Dimensional Analysis


Question 1
The dimension of dynamic viscosity is:
A
ML^{-1}T^{-1}
B
ML^{-1}T^{-2}
C
ML^{-2}T^{-2}
D
ML^{0}T^{-1}
GATE CE 2022 SET-2   Fluid Mechanics and Hydraulics
Question 1 Explanation: 
Unit of dynamic viscosity =\frac{kg}{m.s} \; or \; \frac{Ns}{m^2}=ML^{-1}T^{-1}
Question 2
Kinematic viscosity' is dimensionally represented as
A
\frac{M}{LT}
B
\frac{M}{L^{2} T}
C
\frac{T^{2}}{L}
D
\frac{L^{2}}{T}
GATE CE 2021 SET-1   Fluid Mechanics and Hydraulics
Question 2 Explanation: 
Kinematic viscosity
v=\frac{\mu }{\rho }=\frac{kg/m\cdot s}{kg/m^3}=m^2/s
[v]=\frac{m^2}{s }=\frac{L^2}{T}


Question 3
In a laboratory, a flow experiment is performed over a hydraulic structure. The measured values of discharge and velocity are 0.05 m^{3}/s and 0.25 m/s, respectively. If the full scale structure (30 times bigger) is subjected to a discharge of 270 m^{3}/s, then the time scale (model to full scale) value (up to two decimal places) is ______
A
0
B
0.18
C
0.55
D
0.75
GATE CE 2018 SET-1   Fluid Mechanics and Hydraulics
Question 3 Explanation: 
\begin{aligned} \text { Froude Law }(F r)_{m}&=(F r)_{p} \\ \left(\frac{V}{\sqrt{L \mathrm{g}}}\right)_{m} &=\left(\frac{v}{\sqrt{L g}}\right)_{p} \quad\left(g_{m}=g_{p}\right) \\ v_{r} &=\sqrt{L_{r}} \\ \text{or}\quad \frac{L_{r}}{T_{r}} &=\sqrt{L} \\ T_{r} &=\sqrt{L_{r}} \\ T_{r} &=\sqrt{\frac{1}{30}}=0.1826 \end{aligned}
Question 4
A 1:50 model of a spillway is to be tested in the laboratory. The discharge in the prototype spillway is 1000 m^{3}/s. The corresponding discharge (in m^{3}/s, up to two decimal places) to be maintained in the model, neglecting variation in acceleration due to gravity, is ______
A
0.01
B
0.06
C
0.50
D
0.10
GATE CE 2018 SET-1   Fluid Mechanics and Hydraulics
Question 4 Explanation: 
Froude law is valid
\begin{aligned} Q_{r} &=L_{r}^{25} \\ \frac{Q_{m}}{Q_{p}} &=\left(\frac{1}{50}\right)^{25} \\ \frac{Q_{m}}{1000} &=\left(\frac{1}{50}\right)^{25} \\ Q_{m} &=0.0566 \mathrm{m}^{3 / \mathrm{s}} \\ \text{So},\quad Q_{m} & \simeq 0.06 \mathrm{m}^{3 / \mathrm{s}} \end{aligned}
Question 5
The relationship between the length scale ratio (L_{r}) and the velocity scale ratio (V_{r}) in hydraulic models, in which Froude dynamic similarity is maintained, is:
A
V_{r}=L_{r}
B
L_{r}=\sqrt{V_{r}}
C
V_{r}=L_{r}^{1.5}
D
V_{r}=\sqrt{L_{r}}
GATE CE 2015 SET-2   Fluid Mechanics and Hydraulics
Question 5 Explanation: 
As per Froude Law
\begin{aligned} \left(\frac{V}{\sqrt{L g}}\right)_{m}&=\left(\frac{V}{\sqrt{L g}}\right)_{p} \\ \Rightarrow \quad V_{r}&=\sqrt{L_{r}} \end{aligned}


There are 5 questions to complete.

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