Methods of solution of flow problems in fluid mechanics

1. “Method Of Solution Of Flow
Problems”
Prepared By: Muhammad Meezan
(UE-16058)
FLUID MECHANICS
UE-214
Department of Urban and Infrastructure Engineering, NED University of Engineering and Technology,
Karachi

2. Method Of Solution Of Flow
Problems:
For the solutions of problems of fluid stream there
are two essential equations:
The equation of continuity.
The energy equation.

3. THE EQUATION OF
CONTINUITY
The Law Of Conservation Of Mass states that mass
can be neither created or destroyed. Using The
Mass Conservation Law on a steady flow process -
flow where the flow rate do not change over time -
through a control volume where the stored mass in
the control volume do not change - implements that
 Inflow equals outflow
This statement is called the Equation of Continuity.

4. Mathematical Expression:
The Equation of Continuity and can be expressed
as:
m = ρi1 vi1 Ai1 + ρi2 vi2 Ai2 + .... + ρin vin Ain
= ρo1 vo1 Ao1 + ρo2 vo2 Ao2 +....+ ρom vom Aom
(1)
where
m = mass flow rate (kg/s)
ρ = density (kg/m3)
v = speed (m/s)
A = area (m2)

5. With uniform density equation (1) can be modified to
q = vi1 Ai1 + vi2 Ai2 + .... + vin Ain
= vo1 Ao1 + vo2 Ao2 + .... + vom Aom (2)
where
q = flow rate (m3/s)
ρi1 = ρi2 = . . = ρin = ρo1 = ρo2 = .... = ρom
For a simple reduction (or expansion) as indicated in the
figure behind the equation of continuity for uniform density
can be transformed to
vin Ain = vout Aout (Continuity Equation) (3)
or
vout = vin Ain / Aout (3b)

6. Application Of Continuity
Equation
Common application where the Equation of
Continuity are used are:
 Pipes
 Tubes and ducts with flowing fluids or gases
 Rivers
 Overall processes as power plants
 Diaries
 Logistics in general
 Roads
 Computer networks and semiconductor
technology and more.

7. The Energy Equation
In fluid dynamics, Bernoulli's principle states that an
increase in the speed of a fluid occurs
simultaneously with a decrease in pressure or a
decrease in the fluid's potential energy. The
principle is named after Daniel Bernoulli.
Bernoulli's principle can be derived from the
principle of conservation of energy. This states that,
in a steady flow, the sum of all forms of energy in a
fluid along a streamline is the same at all points on
that streamline. This requires that the sum of kinetic
energy, potential energy and internal
energy remains constant.

8. Mathematical Expression:
By assuming that fluid motion is governed only by
pressure and gravity forces, applying Newton’s
second law, F = ma, leads us to the Bernoulli
Equation.
P/ g + V2/2g + z = constant along a streamline
Where,
P=pressure, g =specific weight, V=velocity,
g=gravity z=elevation

9. A streamline is the path of one particle of water.
Therefore, at any two points along a streamline, the
Bernoulli equation can be applied and, using a set
of engineering assumptions, unknown flows and
pressures can easily be solved for.
At any two points on a streamline:
P1/ g + V1
2/2g + z1 = P2/ g + V2
2/2g + z2

10. Application Of Bernoulli's
Equation
 Sizing of Pumps
 Flow sensors
 Ejectors
 Carburetor
 Siphon
 Pitot tube

11. PROCEDURE TO SOLVE FLOW
PROBLEMS
1. Choose a datum plane through a convenient point.
2. Note at what sections the velocity is known or is to
be assumed. If at any point the section area is great
as compared with its value elsewhere, the velocity
head is so small that it may be disregarded.
3. Note at what points the pressure is known or is to
be assumed. In a body of liquid at rest with a free
surface the pressure is known at every point with in
the body. The pressure in a jet is the same as that of
the medium surrounding the jet.
4. Note whether or not there is any point where all
three term pressure, velocity and elevation are
known.
5. Note whether or not there is any point where there
is only one unknown quantity.

12. EXAMPLEPROBLEMS
#1

13. EXAMPLEPROBLEMS
#2

14. EXAMPLEPROBLEMS
#3