PHYSICAL PROPERTIES

1
PHYSICAL PROPERTIES

• The main distinction between fluids is according
  to their aggregate state
• Liquids (possess the property of free surface)
• Gases (expand to completely fill available space
  limited by impermeable boundaries)

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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.
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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.
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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 lt 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. (dimension M/L3 ?
kg/m3) (Generally ? depends on pressure p (N/m2)
and temperature T (K), writing ? f(p,T)).
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PHYSICAL PROPERTIES
The total change in the density of liquids can
be described with partial changes at constant
temperature and constant pressure ?T
isothermal compressibility coefficient(1/Pa) ?P
thermal elongation coefficient(1/K) For
incompressible fluids the relationship ?T ?P
0 is valid.

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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 ? 2109 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.
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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.
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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).
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PHYSICAL PROPERTIES
Rheological diagram defines the relationship
between shear stress and velocity changes
perpendicular to solid boundary. Newtoni
an liquids (1a), Bingman liquids (1b),
structurally viscous fluids (2a,b), dilatation
liquids (3a,b).
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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).
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PHYSICAL 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 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?.
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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).
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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.
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PHYSICAL PROPERTIES
Net force on liquid molecule at the distance a gt
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 lt 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.
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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.
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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
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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
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PHYSICAL PROPERTIES
T ? ? ?

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
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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).
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HYDROSTATIC
On a fluid particle in z axis direction acts mass
force (weight) and surface force (pressure)
Equilibrium is achieved if all external
forces cancel out Adopting that pressure
difference between upper and lower surface is
given by ?p one gets
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HYDROSTATIC
After the transition ?z?0 one gets the equality
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.
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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
1,013105 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
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HYDROSTATIC
Pressure distribution diagram with adoption of
atmospheric pressure at free surface (integration
constant p0 patm) can be drawn for horizontal
and vertical components.
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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.
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HYDROSTATIC
If liquid density is not uniform along the
vertical column it is necessary to carry out the
integration as given below
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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). In addition to the
intensity of hydrostatic pressure force we are
also interested in position of force
vertex. Lets 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.

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HYDROSTATIC
Water depth can be defined as a function of
coordinate ? Total force is calculated with
the aim of integration Moment of the
surface is expressed by integral
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HYDROSTATIC
This gives an expression for the total pressure
force 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 .
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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 )
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HYDROSTATIC
Applying the Steiner rule one gets 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)
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HYDROSTATIC
In solving some practical problems one would
benefit from using the force components instead
of total force 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
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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
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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) The division of total pressure force
on horizontal and vertical components is an
engineering adaptation in solving problems with
pressure action on curved surfaces.
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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.
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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.
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HYDROSTATIC - relative equilibrium
Partial change of pressure in x-direction is
given by Analog, for 3-D valid notation
is If we are moving through the space along
the line of the same pressure (isobar), total
change in pressure is equal zero
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HYDROSTATIC - relative equilibrium
For the moving system at relative equilibrium
holds Free surface (water table) always
represents the surface of the same pressure
(pkonst.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.
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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 which after integration gives the
equation of the free surface
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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 To determine the pressure in an
arbitrary point of liquid domain, at vertical
distance h from the free surface, one can use
equations
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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.
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HYDROSTATIC - relative equilibrium
After the integration we get the
equation After dividing by g and adoption of
z2 as integration constant 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
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HYDROSTATIC - relative equilibrium
Adopting the relation z1 z2 z one
gets Finally, equation of free surface
for the liquid in vessel that rotates with
constant angular velocity ? around its axis is
given by