COURSE NUMBER: ME 321
Fluid Mechanics I
Fluid: Concept and Properties
Course teacher
Dr. M. Mahbubur Razzaque
Professor
Department of Mechanical Engineering
BUET
1
What is Fluid Mechanics?
Fluid mechanics is a branch of mechanics. It is the study of
fluids either in motion (fluid dynamics) or at rest (fluid
statics) and the subsequent effects of the fluid upon the
boundaries, which may be either solid surfaces or interfaces
with other fluids.
Both gases and liquids are classified as fluids, and the number
of fluids engineering applications is enormous: breathing,
blood flow, swimming, pumps, fans, turbines, airplanes, ships,
rivers, windmills, pipes, missiles, icebergs, engines, filters,
jets, and sprinklers, etc.
Everything on this planet either is a fluid or moves within or
near a fluid.
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What is Fluid?
From the point of view of fluid mechanics, all matter consists
of only two states, fluid and solid.
The technical distinction lies with the reaction of the two to an
applied shear or tangential stress. A solid can resist a shear
stress by a static deformation; a fluid cannot.
Any shear stress applied to a fluid, no matter how small, will
result in motion of that fluid. The fluid moves and deforms
continuously as long as the shear stress is applied.
As a corollary, we can say that a fluid at rest must be in a state
of zero shear stress, a state often called the hydrostatic stress
condition in analysis.
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What is Fluid? …
There are two classes of fluids: liquids and gases.
The distinction technically lies in the effect of cohesive forces.
A liquid, being composed of relatively close-packed molecules
with strong cohesive forces, tends to retain its volume and will
form a free surface in a gravitational field if unconfined from
above.
Gas molecules are widely spaced with negligible cohesive
force, a gas is free to expand until it encounters confining
walls. A gas has no definite volume and when left to itself
without confinement, a gas forms an atmosphere which is
essentially hydrostatic. Gases cannot form a free surface.
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What is Fluid? … …
5
Model of Fluids
Fluids are aggregations of molecules, widely spaced for a gas, closely
spaced for a liquid. The distance between molecules is very large compared
with the molecular diameter.
The molecules are not fixed in a lattice but move about freely relative to
each other. Thus fluid density or mass per unit volume, has no precise
meaning because the number of molecules occupying a given volume
continually changes.
This effect becomes unimportant if the unit volume is large compared with
the molecular spacing, when the number of molecules within the volume
will remain nearly constant in spite of the enormous interchange of particles
across the boundaries.
If, however, the chosen unit volume is too large, there could be a noticeable
variation in the bulk aggregation of the particles.
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Fig. The limit definition of fluid density: (a) an elemental volume in a fluid
region of variable density; (b) calculated density versus size of the
elemental volume.
The limiting volume δυ* is about 10-9 mm3 for all liquids and
for gases at atmospheric pressure.
10-9 mm3 of air at standard conditions contains approximately
3x107 molecules, which is sufficient to define a nearly constant
density.
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Fluid as a Continuum
Most engineering problems are concerned with physical dimensions
much larger than this limiting volume, so that fluid density is
essentially a point function and fluid properties can be thought of as
varying continually in space. Such a substance is called a
continuum, which simply is a mathematical idealization of fluids.
Although any matter is composed of several molecules, the concept
of continuum assumes a continuous distribution of mass within the
matter or system with no empty space and the properties of the matter
are continuous functions of space variables. It means that the
variation in properties is so smooth that the differential calculus
can be used to analyze the substance.
This approximation is invalid for low pressure gases where the
molecular spacing and mean free path are comparable to, or larger
than, the physical size of the system. Classical fluid mechanics is not
applicable in such cases.
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Fluid Properties
Density and Specific Weight
Fluid density is defined as mass per unit volume. The units
of density are Kg/m3 or slug/ft3.
r = m/V
A fluid property directly related to density is the specific
weight. Specific weight is defined as the weight per unit
volume.
g = W/V = mg/V = (m/V)g = rg
Where g is the local gravitational acceleration. The units of
specific weight are N/m3 or lb/ft3.
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Fluid Properties…
Specific gravity
The specific gravity is used to determine the specific weight
or density of a fluid (usually a liquid). It is defined as the ratio
of the density of a substance to that of water at a reference
temperature of 4oC.
For example, the specific gravity of mercury is 13.6, a
dimensionless number; means the mass of mercury is 13.6
times that of water for the same volume.
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Fluid Properties…
The density, specific weight and specific gravity of air and
water at standard conditions are given in Table 1.4.
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12
Fluid Properties…
Viscosity
Viscosity is the most important fluid property in the study of
fluid flows.
-It can be thought of as the internal stickiness of a fluid.
-It is one of the properties that controls the fluid flow rate in a
pipeline.
-It accounts for the energy losses associated with the transport
of fluids in ducts, channels, and pipes.
-It plays a primary role in the generation of turbulence.
-The rate of deformation of a fluid is directly linked to the
viscosity of the fluid. For a given stress, a highly viscous
fluid deforms at a slower rate rhan d fluid with a low
viscosity.
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Viscosity
Consider a flow in which the fluid particles move in the x-direction
at different speeds, so that particle velocities u vary with the ycoordinate. The Figure shows two particle positions at different
times. For such a simple flow field, in which u = u(y), we can
define the viscosity m of the fluid by the relationship
where t is the shear stress and u is the velocity in the x-direction.
The units of are N/m2 or Pa, and of m are N.s/m2. The quantity
du/dy is a velocity gradient and can be interpreted as a strain rate.
This equation is known as the Newton’s Law of Viscosity.
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Example
Consider a fluid within the small gap between two concentric
cylinders. A torque is necessary to rotate the inner cylinder at
constant speed while the outer cylinder remains stationary.
This resistance to the rotation of the cylinder is due to
viscosity.
The shear tress that resists the applied torque for this simple
flow depends directly on the velocity gradient in the fluid
film in the gap between the cylinders, i.e.
15
Example
For a small gap h<<R, this gradient can be approximated by
assuming a linear velocity distribution in the gap.
Thus using the Newton’s Law of viscosity, the shear stress on the
surface of the inner cylinder may be written as
16
Example
We can then relate the applied torque T to the viscosity and other
parameters by the equation
Here the shearing stress acting on the ends of the cylinder is
neglected; L represents the length of the rotating cylinder. Note that
the torque depends directly on the viscosity, thus the cylinders
could be used as a viscometer, a device that measures the viscosity
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of a fluid.
Fluids which follow the
linear pattern of the
Newton’s
law
of
viscosity are called
Newtonian fluids.
There are many nonNewtonian fluids and
they are treated in
rheology.
The figure compares
four examples of nonNewtonian fluids with
the
behaviour
of
Newtonian fluids.
Figure: Rheological behavior of various materials
Stress vs. Strain
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A dilatant (or shear-thickening)
fluid increases resistance with
increasing applied stress.
A pseudoplastic
(or
shearthinning) fluid decreases resistance
with increasing stress.
If the thinning effect is very strong
(the dashed curve) the fluid is
termed plastic.
The limiting case of a plastic substance is one which requires a finite
yield stress before it begins to flow. The linear-flow Bingham plastic
idealization is shown. The flow behavior after yield may also be
nonlinear.
An ex ample of a yielding fluid is toothpaste, which will not flow out
of the tube until a finite stress is applied by squeezing.
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A further complication of non-newtonian behavior is the transient
effect shown in the following figure.
Figure: Rheological behavior of
various materials
Effect of time on applied stress
Some fluids require a gradually increasing shear stress to maintain
a constant strain rate and are called rheopectic.
The opposite case of a fluid which thins out with time and requires
decreasing stress is termed thixotropic.
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Example:
A 60-cm-wide belt moves as shown. Calculate the horsepower
requirement assuming a linear velocity profile in the 10oC water.
Sol:
du/dy = 10*1000/2 = 5000 s-1
m for 10oC water = 1.308 x 10-3 N.s/m2
t = m. du/dy = 5000*1.308 x 10-3 N/m2
F = t.A = 5000*1.308 x 10-3 *4*0.6 N
Power = F.U = 5000*1.308 x 10-3 *4*0.6*10 Nm/s
= 0.21 Hp
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Example:
A 1.2 m long, 2 cm diameter shaft rotates inside an equally long
cylinder that is 2.06 cm in diameter. Calculate the torque required
to rotate the inner shaft at 2000 rpm if SAE-30 oil at 20oC fills the
gap. Also, calculate the horsepower required. Assume symmetric
motion.
Sol:
N = 2000 rpm
w = 2*3.142*N/60 = 209.5 rad/s
du = wr - 0 = 209.5 * 0.01 = 2.095 m/s
dr = 0.06/2 = 0.03 cm = 0.03 x 10-2 m
du/dr = 6982.22 s-1
m for SAE-30 oil at 20oC = 0.4 N.s/m2
t = m. du/dr = 0.4* 6982.22 = 2792.89 N/m2
F = t.A = 2792.89 *3.142*2 x 10-2 *1.2 = 210.61 N
T = r . F = 0.01* 210.61 = 2.1061 Nm
Power = T. w = 2.1061 * 209.5 Nm/s = 441.15 watt
= 0.6 Hp
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Example:
A 25-cm-diameter horizontal disk rotates a distance of 2 mm above
a solid surface. Water at 10oC fills the gap. Estimate the torque
required to rotate the disk at 400 rpm.
w
Sol:
N = 400 rpm, h = 0.002 m
w = 2*3.142*N/60 = 41.9 rad/s
m for 10oC water = 1.308 x 10-3 N.s/m2
du = wr - 0 = wr; du/dy = wr /h
t = m. du/dy = mwr /h
dF = t.dA = (mwr /h)*2prdr = 2pmwr2dr/h
dT = r*dF = 2pmwr3dr/h

2pm 3
2pm r
r dr 
h
h 4
0
R
T

4 R
0

dA
r
pm 4
R
2h
= 3.142*1.308 x 10-3 *41.9/(2*2 x 10-3)* 0.1254 = 0.0105 Nm
Power = T. w = 0.44 watt
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Viscosity
Examples of Newtonian fluids: air, water, and oil, etc. Examples
of non-Newtonian fluids: liquid plastics, blood, slurries, paints,
and toothpaste.
An important effect of viscosity is to cause the fluid to adhere to
the surface; this is known as the no-slip condition. This was
assumed in the previous examples.
The viscosity is very dependent on temperature in liquids.
Viscosity of liquids decreases with increased temperature. For a
gas, the viscosity increases as the temperature increases.
The CGS physical unit for viscosity or dynamic viscosity is the
poise (P), named after Jean Leonard Marie Poiseuille. It is more
commonly expressed, as centipoise (cP). Water at 20 °C has a
viscosity of 1.0020 cP.
1 P = 0.1 Pa·s, 1 cP = 1 mPa·s = 0.001 Pa·s = 0.001 N·s/m2.
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Kinematic Viscosity
Viscosity is often divided by the density in the derivation of
equations, it has become useful and customary to define
kinematic viscosity to be
n = m/r
Where the units of n are m2/s (ft2/sec). Note that for a gas, the
kinematic viscosity will also depend on the pressure since the
density is pressure sensitive.
The SI unit of kinematic viscosity is m2/s. The CGS physical unit
for kinematic viscosity is the stokes (St), named after George
Gabriel Stokes. It is sometimes expressed in terms of centistokes
(cSt).
1 St = 1 cm2·s−1 = 10−4 m2·s−1. 1 cSt = 1 mm2·s−1 = 10−6 m2·s−1.
Water at 20 °C has a kinematic viscosity of about 1 cSt.
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EXAMPLE:
A viscometer is constructed with two 30-cm-long concentric
cylinders, one 20.0 cm in diameter and the other 20.2 cm in
diameter. A torque of 0.13 N-m is required to rotate the inner
cylinder at 400 rpm (revolutions per minute). Calculate the
viscosity.
Ans: 0.00165 Ns/m2
EXAMPLE:
Express the above result in cP and cSt.
Ans: 1 cP = 1 mPa·s = 1cSt
0.00165 Ns/m2 = 1.65 mPa·s = 1.65 cP = 1.65 cSt
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Compressibility
In the preceding section we discussed the deformation of fluids that
results from shear stresses. In this section, we discuss the
deformation that results from pressure changes.
All fluids compress if the pressure increases, resulting in an
increase in density. A common way to describe the compressibility
of a fluid is by the following definition of the bulk modulus of
elasticity B:
In words, the bulk modulus is defined as the ratio of the change in
pressure (Dp) to relative change in density (Dr/r) while the
temperature remains constant. The bulk modulus obviously has the
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same units as pressure.
Compressibility
The bulk modulus for water at standard conditions is approximately
2100 MPa (310,000 psi), or 21 000 times the atmospheric pressure.
For air at standard conditions, B is equal to 1 atm. In general, B for
a gas is equal to the pressure of the gas.
To cause a 1% change in the density of water a pressure of 21 MPa
(210 atm) is required. This is an extremely large pressure needed to
cause such a small change; thus liquids are often assumed to be
incompressible.
For gases, if significant changes in density occur, say 4%, they
should be considered as compressible; for small density changes
they may also be treated as incompressible.
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Compressibility
Small density changes in liquids can be very significant when large
pressure changes are present.
For example, they account for "water hammer," which can be heard
shortly after the sudden closing of a valve in a pipeline.
When the valve is closed an internal pressure wave propagates
down the pipe, producing a hammering sound due to pipe motion
when the wave reflects from the closed valve.
The bulk modulus can also be used to calculate the speed of sound
in a liquid; it is given by
This yields approximately 1450 m/s (4800 ft/sec) for the speed of
sound in water at standard conditions.
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Vapor Pressure
When a small quantity of liquid is placed in a closed
container, a certain fraction of the liquid will vaporize.
Vaporization will terminate when equilibrium is reached
between the liquid and gaseous states of the substance in the
container - in other words, when the number of molecules
escaping from the water surface is equal to the number of
incoming molecules. The pressure resulting from molecules in
the gaseous state is the vapor pressure.
The vapor pressure is different from one liquid to another. For
example, the vapor pressure of water at standard conditions
(15oC, 101.3 kPa) is 1.70 kPa absolute and for ammonia it is
33.8 kPa absolute.
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Vapor Pressure
The vapor pressure is highly dependent on pressure and
temperature; it increases significantly when the temperature
increases. For example, the vapor pressure of water increases to
101.3 kPa (14.7 psi) if the temperature reaches 100oC.
In general, a transition from the liquid state to the gaseous state
occurs if the local absolute pressure is less than the vapor pressure
of the liquid.
In liquid flows, conditions can be created that lead to a pressure
below the vapor pressure of the liquid. When this happens, bubbles
are formed locally. This phenomenon is called cavitation.
Cavitation in a flow can be very damaging when bubbles are
transported by the flow to high pressure regions and collapse. It has
the potential of damaging a pipe wall or a ship‘s propeller.
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Surface Tension
Suface tension is a property that results from the attractive forces
between molecules. As such, it manifests itself only in liquids. The
forces between molecules in the bulk of a liquid are equal in all
directions, and as a result, no net force is exerted on the molecules.
However, at the surface, the molecules exert a force that has a
resultant in the surface layer. This force holds a drop of water
suspended on a rod and limits the size of the drop that may be held.
It also causes the small drops from a sprayer or atomizer to assume
spherical shapes.
Surface tension has units of force per unit length, N/m (lb/ft). The
force due to surface tension results from a length multiplied by the
surface tension; the length to use is the length of fluid in contact
with a solid, or the circumference in the case of a bubble.
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Surface Tension
A surface tension effect can be illustrated by considering the free
body diagrams of half a droplet and half a bubble as shown in Fig.
1.11. The droplet has one surface and the bubble is composed of a
thin film of liquid with an inside surface and an outside surface. The
pressure inside the droplet and bubble can now be calculated.
The pressure force ppR2 in the droplet balances the surface tension
force around the circumference. Hence
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Surface Tension
Similarly, the pressure force in the bubble is balanced by the surface
tension forces on the two circumferences. Therefore,
So, we can conclude that the
internal pressure in a bubble is
twice as large as that in a droplet
of the same size.
Figure 1.12 shows the rise of a liquid
in a clean glass capillary tube due to
surface tension. The liquid makes a
contact angle b with the glass tube.
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Surface Tension
Experiments have shown that
this angle for water and most
liquids is zero.
There are also cases for which
this angle is greater than 90o (e.g.
mercury); such liquids have a
capillary drop.
If h is the capillary rise, D the
diameter, and r the density, s
can be determined from equating
the surface tension force to the
weight of the liquid column.
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EXAMPLE
A 2-mm-diameter clean glass tube is inserted, as shown, in water at
l5oC. Determine the height that the water will climb up the tube. The
water makes a contact angle of 0o with the clean glass.
Solution
A free-body diagram of
the water shows that
the upward surfacetension force is equal
and opposite to the
weight. Writing the
surface-tension force
as surface tension
times distance, we
have
=>
36
EXAMPLE
Solving for h, we get,
The numerical values for s and r were obtained from Table of water
properties. Note that the nominal value used for the density of water
is 1000 kg/m3.
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Contact Angle
b
b
Another important surface effect is the contact angle b which
appears when a liquid interface intersects with a solid surface, as in
the above Fig. The force balance would then involve both s and b. If
the contact angle is less than 90o, the liquid is said to wet the solid;
if b > 90o, the liquid is termed nonwetting.
For example, water wets soap but does not wet wax. Water is
extremely wetting to a clean glass surface, with b = 0o. Like s, the
contact angle b is sensitive to the actual physico-chemical conditions
of the solid-liquid interface. For a clean mercury-air-glass interface,
b = 130o.
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