Buckingham’s pi theorem
If there are n no. of total variables, and m no. of fundamental quantities, then given systems can be grouped into
n-m pi terms.
The resistance force F of a ship is a function of length ‘l’, velocity ‘v’, acceleration due to gravity ‘g’ and fluid
properties like density ‘ρ’, viscosity ‘μ’, and write the relationship in dimensionless form using buckingham’s pi
theorem.
�=??????(�,�,�,�,??????)
�=6,�=3
[�→���
−2
,�→��
−2
,�→�
−3
,??????→��
−1
�
−1
,�→��
−1
]
�ℎ� ����� ������ ��� �� ������� ���� 6−3 =3,3 � �����
Selection of repeated variables
1. Repeated variables must be selected from independent variables.
2. Number of repeated variables is equal to number of fundamental quantities.
3. Each repeated variable must have its own dimension.
4. Repeated variable group must contain all fundamental quantities.
5. Most fundamental quantity must be selected as repeated variable.
→�=??????(�,�,�,�,??????)
→�
1=�⋅�
�1
⋅�
�1
⋅�
�1
→�
2=�⋅�
�2
⋅�
�2
⋅�
�2
→�
3=??????⋅�
�3
⋅�
�3
⋅�
�3
�
�→
�
0
�
0
�
0
=���
−2
⋅�
�1
⋅(��
−1
)
�1
⋅(��
−3
)
�1
�
0
�
0
�
0
=�
1+�1⋅�
1+�1+�1−3�1⋅�
−2−�1
�
1=−1 �
1=−2 �
1=−2
�
1=
�
�
2
�
2
�
���������,�
2=
��
�
2
�
3=
??????
���
→�=??????(�,�,�,�,??????)
→�
1=??????(�
2,�
3)→
�
�
2
�
2
�
=??????(
��
�
2
,
??????
���
)→�=��
2
�
2
⋅??????(
��
�
2
,
??????
���
)
Various forces in fluid mechanics
Inertia force
�
??????=�⋅�
[�=
�
�
3
=
�
�
3
→�=�⋅�
3
,�=
�
�
]
�
??????=�⋅�
3
⋅
�
�
=�⋅�
2
⋅
�
�
⋅� →�
??????=�⋅�
2
⋅�
2
Pressure Force
�=
�
�
�
→�
�=�⋅�
�
�=�⋅�
2