PHYSICAL PROPERTIES
The main distinction between fluids is according to their
aggregate state :
A) Liquids (possess the property of free surface)
A) Gases (expand to completely fill available space limited by
impermeable boundaries)
PHYSICAL PROPERTIES
In fluid mechanics we imagine continuously distributed mass
of arbitrary substance within space - continuum hypothesis
(Mass exists in each point of space).
It is possible to divide continuum into infinitesimal volumes,
and not to lose its physical properties.
Unlike solids, fluids will deform independent of extent of
applied force.
PHYSICAL PROPERTIES
Fluids are also divided into homogeneous and heterogeneous.
Homogeneous are those that in each point of space have the
same value of a given physical quantity.
Liquids even at very high pressures achieve only very little
change in volume. In most practical engineering cases liquids
are treated as incompressible (changes in pressure of 1 bar at
room temperature results in a change of water volume of only
0.005%).
In some particular engineering problems (e.g.. “water
hammer”) it plays an important role and can not be ignored.
PHYSICAL PROPERTIES
Generally, gases should be treated as compressible (e.g.. air).
In some engineering problems (air flow under atmospheric
conditions and room temperature, with the speed < 50 m/s)
gases can be observed as incompressible.
Density is by definition related to “mass” density  , meaning
uniformly distributed mass of fluid m in volume V.
m
a
s
s 
m
d
m1

 
l
i
m
;


v
o
l
u
m
e
V
d
V v

V

0
V
(dimension M/L3  kg/m3)
(Generally  depends on pressure p (N/m2) and temperature T
(K), writing  = f(p,T)).
PHYSICAL PROPERTIES
The total change in the density of liquids can be described
with partial changes at constant temperature and constant
pressure:
d



d
p


d
T
T
P


1










T


1


'
T


p

T
p
P
T – isothermal compressibility coefficient(1/Pa)
P – thermal elongation coefficient(1/K)
For incompressible fluids the relationship T = P = 0 is valid.
PHYSICAL PROPERTIES
Between elasticity modulus of liquid EF and compressibility
coefficient relation EF = 1/T is also valid.
In solving engineering problems one uses EF  2*109 Pa for
water.
Barotropic fluids - compressible fluids with the relation given
by  = f(p).
Adiabatic process – exactly defined mass of fluid that is
insulated from the environment so there is no heat exchange
with environment.
PHYSICAL PROPERTIES
It is known from experience that any solid body immersed in
fluid flow experiences force ; equally any solid body moving
through fluid in repose.
That force is a consequence of viscosity, or internal friction, as
basic properties of the real fluids.
Due to interaction of neighboring fluid particles deformation
takes place as a result of stresses caused by friction.
PHYSICAL PROPERTIES
Friction is the cause of mechanical energy losses.
In some cases of real fluid flow, energy losses can be ignored
due to their minor influence (ideal or inviscid fluids).
Real liquids are divided into Newtonian fluids and anomalous
viscous liquids.
In fluid mechanics, Rheology is the study of the relationship
between stresses in the fluid and the speed of deformation
(caused by friction).
PHYSICAL PROPERTIES
Rheological diagram defines the relationship between shear
stress and velocity changes perpendicular to solid boundary.
Newtonian liquids (1a), Bingman liquids (1b), structurally
viscous fluids (2a,b), dilatation liquids (3a,b).
PHYSICAL PROPERTIES
During the movement of fluid there is internal friction
(viscosity) between adjacent particles of liquid.
That friction is approximately independent on existing normal
stresses and proportional to the velocity difference between
two adjacent fluid particles.
Velocity profile between two infinitely long parallel plates
separated by constant and small vertical distance h. The lower
plate is at rest, while the one above moves with the velocity v0
(pressure is constant everywhere, fluid particle velocity at the
contact with lower and upper plate are v = 0 and v = v0).
YSICAL PROPERTIES
Due to friction (internal resistance) upper “faster” fluid particle
is decelerated and lower “slower” fluid particle is accelerated.
The corresponding tangential stress is defined with
constitutive Newtonian equation for fluids:

v
v



l
i
m


(
N
e
w
t
o
n
)

nn


n

0
Proportionality coefficient  is called dynamic viscosity and has
the unit Pas.
Dividing the coefficient of dynamic viscosity with the density
gives coefficient of kinematic viscosity (independent on
density)  =  /  m2/s.
PHYSICAL PROPERTIES
Attractive molecular forces of both liquids act on the contact
surface of two immiscible liquids, or a liquid and a gas.
If a drop of lower density fluid is set above the higher density
liquid, the drop will retain its shape, or will be spilled in a thin
film over the surface of the denser fluid (water drop on
mercury or oil drop on water).
PHYSICAL PROPERTIES
Molecules of liquid in rest are exposed to mutually attractive
force with the influence radius rM =10-7 cm. Attraction forces
of gas molecules on liquid molecules are negligibly small.
PHYSICAL PROPERTIES
Net force on liquid molecule at the distance a > rM from the
contact surface with gas is zero (Forces acting with the same
magnitude in all direction FM = 0).
On the other hand, a liquid molecule at distance a < rM
experiences net force FM  0
FM increase as a decrease.
Finally, at the contact surface between liquid and gas remain
only minimum number of molecules required for the
formation of free surface.
PHYSICAL PROPERTIES
Minimum surface required to envelop some volume of liquid is
achieved in the form of a droplet.
Retention of the droplet form is possible only if there is a
certain state of stress in the contact surface with the gas
(surface tension).
The stresses at the contact surface are called capillary stresses,
labeled with symbol  and having the unit N/m.
Thin tubes in which the effect of capillarity is highly
expressed are called capillaries.
PHYSICAL PROPERTIES
Stresses  on a curved segment of contact surface dA have
resultant force dFn that is perpendicular to that surface.
Resultant dFn is proportional to the
surface curvature.
Resultant pressure pK Pa is
defined with relation pK = dFn /dA
and equation:




1
1
2
p


,p

r

r

r

K
K
1
2
K


r
r
r
1
2
K


PHYSICAL PROPERTIES
If a liquid is in contact with a solid boundary its molecules are
under the influence of both fluids (gas and liquids), as well as
under the influence of solid boundary (adhesive force).
If the attractive force between solid wall and fluid molecules is
much stronger than the attractive forces between the
molecules of the liquid, the liquid near the solid walls has a
tendency to spread on it.
a) water and glass
b) mercury and glass
YSICAL PROPERTIES
T
ρ
η
ν
0
0,999840
1,7921
1,7924
0,06
5
0,999964
1,5108
1,5189
0,09
10
0,999700
1,3077
1,3081
0,12
15
0,999101
1,1404
1,1414
0,17
20
0,998206
1,0050
1,0068
0,24
30
0,995650
0,8007
0,8042
0,43
40
0,992219
0,6560
0,6611
0,75
50
0,988050
0,5494
0,5560
1,25
60
0,983210
0,4688
0,4768
2,02
70
0,977790
0,4061
0,4153
3,17
80
0,971830
0,3565
0,3668
4,82
90
0,965320
0,3165
0,3279
7,14
100
0,958350
0,2838
0,2961
10,33
HYDROSTATIC
The state of equilibrium is related to fluid at absolute or
relative rest. According to fluid definition, it is possible only if
shear stresses are absent and normal stresses (pressure) are
present.
Pressure is a scalar with the magnitude dependent on
position p(x,y,z).
As a first step, we analyze
pressure distribution on
fluid particle at rest
(equilibrium of
external forces).
HYDROSTATIC
On a fluid particle in z axis direction acts mass force (weight)
and surface force (pressure) :
F


m

g


ρ
g
Δ
x
Δ
y
Δ
z
Fp

Δ
x
Δ
yFp

Δ
x
Δ
y
m
p
d
p
g
d
g
Equilibrium is achieved if all external forces cancel out:
Σ
F
0 F

F

F

0
z
m
p
p
i
d
g

ρ
g
Δ
x
Δ
y
Δ
z

p
Δ
x
Δ
y

p
Δ
x
Δ
y

0
d
g
Adopting that pressure difference between “upper” and
“lower” surface is given by p one gets:

ρ
g
Δ
x
Δ
y
Δ
z

p

p

Δ
p
Δ
x
Δ
y

0




d
d

ρ
g
Δ
z

Δ
p

0
HYDROSTATIC
After the transition z0 one gets the equality :
p
ρg
z
In hydrostatic condition there is a pressure gradient in vertical
direction and it acts against the direction of gravity.
Therefore, pressure increases with increasing depth, linearly
dependent on fluid density .
Mass force is not present in horizontal direction, so pressure
change in that direction does not exist.
To obtain the absolute amount of pressure it is necessary to
integrate the above expression by variable z.
HYDROSTATIC
For the constant of integration one can choose the absolute
zero pressure p0= 0 Pa (vacuum) or relative zero pressure patm
(standard atmospheric pressure) that is generally used in
technical application (prel= paps - patm ).
Standard atmospheric pressure is defined at 15 0C and zero
elevation (sea surface) paps = 1,013 bar = 1,013*105 Pa.
In many engineering problems the fluid layer is so thin that
density can be accepted as constant along the vertical.
Consequently, pressure increase is linear :

p
d
z
ρ
g
d
z


z z0
z
0
z
z
p
z

p


ρ
g
z

z




0
0
p
z

p

ρ
g
z

z




0
0
HYDROSTATIC
Pressure distribution diagram with adoption of atmospheric
pressure at free surface (integration constant p0= patm) can be
drawn for horizontal and vertical components.
HYDROSTATIC
Dividing p with g one gets the so-called pressure head .
In the sum with geodetic datum z (from some referential
point) one gets the so-called piezometric head.
Piezometric head is constant for arbitrary point within the
fluid domain as long as fluid is at rest.
p
p
z 0 z
ko
n
st.
0
ρg
ρg
p
p
0
1
h z

z
0
1
ρg
ρg
HYDROSTATIC
If liquid density is not uniform along the vertical column it is
necessary to carry out the integration as given below:
p
z zdz  z  ρ  z  gdz
0
0
z
z
z1
z2
p
p
z zdz  z zdz 
0
1
z1
z2
z0
z1
   ρ1gdz   ρ2 gdz
HYDROSTATIC
Pressure is - according to definition - infinitesimal force dF
that acts on infinitesimal surface dA. The total pressure force
is obtained by integration over entire surface A (made up of
infinitesimals dA).
d
F

p

d
A

ρ
g
z

z
d
A


0
In addition to the intensity of hydrostatic pressure force we
are also interested in position of force vertex.
Let’s analyze the general case of arbitrary surface area A in a
plane at an angle  to the free surface horizontal plane. With
h we label depth (vertical distance) from free surface up to
some point, and with  the coordinate in the plane where
observed surface is situated.
HYDROSTATIC
Water depth can be defined
as a function of coordinate :
h

h

ζ

s
i
n
α
0
Total force is calculated with the aim of integration:
F

p
d
A

ρ
g
h
d
A

ρ
g
h
d
A

ρ
g
h
A

ρ
g
s
i
n
α

ζ
d
A
p
0




A
A
A
A
k
o
n
s
t
.
Moment of the surface is expressed by integral:
A

ζ A
ζd
T
A
HYDROSTATIC
This gives an expression for the total pressure force :
F

p

A
ρ
g
h
A
p
T
T
where: hT = T sin  the depth of observed surface A centroid,
pT pressure in the point of observed surface A centroid at the
depth hT.
The acting point of
total force FP is derived
from condition of
moment balance .
HYDROSTATIC
The sum of infinitesimal moments dM (pressure forces dF
multiplied with corresponding arms) is equal to resultant
moment (total pressure force FP multiplied with resultant
arm H ):
MP   ζ dF  ζ H  FP
A
dF  ρgh  dA  ρg  ζ sin α  dA
ζ H  FP   ρg  ζ 2 sin α  dA
A
ρg sin α 2
ζH 
ζ  dA

ρghT A A
Iζ
1
2
ζH 
ζ  dA 

ζT  A A
ζT  A
F
g
h
Pρ
TA
h
ζTs
inα
T
HYDROSTATIC
Applying the Steiner rule:
2
I

I

ζ

A
ζ T T
one gets:
2
I

ζ

AI
T T
T
ζ



ζ
H
T
ζ

A
ζ

A
T
T
where:
I moment of inertia for surface A around  axes (through
the origin of coordinate system)
IT moment of inertia for surface A around  axes (through the
centroid of surface A)
HYDROSTATIC
In solving some practical problems one would benefit from
using the force components instead of total force:
 Fx
F   Fy

F
 z
F 





F x2  F y2
Total pressure force in horizontal direction FPx is obtained
multiplying the pressure pTx = ghTx at the point of surface
projection AX centroid and surface projection area AX :
Fg

ρ
h

A
x
T
x
x
HYDROSTATIC
On the upper side of the immersed body the pressure acts
with intensity gh , while on the lower side a pressure has
intensity g(h + h).
Pressure difference at the immersed body surface is present
everywhere , so after the integration over the entire surface
one gets the so-called buoyancy force:
Fg

ρ
V

F
u
g
HYDROSTATIC
Vertical component of total pressure force is equal to the
weight of water column above the observed surface (up to
the water free surface):
F
ρ
g
V
z
z
The division of total pressure force on horizontal and vertical
components is an engineering adaptation in solving problems
with pressure action on curved surfaces.
HYDROSTATIC - relative equilibrium
If a fluid is contained in a vessel which is at rest, or moving
with constant linear velocity, it is not affected by the motion
of the vessel; but if the container is given a continuous
acceleration, this will be transmitted to the fluid and affect
the pressure distribution within.
Since the fluid remains at rest relative to the container, there
is no relative motion of the particles of the fluid and,
therefore, no shear stresses, fluid pressure being everywhere
normal to the surface on which it acts.
Under these conditions the fluid is said to be in relative
equilibrium.
HYDROSTATIC - relative equilibrium
Although it sounds paradoxical, analysis of the systems in
relative equilibrium belongs thematically to hydrostatic
chapter.
An observer who travels with the liquid in the relative
equilibrium observes the fluid as if it were in rest.
Accordingly, external forces on the liquid are in equilibrium.
External forces again
consist of pressure force
and weight. Novelty is
the existence of pressure
gradient in horizontal
Direction.
HYDROSTATIC - relative equilibrium
Partial change of pressure in x-direction is given by:

p
p
d
F

d
m

a


ρ
a

ρ
a
x 
x

x 
x
Analog, for 3-D valid notation is:
ax 
p  ρay 
 
a 
 z
If we are moving through the space along the line of the same
pressure (isobar), total change in pressure is equal zero:

p
p
p
d
p

d
x

d
y

d
z

0

x
y 
z
HYDROSTATIC - relative equilibrium
For the moving system at relative equilibrium holds :
ρ

a

d
x

ρ

a

d
y

ρ

a

d
z

0
x
y
z
Free surface (water table) always represents the surface of
the same pressure (p=konst.=p0 = patm).
Acceleration vector is always perpendicular to that surface.
We should remind ourselves that liquid at absolute rest (non
movable cane filled with liquid) has horizontal water table
due to the presence of only one acceleration vector (gravity)
acting in vertical direction.
HYDROSTATIC - relative equilibrium
At a constant change of vehicle speed in time and direction
beside the mass force in the vertical direction (gravitational
acceleration aZ = g ) coexist another mass force in horizontal
direction (aX  0).
For the free surface one has to apply:
a
d
x

g
d
z

0
x
which after integration gives the equation of the free surface:
a

x

g

z

k
o
n
s
t
.

0
x
HYDROSTATIC - relative equilibrium
By setting the coordinate system origin at free surface in the
middle of the vehicle (intersection of free surface lines in total
and relative equilibrium) equation of water table reads:
a

x
g

z
0
x
To determine the pressure in an arbitrary point of liquid
domain, at vertical distance h from the free surface, one can
use equations:
p

ρ
g
h
p

ρ

a

h
c
o
s
α
HYDROSTATIC - relative equilibrium
Next example of relative
equilibrium is the case of vessel
that rotates around its axis at a
constant angular velocity .
The function of free surface is
one more time obtained from the condition that the resultant
vector of mass forces is perpendicular to it.
ar  dr az  dz 0
az  g
ar  ω2r
ω2r  dr g z 0
HYDROSTATIC - relative equilibrium
After the integration we get the equation:
22
ω

r

g

z
k
o
n
s
t
.0

2
After dividing by g and adoption of z2 as integration constant:
2 2
ω
r
z



z
2
2
g
Constant z2 is defined according to adopted position of
coordinate system origin (at intersection of free surface and
vessel axis before the rotation). It means that after the onset
of rotation volume integral holds:
z
2

R
1
V

r
d
rd

d
z

0

z
0
20
HYDROSTATIC - relative equilibrium
Adopting the relation z1 – z2 = z one gets:
R 2
V    z  r  d  dr  0
0 0
 r

0   2g  z2r dr  0
 2R 4 z2R 2


0
8g
2
R
2 3
 ω2r 2

2π   
 z2 rdr  0
2g

0
R
R
 ωr
z2r 
 8g  2   0

0
2 4
2
ω2R2
z2 
4g
Finally, equation of free surface for the liquid in vessel that
rotates with constant angular velocity  around its axis is
given by:
22
22
ω
r ω
R
z

 
2
g 4
g
22
R
ω
Δ
h

z

z

2
1
2
g