Advanced Fluid Mechanics
Principles of Fluid Mechanics
➢Ventilation is the application of the principles of fluid dynamics
to the flow of air
➢Airflow is induced from the atmosphere, through the intake
opening
➢and back to the atmosphere again by a differential pressure
between
intake
and return openings
➢Airflow is also generally treated as steady state, turbulent, and
incompressible.
➢However, in situations where the pressure, temperatures, and
humidity changes are large and air conditioning processes are
involved, calculations must incorporate compressibility effect of
air
Definition of A Fluid
➢ A fluid may be defined broadly as a substance which deforms
continuously when subjected to shear stress . This fluid can be
made to flow if it is acted
upon by a source of energy
➢ If the neighboring layers offer no resistance to the movement of
fluid, this fluid is said to be frictionless fluid or ideal fluid.
(Practically speaking, ideal fluids do not exist in nature, but in many
practical problems the resistance is either small or is not important,
therefore can be ignored.)
❖The properties of a fluid,
e.g., density, may, however, vary from place to place in the fluid
❖In addition to shear force, fluid may also be subjected to compressive
forces. These compressive forces tend to change the volume of the fluid
and in turn its density.
❖If the fluid yields to the effect of the compressive forces and changes its
volume, it is compressible, otherwise it is incompressible
Turbulent and Laminar Flows
➢ In a flowing fluid, each particle changes its position with a certain velocity.
The magnitudes and directions of the velocities of all particles may vary
with position as well as with time
➢ Reynold’s number: is an important dimensionless quantity in predict flow
patterns in different fluid situations.
➢ Laminar flow: Re<2300
➢ Transient flow: 2300<Re<4000
➢ Turbulent flow: 4000<Re
Streamlines
➢
A streamline is an imaginary line in a fluid, the tangent to which gives the direction of the
flow velocity at that position, as shown in Figure 4-2, where the distance between two
streamlines is an inverse measure of the magnitude of the velocity.
Streamlines
➢ If the streamlines are smoothly curved and almost parallel to each other, as illustrated in
Figure 4-3, the flow is known as streamlined flow or laminar flow
➢ A streamline is an imaginary line in a fluid, the tangent to which gives the direction of the flow
velocity at that position, as shown in Figure 4-2, where the distance between two streamlines
is an inverse measure of the magnitude of the velocity.
➢ If the streamlines are smoothly curved and almost parallel to each other, as illustrated in
Figure 4-3, the flow is known as streamlined flow or laminar flow .
➢ On the other hand, if the streamlines
are arranged haphazardly as illustrated
in Figure 4-4, the flow is known as
turbulent flow .
Reynold’s number (NRe) is a
dimensionless number
Example
Equation of motion
➢ According to Newton's Second Law, the force F used to accelerate a
body must be equal to the product of the mass m of the body and its
acceleration a.
Mathematically: Fx = m.ax
i. Fg: gravity force
ii. Fp: the pressure force
iii. Fv: force due to viscosity
iv. Ft: force due to turbulence
v. Fc: force due to compressibility
Fx = (Fg)x + (Fp)x + (Fv)x + (Ft)x + (Fc)x
❖ If the force due to compressibility , “Fc” is negligible the
resulting net force
Fx = (Fg)x + (Fp)x + (Fv)x + (Ft)x
➢ And the equations of motions are called Reynold’s equations
of motion
❖ For flow where (Ft) is negligible the resulting equations of
motion are known as Navier-Stokes Equation.
❖ If the flow is assumed to be ideal, viscous force (Fv) is zero
and equations of motions are known as Euler’s equation of
motion.
Euler’s equation of motion
Consider a cylindrical element of cross-section dA and
length dS. The forces acting on the cylindrical element are:
Bernoulli’s equation from Euler’s equation
We have got Bernoulli's equation where
Example 01
❑ Water is flowing through a pipe of 5cm diameter under a pressure of
29.43 N/cm2 (gauge) and with a mean velocity of 2.0 m/s. Find the total
head or total energy per unit weight of the water at a cross-section, which
is 5m above the datum line.
Example 02
❑ A pipe through which water is flowing is having diameters
20cm and 10cm at the cross-section 1 and 2 respectively. The
velocity of water at section at section 1 is given 4.0m/s. find
the velocity head at sections 1 and 2 and also rate of
discharge.
Exercise
Water is flowing through a pipe having diameter 300mm and
200mm at the bottom and upper end respectively. The intensity of
pressure at the bottom end is 24.525 N/cm2 and the pressure at
the upper end is 9.81N/cm2. Determine the difference in datum
head if the rate of flow through pipe is 40 lit/s
Bernoulli’s equation for real fluid
➢ Bernoulli’s equation was derived on the assumption that fluid is inviscid (nonviscous) and therefore frictionless.
➢ But all the real fluids are viscous and hence offer resistance to flow.
➢ Thus there are always some losses in fluid flows and hence in the application
of Bernoulli’s equation.
➢ Thus the Bernoulli’s equation for real fluids between points 1 and 2 is given
as
Example
Practical applications of Bernoulli’s equation
Venturimeter
A venturimeter is a device used to measure the rate of flow
of a fluid flowing through a pipe. It consists of three parts:
Example
Example
Moment of momentum equation