1
FLUID MECHANICS
1. Division of Fluid Mechanics
Hydrostatics Aerostatics
Hydrodynamics Gasdynamics
v velocity
p pressure
 density
2. Properties of fluids
Comparison of solid substances and fluids
solid
fluid
  F A [Pa] shear stress
 (deformation) is proportional to  shear stress
Solid
Fluids
(Newtonian)
d/dt (rate of deformation, strain rate) is proportional to  shear stress
non-Newtonian fluids
Fluids:




no slip condition
no change in internal structure at any deformation
continuous deformation when shear stress exists
no shear stress in fluids at rest
Viscosity
Velocity distribution: line or surface connecting the tips of velocity vectors the foot-end of which
lies on a straight line or on a plane.
2
Turn of the bar M: d
d dv x

.
dt dy
 xy  


   dy  
 dv x 


m2 / s



dv x
d
  . Newton's law of viscosity
dy
dt
kgm m
kg

. Dynamic viscosity
2
2
s m m / s ms
Kinematic viscosity
Compression of water vapor
heat exchanger
T=const
If T  Tkrit : gas O 2 and N 2  Tkrit 154 [K] and 126 [K]
pv 
p
 RT ideal-gas law

where p[Pa],  [kg/m3], T [K], R  R u / M , Ru = 8314.3 J/kmol/K universal gas constant,
M kg/kmol molar mass, for air: M=29 [kg/kmol], therefore R=287J/kg/K.
Cavitation
saturated steam pressure (vapor pressure) - temperature. Water 15 0C, pv = 1700Pa, 100 0C, pv =
1.013*105 Pa standard atmospheric pressure
3
pv
Cavitation erosion
Interactions between molecules (attraction and repulse)
repulsion
attraction
Comparison f liquids and gases
liquids
gases
distance between molecules small  d 0
large  10 d 0
role of interactions of
molecules
significant free
surface
small  fill the available
space
effect of change of pressure
on the volume
small  1000 bar
causes 5% decrease in
V
large in case of T=const V
proportional to 1/p
cause of viscosity
attraction among
molecules
momentum exchange among
molecules
relation between
viscosity and
T increases  decreases T increases 
temperature
independent
pressure
Comparison of real and perfect fluids
real fluids
perfect fluids
viscosity
viscous
inviscid
density
compressible
incompressible
structure
molecular
continuous
increases
independent
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3. Description of flow field
Scalar fields


m
kg / m 3 V incremental volume    (mean free path)
V   V
continuum =(r,t) =(x,y,z,t)
Density
 v  lim 3
Pressure
p =F/A [N/m2], [Pa].
p=p(r,t), p=p(x,y,z,t)
Temperature
T=T(r,t)
Vector fields
Velocity
v  vr, t  Eulerian description of motion

Fields (of force) g  N / kg  m / s 2 .
gravity field: g  g g k g g = 9.81 N/kg
field of inertia: accelerating coordinate system ( a  ai ) g t  ai .
centrifugal field: rotating coordinate system g c  r 2
Characterization of fields
Characterization of scalar fields:
grad p 
p
p
p
p
i
j
k
gradient vector
x
y
z
r
4 characteristics of the vector:
it is parallel with the most rapid change of p
it points towards increasing p
its length is proportional to the rate of the change of p
it is perpendicular to p = constant surfaces
Change of a variable: e.g. increment of pressure
p
p
p
p  p B  p A  grad ps 
x  y  z
x
y
z
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v  vx i  v y j  vz k  vr, t  .
Characterization of vector fields:
v x  v x ( x, y, z, t ), v y  v y ( x, y, z, t ), v z  v z ( x, y, z, t ) . vector field = 3 scalar fields
v x  grad v x  r 
v x
v
v
x  x y  x z .
x
y
z
v x
v x
 v x

 x x  y y  z z 


v y
v y
v y

v 
x 
y 
z 
 x

y
z
 v

v
v
 z x  z y  z z 
y
z
 x

Divergence: div v 
v x v y v z
,


x
y
z

dq v  vdA  v dA cos m3 / s
 vdA   div vdV
A

Gauss-Osztrogradszkij theorem
V
i

Rotation, vorticity: rot v    v 
x
vx
j

y
vy
rot v  2 .
   vds   rot vd A
G
A
Stokes theorem
 v z v y 



k

y
z 


v
v
 x  z
z  z
x 


v
v 
vz
y
 x

y 
 x
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Potential flow
v  grad  condition:    vds  0 , or rot v  0
G
Example: fields of force
for gravity force  gds  0 work of the field
G
U [m2/s2] potential of the field
v   grad U
gravity field: g  g g k U g  g g z  konst .
field of inertia: accelerating coordinate system ( a  a i ) g t  ai U t  a x  konst .
centrifugal field: rotating coordinate system g c  r 2 Uc  
r 22
 kost.
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4. Kinematics
Definitions
Pathline: loci of points traversed by a particle (photo: time exposure)
Streakline: a line whose points are occupied by all particles passing through a specified point of the
flow field (snapshot). Plume arising from a chimney, oil mist jet past vehicle model
Streamline: v x ds = 0 velocity vector of particles occupying a point of the streamline is tangent to
the streamline.
Stream surface, stream tube: no flow across the surface.
Time dependence of flow: Unsteady flow: v = v(r, t) Steady flow: v = v(r)
In some cases the time dependence can be eliminated through transformation of coordinate system.
In steady flows pathlines, streaklines and streamlines coincide, at unsteady flows in general not.
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Flow visualization:
quantitative and/or qualitative information
a) Transparent fluids, light-reflecting particles (tracers) moving with the fluid: particles of the
same density, or small particles (high aerodynamic drag). Oil mist, smoke, hydrogen
bubbles in air and in water, paints, plastic spheres in water, etc. PIV (Particle Image
Velocymetry), LDA Laser Doppler Anemometry),
b) Wool tuft in air flow shows the direction of the flow.
1,0
1,1
1,2
1,4
1,6
0,9
0°
0,8
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Irrotational (potential) vortex
Concept pf two-dimensional (2D), plane flows:
vz  0
and
v x v y

 0.
z
z
Because of continuity consideration at vortex flow v = v(r) v(r) = ?
Calculation of rotv using Stokes theorem:    vds   rot vd A
G
2
3
1
2
 vd s   vd s   vd s
G
4
1
3
4
A
  vd s   vd s
0
0
Since vds at 2nd and 4th integrals, and at 1st and 3rd integral v and ds include an angle of 00 and
1800:
 vd s  r  dr d vr  dr rdvr 
G
Since
vr  dr vr  
dv
dr
dr
after substitution
dv
dv
G vd s  r d dr dr  dr dvr   drd dr dr 0
In plane flow only (rotv)z differs from 0.
 rot vd A  rot v  r d dr
z
dA
rot v z

dv v
 .
dr r
Example: v    r  rot v z  2
In case of rotv = 0
K
dv
dr

 ln v   ln r  ln Konst.  v  . Velocity distribution in an irrotational (potential)
r
v
r
vortex.
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Motion of a small fluid particle
The motion of a FLID particle can be put together from parallel shift, deformation and rotation.
In case of potential flow no rotation occurs.
5. Continuity equation
dq m  vdA   v dA cos kg / s
integral form of continuity equation:

 vdA   t dV  0
A
differential form:
V

 div  v   0 , if the flow is steady: v = v(r)  div v  0 ,
t
if the fluid is incompressible = const. div v  0
Application of continuity equation for a stream tube
Steady flow, no flow across the surface.
Integral form of continuity equation for steady flow:  vdA  0 . "A" consists of the mantle Ap (v
A
 dA) and A1 and A 2 in- and outflow cross sections.  vd A   vd A  0 . Since
A1
vdA  v dA cos ,
A2
  v dA cos    v dA cos  0 Assumptions: over A
1
A1
A2
and A 2 (v  A) and
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over A1  = 1 =const., over A 2  = 2 =const'.
vA = Const., where v mean velocity at
changing cross section of a pipeline: 1 v1 A1  2 v2 A2  v 2  v1
1D1
2
2 D2
2
6. Hydrostatics
Static fluid: forces acting on the mass (e.g. gravity) and forces acting over the surface (forces
caused by pressure and shear stresses) balance each other (no acceleration of fluid).
p 

 dx dy dz g x  dy dz p( x )  dy dz  p( x ) 
dx   0
x 

p
 gx 
 grad p   g fundamental equation of hydrostatics.
x
Assumption: g  grad U (potential field of force)
grad p   grad U  p=const. surfaces coincide with U = Const. (equipotential surfaces)
The surface of a liquid coincides with one of the U = Const. equipotential surfaces  the
surface is perpendicular to the field of force.
Assumptions g  grad U (potential field of force),
 = const. (incompressible fluid)
p
p

p
1
grad p  grad   grad U  grad   U   0   U  const .





p1
p
 U1  2  U 2 incomplete Bernoulli equation
1
2
Pressure distribution in a static and accelerating tank
g g  g k , where g  9.81 N kg .
p p
p
i
j  k  g k
x
y
z
dp / dz  g , =áll. p  g zConst .
If z  0 , then p  p0 .  Const .  p 0  p  p0  gz . In z = H point p  p 0  gH
p1
p
 U1  2  U 2 point 1 on the surface =0), point 2 at the bottom (z = H). At z coordinate
1
2
pointing downwards U  gz , p1  p 0 , z 1  0, p 2  ?, z 2  H .
p 2  p 0  gH
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If the tank accelerates upwards, the fluid is static only in an upwards accelerating coordinate
system. Here additional (inertial) field of force should be considered: g i  a k
U i  a z U  U g  U i  g  a z . After substitution:
p2  p0  g  a H
7. Calculation of mean velocity in a pipe of circular cross section
v = ? mean velocity
In cross section of diameter D the velocity distribution is described by a paraboloid. The difference
n
of vmax and v(r) depends on the nth power of r vr   vmax 1  r / R  .

Mean velocity: v 



4q v
m / s where q v m 3 / s is the flow rate.
2
D 
The flow rate through an annulus of radius r thickness dr, cross section 2rdr is dqv = 2r
R


v(r)dr  q v   2rvmax 1  r / R  dr .
n
0
Integration yields: q v  R 2 v max
v
n
, so the mean velocity is:
n2
n
v max .
n2
In case of paraboloid of 2nd degree (n = 2) the mean velocity is half of the maximum velocity.
8. Local and convective change of variables


 div v   0 
 vgrad   div v   0
t
t
In point P the velocity is v, the variation of density in space is characterized by grad. Unsteady
flow:  / t  0 . Variation of density d in time dt?
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Two reasons for variation of :
a) Because of time dependence of density (  / t  0 ), the variation of density in point P:

d l  dt
t
b) In dt time the fluid particle covers a distance ds  vdt and gets in P' point, where the
density differs dc  grad  ds  grad  v dt from that of in point P.
d l local variation of density (only in unsteady flows)
d c convective variation of density is caused by the flow and the spatial variation of the density

The substantial variation of the density is time dt: d  d l  d c  dt  v grad  dt ,
t
d 
d

 vgrad  
 div v  0
The variation in time unit:
dt t
dt
9. Acceleration of fluid particles
The variation of vx in unit time.
dvx vx

 vgradvx .
dt
t
Acceleration of fluid particle in x direction.
The first term: local acceleration, the second term: convective acceleration.
v x
v x
v x
dv x v x

 vx
 vy
 vz
t
x
y
z
dt
dvy vy
vy
vy
vy

 vx
 vy
 vz
dt
t
x
y
z
dv z v z
v z
v z
v z

 vx
 vy
 vz
dt
t
x
y
z
Determining the differential of v(r,t): d v 
it by dt:
v
 v r
dt 
dt . Referring dv to unit time, i.e. dividing
t
 r t
r
dv  v  v r
v
, where


t
dt t  r t
Local acceleration is different from 0 if the flow is unsteady. The convective acceleration exists, if
the magnitude and/or direction of flow alter in the direction of the motion of the fluid.
The formula for acceleration can be transformed:
dv v
v2

 grad
 v  rot v .
dt t
2