# Fluid Mechanics and Hydraulics

 Question 1
Two discrete spherical particles (P and Q) of equal mass density are independently released in water. Particle P and particle Q have diameters of 0.5 mm and 1.0 mm, respectively. Assume Stokes' law is valid.
The drag force on particle Q will be________ times the drag force on particle P. (round off to the nearest integer)
 A 4 B 8 C 10 D 12
GATE CE 2022 SET-2      Boundary Layer Theory, Drag and Lift
Question 1 Explanation:
In case of discrete particle settling and Stoke's law valid, at terminal velocity, since there is no change in velocity, the net force on the body is zero. Hence,

$F_D=\left ( \frac{\pi}{6}D^3\rho _sg-\frac{\pi}{6}D^3\rho _sg \right )=\frac{\pi}{6}gD^3(\rho _s-\rho _f)$
For density of medium $(\rho _f)$ and mass density of sphere $(\rho _s)$ constant,
Drag force $(F_D)\propto D^3$
For particle P, diameter $(D_P)=0.5 mm$
For particle Q, diameter $(D_Q)=1 mm$
$\Rightarrow \frac{(F_D)_Q}{(F_D)_P}=\frac{(D_Q)^3}{(D_P)^3}=\left ( \frac{1}{0.5} \right )^3=8$
$\Rightarrow (F_D)_Q=8 \times (F_D)P$
 Question 2
A pump with an efficiency of 80% is used to draw groundwater from a well for irrigating a flat field of area 108 hectares. The base period and delta for paddy crop on this field are 120 days and 144 cm, respectively. Water application efficiency in the field is 80%. The lowest level of water in the well is 10 m below the ground. The minimum required horse power (h.p.) of the pump is ________. (round off to two decimal places)
(Consider 1 h.p. = 746 W; unit weight of water = 9810 $N/m^3$)
 A 25.64 B 36.25 C 30.82 D 48.32
GATE CE 2022 SET-2      Hydraulic Pumps
Question 2 Explanation:
Base period (B) = 120 days
\begin{aligned} Delta (\Delta )&=144 cm=1.44m\\ Duty(D)&=\frac{8.64B}{\Delta }=\frac{8.64 \times 120}{1.44}\\ D&=720\frac{hec}{cumec}\\ Q&=\frac{Area}{D}=\frac{108}{720}=\frac{3}{20}m^3/s\\ Q_{applied}&=\frac{Q}{\eta _a}=\frac{3}{20 \times 0.8}=\frac{3}{16}\\ \text{Power required}&=\frac{mgh}{t}\\ &=\rho _w \times \left ( \frac{volume}{t} \right )g \times h\\ &=\rho _w Qgh=\gamma Qh\\ &=9810 \times \frac{3}{16} \times 10\\ &=\frac{73575}{4}watt \end{aligned}
Horse power of pump $= \frac{\text{Power required}}{746 \times \text{efficiency}}=\frac{73575}{4 \times 746 \times 0.8}=30.82hp$
 Question 3
A hydraulic jump takes place in a 6 m wide rectangular channel at a point where the upstream depth is 0.5 m (just before the jump). If the discharge in the channel is 30 $m^3/s$ and the energy loss in the jump is 1.6 m, then the Froude number computed at the end of the jump is ___________. (round off to two decimal places) (Consider the acceleration due to gravity as 10 $m/s^2$.)
 A 0.4 B 0.85 C 0.65 D 0.75
GATE CE 2022 SET-2      Open Channel Flow
Question 3 Explanation:

$Q=30m^3/sec$
$B=6m$
$y_1=0.5m$
$E_L=1.6m$
$q= \frac{Q}{B} =\frac{30}{6}=5m^3/sec/m$
We know that,
\begin{aligned} E_L=\frac{(y_2-y_1)^3}{4y_1y_2}&=1.6m\\ \frac{(y_2-0.5)^3}{4 \times 0.5 \times y_2}&=1.6\\ y_2^3-1.5y_2^2+0.75y_2-0.125&=3.2y_2\\ y_2=2.5m,-0.0527m,&-0.0947m \end{aligned}
Hence, $y_2=2.5m$
Post jump Froude's No.
$(F_2)=\frac{V_2}{\sqrt{\sqrt{gy_2}}}=\frac{\left ( \frac{30}{6 \times 2.5} \right )}{\sqrt{10 \times 2.5}}=0.4$

 Question 4
Water is flowing in a horizontal, frictionless, rectangular channel. A smooth hump is built on the channel floor at a section and its height is gradually increased to reach choked condition in the channel. The depth of water at this section is $y_2$ and that at its upstream section is $y_1$. The correct statement(s) for the choked and unchoked conditions in the channel is/are
 A In choked condition, $y_1$ decreases if the flow is supercritical and increases if the flow is subcritical. B In choked condition, $y_2$ is equal to the critical depth if the flow is supercritical or subcritical. C In unchoked condition, $y_1$ remains unaffected when the flow is supercritical or subcritical. D In choked condition, $y_1$ increases if the flow is supercritical and decreases if the flow is subcritical.
GATE CE 2022 SET-2      Open Channel Flow
Question 4 Explanation:

 Question 5
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      Dimensional Analysis
Question 5 Explanation:
Unit of dynamic viscosity $=\frac{kg}{m.s} \; or \; \frac{Ns}{m^2}=ML^{-1}T^{-1}$
 Question 6
Match Column X with Column Y:
$\begin{array}{|l|l|}\hline \text{Column X}&\text{Column Y} \\ \hline \text{(P) Horton equation} & \text{((I) Design of alluvial channel} \\ \hline \text{(Q) Penman method} & \text{(II) Maximum flood discharge}\\ \hline \text{(R) Chezys formula}& \text{(III) Evapotranspiration}\\ \hline \text{(S) Lacey's theory}& \text{(IV) Infiltration}\\ \hline \text{(T) Dicken's formula}& \text{(V) Flow velocity}\\ \hline \end{array}$
Which one of the following combinations is correct?
 A (P)-(IV), (Q)-(III), (R)-(V), (S)-(I), (T)-(II) B (P)-(III), (Q)-(IV), (R)-(V), (S)-(I), (T)-(II) C (P)-(IV), (Q)-(III), (R)-(II), (S)-(I), (T)-(V) D (P)-(III), (Q)-(IV), (R)-(I), (S)-(V), (T)-(II)
GATE CE 2022 SET-2      Fluid Dynamics and Flow Measurements
 Question 7
Depth of water flowing in a 3 m wide rectangular channel is 2 m. The channel carries a discharge of 12 $m^3/s$. Take g = 9.8 $m/s ^2$.
The bed width (in m) at contraction, which just causes the critical flow, is _________ without changing the upstream water level. (round off to two decimal places)
 A 2.85 B 4.25 C 2.15 D 1.55
GATE CE 2022 SET-1      Open Channel Flow
Question 7 Explanation:
Given: $B = 3m, y = 2m, Q = 12 m^3 /sec$
$Velocity(v)=\frac{Q}{A} =\frac{12}{2 \times 3}=2m/sec$

Specific energy at section (1-1)
$E=y+\frac{v^2}{2g}=2+\frac{2^2}{2 \times 9.8}=\frac{108}{49}m$
When channel section is contracted to minimum width and for constant discharge Q, the flow over contracted section will be critical flow and under the assumption that no energy loss has taken place.
$E=E_c=\frac{108}{49}$
We have that, $E_c=\frac{3}{2}y_c$ (for rectangular crosssection)
$y_c=\frac{2}{3}E_c=\frac{2}{3} \times \frac{108}{49}=\frac{72}{49}$
For critical flow condition,
\begin{aligned} \frac{Q^2T}{gA^3}&=1\\ \frac{Q^2B_{min}}{g(B_{min} \times y_c)^3}&=1\\ (B_{min})^2&=\frac{Q^2}{gy_c^3}\\ B_{min}&=\left ( \frac{(12)^2}{9.8 \times \left ( \frac{72}{49} \right )^3} \right )^{1/2}\\ &=2.152m \end{aligned}
 Question 8
Two reservoirs are connected by two parallel pipes of equal length and of diameters 20 cm and 10 cm, as shown in the figure (not drawn to scale). When the difference in the water levels of the reservoirs is 5 m, the ratio of discharge in the larger diameter pipe to the discharge in the smaller diameter pipe is ____________. (round off to two decimal places)
(Consider only loss due to friction and neglect all other losses. Assume the friction factor to be the same for both the pipes)

 A 2.25 B 6.32 C 4.22 D 5.66
GATE CE 2022 SET-1      Fluid Dynamics and Flow Measurements
Question 8 Explanation:
\begin{aligned} h_{f1} &=h_{f2} \\ \frac{8Q_1^2}{\pi ^2 g} \times \frac{fl}{D_1^5}&= \frac{8Q_2^2}{\pi ^2 g} \times \frac{fl}{D_2^5} \\ \left ( \frac{Q_1}{Q_2} \right )^2&= \left ( \frac{D_1}{D_2} \right )^5\\ \frac{Q_1}{Q_2} &= \left ( \frac{D_1}{D_2} \right )^{5/2}\\ &= \left ( \frac{0.2}{0.1} \right )^{5/2}\\&=5.66 \end{aligned}
 Question 9
A rectangular channel with Gradually Varied Flow (GVF) has a changing bed slope. If the change is from a steeper slope to a steep slope, the resulting GVF profile is
 A $S_3$ B $S_1$ C $S_2$ D either $S_1$ or $S_2$, depending on the magnitude of the slopes
GATE CE 2022 SET-1      Open Channel Flow
Question 9 Explanation:

 Question 10
With respect to fluid flow, match the following in Column X with Column Y:
$\begin{array}{|l|l|}\hline \text{Column X}& \text{Column Y}\\ \hline \text{(P) Viscosity} & \text{(I) Mach number}\\ \hline \text{(Q) Gravity}&\text{(II) Reynolds number}\\ \hline \text{(R) Compressibility}&\text{(III) Euler number}\\ \hline \text{(S) Pressure} &\text{(IV) Froude number}\\ \hline \end{array}$
Which one of the following combinations is correct?
 A (P) - (II), (Q) - (IV), (R) - (I), (S) - (III) B (P) - (III), (Q) - (IV), (R) - (I), (S) - (II) C (P) - (IV), (Q) - (II), (R) - (I), (S) - (III) D (P) - (II), (Q) - (IV), (R) - (III), (S) - (I)
GATE CE 2022 SET-1      Flow Through Pipes
Question 10 Explanation:
Reynold's number ($R_e$) is defined when apart from inertial force, viscous forces are dominant.
$R_e=\frac{\text{Inertial force}}{\text{Viscous force}}$
Froude?s number ($F_e$): It is used when in addition to inertial force, gravity forces are important.
$F_e=\frac{\text{Inertial force}}{\text{Gravity force}}$
Euler number ($E_u$): It is used when apart from inertial force, only pressure forces are dominant.
$E_u=\frac{\text{Inertial force}}{\text{Pressure force}}$
Mach number ($M$): It is used when in addition to inertial force, compressibility forces are dominant
$M=\frac{\text{Inertial force}}{\text{Elastic force}}$

There are 10 questions to complete.