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** Chapter 8 Dimensional Analysis**

- Velocity potential = [L
^{2}T^{â€“1}]

Stream function = [L^{2}T^{â€“1}]

Acceleration = [LT^{â€“2}]

Vorticity = [T^{â€“1}] - Total no. of variables influencing the problem is equal to the no. of independent variables plus one, one being the no. of dependent variable.
- Buckingham p theorem states that if all the n-variable are described by m fundamental dimensions, they may be grouped into (n - m) dimensions p terms.
- Selection of 3 repeating variables from the geometry of flow, fluid properties and fluid motion.
- Geometric similarity - similarity of shape

Kinematic similarity - similarity of motion

dynamic similarity - similarity of forces

Number | Equation | Significance |

Reynolds No. | Flow in closed conduit pipe | |

Froude No | where a free surface is present,structure eg. weirs spillway, channels, etc. w here gravity force is predominant. | |

Eulers No. | In cavitation studies. | |

Mach No. | where fluid compressibility is important. | |

Weber No | In capillary studies. |

**Reynolds Model Law :**

(i) Velocity ratio

(ii) Time ratio

(iii) Acceleration ratio,

(iv) Force ratio

(v) Power ratio

(vi) Discharge ratio

**Applications of Reynoldâ€™s Model Law :-**

- Flow through small sized pipes
- Low velocity motion around automobiles and aeroplane.
- Submarines completely under water.
- Flow through low speed trubo machines.

**Froudeâ€™s Model law :**

(F_{r}) m = (F_{r})

(i) Time ratio

T_{r} = L_{r}^{1/2} g_{r}^{â€“1/2} = L_{r}^{1/2}

(ii) Acceleration ratio a_{r} = g_{r} = 1

(iii) Force ratio

(iv) Power ratio

(v) Discharge ratio

**Applications : **

- Open channels
- Notches & weirs
- Spill ways & dams
- Liquid jets from orifice
- Ship partially submerged in rough & turbulent sea

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