FLUIDS 2.pdf

Outline

• Part 1: Fluid Statics

• 1. Density and Specific Gravity • 2. Pressure in a Fluid

• 3. Pascal's Principle

• 4. Buoyancy and Archimedes’ Principle

Fluids In Motion

• The Behavior of fluids in motion is complex

• Consider for example, the rise of smoke from a burning

cigarette.

• First the smoke rises in a

regular stream (laminar flow), but the simple streamlined flow quickly becomes

turbulent and the smoke begins to swirl irregularly.

1. Laminar Flow - orderly flow of neighboring layers of fluid. Streamlines do not cross over each

other.

2. Turbulent Flow -

irregular, complex flow of fluids.

Fluids In Motion

• In this part, we only consider

– Non-turbulent flows

– Steady-state flow

– Non-viscous fluid (flow with no dissipation of mechanical

energy)

– Incompressible fluids (density is constant throughout the

fluid)

Consider the case of a fluid moving from a cross-sectional area A₁ to a region of A₂. Because the fluid is

incompressible, the same amount of it leaves each region during the same time interval.

The volume of the fluid that flows into the tube in a time interval t is:

V₁ = A₁v₁ t

where v₁ the velocity of the fluid at A₁.

If the density of the fluid is , then the mass of the

fluid that flows into the tube in a given time interval is: V₁ =  A₁v₁ t

5.Equation of Continuity

• Describes the flow of a fluid through a tube with varying cross-sectional area.

• A₁v₁ = A₂v₂ (1)

• A is the cross-sectional area of the tube, v is the velocity of the fluid that is flowing

• The quantity Av is called I_V (volume flow rate)

• I_V = Av = constant (2)

- the flow of material (mass) through a tube of changing cross section is constant

when the density of the fluid does not change.

- statement of conservation of mass.

A horizontal pipe of 25 cm2 _{cross-section carries water}

at a velocity of 3.0 m/s. The pipe feeds into a smaller pipe with a cross section of only 15 cm2_{. What is the}

velocity in the smaller pipe?

Using equation of continuity:

v₂ = (v₁A₁)/A₂

v2 = (3.0 m/s)(25 cm2)/(15 cm2)

v₂ = 5.0 m/s

KAPUSO PA RIN!!!

• Blood flows in an aorta of radius 1.0 cm at 30 cm/s. What is the volume flow rate?

• (Answer: I_V = Av = 9.42 x 10-5 _m3_/s)

• Blood flows from an artery of

radius of 0.3 cm, where its speed is 10 cm/s into a region where the radius is reduced to 0.2 cm because of the thickening of arterial walls (arteriosclerosis). What is the speed of the blood in the narrower region?

The relationship between fluid speed, pressure, and elevation was first

derived in 1738 by the Swiss physicist Daniel Bernoulli.

The speed of water spraying from the end of a hose

6. Bernoulli’s Equation

• Bernoulli’s Equation relates the pressure, elevation and speed of an incompressible fluid in steady flow.

• It follows Newton’s Laws and is most easily derived by applying the work-energy theorem to a segment of a fluid.

• P + ρgy + ½ ρv2 _{= constant}

• Relates pressure to fluid speed and elevation • Bernoulli’s equation is a consequence of

Conservation of Energy applied to an ideal fluid • States that the sum of the pressure, kinetic energy

per unit volume, and the potential energy per unit volume has the same value at all points along a streamline

constant







v

gy

P





2

1

6. Bernoulli’s Equation

• P + ρgy + ½ ρv2 _{= constant}

• P₁ + ρgy₁ + ½ ρv₁2 _{= P}

2 + ρgy2 + ½ ρv22

• Special case when fluid is at rest v1=v2 = 0

6. Bernoulli’s Equation

• P + ρgy + ½ ρv2 _{= constant}

• P₁ + ρgy₁ + ½ ρv₁2 _{= P}

2 + ρgy2 + ½ ρv22

If h1=h2

Application:

How Airplanes Fly

V₂

V_{1 P} V2< V1

1

P₂

6. Bernoulli’s Equation

small hole a distance h below the water surface. Find the speed of the water as it flows out the hole.

• Picture the Problem: • We apply Bernoulli’s

Equation to points a and b in the figure. Since the

6. Bernoulli’s Equation

• 1. Bernoulli’s Equation with v_a = 0 gives:

• P_a + ρgy_a + 0 = P_b + ρgy_b + ½ ρv_b2

• 2. The pressure at point a and point b is the same, P_at, since both points are open to the air:

• Pa = Pb = P_at

• P_at + ρgy_a + 0 = P_at + ρgy_b + ½ ρv_b2

6. Bernoulli’s Equation

• 3. Solve for the speed v_b of the water flowing from the hole.

• v_b2 = 2g (ya – yb) =2gh

7. Torricelli’s Law

Bernoulli’s Equation.

• From the previous example, the water emerges from the hole with speed equal to the speed it would have if it dropped in free fall a

distance h.

Torricelli’s Theorem

Relates the

speed of fluid flowing

out of an opening

to the height of

fluid above the opening.

•

SPEED of efflux

v

= v

+ 2(P-P

/



) + 2gh

• Ordinarily v

is very much smaller

than v

:

v

= 2(P-P

0

/



) + 2gh

•

If P=P

gh

-Relates the speed of a fluid flowing out of an

opening, which was later shown to be a particular case of Bernoulli's

principle.

The speed of efflux of a

liquid from an opening in a reservoir equals the speed that the liquid

would acquire if allowed to fall from rest from the surface of the reservoir to the opening.

Torricelli’s Theorem

•

The effect of the decrease in pressure

with the increase in the speed of fluid in a

horizontal pipe.

•

This follows from Bernoulli’s Equation, in

which the

elevations (h) are the same

.

P + ½ ρv

= constant

BACKGROUND: “The fluid velocity must increase through the constriction to satisfy the equation of continuity, while its

pressure must decrease due to conservation of energy: the gain in kinetic energy is balanced by a drop in pressure or a pressure gradient force.”

Other Application of

Bernoulli’s Principle

• Pitot Tube

• used in wind tunnel experiments and on airplanes to measure flow speed. It's a slender tube that has two holes on it. The front hole is placed in the airstream to measure what's called the stagnation pressure. The side hole measures the static pressure. By measuring the

difference between these pressures, you get the

dynamic pressure, which can be used to calculate airspeed.

• Why is measuring airspeed important?

Air France Flight 447

concluded that the aircraft crashed after temporary

inconsistencies between the airspeed measurements –

likely due to the aircraft's pitot tube being obstructed by ice crystals – caused

the autopilot to disconnect, after which the crew reacted

incorrectly and ultimately led the aircraft to an aerodynamic

Booeing 757

Accidents attributed to pilot

disorientation due to improperly maintained instruments on

February 6, 1996 in Puerto Plata, Dominican Republic, with the

loss of all 189 passengers and crew and on

Investigators found that the

aircraft had been stored without the necessary covers for its pitot tube sensors, thus allowing

A circular hole 2.00 cm in diameter is cut in the side of a large water tank, 14.0 m below the water level in the tank. The top of the tank is open to the air. Find:

a) the speed of efflux

b) the volume discharged per unit time

v₂ = 2gh

v₂ = 2(9.8)(14) v₂ = 16.57 m/s

Volume discharged per unit time can be calculated from equation of continuity:

I= Av = volume rate of flow = r2_v

= (0.01 m)2_{(16.57 m/s)}

• Water enters a house through a pipe with an

inside diameter of 2.0 cm at an absolute

pressure of 4.0x10

Pa (about 4 atm). A 1.0

cm diameter pipe leads to the second floor of

the bathroom 5.0 m above. When the flow

speed at the inlet pipe is 1.5 m/s. Find the

flow speed, pressure and volume flow rate in

the bathroom.