Fluid Flow or Discharge

1
Fluid Flow or Discharge

• When a fluid that fills a pipe flows through a
  pipe of cross-sectional area A with an average
  velocity v, the flow or discharge Q is determined
  by

2
Equation of Continuity

• Suppose an incompressible (constant density)
  fluid fills a pipe and flows through it.
• If the cross-sectional area of the pipe is A1 at
  one point and A2 at another point, the flow
  through A1 must equal the flow through A2.

3
Equation of Continuity

• The equation of continuity in the form
• applies only when the density of the fluid is
  constant. If the density of the fluid is not
  constant, the equation of continuity is

4
Equation of Continuity

• The equation of continuity shows that where the
  cross-sectional area is large, the fluid speed is
  slow and that where the cross-sectional area is
  small, the fluid speed is large.
• This explains why water shoots out of a hose
  faster when you place your thumb across the
  opening, reducing the cross-sectional area
  through which the water can come out.

5
Viscosity and Viscous Flow

• Viscosity h of a fluid is a measure of how
  difficult it is to cause the fluid to flow.
• In an ideal fluid there is no viscosity to hinder
  the fluid layers as they slide past one another.
• Within a pipe of uniform cross-sectional area,
  every layer of an ideal fluid moves with the same
  velocity,
• even the layer next to
• the wall.

6
Viscosity and Viscous Flow

• When viscosity is present, the fluid layers do
  not all have the same velocity.
• The fluid closest to the wall does not move at
  all, while the fluid at the center of the pipe
  has the greatest velocity.
• The fluid layer next to the wall surface does not
  move because it is held tightly by intermolecular
  forces.
• The intermolecular forces
• are so strong that if a
• solid surface moves,
• the adjacent fluid layer
• moves along with it and
• remains at rest relative
• to the moving surface.

7
Viscosity and Viscous Flow

• This is why a layer of dust lies on the surface
  of fan blades even at high speeds. The layer of
  air in contact with the fan blade has no velocity
  relative to the fan blade and does not blow off
  the dust.
• Force F needed to move a layer of viscous fluid
  with constant velocity
• ? viscosity
• A area
• v velocity
• d distance from the immobile surface

8
Viscosity and Viscous Flow

• Viscosity of liquids and gases depend on
  temperature.
• Usually, the viscosities of liquids decrease as
  the temperature increases.
• The viscosities of gases increase as the
  temperature increases.
• Viscous fluids have a high viscosity, such as tar
  and molasses.

9
Forces Exerted By a Fluid

• If a fluid were subjected to a tangential force
  F, the layers of the fluid would slide past one
  another without friction.
• This means that a fluid can sustain only a
  perpendicular force, and conversely, can exert
  only a force perpendicular to the surface.

• If a fluid were subjected to a tangential force
  F, the layers of the fluid would slide past one
  another without friction.
• This means that a fluid can sustain only a
  perpendicular force, and conversely, can exert
  only a force perpendicular to the surface.

10
Forces Exerted By a Fluid

• Suppose that a nonelastic fluid is between 2
  plates. If the velocity v of the upper plate is
  not too large, the fluid shears in the way
  indicated. The viscosity h is related to the
  force F required to produce the velocity v by

11
Forces Exerted By a Fluid

• A area of either plate
• d distance between plates
• units for h Ns/m2 or kg/ms or lb s/ft2
• 1 poiseuille (Pl) 1 Ns/m2 1 kg/ms
• 1 poise (P) 0.1 kg/ms

12
Poiseuilles Law

• The fluid flow through a cylindrical pipe of
  length l and cross-sectional radius r is given
  by
• P1 - P2 is the pressure difference between the
  two ends of the pipe.

13
Work Done by a Piston

• Work done by a piston in forcing a volume V of
  fluid into a cylinder against an opposing
  pressure P is given by W PV

14
Bernoullis Principle

• If a fluid is incompressible - a change in
  pressure does not cause a change in volume - the
  volume of fluid entering per second must equal
  the volume leaving per second.

15
Volumetric flow rate 1 entering 2 leaving
Let v represent the speed with which a liquid
moves in a cylindrical pipe, so that during the
time t the liquid moves a distance equal to vt
(where v is velocity.
16
Bernoullis Principle

• The volume of liquid passing a cross-sectional
  area A is given by
• Volumetric flow rate Q
• Because the liquid is incompressible, the
  volumetric flow rate is the same entering and
  leaving the system.
• Volumetric flow rate (system)

17
Bernoullis Principle

• The quantity has the same value at every
  point in an incompressible fluid moving in
  streamline (non-turbulent) flow.
• Bernoullis equation

18
Bernoullis Principle

• If the fluid is not moving, then both speeds are
  zero. The fluid is static. If the height at the
  top of the column is h1 is defined as zero, and
  h2 is the depth, then Bernoullis equation
  reduces to the equation for pressure as a
  function of depth
• If the fluid is flowing through a horizontal pipe
  with a constriction, as shown in the figure on
  the next slide, there is no change in height and
  the gravitational potential energy does not
  change. Bernoullis equation reduces to

19
Bernoullis Principle

• The flow rate Q in the tube has to be constant,
  therefore, the fluid has to move faster through
  the constriction to maintain the constant flow
  rate Q.
• The velocity at point a is greater than at either
  the meter entrance or the meter exit.
• The pressure in a fluid is related to the speed
  of flow, therefore the pressure in the fluid is
  less at point a and greater at the meter
  entrance, as illustrated by the liquid levels in
  the U-tube manometer.
• The pressure difference is equal to

20
Bernoullis Principle

• Bernoulli's principle describes the relationship
  between pressure and velocity in a fluid and
  describes the conservation of energy as it
  applies to fluids.
• Bernoullis principle also explains why a roof
  blows off of a house in violent winds.
• Wind creates a low pressure region above the peak
  of the roof, creating a pressure difference
  inside and outside the house which results in the
  loss of the roof.

21
Bernoulli's Equation and Lift

• The shape of a wing forces air to travel faster
  over the curved upper surface than it does over
  the flatter lower surface.
• According to Bernoullis equation, the pressure
  above the wing is lower (faster moving air),
  while the pressure below the wing is higher
  (slower moving air).
• The wing is lifted upward due to the higher
  pressure on the bottom of the wing.

22
Bernoulli's Equation and Lift

• Air flows over the top of an airplane wing of
  area A with speed vt, and past the underside of
  the wing (also of area A) with speed vu.
• the magnitude FL of the upward lift force on the
  wing will be
• Ski jumpers use this same principle to help
  themselves stay in the air longer during jumps.
• A boomerang with a curved surface will turn in
  the direction of the curved face due to pressure
  differences created by the different air
  velocities over the two surfaces.

23
Torricellis Theorum

• If an opening exists in a tank containing a
  liquid at a distance h below the top of the
  liquid, then the velocity v of outflow from the
  opening is
• provided the liquid obeys Bernoullis equation
  and the top of the liquid may be regarded as
  motionless (v 0 m/s).

24
Tidal Waves

• Tidal waves are the dissipation of energy in a
  viscous fluid over an inclined plane tidal
  waves have nothing to do with tides.
• The energy source is usually an under-sea
  earthquake (it could also be an under-sea
  explosion or a meteor strike) the viscous fluid
  is the ocean the inclined plane is the ocean
  floor sloping upward toward land.
• Earthquake
• When the Earth moves up and down it also moves
  the ocean water up and down. This generates a
  huge wave traveling outward in a series of
  concentric rings.
• In deep water, most of the tidal wave (tsunami)
  remains hidden beneath the surface. But as the
  tidal wave moves toward more shallow water, its
  enormous energy is forced to the surface.
• In the open ocean, tidal waves are hundreds of
  miles wide and travel at jetliner speeds. Near
  land they slow down to freeway speeds.
• What makes a tidal wave so destructive is the
  speed and tremendous volume of water delivered
  onto a coastal or island environment as the tidal
  wave is forced by the inclining ocean floor onto
  the land.

25
Bernoullis Principle and Syringes

• The force applied to the plunger is equal to the
  pressure times the area of the plunger.
• Viscous flow will occur within the barrel of the
  syringe and only a little pressure difference is
  needed to move the fluid through the barrel to
  point 2, where the fluid will enter the
• narrow needle.
• The pressure applied
• to the plunger is nearly
• equal to the pressure
• P2 at point 2.
• The pressure at point 1, P1,
• is also known as the gauge
• pressure.

26
Bernoullis Principle and Syringes

• Apply Bernoullis principle,
• and if the needle is held horizontally,
• Poiseuilles Law may also be needed to solve this
  type of problem.

27
Bernoullis Principle and Siphons

• A siphon is an inverted U-shaped pipe or tube
  that can transfer water from a higher container
  to a lower container by lifting the water upward
  from the higher container and then lowering it
  into the lower container. The water is simply
  seeking its level, just as it would if you
  connected the two containers with a pipe at their
  bottoms. In that case, the water in the higher
  container would flow out of it and into the lower
  container, propelled by the higher water pressure
  at the bottom of the higher container. In the
  case of a siphon, it's still the higher water
  pressure in the higher container that causes the
  water to flow toward the lower container, but in
  the siphon the water must temporarily flow above
  the water level in the higher container on its
  way to the lower container.

28
Bernoullis Principle and Siphons

• Two means of initiating the
• liquid flow (assume the liquid is water)
• You can make a siphon using a rubber
• hose and gravity is the key to getting it to
  work. A siphon needs to have the "dry" end of the
  hose lower than the end that is stuck in the
  water (the "wet" end). You can get the siphon
  started by first filling the hose with water.
  Once the hose is full, use your thumb to plug the
  end of the hose that will be removed from the
  water. Place the dry end into the second
  container and then remove your thumb. Gravity
  does all the work from there...

29

• How does it work? Think of the water in
  terms of distinct packets". Since the dry end
  of the hose is lower than the wet end, there are
  more water "packets" towards the dry end. As
  such, the column of water being pulled downward
  by gravity is heavier than the column of water at
  the wet end of the tube. Gravity pulls on one
  packet" of water on the dry end of the tube
  causing it to move down the tube. As it moves, it
  creates a small vacuum behind itself. This vacuum
  pulls the next packet forward (downward) as
  well. This suction is strong enough to pull
  other packets up the tube (against gravity) at
  the wet end. Once a given packet passes the
  highest point in the tube, gravity pulls it
  downward and the
• process continues. The siphon will work
• as long as the vertical (up and down)
• column of water outside the container is
• larger than the vertical column inside the
• container. If the two ends of the hose are
• exactly the same height (the columns are
• equal), the pull of gravity will be the same
• on each side and the flow of water will stop.
• If you then lower the free end, the flow of
• water will begin once again.

30
Bernoullis Principle and Siphons

• Sucking on the lower end of the tube causes a
  partial vacuum (a region of space with a pressure
  that's less than atmospheric pressure) at the top
  of the siphon. The partial vacuum results in a
  difference in pressure between the bottom of
  the tube and the top of the tube. With greater
  fluid pressure at the top than the bottom, the
  water is pushed up into the tube and over to the
  lower container. The same kind of partial vacuum
  exists in a drinking straw when you suck on it
  and is what allows atmospheric pressure to push
  the beverage up toward your mouth.

31
Bernoullis Principle and Siphons

• The maximum height h1 between the surface of the
  liquid and the top of the siphon is the gauge
  pressure, with the gauge pressure being equal to
  the atmospheric pressure.
• To determine the speed of the liquid
• flow at the bottom of the siphon,
• start with Bernoullis equation

32
Bernoullis Principle and Siphons

• Atmospheric pressure is found at the top of the
  liquid and at the bottom of the siphon,
  therefore, Pt and Pb are equal and cancel out.
• Consider the velocity vt at the top of the liquid
  to be
• 0 m/s.
• Consider the lower end of the
• siphon to be the point at which
• the height is 0 m. From the figure,
• the distance from the bottom of the
• to the siphon to the upper level of
• the liquid is d h2.

33
Bernoullis Principle and Siphons

• The density cancels out
• Solve for vb

34
Equation of Continuity Example

• What is the flow rate of water in a pipe whose
  diameter is 10 cm when the water is moving with a
  velocity of 0.322 m/s?

35
Equation of Continuity Example

• If the diameter of the pipe to the right is
  reduced to 4 cm, what is the velocity of the
  fluid in the right-hand side of the pipe?

36
Bernoullis Example

• The pressure P1 53913.24 N/m2, whereas the
  velocity of the water v1 0.322 m/s. The
  diameter of the pipe at location 1 is 10 cm and
  it is at ground level. If the diameter of the
  pipe at location 2 is 4 cm, and the pipe is 5 m
  above the ground, find the pressure P2 of the
  water at position 2.
• From the previous example, we know that the
  velocity of the water at location 2 is 2.015 m/s.

37
Bernoullis Example
38
Helpful Online Links

• Hyperphysics Fluids
• Work-Energy Applet (to determine the power needed
  in the pump for the water-jet to pass over the
  wall)
• Gallery of Fluid Mechanics