v2t

y2

ground

Work is also done by GRAVITY as the water travels a vertical displacement UPWARD. As the water moves UP the force due to gravity is DOWN. So the work is NEGATIVE.

y1

Bernoulli's EquationNow let’s find the NET WORK done by gravity and the water acting on itself.

WHAT DOES THE NET WORK EQUAL TO? A CHANGE IN KINETICENERGY!

Bernoulli's Equation

Consider that Density = Mass per unit Volume AND that VOLUME isequal to AREA time LENGTH

Bernoulli's Equation

We can now cancel out the AREA and LENGTH

Leaving:

Bernoulli's Equation

Moving everything related to one side results in:

What this basically shows is that Conservation of Energy holds true within a fluid and that if you add the PRESSURE, the KINETIC ENERGY (in terms of density) and POTENTIAL ENERGY (in terms of density) you get the SAME VALUE anywhere along a streamline.

ExampleWater circulates throughout the house in a hot-water

heating system. If the water is pumped at a speed of 0.50 m/s through a 4.0 cm diameter pipe in the basement under a pressure of 3.0 atm, what will be the flow speed and pressure in a 2.6 cm-diameter pipe on the second floor 5.0 m above?

2

222

2221

21

2111

)026.0(50.0)04.0(

v

v

vrvr

vAvA

1.183 m/s

1 atm = 1x105 Pa

P

Px

ghvPghvP ooo

)5)(8.9)(1000()183.1)(1000(2

1)0)(8.9)(1000()50.0)(1000(

2

1103

2

1

2

1

225

22

2.5x105 Pa(N/m2) or 2.5 atm