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1. Density
y
Volume, 
Mass, m
C
z
Elemental
Volume,  
Mass, m
x
2. Fluid Shear
M
y
M’
Fluid Element
at time, t
P
y
P’
Fluid Element
at time, t+t
x
N
y
O
 Fx d Fx du
 yx  lim


 A 0  A
d Ay d y
y
y
Force, Fx
Velocity u
3. Viscosity
In some sense measures fluidity of a fluid. Actually it
is the resistance offered by a layer of fluid to the
motion of an adjacent one. Consider the two-plate
experiment. In case of a fluid in between them, we
know that the upper plate moves with a speed U
whereas the lower plate does not move. This sets up a
velocity gradient in a direction normal to flow.
4. Newtonian Fluids
du
 yx 
dy
• For a Newtonian fluid
In general
ie.,  yx
du

dy
u

y
 is called
•absolute or dynamic viscosity.
•Its dimensions are ML-1 T-1
•Air and water are common examples
Kinematic
viscosity (n) is
defined as ( /r) .
Its dimensions
are M L-3
5. Non-Newtonian Fluids
• Shear stress not proportional to deformation rate
•Toothpaste, paint are common examples
 du 
  k  
 dy 
Deformation rate
du
dy
n
6. Temperature Effect
It is observed
that viscosity of
a liquid decreases
with temperature
where as that of
a gas increases
with temperature.
Find out why.
Values of viscosity  and
kinematic viscosity n for
various fluids are tabulated in
handbooks and textbooks. For
air viscosity may be calculated
using
1.5
T
  1.458 106
T  110.4
the Sutherland formula,
where T is in Kelvin and  is in
kg/s m.
7. Velocity Field
•Velocity at a point may be defined as the instantaneous
velocity of a fluid particle passing through that point.
V  V(x, y, z, t) or V  uiˆ  vˆj  wkˆ
For a steady flow the properties do not change with time -
V
 0 or V  V(x, y, z)
t
If S is any property,
S
 0 or S  S(x, y, z)
t
8. 1, 2, 3 Dimensional Flows
One, Two, Three Dimensional Flows
-- One, Two, Three Space Coordinates required to specify Velocity Field
r
u
R
One Dimensional flow. u = u(r)
x
umax
y
x
u=u(x,y)
u=u(x,y)
Two Dimensional flow. u = u(x,y)
9. Surface Tension
It is the apparent interfacial stress that acts when a
liquid has a density interface
like liquid-gas, liquid-solid, liquid-liquid
2R
2R
q<90
0
q
Dh
q>90 0
2σ cosθ
Δh 
λR
Dh
q
10. Surface Shapes
water
water
soap
Wetting
wax
Non-wetting
11. Forces on half a fluid drop
pπ R 2
s
2R
2
p
R
12. Continuum Flow
For most engineering applications we consider fluid to be
continuous.
But we do know that matter consists of molecules.
To be considered continuous a fluid must have a large
number of molecules in a tiny place which is small compared
to the body dimensions.
Under ordinary conditions this is true.
For eg., A cubic metre of Air at STP contains
2.5 x 10 25 molecules.
Its mean free path is like 6.6 x 10-8 m.
13. Rarefied Flows
At great heights from the sea level it is not
possible to consider air to be continuous.
The molecular mean free path may be of the
same order of magnitude as the body dimensions.
Eg., at an altitude of 130 km the mean free path
of air is 10.2m.
Then it becomes important to consider individual
or groups of molecules. This leads to the
discipline of Rarefied Gas Dynamics.
14. Bulk Modulus of Elasticity
Compressibility of a fluid may be expressed in terms
of Bulk Modulus of Elasticity.
dp
k
dv
v
For air k is equal to g (adiabatic conditions) and p (isothermal)
For water, k =2.2 Gpa, meaning that when a pressure of
0.1Mpa acts upon a cubic metre of water, the change in volume
resulting is 1/22000 m 3.