Transcript Fluid Mechanics

Fluid Mechanics
A fluid is a substance that has the ability to flow, and therefore, does not
maintain a specific shape. It can either be a liquid or a gas. A valuable
measurement of a fluid is its density.
Density is mass per unit of
volume.
ρ=m/V
ρ = density
m = mass (kg)
0.50
0.73
0.82
V = volume (m3)
Things with low densities
float in things with higher
densities.
Specific gravity = the ratio
of a substance’s density to
the density of water
1.00
(water)
1.34
Fluids exert pressure on their surroundings.
Pressure is force per unit of area.
P=F/A
P = pressure (Pascal)
F = force (N)
A = surface area (m2)
Which book exerts more
pressure on the table?
Why can she lay on a bed
of nails and be struck with a
sledgehammer without
getting hurt?
The Forces Exerted by Fluids
Fluids exert pressure in all directions.
Forces acting in all directions on the cube
are equal (assuming cube is very small)
Fluids exert forces that are
perpendicular to the solid surfaces
they are in contact with.
If they are at rest, they do NOT exert
forces parallel to the solid surface.
Pressure varies with depth.
As you go deeper, the amount of
water above you increases exerting
more force on you because of
gravity.
P=F/A
F (cube of water) = mg
m = ρV = ρAh
F = ρAhg
P = ρAhg / A
P = ρgh
P = pressure (Pa)
ρ = density (kg/m3)
g = gravitational acceleration (m/s2)
Furthermore,
∆P = ρg∆h
Assuming that the difference between the surface of the water in the tank
and the household faucet is 30 m, calculate the difference in water pressure
between the surface and the faucet.
A manometer is used to measure
pressure. A fluid is pushed on by two
pressures – the atmospheric pressure
from the outside (P0) and the pressure
being measured (P).
The fluid adjusts itself to an equilibrium
point based on the pressure difference
which produces a height difference (Δh).
∆P = ρg∆h
P - P0 = ρg∆h
The term ρg∆h is known as the gauge
pressure. It is the difference between the
atmospheric pressure and the pressure
being measured (or the pressure relative
to the atmospheric pressure)
Units for P  Pa (N/m2), mm Hg (torr), atm
Intravenous infusions are often made under gravity. Assuming the fluid
has a density of 1.00 g/cm3, at what height should the bottle be placed
so that the liquid pressure is 55 mm-Hg?
If the blood pressure is 18 mm-Hg above atmospheric pressure, how
high should the bottle be placed so that the fluid just barely enters the
vein?
Pascal’s Principle
If an external pressure is applied
to a confined fluid, that extra
pressure is transmitted
throughout the fluid
Pin = Pout
Fin / Ain = Fout / Aout
Fout / Fin = Aout / Ain
(mechanical advantage)
Archimedes’ Principle
Why do people rehabilitate
injuries by training in a pool?
There is less stress on their
joints.
Why?
Buoyant Force – an upward force exerted by a
liquid on an object submerged in the liquid
Picture shows an object submerged in a fluid.
The buoyant force is the net force in the vertical
direction (we know F2 is greater than F1
because of the greater fluid pressure)
F b = F2 – F 1
Since P = F / A
= P2A2 – P1A1
= A(P2 – P1)
Since P = ρgh
Fb = A(ρFgh2 – ρFgh1)
Fb= mFg
= ρFgA (h2 – h1)
Fb = buoyant force (N)
= ρFgA∆h
mF = mass of fluid displaced (kg)
= ρFgV
g = gravitational acceleration (m/s2)
Since ρF = mF / V
Archimedes’ Principle – The buoyant
force on an object immersed in a fluid
is equal to the weight of the fluid
displaced by that object.
An object’s apparent weight is the weight that object experiences when
submerged in a liquid
W ´ = W – Fb
W ´ = apparent weight
W = actual weight
Fb = buoyant force
There is a buoyant
force of:
Fb = W – W ’
14.7(9.8)-13.4(9.8)
= 12.7 N
(equal to the weight of
the liquid displaced)
If this 1200-kg log was fully submerged, the water it would displace
would have a greater mass than the log itself (2000 kg)
It rises until the fluid it displaces matches the weight of the log.
A scuba diver and her gear displace a volume of 65.0 L and have a total
mass of 68.0 kg.
(a)What is the buoyant force on the diver in water?
(b)Will the diver sink or float?
(c)What is her apparent weight?
Fluids in Motion
If fluid flow (gas or liquid) is smooth as in figure (a), it is called
streamline or laminar flow. In this flow, each particle follows a
smooth path or streamline, not crossing paths with other particles.
Turbulent flow exists with the development of eddies, which are
whirlpool-like circles along the path of the streamlines. In this flow,
particles cross paths and lose a great deal of energy.
To express the rate of laminar flow, the
mass flow rate is useful:
m / ∆t
m = mass (kg)
∆t = time interval (s)
Fluids typically flow through pipes which have varying diameters. When the
diameter changes, the flow changes.
Since ρ = m / V
m / ∆t = ρV / ∆t
Since V = AΔl
m / ∆t = ρAΔl/ ∆t
m / ∆t = ρAv
(v = fluid velocity)
In a pipe, flow rate is constant, so
ρA1v1 = ρA2v2
Since ρ is constant,
A1v1 = A2v2
(Continuity Equation)
What area must a heating duct have if air moving 3.0 m/s along it can replenish
the air every 15 minutes in a room of volume 300 m3? Assume the air’s density
remains constant.
Bernoulli’s Principle
Ever wondered…
• Why a curve ball curves?
• Why a sailboat can move against
the wind?
• Why an airplane wing has lift?
Bernoulli’s Principle – Where the velocity of the fluid is high, the pressure is
low and where the velocity is low, the pressure is high
Why is that the case?
Consider the amount of work
necessary to move blue-colored
fluid from position in (a) to position
in (b)
At point 1(left end), work is done on
blue fluid to move it:
W1 = F1Δl1
= P1A1Δl1
At point 2 (right end), negative work
is done by white fluid against
motion of blue fluid:
W2 = -P2A2Δl2
Also, negative work is done to lift
the fluid against gravity.
W3 = -mg(y2 - y1)
Net work = W1 + W2 + W3
= P1A1Δl1 - P2A2Δl2 - mgy2 + mgy1
Since W = ∆KE
½m2v22 – ½m1v12 = P1A1Δl1 - P2A2Δl2 –
m2gy2 + m1gy1
Since m = ρV = ρAΔl
½ρA2Δl2v22 – ½ρA1Δl1v12 = P1A1Δl1 P2A2Δl2 – ρA2Δl2gy2 + ρA1Δl1gy1
Since A1Δl1 = A2Δl2
½ρv22 – ½ρv12 = P1 - P2 - ρgy2 + ρgy1
P1 + ½ρv12 + ρgy1 = P2 + ½ρv22 + ρgy2
P + ½ρv2 + ρgy = constant
P = Pressure (Pa)
ρ = density (kg/m3)
v = fluid velocity (m/s)
g = gravitational acceleration (m/s2)
y = vertical height above reference point (m)
Water circulates throughout a 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 2nd floor 5.0 m above?
Returning to the original questions…
When pitch is thrown with a
spin on it, spin on side A
slows down air molecules
causing higher pressure.
Spin on side B speeds up air
molecules causing lower
pressure. Net force in left
direction
When air moves over the top of the wing, the streamlines get forced
together causing them to speed up (Av = Av). The streamlines
underneath are moving slower. Slower molecules underneath have
higher pressure than molecules above wing causing net for up.
When sails are set at angle, the
wind hits the front of the sail
lowering the pressure, and still air
behind the sail has higher pressure.
Net force pushes sail to left. Water
pushes against keel to the right
giving a net force into the wind.
What is the speed at which the water leaves the spigot?
Assumptions
• the size of the spigot is
small compared to the
diameter of the jar, so v2
will be insignificant
compared to v1
• P1 = P2 because both are
open to atmosphere
P1 + ½ρv12 + ρgy1 = P2 + ½ρv22 + ρgy2