Dimensional analysis

1. FLUID MECHANICS
(UE-214)
TOPIC: DIMENSIONAL ANALYSIS
SUBMITTED FROM:
MUHAMMAD MUDDASIR (UE-16063)
SUBMITTED TO:
SIR FAHAD ABDULLAH
Department of Urban and Infrastructure Engineering, NED University of Engineering and Technology,
Karachi

2. Dimensional Analysis:
Dimensional analysis is a means of simplifying a
physical problem by appealing to dimensional
homogeneity to reduce the number of relevant
variables. It is particularly useful for:
 Presenting and interpreting experimental
data.
 Attacking problems not amenable to a direct.
theoretical solution.
 Checking equations.
 Establishing the relative importance of
particular physical phenomena.
 Physical modeling.

3. Dimensional Analysis refers to the physical nature
of the quantity (Dimension) and the type of unit
used to specify it.
 Distance has dimension L.
 Area has dimension L2.
 Volume has dimension L3.
 Time has dimension T.
 Speed has dimension L/T
Application of Dimensional Analysis:
 Development of an equation for fluid
phenomenon.
 Conversion of one system of units to another.
 Reducing the number of variables required in an
experimental program.
 Develop principles of hydraulic similitude for
model study.

4. Fundamental Dimensions

5. Methods for Dimensional
Analysis:
1. Rayleigh’s Method (Named after
Lord Rayleigh)
2. Buckingham’s ∏-Method (Although named
for Edgar Buckingham, the π theorem was
first proved by French mathematician Joseph
Bertrand)

6. RAYLEIGH’S METHOD
In this method, the expression for the variables in
form of exponential equation and dimensionally
homogeneous. Let, Y is a variable, which depends on
x1,x2,x3……. Variables, then functional relationship may
be written as:
Where , Y = dependent variable, x1,x2,x3……. =
independent variables, f = function
This method is used for determining expressions for a
variables which depends upon maximum three or four
variables only. If the number of independent variables
becomes more than four, then it is very difficulty to
find expression for the dependent variables.
),.......,,( 321 nxxxxfY 

7. Method Involves The Following Steps:
1.Gather all the independent variables which
govern variation of dependent variables.
2.Write the functional relationship with the given
data
3.Write the equation in terms of a constant with
exponents(power) a,b,c.....
Where , K is a dimensionless co- efficient and
a,b,c....are the arbitrary powers.
4. Apply principal of dimensional homogeneity, and
),.......,,( 321 nxxxxfY 
,.......])(,)(,)[( 321
cba
xxxKY 

8. 5. Find out the values of exponents (a,b,c,...) By
obtaining simultaneous equation.
6. Put the value of exponents (a,b,c...) In the main
equation and form the dimensionless parameter by
grouping the variables with similar exponents.

9. Buckingham’s ∏ Method
A more generalized method of dimension
analysis developed by E. Buckingham and
others and is most popular now. This arranges
the variables into a lesser number of
dimensionless groups of variables. Because
Buckingham used ∏ (pi) to represent the product
of variables in each groups, we call this method
Buckingham pi theorem.
“If ‘n’ is the total number of variables in a
dimensionally homogenous equation containing
‘m’ fundamental dimensions, then they may be
grouped into (n-m) ∏ terms.
f(X1, X2, ……Xn) = 0
then the functional relationship will be written as
Ф (∏ , ∏ ,………….∏ ) = 0

10. The final equation obtained is in the form of:
∏1= f (∏2,∏3,………….∏n-m) = 0
Suitable where n ≥ 4
Not applicable if (n-m) = 0

11. Method Involves The Following
Steps:
 List all physical variables and note ‘n’ and ‘m’.
n = Total no. of variables
m = No. of fundamental dimensions (That is, [M], [L],
[T])
 Compute number of ∏-terms by (n-m)
 Write the equation in functional form.
 Write equation in general form.
 Select repeating variables. Must have all of the ‘m’
fundamental dimensions and should not form a ∏
among themselves.
 Solve each ∏-term for the unknown exponents by
dimensional homogeneity.

15. DISCUSSION &
CONCLUSION
We must emphasize that dimensional analysis does
not provide a complete solution to fluid problems. It
provides a partial solution only. The success of
dimensional analysis depends entirely on the ability
of the individual using it to define the parameters
that are applicable. If we omit an important variable.
The results are incomplete, and this may lead to
incorrect conclusions. Thus, to use dimensional
analysis successfully, one must be familiar with the
fluid phenomena involved.