Chapter 1 FLUID PROPERTIES

1
Chapter 1FLUID PROPERTIES
2

• The engineering science of fluid mechanics
  fluid properties, the application of the basic
  laws of mechanics and thermodynamics, and orderly
  experimentation.
• The properties of density and viscosity play
  principal roles in open- and closed-channel flow
  and in flow around immersed objects.
• Surface-tension effects are important in the
  formation of droplets, in the flow of small jets,
  and in situations where liquid-gas-solid or
  liquid-liquid-solid interfaces occur, as well as
  in the formation of capillary waves.
• The property of vapor pressure, which accounts
  for changes of phase from liquid to gas, becomes
  important when reduced pressures are encountered.
  
• This chapter liquid is defined and the
  International System of Units (SI) of force,
  mass, length, and time units are discussed before
  the discussion of properties and definition of
  terms is taken up.

3
1.1 DEFINITION OF A FLUID

• Fluid substance that deforms continuously when
  subjected to a shear stress, no matter how small
  that shear stress may be.
• Shear force is the force component tangent to a
  surface, and this force divided by the area of
  the surface is the average shear stress over the
  area. Shear stress at a point is the limiting
  value of shear force to area as the area is
  reduced to the point.
• Fig. 1.1

4

• Figure 1.1 Deformation resulting from application
  of constant shear force.
• A substance is placed between two closely spaced
  parallel plates so large that conditions at their
  edges may be neglected. The lower plate is fixed,
  and a force F is applied to the upper plate,
  which exerts a shear stress F/A on any substance
  between the plates. A is the area of the upper
  plate. When the force F causes the upper plate to
  move with a steady (nonzero) velocity, no matter
  how small the magnitude of F, one may conclude
  that the substance between the two plates is a
  fluid.

5

• The fluid in immediate contact with a solid
  boundary has the same velocity as the boundary
  (no slip at the boundary)
• Fig.1.1 fluid in the area abcd flows to the new
  position ab'c'd, each fluid particle moving
  parallel to the plate and the velocity u varying
  uniformly from zero at the stationary plate to U
  at the upper plate.
• Experiments other quantities being held
  constant, F is directly proportional to A and to
  U and is inversely proportional to thickness t.
  In equation form
• µ is the proportionality factor and includes the
  effect of the particular fluid

6

• If t F/A for the shear stress,
• The ratio U/t angular velocity of line ab, or it
  is the rate of angular deformation of the fluid
  (rate of decrease of angle bad)
• The angular velocity may also be written du/dy
  more general
• The velocity gradient du/dy may also be
  visualized as the rate at which one layer moves
  relative to an adjacent layer, in differential
  form,
• (1.1.1)
• - Newton's law of viscosity
• - proportionality factor µ viscosity of the
  fluid

7

• Materials other than fluids cannot satisfy the
  definition of a fluid
• A plastic substance will deform a certain amount
  proportional to the force, but not continuously
  when the stress applied is below its yield shear
  stress.
• A complete vacuum between the plates would cause
  deformation at an ever-increasing rate.
• If sand were placed between the two plates,
  Coulomb friction would require a finite force to
  cause a continuous motion.
• ? plastics and solids are excluded from the
  classification of fluids.

8

• Fluids
• Newtonian
• non-Newtonian
• Newtonian fluid linear relation between the
  magnitude of applied shear stress  and the
  resulting rate of deformation µ constant in Eq.
  (1.1.1) (Fig. 1.2)
• Non-Newtonian fluid nonlinear relation between
  the magnitude of applied shear stress and the
  rate of angular deformation
• An ideal plastic has a definite yield stress and
  a constant linear relation of t to du/dy.
• A thixotropic substance, such as printer's ink,
  has a viscosity that is dependent upon the
  immediately prior angular deformation of the
  substance and has a tendency to take a set when
  at rest.
• Gases and thin liquids tend to be Newtonian
  fluids, while thick, long-chained hydrocarbons
  may be non-Newtonian.

9
Figure 1.2 Rheological diagram.
10

• For purposes of analysis, the assumption is
  frequently made that a fluid is nonviscous
• With zero viscosity the shear stress is always
  zero, regardless of the motion of the fluid.
• If the fluid is also considered to be
  incompressible, it is then called an ideal fluid
  and plots as the ordinate in Fig. 1.2.

11
1.2 FORCE, MASS, LENGTH, AND TIME UNITS

• Force, mass, length, and time consistent units
• greatly simplify problem solutions in mechanics
• derivations may be carried out without reference
  to any particular consistent system
• A system of mechanics units consistent when unit
  force causes unit mass to undergo unit
  acceleration
• The International System (SI)
• newton (N) as unit or force
• kilogram (kg) as unit of mass
• metre (m) as unit of length
• the second (s) as unit of time

12

• With the kilogram, metre, and second as defined
  units, the newton is derived to exactly satisfy
  Newton's second law of motion
• (1.2.1)
• The force exerted on a body by gravitation is
  called the force of gravity or  the gravity
  force. The mass m of a body does not change with
  location the  force of gravity of a body is
  determined by the product of the mass and the 
  local acceleration of gravity g
• (1.2.2)
• For example, where g 9.876 m/s2, a body with
  gravity force of 10 N has a mass m 10/9.806 kg.
  At the location where g 9.7 m/s2, the force of
  gravity is
• Standard gravity is 9.806 m/s2. Fluid properties
  are often quoted at  standard conditions of 4oC
  and 760 mm Hg.

13
Table 1.1 Selected prefixes for powers of 10 in
SI units
14
1.3 VISCOSITY

• Viscosity requires the greatest consideration in
  the  study of fluid flow.
• Viscosity is that property of a fluid by virtue
  of which it offers resistance to  shear.
• Newton's law of viscosity Eq. (1.1.1) states
  that for a given rate of  angular deformation of
  fluid the shear stress is directly proportional
  to the  viscosity.
• Molasses and tar are examples of highly viscous
  liquids water and  air have very small
  viscosities.

15

• The viscosity of a gas increases with
  temperature, but the viscosity of a  liquid
  decreases with temperature it can be explained
  by examining the causes of viscosity.
• The resistance of a fluid to shear depends upon
  its cohesion and upon its rate of transfer of
  molecular momentum.
• A liquid, with molecules much more closely spaced
  than a gas, has cohesive forces much larger than
  a gas. Cohesion - predominant cause of viscosity
  in a liquid and since cohesion decreases with
  temperature, the viscosity does likewise.
• A gas, on the other hand, has very small cohesive
  forces. Most of its resistance to shear stress is
  the result of the transfer of molecular momentum.

16

• Fig.1.3 rough model of the way in which momentum
  transfer gives rise to an apparent shear stress,
  considering two idealized railroad cars loaded
  with sponges and on parallel tracks
• Assume each car has a water tank and pump so
  arranged that the water is directed by nozzles at
  right angles to the track. First, consider A
  stationary and B moving to the right, with the
  water from its nozzles striking A and being
  absorbed by the sponges. Car A will be set in
  motion owing to the component of the momentum of
  the jets which is parallel to the tracks, giving
  rise to an apparent shear stress between A and B.
  Now if A is pumping water back into B at the same
  rate, its action tends to slow down B and equal
  and opposite apparent shear forces result. When
  both A and B are stationary or have the same
  velocity, the pumping does not exert an apparent
  shear stress on either car.

17
Figure 1.3 Model illustrating transfer of
momentum.

• Within fluid there is always a transfer of
  molecules back and forth across any fictitious
  surface drawn in it. When one layer moves
  relative to an adjacent layer, the molecular
  transfer of momentum brings momentum from one
  side to the other so that an apparent shear
  stress is set up that resists the relative motion
  and tends to equalize the velocities of adjacent
  layers in a manner analogous to that of Fig. 1.3.
  The measure of the motion of one layer relative
  to an adjacent layer is du/dy.

18

• Molecular activity gives rise to an apparent
  shear stress in gases which is more important
  than the cohesive forces, and since molecular
  activity increases with temperature, the
  viscosity of a gas also increases with
  temperature.
• For ordinary pressures viscosity is independent
  of pressure and depends upon temperature only.
  For very great pressures, gases and most liquids
  have shown erratic variations of viscosity with
  pressure.
• A fluid at rest or in motion so that no layer
  moves relative to an adjacent layer will not have
  apparent shear forces set up, regardless of the
  viscosity, because du/dy is zero throughout the
  fluid
• ? fluid statistics - no shear forces considered,
  and the only stresses remaining are normal
  stresses, or pressures ? greatly simplifies the
  study of fluid statics, since any free body of
  fluid can have only gravity forces and normal
  surface forces acting on it

19

• Dimensions of viscosity from Newton's law of
  viscosity solving for the viscosity µ
• and inserting dimensions F, L, T for force,
  length, and time,
• shows that µ has the dimensions FL-2T.
• With the force dimension expressed in   terms of
  mass by use of Newton's second law of motion, F
  MLT-2, the  dimensions of viscosity may be
  expressed as ML-1T-1.
• The SI unit of viscosity which is the pascal
  second (symbol Pa?s) has no name.

20
Kinematic Viscosity

• µ - absolute viscosity or the dynamic viscosity
• ? - kinematic viscosity (the ratio of viscosity
  to mass density)
• (1.3.1)
• - occurs in many applications (e.g., in the
  dimensionless Reynolds number for motion of a
  body through a fluid, Vl/?, in which V is the 
  body velocity and l is a representative linear
  measure or the body size)
• The  dimensions of ? are L2T-1.
• SI unit 1 m2/s, has no name.
• Viscosity is practically independent of pressure
  and depends upon temperature only.
• The kinematic viscosity of liquids, and of gases
  at a given pressure, is substantially a function
  of temperature.

21

• Example 1.1
• A liquid has a viscosity or 0.005 Pa?s and a
  density or 850 kg/m3. Calculate the kinematic
  viscosity

22

• Example 1.2
• In Fig. 1.4 the rod slides inside a concentric
  sleeve with a reciprocating motion due to the
  uniform motion of the crank. The clearance is d
  and the viscosity µ. Write a program in BASIC to
  determine the average energy loss per unit time
  in the sleeve. D 0.8 in, L 8.0 in, d 0.001
  in, R 2 ft, r 0.5 ft, µ 0.0001 lb s/ft2,
  and the rotation speed is 1200 rpm.
• The energy loss in the sleeve in one rotation is
  the product of resisting viscous (shear) force
  times displacement integrated over the period of
  the motion. The period T is 2p/w, where w
  d?/dt. The sleeve force depends upon the
  velocity. The force Fi and position xi are found
  for 2n equal increments of the period. Then by
  the trapezoidal rule the work done over the half
  period is found
• Using the law of sines to eliminate f, we get
• Figure 1.5 lists the program, in which the
  variable RR represents the crank radius r.

23
Figure 1.4 Notation for sleeve motion
24
Figure 1.5 BASIC program to determine loss in
sleeve motion
25
1.4 CONTINUUM

• In dealing with fluid-flow relations on a
  mathematical or analytical basis consider that
  the actual molecular structure is replaced by a
  hypothetical continuous medium - continuum.
• Example velocity at a point in space is
  indefinite in a molecular medium, as it would be
  zero at all times except when a molecule occupied
  this exact point, and then it would be the
  velocity of the molecule and not the mean mass
  velocity of the particles in the neighborhood.
• This is avoided if consider velocity at a point
  to be the average or mass velocity of all
  molecules surrounding the point. With n molecules
  per cubic centimetre, the mean distance between
  molecules is of the order n-1/3 cm.

26

• Molecular theory, however, must be used to
  calculate fluid properties (e.g., viscosity)
  which are associated with molecular motions, but
  continuum equations can be employed with the
  results of molecular calculations.
• Rarefied gases (the atmosphere at 80 km above sea
  level) the ratio of the mean free path the mean
  free path is the average distance a molecule
  travels between collisions of the gas to a
  characteristic length for a body or conduit is
  used to distinguish the type of flow.
• The flow regime is called gas dynamics for very
  small values of the ratio the next regime is
  called slip flow and for large values of the
  ratio it is free molecular flow.
• In this text only the gas-dynamics regime is
  studied.

27
1.5 DENSITY, SPECIFIC VOLUME, UNIT GRAVITY FORCE,
RELATIVE DENSITY, PRESSURE

• The density ? its mass per unit volume.
• Density at a point the mass ?m of fluid in a
  small volume ?V surrounding the point
• (1.5.1)
• For water at standard pressure (760 mm Hg) and
  4oC, ? 1000 kg/m3.
• The specific volume vs the volume occupied by
  unit mass of fluid
• (1.5.2)

28

• The unit gravity force, ? the force of gravity
  per unit volume. It changes with location
  depending upon gravity
• (1.5.1)
• Water ? 9806 N/m3 at 5oC, at sea level.
• The relative density S of a substance the ratio
  of its mass to the mass of an equal volume of
  water at standard conditions. (may also be
  expressed as a ratio or its density to that of
  water).
• The average pressure the normal force pushing
  against a plane area divided by the area.
• The pressure at a point is the ratio of normal
  force to area as the area approaches a small
  value enclosing the point.
• If a fluid exerts a pressure against the walls or
  a container, the container will exert a reaction
  on the fluid which will be compressive.
• Liquids can sustain very high compressive
  pressures, but are very weak in tension ?
  absolute pressures in this book are never
  negative (otherwise fluid would be sustaining a
  tensile stress)
• Units force per area, which is newtons per square
  metre, called pascals (Pa).
• Absolute pressure P, gage pressures p.

29
1.6 PERFECT GAS

• This treatment thermodynamic relations and
  compressible-fluid-flow cases are limited
  generally to perfect gases (defined in this
  section)
• The perfect gas substance that satisfies the
  perfect-gas-law
• (1.6.1)
• and that has constant specific heats. P is the
  absolute pressure vs is the specific volume R
  is the gas constant T is the absolute
  temperature.

30

• The perfect gas must be carefully distinguished
  from the ideal fluid. An ideal fluid frictionless
  and incompressible. The perfect gas has viscosity
  and can therefore develop shear stresses, and it
  is compressible according to Eq. (1.6.1).
• Eq.(1.6.1) the equation of state for a perfect
  gas may be written
• (1.6.2)
• The units of R can be determined from the
  equation

31

• Real gases below critical pressure and above the
  critical temperature tend to obey the perfect-gas
  law. As the pressure increases, the discrepance
  increases and becomes serious near the critical
  point.
• The perfect-gas law encompasses both Charles' law
  and Boyle's law.
• Charles' law for constant pressure the volume of
  a given mass of gas varies as its absolute
  temperature.
• Boyle's law (isothermal law) for constant
  temperature the density varies directly as the
  absolute pressure.

32

• The volume v of m mass units of gas is mvs ?
• (1.6.3)
• With being the volume per mole
• (1.6.4)
• If n is the number of moles of the gas in volume
  ?
• (1.6.5)
• The product MR, called the universal gas
  constant, has a value depending only upon the
  units employed
• (1.6.6)
• The gas constant R can then be determined from
• (1.6.7)
• ? knowledge of relative molecular mass leads to
  the value of R

33

• The specific heat cv of a gas number of units
  of heat added per unit mass to raise the
  temperature of the gas one degree when the volume
  is held constant.
• The specific heat cp the number of heat units
  added per unit mass to raise the temperature one
  degree when the pressure is held constant.
• The specific heat ratio k cp/cv.
• The intrinsic energy u (dependent upon P, ? and
  T) the energy per unit mass due to molecular
  spacing and forces.
• The enthalpy h important property of a gas
  given by huP/?.
• cv and cp units joule per kilogram per kelvin
  (J/kg?K)
• 4187 J of heat added raises the temperature of
  one kilogram of water one degree Celsius at
  standard conditions
• R is related to cv and cp by

34

• Example 1.2
• A gas with relative molecular mass of 44 is at a
  pressure or 0.9 MPa and a temperature of 20oC.
  Determine its density.
• From Eq.(1.6.7),
• Then, from Eq.(1.6.2)

35
1.7 BULK MODULUS OF ELASTICITY

• For most purposes a liquid may be considered as
  incompressible, but for situations involving
  either sudden or great changes in pressure, its
  compressibility becomes important also when
  temperature changes are involved, e.g., free
  convection.
• The compressibility of a liquid is expressed by
  its bulk modulus of elasticity.
• If the pressure of a unit volume of liquid is
  increased by dp , it will cause a volume decrease
  -dV the ratio dp/dV is the bulk modulus of
  elasticity K
• For any volume V
• (1.7.1)
• Expressed in units of p. For water at 20oC K
  2.2. GPa.

36

• Example 1.3
• A liquid compressed in a cylinder has a volume
  of 1 L (1000cm3) at 1 MN/m2 and volume of 995 cm3
  at 2 MN/n2. What is its bulk modulus of
  elasticity?

37
1.8 VAPOR PRESSURE

• Liquids evaporate because or molecules escaping
  from the liquid surface vapor molecules exert
  partial pressure in the space - vapor pressure.
• If the space above the liquid is confined, after
  a sufficient time the number of vapor molecules
  striking the liquid surface and condensing is
  just equal to the number escaping in any interval
  of time, and equilibrium exists.
• Depends upon temperature and increases with it.
  Boiling when the pressure above a liquid equals
  the vapor pressure of the liquid
• 20oC water 2.447 kPa, mercury 0.173 Pa
• When very low pressures are produced at certain
  locations in the system, pressures may be equal
  to or less than the vapor pressure ? the liquid
  flashes into vapor - cavitation.

38
1.9 SURFACE TENSION

• Capillarity
• At the interface between a liquid and a gas, or
  two immiscible liquids, a film or special layer
  seems to form on the liquid, apparently owing to
  attraction of liquid molecules below the surface
• The formation or this film may be visualized on
  the basis of surface energy or work per unit area
  required to bring the molecules to the surface.
  The surface tension is then the stretching force
  required to form the film, obtained by dividing
  the surface-energy term by unit length of the
  film in equilibrium.
• The surface tension of water varies from about
  0.074 N/m at 20oC to 0.059 N/m at 100oC (Table
  1.2)

39
Table 1.2 Approximate properties of common
liquids at 20oC and standard atmospheric pressure
40

• The action of surface tension is to increase the
  pressure within a droplet of liquid or within a
  small liquid jet.
• For a small spherical droplet of radius r the
  internal pressure p necessary to balance the
  tensile force due to the surface tension s
  calculated in terms of the forces which act on a
  hemispherical free body (see Sec. 2.6),
• For the cylindrical liquid jet of radius r, the
  pipe-tension equation applies
• Both equations the pressure becomes large for a
  very small radius of droplet or cylinder

41

• Capillary attraction is caused by surface tension
  and by the relative value of adhesion between
  liquld and solid to cohesion of the liquid.
• A liquid that wets the solid has a greater
  adhesion than cohesion. Surface tension in this
  case causes the liquid to rise within a small
  vertical tube that is partially immersed in it
• For liquids that do not wet the solid, surface
  tension tends to depress the meniscus in a small
  vertical tube. When the contact angle between
  liquid and solid is known, the capillary rise can
  be computed for an assumed shape of the meniscus.
• Figure 1.4 the capillary rise for water and
  mercury in circular glass tubes in air

42
Figure 1.5 Capillarity in circular glass tubes