In dimensional analysis, a dimensionless quantity is a quantity to which no physical dimension is assigned. It is also known as a bare number or pure number or a quantity of dimension one[1] and the corresponding unit of measurement in the SI is one (or 1) unit[2][3] and it is not explicitly shown. Dimensionless quantities are widely used in many fields, such as mathematics, physics, chemistry, engineering, and economics. Examples of quantities, to which dimensions are regularly assigned, are length, time, and speed, which are measured in dimensional units, such as meter , second and meter per second. This is considered to aid intuitive understanding. However, especially in mathematical physics, it is often more convenient to drop the assignment of explicit dimensions and express the quantities without dimensions, e.g., addressing the speed of light simply by the dimensionless number 1.
2. SUB:- ADVANCE FLUID MECHANICS
(2160602)
GUIDED BY: CHIRAG SIR
SR. NO. NAME ENROLLMENT NO.
1 NILESH PRAJAPATI 151240106065
2 TIRATH PRAJAPATI 151240106066
TOPIC: DIMENSION LESS NUMBERS
3. Dimensionless number
These are numbers which are obtained by dividing the inertia force by
viscous force or gravity force or pressure force or surface tension force
or elastic force.
As this is ratio of once force to other, it will be a dimensionless
number. These are also called non-dimensional parameters.
Dimensionless numbers are widely applied in mechanical &
chemical engineering.
4. Properties
A dimensionless number has no physical unit associated with it. However, it is
sometimes helpful to use the same units in both the numerator and denominator,
such as kg/kg, to show the quantity being measured.
A dimensionless number has the same value regardless of the measurement units
used to calculate it. It has the same value whether it was calculated using the metric
measurement system or the imperial measurement system.
However, a number may be dimensionless in one system of units (e.g., in a
nonrationalized cgs system of units with the electric constant ε0 = 1), and not
dimensionless in another system of units (e.g., the rationalized SI system, with ε0 =
8.85419×10-12 F/m).
5. The following are most important dimensionless
numbers.
Reynold’s Number
Froude’s Number
Euler’s Number
Mach’s Number
Weber’s Number
6. Reynolds number
Sir George Stokes introduced Reynolds numbers. Osbome Reynolds popularised the concept.
The concept was introduced by Sir George Stokes in 1851, but the Reynolds number was named
by Arnold Sommerfeld in 1908after Osborne Reynolds (1842–1912), who popularized its use in
1883.
7. Dimensionless Numbers
Reynold’s Number,Re:It is the ratio of inertia force to the viscous force
of flowing fluid.
. .
Re
. .
. . .
. . .
Velocity Volume
Mass Velocity
Fi Time Time
Fv Shear Stress Area Shear Stress Area
QV AV V AV V VL VL
du VA A A
dy L
The Reynolds Number can be used to determine if flow is
laminar, transient or turbulent. The flow is
•laminar when Re < 2300
•transient when 2300 < Re < 4000
•turbulent when Re > 4000
8. Example - Calculating Reynolds Number
A Newtonian fluid with a dynamic or absolute viscosity of 0.38 Ns/m2 and a specific gravity
of 0.91 flows through a 25 mm diameter pipe with a velocity of 2.6 m/s.
The density can be calculated using the specific gravity like
ρ = 0.91 (1000 kg/m3)
= 910 kg/m3
The Reynolds Number can then be calculated using equation (1) like
Re = (910 kg/m3) (2.6 m/s) (25 mm) (10-3 m/mm) / (0.38 Ns/m2)
= 156 (kg m / s2)/N
= 156 ~ Laminar flow
(1 N = 1 kg m / s2)
9. Froude’s number
named after William Froude is a dimensionless number defined as the ratio of
characteristic velocity to the gravity wave velocity.
The Froude number in terms of gravity is expressed as,
It is used to determine the resistance of an body which is submerged partially moving
along with water.
10. Where,
Fr is Froude number,
v is velocity,
g is gravity,
l is characteristic length.
2
. .
. .
. .
. .
Velocity Volume
Mass Velocity
Fi Time TimeFe
Fg Mass Gavitational Acceleraion Mass Gavitational Acceleraion
QV AV V V V
Volume g AL g gL gL
11. Example – Calculating frodue’s Number
Question 1: Find the Froude number if the length of the boat is 2m and velocity is 10 m/s.
Solution:
Given: length l = 2m, velocity v = 10 m/s
The froude number is given by,
Fr = v(gl)1/2v(gl)1/2
Fr = 2m/s(9.8m/s2×2m)1/22m/s(9.8m/s2×2m)1/2
Fr = 0.451
Therefore, the froude number of the boat is 0.451.
12. Question 2: Calculate the velocity of the moving fish in the water if its froude
number is 0.72 and length 0.5 m.
Solution:
Given: length l = 0.5m, froude number Fr = 0.72
The froude number is given by,
Fr = v(gl)1/2v(gl)1/2.
The velocity of the moving fish is,
v = Fr ×× (gl)1/2(gl)1/2
v = 0.72 ×× (9.8×0.5)1/2(9.8×0.5)1/2
v = 1.59 m/s.
13. Euler’s Number
Euler's Number, E:It is the ratio of inertia force to the pressure force of
flowing fluid.
2
. .
Pr . Pr .
. .
. . / /
u
Velocity Volume
Mass Velocity
Fi Time TimeE
Fp essure Area essure Area
QV AV V V V
P A P A P P
14. Mach’s Number, M:
It is the ratio of inertia force to the elastic force of flowing fluid.
2 2
2
. .
. .
. .
. . /
: /
Velocity Volume
Mass Velocity
Fi Time TimeM
Fe Elastic Stress Area Elastic Stress Area
QV AV V L V V V
K A K A KL CK
Where C K
Mach’s number
15. Weber’s Number
The Weber Number is a dimensionless value useful for analyzing fluid
flows where there is an interface between two different fluids.
The Weber Number is the ratio between the inertial force and the
surface tension force and the Weber number indicates whether the
kinetic or the surface tension energy is dominant. It can be expressed
as
We = ρ v2 l / σ
16. where
We = Weber number (dimensionless)
ρ = density of fluid (kg/m3, lb/ft3)
v = velocity of fluid (m/s, ft/s)
l = characteristic length (m, ft)
σ = surface tension (N/m)
Since the Weber Number represents an index of the inertial force to
the surface tension force acting on a fluid element, it can be useful
analyzing thin films flows and the formation of droplets and bubbles.
