The document discusses dimensional analysis and dimensionless numbers. It defines dimensionless numbers as ratios of different forces that give a dimensionless quantity. Some important dimensionless numbers discussed are Reynolds number, Froude number, Euler number, Weber number, and Mach number. Equations for each of these numbers are provided showing the ratio of inertia force to viscous, gravity, pressure, surface tension, and elastic forces respectively. Methods of dimensional analysis including Rayleigh's method and Buckingham π-theorem are also summarized. Repeating variables, their selection criteria, and example problems applying Buckingham π-theorem are outlined in brief.

1. FLUID MECHANICS-II
LEC #2: SIMILITUDE AND DIMENSIONAL
ANALYSIS
Dr. M. Mubashir
Qureshi

2. DIMENSIONLESS NUMBERS
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 one force to other, it will be a
dimensionless number. These are also called non-
dimensional parameters.
 The following are most important dimensionless
numbers.
 Reynold’s Number
 Froude’s Number
 Euler’s Number
 Weber’s Number
 Mach’s Number

3. 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

   
   
  
    
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

 
 
  
   
 Froude’s Number, Re: It is the ratio of inertia force to the gravity
force of flowing fluid.

4. DIMENSIONLESS NUMBERS
Eulers’s Number, Re: 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

 
 
  
   
2 2
. .
. .
. .
. . .
Velocity Volume
Mass Velocity
Fi Time TimeWe
Fg Surface Tensionper Length Surface Tensionper Length
QV AV V L V V
L L L
L

  
   

  
   
 Weber’s Number, Re: It is the ratio of inertia force to the surface
tension force of flowing fluid.

5. DIMENSIONLESS NUMBERS
Mach’s Number, Re: 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

  


  
    


6. DIMENSIONAL ANALYSIS
Introduction: Dimensional analysis is a mathematical
technique making use of study of dimensions.
This mathematical technique is used in research work for
design and for conducting model tests.
It deals with the dimensions of physical quantities
involved in the phenomenon. All physical quantities are
measured by comparison, which is made with respect to
an arbitrary fixed value.
In dimensional analysis one first predicts the physical
parameters that will influence the flow, and then by,
grouping these parameters in dimensionless combinations
a better understanding of the flow phenomenon is made
possible.
It is particularly helpful in experimental work because it
provides a guide to those things that significantly
influence the phenomena; thus it indicates the direction in
which the experimental work should go.

7. TYPES OF DIMENSIONS
There are two types of dimensions
 Fundamental Dimensions or Fundamental Quantities
 Secondary Dimensions or Derived Quantities
Fundamental Dimensions or Fundamental
Quantities: These are basic quantities. For Example;
 Time, T
 Distance, L
 Mass, M

8. TYPES OF DIMENSIONS
Secondary Dimensions or Derived Quantities
The are those quantities which possess more than
one fundamental dimension.
For example;
 Velocity is denoted by distance per unit time L/T
 Acceleration is denoted by distance per unit time square L/T2
 Density is denoted by mass per unit volume M/L3
Since velocity, density and acceleration involve more
than one fundamental quantities so these are called
derived quantities.

9. METHODOLOGY OF DIMENSIONAL ANALYSIS
The Basic principle is Dimensional Homogeneity, which
means the dimensions of each terms in an equation on
both sides are equal.
So such an equation, in which dimensions of each term on
both sides of equation are same, is known as
Dimensionally Homogeneous equation. Such equations are
independent of system of units. For example;
Lets consider the equation V=(2gH)1/2
 Dimensions of LHS=V=L/T=LT-1
 Dimensions of RHS=(2gH)1/2=(L/T2xL)1/2=LT-1
 Dimensions of LHS= Dimensions of RHS
So the equation V=(2gH)1/2 is dimensionally homogeneous
equation.

10. METHODS OF DIMENSIONAL ANALYSIS
If the number of variables involved in a physical phenomenon are
known, then the relation among the variables can be determined by
the following two methods;
 Rayleigh’s Method
 Buckingham’s π-Theorem
Rayleigh’s Method:
It is used for determining expression for a variable (dependent)
which depends upon maximum three to four variables
(Independent) only.
If the number of independent variables are more than 4 then it is
very difficult to obtain expression for dependent variable.
Let X is a dependent variable which depends upon X1, X2, and X3 as
independent variables. Then according to Rayleigh’s Method
X=f(X1, X2, X3) which can be written as
X=K X1
a, X2
b, X3
c
Where K is a constant and a, b, c are arbitrary powers which are obtained by
comparing the powers of fundamental dimensions.

11. METHODS OF DIMENSIONAL ANALYSIS
Rayleigh’s Method
Using Rayleigh’s Method determine the rational formula for the power developm
The Pump when power (P) Depends upon the Head (H), discharge (Q)
and specific weight of the fluid.

12. RAYLEIGH’S METHOD
Q. The resisting force R of a supersonic plane during flight can be
considered as dependent upon the length of the aircraft l, velocity V, air
viscosity μ, air density ρ, and bulk modulus of air k. Express the
functional relationship between the variables and the resisting force.
 Solution:

13. BUCKINGHAM’S Π-THEOREM:
Buckingham’s π-Theorem: Since Rayleigh’s Method becomes
laborious if variables are more than fundamental dimensions (MLT),
so the difficulty is overcome by Buckingham’s π-Theorem which
states that
“If there are n variables (Independent and Dependent) in a physical
phenomenon and if these variables contain m fundamental
dimensions then the variables are arranged into (n-m)
dimensionless terms which are called π-terms.”
Let X1, X2, X3,…,X4, Xn are the variables involved in a physical
problem. Let X1 be the dependent variable and X2, X3, X4,…,Xn are
the independent variables on which X1 depends. Mathematically it
can be written as
X1=f(X2 ,X3 ,X4 ,Xn) which can be rewritten as
f1(X1,X2 X3 X4 Xn)=0
Above equation is dimensionally homogenous. It contain n variables
and if there are m fundamental dimensions then it can be written in
terms of dimensions groups called π-terms which are equal to (n-
m)
Hence f1(π1 π2 π3,… πn-m)=0

14. BUCKINGHAM’S Π-THEOREM:
Properties of π-terms:
 Each π-term is dimensionless and is independent of system of units.
 Division or multiplication by a constant does not change the character of
the π-terms.
 Each π-term contains m+1 variables, where m is the number of
fundamental dimensions and also called repeating variable.
Let in the above case X2, X3, X4 are repeating variables and if fundamental
dimensions m=3 then each π-term is written as
Π1=X2
a1. X3
b1. X4
a1 .X1
Π2=X2
a2. X3
b2. X4
a2 .X5
.
.
Πn-m=X2
a(n-m). X3
b(n-m). X4
a(n-m) .Xn
Each equation is solved by principle of dimensionless homogeneity and
values of a1, b1 & c1 etc are obtained. Final result is in the form of
Π1=(Π2, Π3, Π4 ,…, Π(n-m))
Π2=(Π1, Π3, Π4 ,…, Π(n-m))

15. METHODS OF SELECTING REPEATING VARIABLES
The number of repeating variables are equal to
number of fundamental dimensions of the problem.
The choice of repeating variables is governed by
following considerations;
 As far as possible, dependent variable should’t be selected as
repeating variable
 The repeating variables should be chosen in such a way that one
variable contains geometric property, other contains flow property
and third contains fluid property.
 The repeating variables selected should form a dimensionless group
 The repeating variables together must have the same number of
fundamental dimension.
 No two repeating variables should have the same dimensions.
Note: In most of fluid mechanics problems, the choice of
repeating variables may be (i) d,v ρ, (ii) l,v,ρ or (iii) d, v, μ.

16. BUCKINGHAM’S Π-THEOREM:

17. BUCKINGHAM’S Π-THEOREM:
Q. The resisting force R of a supersonic plane during flight can be
considered as dependent upon the length of the aircraft l, velocity V,
air viscosity μ, air density ρ, and bulk modulus of air k. Express the
functional relationship between the variables and the resisting force.

18. BUCKINGHAM’S Π-THEOREM:
Q. The drag force FD of a sphere in a fluid flowing past
the sphere is a function of the viscosity (µ),the mass
density (p), the velocity of the flow (V) and the
diameter of the sphere (D) use the step by step
method to find 𝜋 groups.
Solve…??
U,p