Physics 201 : Lecture 25

1
Physics 201 Lecture 25

• Fluid Dynamics
• Continuity Equation
• Bernoulli Equation
• Poissieules Equation

2
Types of Fluid Flow

• Laminar flow
• Steady flow
• Each particle of the fluid follows a smooth path
• The paths of the different particles never cross
  each other
• The path taken by the particles is called a
  streamline
• Turbulent flow
• An irregular flow characterized by small
  whirlpool like regions
• Turbulent flow occurs when the particles go above
  some critical speed

3
Viscosity

• Characterizes the degree of internal friction in
  the fluid
• This internal friction, viscous force, is
  associated with the resistance that two adjacent
  layers of fluid have to moving relative to each
  other
• It causes part of the kinetic energy of a fluid
  to be converted to internal energy

4
Ideal Fluid Flow

• There are four simplifying assumptions made to
  the complex flow of fluids to make the analysis
  easier
• The fluid is nonviscous internal friction is
  neglected
• The flow is steady the velocity of each point
  remains constant
• The fluid is incompressible the density remains
  constant
• The flow is irrotational the fluid has no
  angular momentum about any point

5
Streamlines

• The path the particle takes in steady flow is a
  streamline
• The velocity of the particle is tangent to the
  streamline
• A set of streamlines is called a tube of flow

6
Fluid Flow
Fluid flow without friction

• Volume flow rate ?V/?t A ?d/?t Av (m3/s)
• Continuity A1 v1 A2 v2
• i.e., flow rate the same everywhere
• e.g., flow of river

7
Bernoullis Equation

• As a fluid moves through a region where its speed
  and/or elevation above the Earths surface
  changes, the pressure in the fluid varies with
  these changes
• The relationship between fluid speed, pressure
  and elevation was first derived by Daniel
  Bernoulli
• Consider the two shaded segments
• The volumes of both segments are equal
• The net work done on the segment is W (P1 P2)
  V
• Part of the work goes into changing the kinetic
  energy and some to changing the gravitational
  potential energy

8
Bernoullis Equation

• The change in kinetic energy
• DK 1/2 mv22 - 1/2 mv12
• There is no change in the kinetic energy of the
  unshaded portion since we are assuming streamline
  flow
• The masses are the same since the volumes are the
  same
• The change in gravitational potential energy
• DU mgy2 mgy1
• The work also equals the change in energy
• Combining
• W (P1 P2)V 1/2 mv22 - 1/2 mv12 mgy2
  mgy1
• Rearranging and expressing in terms of density
• P1 1/2 rv12 mgy1 P2 1/2 rv22 mgy2

9
Bernoullis Equation

• Pressure drops in a rapidly moving fluid
• whether or not the fluid is confined to a tube
• For incompressible, frictionless fluid

10
Problem

• A large bucket full of water has two equal
  diameter drains. The water level in the bucket is
  kept constant by constantly refilling it. One is
  a hole in the side of the bucket at the bottom,
  and the other is a pipe coming out of the bucket
  near the top, which is bent downward such that
  the bottom of this pipe even with the other hole,
  like in the picture below
• Though which drain is the water spraying out with
  the highest speed?
• 1. The hole
• 2. The pipe
• 3. Same

Since the pressures at the two drains are the
same and the liquid leaves the pipe at the same
height, their speeds are the same.
11
Applications of Bernoullis Principle

• Wings and sails
• Higher velocity on one side of sail versus the
  other results in a pressure difference that can
  even allow the boat to sail into the wind
• Entrainment
• Reduced pressure in high velocity fluid pulls in
  particles from static or lower velocity fluid
• Bunsen burner, Aspirator,
• Velocity measurement

12
Problem
(a) Calculate the approximate force on a square
meter of sail, given the horizontal velocity of
the wind is 6 m/s parallel to its front surface
and 3.5 m/s along its back surface. Take the
density of air to be 1.29 kg/m3. (b) Discuss
whether this force is great enough to be
effective for propelling a sail boat.
13
Applications of Fluid Dynamics

• Streamline flow around a moving airplane wing
• Lift is the upward force on the wing from the air
• Drag is the resistance
• The lift depends on the speed of the airplane,
  the area of the wing, its curvature, and the
  angle between the wing and the horizontal

14
Lift General

• In general, an object moving through a fluid
  experiences lift as a result of any effect that
  causes the fluid to change its direction as it
  flows past the object
• Some factors that influence lift are
• The shape of the object
• The objects orientation with respect to the
  fluid flow
• Any spinning of the object
• The texture of the objects surface

15
Golf Ball

• The ball is given a rapid backspin
• The dimples increase friction
• Increases lift
• It travels farther than if it was not spinning

16
Atomizer

• A stream of air passes over one end of an open
  tube
• The other end is immersed in a liquid
• The moving air reduces the pressure above the
  tube
• The fluid rises into the air stream
• The liquid is dispersed into a fine spray of
  droplets

17
Velocity Measurement Pitot tube
18
Problem
(a) What is the pressure drop due to Bernoulli
effect as water goes into a 3 cm diameter nozzle
from a 9 cm diameter fire hose while carrying a
flow of 40 L/s? (b) To what maximum height above
the nozzle can this water rise neglecting air
resistance.
19
Torricellis Theorem
P1, v1, h1
h
P2P1 , v2 , h2
20
Viscosity

• Friction in fluids

L
F
V0
V0
Newtons law Laminar flow -- no turbulence
shearing stress

strain
21
Real fluid flow
At constant velocity net force is zero.
22
More Viscosity
23
Flow and Viscosity
Poiseuilles Eqn.

• no turbulence
• no sized particles
• constant ?