Continuum Hypothesis
• In this course, the assumption is made that the fluid
behaves as a continuum, i.e., the number of molecules
within the smallest region of interest (a point) are
sufficient that all fluid properties are point functions
(single valued at a point).
• For example:
• Consider definition of density ρ of a fluid
• δV* = limiting volume below which molecular variations may be important
and above which macroscopic variations may be important.
Static Fluid
For a static fluid
Shear Stress should be zero.
For A generalized Three dimensional fluid Element, Many forms of shear stress
is possible.
One dimensional Fluid Element
+Y
u=U
u=0
+
+X
Fluid Statics
• Pressure : For a static fluid, the only stress is the normal stress
since by definition a fluid subjected to a shear stress must deform
and undergo motion.
Y
tyx
tzy
tyz
tzx
txz
txy
X
Z
• What is the significance of Diagonal Elements?
• Vectorial significance : Normal stresses.
•Physical Significance : ?
•For the general case, the stress on a fluid element or at a point is a
tensor
Stress Tensor
Y
tyy
tyx
tzy
Z
tzz
txy
txx
tyz
tzx
txz
t
xz
X
First Law of Pascal
Proof ?
Simple Non-trivial Shape of A Fluid Element
Pressure Variation with Elevation
• For a static fluid, pressure varies only with elevation within the fluid.
• This can be shown by consideration of equilibrium of forces on a
fluid element
•Basic Differential Equation:
Newton's law (momentum principle)
applied to a static fluid
ΣF = ma = 0 for a static fluid
i.e., ΣFx = ΣFy = ΣFz = 0
1st order Taylor series estimate for
pressure variation over dz
For a static fluid, the pressure only varies with elevation z and is
constant in horizontal xy planes.
• The basic equation for pressure variation with elevation can be
integrated depending on
• whether ρ = constant i.e., the fluid is incompressible (liquid or lowspeed gas)
• or ρ = ρ(z), or compressible (high-speed gas) since g is constant.
Pressure Variation for a Uniform-Density Fluid
Reading Material
• Fluid Mechanics – Frank M White, McGraw Hill International
Editions.
• Introduction to Fluid Mechanics – Fox & McDOnald, John Wiley &
Sons, Inc.
• Fluid Mechanics – V L Streeter, E Benjamin Wylie & K W Bedfore,
WCB McGraw Hill.
• Fluid Mechanics – P K Kundu & I M Cohen, Elsevier Inc.
Pressure Variation for Compressible Fluids
Basic equation for pressure variation with elevation
Pressure variation equation can be integrated for γ(p,z)
known.
For example, here we solve for the pressure in the
atmosphere assuming ρ(p,T) given from ideal gas law, T(z)
known, and g ≠ g(z).
Draft Required to Establish Air Flow
Air in
Flue gas out
Natural Draft
Zref,,pref
 pg dz
dp 
R T (z )
pA = pref +Dp
p A  pref 
ZA

Z ref
Hchimney
 pg dz
Rair Tair ( z )
pB  pref 
ZB

Z ref
 pg dz
Rgas Tgas ( z )
Tgas
Tatm
A
B
Zref,,pref
 pg dz
dp 
R T (z )
pA = pref +Dp
Hchimney
Tgas
Tatm
A
B
Pressure variations in Troposphere:
T  Tref   ( Z ref  Z )
Linear increase towards earth surface
Tref & pref are known at Zref.
dp  g dz

p
R T (z )
 : Adiabatic Lapse rate : 6.5 K/km
dp  g
dz

p Ratm Tref   (Z ref  Z )
ln p 


g
ln Tref   ( Z ref  Z )  cons tan t
Ratm
Reference condition:
At Zref : T=Tref & p = pref
ln pref
 
g

ln Tref  cons tan t
Ratm


g
g
ln p 
ln Tref   ( Z ref  Z )  ln pref 
ln Tref
Ratm
Ratm
 p 
 Tref   ( Z ref  Z ) 
g


ln

ln 

 p  R
Tref


atm
 ref 
 p   Tref   ( Z ref  Z ) 



p  
Tref

 ref  
g
Ratm
Pressure at A:
 p A   Tref   ( Z ref  Z A ) 



p  
Tref

 ref  
g
Ratm
Pressure variation inside chimney differs from atmospheric pressure.
The variation of chimney pressure depends on temperature variation along
Chimney.
Temperature variation along chimney depends on rate of cooling of hot gas
Due to natural convection.
Using principles of Heat transfer, one can calculate, Tgas(Z).
If this is also linear: T = Tref,gas + gs(Zref-Z).
Lapse rate of gas, gas is obtained from heat transfer analysis.
 pB   Tref , gas   gas ( Z ref  Z B ) 



p  
Tref , gas

 ref  
g
Ratm
Natural Draft
• Natural Draft across the furnace,
 Dpnat = pA – pB
The difference in pressure will drive the exhaust.
•Natural draft establishes the furnace breathing by
–Continuous exhalation of flue gas
–Continuous inhalation of fresh air.
•The amount of flow is limited by the strength of the draft.
Pressure Measurement
Another Application of Fluid Statics
Pressure Measurement
Pressure is an important variable in fluid mechanics and
many instruments have been devised for its
measurement.
Many devices are based on hydrostatics such as
barometers and manometers, i.e., determine pressure
through measurement of a column (or columns) of a
liquid using the pressure variation with elevation equation
for an incompressible fluid.
PRESSURE
• Force exerted on a unit area :
Measured in kPa
• Atmospheric pressure at sea
level is 1 atm, 76.0 mm Hg,
101 kPa
• In outer space the pressure is
essentially zero. The pressure
in a vacuum is called absolute
zero.
• All pressures referenced with
respect to this zero pressure
are termed absolute
pressures.
• Many pressure-measuring
devices measure not
absolute pressure but only
difference in pressure. This
type of pressure reading is
called gage pressure.
• Whenever atmospheric
pressure is used as a
reference, the possibility
exists that the pressure thus
measured can be either
positive or negative.
• Negative gage pressure are
also termed as vacuum
pressures.
Manometers
Inverted U
Tube
Enlarged Leg
U Tube
Two Fluid
Inclined Tube
U-tube or differential manometer
Right Limb fluid statics :
p1  patm
p2  p1  Dh  g mfluid
Left Limb fluid statics :
p3  p4  l  g sysfluid
Point 3 and 2 are at the same elevation and same fluid
 p3  p2
p1  Dh  g mfluid  p4  l  g sysfluid
patm  Dh  g mfluid  p4  l  g sysfluid
psys  patm  Dh  g mfluid  l  g sysfluid
Gauge Pressure:
psys  patm  Dh  g mfluid  l  g sysfluid
Absolute, Gauge & Vacuum Pressures
System Pressure
Gauge Pressure
Atmospheric Pressure
Absolute zero pressure
Absolute
Pressure
Absolute, Gauge & Vacuum Pressures
Atmospheric Pressure
Vacuum Pressure
System Pressure
Absolute
Pressure
Absolute zero pressure
Stress Tensor for A Static Fluid
Y
0
0
0
0
0
tyy
txx
0
tzz
Z
X
An important Property of A Fluid under Motion
ut
tan  
y
Shear stress(t): Tangential force on per unit area of contact
between solid & fluid
Elasticity (Compressibility)
• Increasing/decreasing pressure corresponds to
contraction/expansion of a fluid.
• The amount of deformation is called elasticity.
Surface Tension
• Two non-mixing fluids (e.g., a liquid and a gas) will form an
interface.
• The molecules below the interface act on each other with forces
equal in all directions, whereas the molecules near the surface act
on each other with increased forces due to the absence of
neighbors.
• That is, the interface acts like a stretched membrane, e.g.
Vapour Pressure
• When the pressure of a liquid falls below the vapor pressure it
evaporates, i.e., changes to a gas.
• If the pressure drop is due to temperature effects alone, the process
is called boiling.
• If the pressure drop is due to fluid velocity, the process is called
cavitation.
• Cavitation is common in regions of high velocity, i.e., low p such as
on turbine blades and marine propellers.