LECTURE №1
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1. Introduction to Fluid Mechanics
The Fluid mechanics is a part of mechanics,
that studies the states of motion and rest of
liquids and gases, as well as the interactions
with the immersed bodies.
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History
The faces of fluid mechanics
Archimedes
(287-212 BC)
Navier
(1785-1836)
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Leonardo da Vinci
(1452-1519)
Stokes
(1819-1903)
Newton
(1642-1727)
Reynolds
(1842-1912)
Bernoulli
(1667-1748)
Euler
(1707-1783)
Prandtl
(1875-1953)
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2. Fluid Properties and Types :
density - compressibility, normal force - pressure,
tangential force – stress, viscosity – Newtonian
fluids, non-Newtonian fluids, ideal fluids.
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* density - mass per unit volume

M
kg
   3  3
L
m
 = const. – incompressible fluid (water)
  const. – compressible fluid (air)
  1000 kg m3
3
  1.2 kg m
40С
water at
air at 200С
• relative density


 water 4 C 
0
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

1000
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•specific weight - weight per unit volume
F
N
   3  3
L
m
• relative specific weight
 

 water 4 C 
0
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
g
g water 40 C 
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
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•
compressibility
K – modulus of elasticity (compressibility) at constant temperature Т
dp
K w
 0,
dw T const.
K   Pa
where w - volume, p pressure
K water ~ 109 Pa
K >> 1 – incompressible fluid
K  a ,
2
a
dp
d
- sound speed
a ≈ 300 m/s – air
a ≈ 1500 m/s – water
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a >>1 – incompressible flow
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M = U/a – Mach number
М << 1 – incompressible flow
М < 1 – subsonic flow
М ~ 1 – sonic flow
М > 1 – supersonic flow
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* NORMAL FORCE F
solid body – the force F locally
acts on the contact surface A
through the pressure p
F
p ,
A
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fluid – the pressure p distributes in
all directions; acts normally to each
surface inside the fluid
 p   Pa  N
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* PRESSURE p
pressure at point: dF – elementary normal force, ds – elementary surface
dF
p  lim
 dF  pds
ds0 ds
or


dF  n pds

where n is a unit normal vector to the
elementary surface ds
Pascal’s law (17th century) – the outer pressure is distributed in
every point of the fluid volume and does not depend on the
orientation of the elementary surface ds
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* TANGENTIAL FORCE F
UA
F~
h
F – drag force
UA
F 
h
* VISCOSITY – inner friction
 - coefficient of dynamic viscosity
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* TANGENTIAL STRESS
F
 ,
A

   Pa

u  u   u
u
A ~
A
A
y
y
u
du
~


y 0
y
dy
du
 
dy
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* NEWTON’S LAW FOR VISCOSITY
tg  ~ 
s  tg y ~ y
s  ut
 s y  u
du




  
y ,t 0
t
t
y
dy
 - angular deformation, - velocity of angular deformation
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Newton's law of viscosity: The tangential stress is proportional
to the angular deformation.
   Pa.s SI 
du
 
 
dy
1 P  0.1Pa.s
 water  10 3 Pa.s  1cP
  const . (for most of the liquids at small temperature
differences)
   0   T  T0 
  const .,  0   T0 
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(for gases)
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


- coefficient of kinematic viscosity
   m 2
1St  10
s
4
SI 
2
m s
 water 20 C   10 6 m 2 s  1cSt
0
•NEWTONIAN FLUIDS – fulfill the Newton’s law of viscosity–
water, alcohol, air, some oils; stick (non-slip) on the solid walls and
are moving with the wall velocity
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* NON-NEWTONIAN FLUIDS – do not fulfill the Newton’s law of
viscosity – biological solutions, blood, milk, plasma, dye solutions,
cement, tooth paste, clay ...
* IDEAL FLUIDS – with neglected viscosity and deformation – do
not exist in nature, but at some conditions every fluid could behave
in such manner in the regions far away the solid surfaces, i.e., does
not stick on the solid walls and does not move with their velocities.
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Viscoso-elastic fluids (NON-NEWTONIAN FLUIDS )
Thin strip of a
viscous fluid
suspended on two
supports
Coiling of a honey jet falling onto a
plane: (a) Experiment (b) Numerical
simulation
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Periodical folding of a layer of
glucose syrup with 120 times
bigger viscosity than that of
water
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The effect of Weissenberg
Viscoso-elastic fluid (solution of a polystyrene in an
organic solvent) – two successive moments of the fluid
climbing on the rotation axis
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Fluid Mechanics
Coiling of the same fluid on a bobbin–
application at the synthetic fibers in
the textile industry
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
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Thank you for your attention!
Till the next Tuesday at the same place and
same time...
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