Fluid Mechanics Chapter 1

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Fluid Mechanics Chapter 1

• Introduction

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Scope

• fluid mechanics is the study of fluids at rest or
  in motion.
• It has traditionally been applied in such areas
  as the design of canal, and dam systems
• the design of pumps, compressors, and piping and
  ducting used in the water
• and air conditioning systems of homes and
  businesses, as well as the piping systems needed
  in chemical plants
• the aerodynamics of automobiles and sub- and
  supersonic airplanes
• and the development of many different flow
  measurement devices such as gas pump meters

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Definition of a Fluid

• A fluid is a substance that deforms continuously
  under the application of a shear (tangential)
  stress no matter how small the shear stress may
  be.
• Because the fluid motion continues under the
  application of a shear stress, we can also define
  a fluid as any substance that cannot sustain a
  shear stress when at rest

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Deformation of Fluids

• The amount of deformation of the solid depends on
  the solids modulus of rigidity G
• the rate of deformation of the fluid depends on
  the fluids viscosity µ.
• We refer to solids as being elastic and fluids as
  being viscous

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Basic Equations

• The basic laws, which are applicable to any
  fluid, are
• 1. The conservation of mass
• 2. Newtons second law of motion
• 3. The principle of angular momentum
• 4. The first law of thermodynamics
• 5. The second law of thermodynamics

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Solids and liquids

• A solid can resist a shear stress by a static
  deformation
• Molecular forces are much higher and they tend to
  retain their shapes
• Solids tend to regain their shapes when applied
  stress is removed

• A fluid cannot resist a shear stress
• Any shear stress applied to a fluid, no matter
  how small, will result in motion of that fluid.
• The fluid moves and deforms continuously as long
  as the shear stress is applied
• Fluid at rest must be in a state of zero shear
  stress

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Liquids and Gases

• A liquid, being composed of relatively
  close-packed molecules with strong cohesive
  forces, tends to retain its volume and will form
  a free surface in a gravitational field if
  unconfined from above
• These are incompressible

• A gas has no definite volume, and when left to
  itself without confinement, a gas forms an
  atmosphere which is essentially hydrostatic
• These could be compressed

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Solid Liquids and gases
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System and Control Volume

• For a piston-cylinder assembly the gas in the
  cylinder is the system.
• If the gas is heated, the piston will lift the
  weight
• the boundary of the system thus moves.
• Heat and work may cross the boundaries of the
  system,
• but the quantity of matter within the system
  boundaries remains fixed.
• No mass crosses the system boundaries as on next
  slide

• A system is defined as a fixed, identifiable
  quantity of mass
• the system boundaries separate the system from
  the surroundings.
• The boundaries of the system may be fixed or
  movable
• however, no mass crosses the system boundaries

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System and Control Volume

• For a piston-cylinder assembly the gas in the
  cylinder is the system.
• If the gas is heated, the piston will lift the
  weight
• the boundary of the system thus moves.
• Heat and work may cross the boundaries of the
  system,
• but the quantity of matter within the system
  boundaries remains fixed.
• No mass crosses the system boundaries as on next
  slide

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Equations

• For a closed system

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Control Volume

• A control volume is an arbitrary volume in space
  through which fluid flows.
• The geometric boundary of the control volume is
  called the control surface.
• The control surface may be real or imaginary it
  may be at rest or in motion.
• Figure 1.3 shows flow through a pipe junction,
  with a control surface drawn on it.
• Note that some regions of the surface correspond
  to physical boundaries (the walls of the pipe)
  and others (at locations 1 ,2 , and 3 ) are parts
  of the surface that are imaginary (inlets or
  outlets).

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Control Volume
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Systems and control mass/volumes

• Systems may be considered to be closed or open,
  depending on whether a fixed mass or a fixed
  volume in space is chosen for study.
• A closed system consists of a fixed amount of
  mass, and no mass can cross its boundary. It is
  also called control mass system
• That is, no mass can enter or leave a closed
  system, But energy, in the form of heat or work,
  can cross the boundary
• And the volume of a closed system does not have
  to be fixed.
• If, as a special case, even energy is not allowed
  to cross the boundary, that system is called an
  isolated system.

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Systems and control mass/volumes

• An open system, or a control volume, as it is
  often called, is a properly selected region in
  space. It usually encloses a device that involves
  mass flow such as a compressor, turbine, or
  nozzle.
• Flow through these devices is best studied by
  selecting the region within the device as the
  control volume.
• Both mass and energy can cross the boundary of a
  control volume.

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Systems and control mass/volumes

• A large number of engineering problems involve
  mass flow in and out of a system and, therefore,
  are modeled as control volumes.
• A water heater, a car radiator, a turbine, and a
  compressor all involve mass flow and should be
  analyzed as control volumes (open systems)
  instead of as control masses (closed systems).
• A control volume can be fixed in size and shape,
  as in the case of a nozzle, or it may involve a
  moving boundary,
• Most control volumes, however, have fixed
  boundaries and thus do not involve any moving
  boundaries.
• A control volume can also involve heat and work
  interactions just as a closed system, in addition
  to mass interaction..

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Open system example
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An open system (a control volume)with one inlet
and one exit.
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B
A

• A control volume with real and imaginary
  boundaries
• A control volume with fixed and
• moving boundaries

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PROPERTIES OF A SYSTEM

• Any characteristic of a system is called a
  property.
• Some familiar properties are pressure P,
  temperature T, volume V, and mass M.
• The list can be extended to include less familiar
  ones such as viscosity, thermal conductivity,
  modulus of elasticity, thermal expansion
  coefficient, electric resistivity, and even
  velocity and elevation.

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PROPERTIES OF A SYSTEM

• Any characteristic of a system is called a
  property.
• Some familiar properties are pressure P,
  temperature T, volume V, and mass M.
• The list can be extended to include less familiar
  ones such as viscosity, thermal conductivity,
  modulus of elasticity, thermal expansion
  coefficient, electric resistivity, and even
  velocity and elevation.

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PROPERTIES OF A SYSTEM

• Properties are considered to be either intensive
  or extensive.
• Intensive properties are those that are
  independent of the mass of a system, such as
  temperature, pressure, and density.
• Extensive properties are those whose values
  depend on the sizeor extentof the system. Total
  mass, total volume, and total momentum are some
  examples of extensive properties.
• An easy way to determine whether a property is
  intensive or extensive is to
• divide the system into two equal parts with an
  imaginary partition, as shown
• in Fig.
• Each part will have the same value of intensive
  properties as the original system, but half the
  value of the extensive properties

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PROPERTIES OF A SYSTEM

• Generally, uppercase letters are used to denote
  extensive properties (with mass m being a major
  exception), and lowercase letters are used for
  intensive properties (with pressure P and
  temperature T being the obvious exceptions).
• Extensive properties per unit mass are called
  specific properties.
• Some examples of specific properties are specific
  volume (v V/m) and specific
• total energy (e E/m).

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Dimensions

• A dimension is the measure by which a physical
  variable is expressed quantitatively.
• A unit is a particular way of attaching a number
  to the quantitative dimension. Thus
• Length is a dimension associated with such
  variables as distance, displacement, width,
  deflection, and height, while centimeters and
  inches are both numerical units for expressing
  length
• Systems of units have always varied widely from
  country to country, even after international
  agreements have been reached

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Dimensions

• Engineers need numbers and therefore unit
  systems, and the numbers must be accurate because
  the safety of the public is at stake.
• You cannot design and build a piping system whose
  diameter is D and whose length is L. And U.S.
  engineers have persisted too long in clinging to
  British systems of units based on 12
• France proposed a treaty called the Metric
  Convention, which was signed in 1875 by 17
  countries including the United States.
• It was an improvement over British systems
  because its use of base 10 is the foundation of
  our number system
• To standardize the metric system, a General
  Conference of Weights and Measures attended in
  1960 by 40 countries proposed the International
  System of Units (SI).

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Dimensions

• In fluid mechanics there are only four primary
  dimensions from which all other dimensions can be
  derived mass, length, time, and temperature
• All other variables in fluid mechanics can be
  expressed in terms of M, L, T, and theta.
  
• For example, acceleration has the dimensions
  LT-2.
• The most crucial of these secondary dimensions is
  force, which is directly related to mass, length,
  and time by Newtons second law F ma
• From this we see that, dimensionally,
• F MLT-2.

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Dimensions

• MLtT In SI unit of Mass is Kg, length is meter,
  time is second
• In absolute metric system unit of mass is gram,
  length is centimeter
• FLtT British gravitational system (BG units)
  unit of force is pound (lbf), unit of length is
  foot (ft) and time is second, unit of temperature
  is Rankine R
• FMLtT English Engineering system(EEunits) unit
  of force is pound force(lbf) and unit of mass is
  pound mass(lbm), unit of length is foot(ft)
• 1lbf 1lbm x 32.2 ft/s2 . From Fma

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FIRST LAW APPLICATION TO CLOSED SYSTEM
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FIRST LAW APPLICATION TO CLOSED SYSTEM
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Fluid flow through a pipe junction.
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MASS CONSERVATION APPLIED TO CONTROL VOLUME

• A reducing water pipe section has an inlet
  diameter of 50 mm and exit diameter of 30 mm. If
  the steady inlet speed (averaged across the inlet
  area) is 2.5 m/s, find the exit speed.
• We need to apply law of conservation of mass as

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Application of Newtons 2nd Law

• Mathematically, we can write Newtons second law
  for a system of mass m as
• SF is the sum of all external forces acting on
  the system,
• a is the acceleration of the center of mass of
  the system,
• V is the velocity of the center of mass of the
  system,
• and r is the position vector of the center of
  mass of the system
• relative to a fixed coordinate system.

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Application of Newtons 2nd Law
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Application of Newtons 2nd Law
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Application of Newtons 2nd Law
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Approaches to solution of FM problems

• Lagrangian approach to analyze a fluid flow by
• assuming the fluid to be composed of a very
  large number of particles whose motion must be
  described.
• However, keeping track of the motion of each
  fluid particle is extremely difficult.
• Consequently, a particle description becomes
  unmanageable

• Field, or Eulerian, approach of description
• which focuses attention on the properties of a
  flow at a given point in space as a function of
  time.
• the properties of a flow field are described as
  functions of space coordinates and time

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Experimental uncertainty

• Most consumers are unaware of it but, as with
  most foodstuffs, soft drink containers are filled
  to plus or minus a certain amount, as allowed by
  law. Because it is difficult to precisely measure
  the filling of a container in a rapid production
  process, a 12-fl-oz container may actually
  contain 12.1, or 12.7, fl oz.
• Asupplier of components for the interior of a car
  must satisfy minimum and maximum dimensions
  (called tolerances) Engineers performing
  experiments must measure not just data but also
  the uncertainties in their measurements
• There is always a trade-off in experimental work
  or in manufacturing We can reduce the
  uncertainties to a desired level, but the smaller
  the uncertainty (the more precise the measurement
  or experiment), the more expensive the procedure
  will be

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Estimation of Uncertainty

• 1. Estimate the uncertainty interval for each
  measured quantity.
• 2. State the confidence limit on each
  measurement.
• 3. Analyze the propagation of uncertainty into
  results calculated from experimental data.

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Procedure of measuring uncertainity

• Step 1 Estimate the measurement uncertainty
  interval.
• Designate the measured variables in an experiment
  as x1, x2, . . . , xn.
• One possible way to find the uncertainty interval
  for each variable would be to repeat each
  measurement many times.
• The result would be a distribution of data for
  each variable. Random errors in measurement
  usually produce a normal (Gaussian) frequency
  distribution of measured values.
• The data scatter for a normal distribution is
  characterized by the standard deviation, s.
• The uncertainty interval for each measured
  variable, xi, may be stated as nsi, where n 1,
  2, or 3.

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Procedure of measuring uncertainity

• Step 2 State the confidence limit on each
  measurement. The uncertainty interval of a
  measurement should be stated at specified odds.
  For example, one may write h 752.6 6 /- 0.5 mm
  (20 to 1).
• The confidence interval statement is based on the
  concept of standard deviation for a normal
  distribution.
• Odds of about 370 to 1 correspond to 3s as 99.7
  percent of all future readings are expected to
  fall within the interval
• Odds of about 20 to 1 correspond to 2s and odds
  of 3 to 1 correspond to 1s confidence limits.
• Odds of 20 to 1 typically are used for
  engineering work

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Procedure of measuring uncertainity

• Step 3 Analyze the propagation of uncertainty in
  calculations. Suppose that measurements
• of independent variables, x1, x2, . . . , xn, are
  made in the laboratory.
• The relative uncertainty of each independently
  measured quantity is estimated as ui.
• The measurements are used to calculate some
  result, R, for the experiment. We wish to analyze
  how errors in the xis propagate into the
  calculation of R from measured values

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Mathematically
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Basic Equations

• The basic laws, which are applicable to any
  fluid, are
• 1. The conservation of mass
• 2. Newtons second law of motion
• 3. The principle of angular momentum
• 4. The first law of thermodynamics
• 5. The second law of thermodynamics

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Summary

• On completion of this chapter students have
  studied following concepts
• How fluids are defined, and the no-slip condition
• System/control volume concepts
• Lagrangian and Eulerian descriptions
• Units and dimensions (including SI, British
  Gravitational, and English Engineering systems)
• Experimental uncertainty