# Fluid Mechanics

 Question 1
A uniform wooden rod (specific gravity = 0.6, diameter = 4 cm and length = 8 m) is immersed in the water and is hinged without friction at point A on the waterline as shown in the figure. A solid spherical ball made of lead (specific gravity = 11.4) is attached to the free end of the rod to keep the assembly in static equilibrium inside the water. For simplicity, assume that the radius of the ball is much smaller than the length of the rod.
Assume density of water $= 10^3 kg/m^3$ and $\pi = 3.14$.
Radius of the ball is _______ cm (round off to 2 decimal places). A 2.42 B 3.62 C 8.26 D 3.59
GATE ME 2022 SET-2      Viscous, Turbulent Flow and Boundary Layer Theory
Question 1 Explanation: As the system is in equilibrium,
$\Sigma M_o=0$
$(F_{B1}-W_1) \times \frac{L}{2} \cos \theta +F_{B2}-W_2) \times L \times \cos \theta =0$
$(\rho gV_1-\rho _1gV_1) \times \frac{1}{2}+(\rho gV_2-\rho _2gV_2) =0$
$(\rho -\rho _1) \times g \times \frac{\pi d^2L}{4} \times \frac{1}{2}+(\rho -\rho _2) \times \frac{4}{3} \times \\pi R^3 \times g=0$
$(100-600) \times \frac{0.04^2 \times 8}{8} +(1000-11400) \times \frac{4}{3} \times R^3=0$
$\Rightarrow R=0.03587m=3.59cm$
 Question 2
The steady velocity field in an inviscid fluid of density 1.5 is given to be $\vec{V}=(y^2-x^2)\hat{i}+(2xy)\hat{j}$ Neglecting body forces, the pressure gradient at $(x = 1, y = 1)$ is
 A $10\hat{j}$ B $20\hat{i}$ C $-6\hat{i}-6\hat{j}$ D $-4\hat{i}-4\hat{j}$
GATE ME 2022 SET-2      Fluid Kinematics
Question 2 Explanation:
By Euler's equation of motion,
\begin{aligned} x:-\frac{\partial p}{\partial x}+\rho g_x&=\rho \frac{Du}{Dt}\\ y:-\frac{\partial p}{\partial y}+\rho g_y&=\rho \frac{Dv}{Dt}\\ \end{aligned}
Neglecting body forces (i.e. $g_x=g_y=0$)
\begin{aligned} \frac{\partial p}{\partial x}&=-\rho \frac{Du}{Dt}=-\rho \left [ u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y} \right ]\\ &=-\rho [(y^2-x^2)(-2x)+(2xy)(2y)]\\ &=-\rho[-2xy^2+2x^3+4xy^2]\\ &=-\rho(2xy^2+2x^3)\\ &=-1.5 \times (2 \times 1 \times 1^2 +2\times1^3 )\\ &=-6 Pa/m \end{aligned}
Similarly,
\begin{aligned} \frac{\partial p}{\partial y}&=-\rho \frac{Dv}{Dt}\\ &=-\rho \left [ u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y} \right ]\\ &=-\rho [(y^2-x^2)(2y)+(2xy)(2x)]\\ &=-\rho[2y^3-2x^2y+4x^2y]\\ &=-\rho(2y^3+2x^2y)\\ &=-1.5 \times (2 \times 1^3 +2\times1^2 \times 1 )\\ &=-6 Pa/m \end{aligned}
The pressure gradient vector is given by
$\triangledown p=\frac{\partial p}{\partial x}\hat{i}+\frac{\partial p}{\partial y}\hat{j}=-6\hat{i}-6\hat{j}$
 Question 3
A tube of uniform diameter $D$ is immersed in a steady flowing inviscid liquid stream of velocity $V$, as shown in the figure. Gravitational acceleration is represented by $g$. The volume flow rate through the tube is ______. A $\frac{\pi}{4}D^2V$ B $\frac{\pi}{4}D^2\sqrt{2gh_2}$ C $\frac{\pi}{4}D^2\sqrt{2g(h_1+h_2)}$ D $\frac{\pi}{4}D^2\sqrt{V^2-2gh_2}$
GATE ME 2022 SET-2      Fluid Dynamics
Question 3 Explanation: By applying Bernoulli's equation between (1) and (2)
\begin{aligned} \frac{P_1}{\rho g}+\frac{V_1^2}{2g}+z_1&=\frac{P_2}{\rho g}+\frac{V_2^2}{2g}+z_2\\ \frac{P_{atm}+\rho gh_1}{\rho g}+\frac{V_1^2}{2g}+0&=\frac{P_{atm}}{\rho g}+\frac{V_2^2}{2g}+h_1+h_2 \\ &[\because \;\; V_1=V]\\ \frac{V_1^2}{2g}&=\frac{V}{2g}+h_2-(h_1+h_2)\\ \therefore \; V_2&=\sqrt{V^2-2gh_1}\\ \therefore \; Q&=A_2V_2=\frac{\pi d^2}{4} \times \sqrt{V^2-2gh_1} \end{aligned}
 Question 4
The velocity field in a fluid is given to be
$\vec{V}=4(xy)\hat{i}+2(x^2-y^2)\hat{j}$
Which of the following statement(s) is/are correct?

MSQ
 A The velocity field is one-dimensional. B The flow is incompressible C The flow is irrotational D The acceleration experienced by a fluid particle is zero at (x = 0, y = 0).
GATE ME 2022 SET-2      Fluid Kinematics
Question 4 Explanation:
For given flow,
$u=4xy, v = 2(x^2- y^2)$
As velocity field is function of two space variables, flow is two dimensional.
$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=4y-4y$
Therefore, flow is incompressible.
$\omega _z=\frac{1}{2}\left ( \frac{\partial v}{\partial x} -\frac{\partial u}{\partial y}\right )=\frac{1}{2}(4x-4x)=0$
Therefore, flow is irrotational.
\begin{aligned} a_x &=u\frac{\partial u}{\partial x} +v\frac{\partial u}{\partial y}\\ &= 4xy(4y)+2(x^2-y^2)(4x)\\ &= 16xy^2+8x^3-8xy^2 \\ &=16 \times 0 \times 0^2+8 \times 0^3-8 \times 0 \times 0^2 \\ &= 0 \end{aligned}
\begin{aligned} a_y &=u\frac{\partial v}{\partial x} +v\frac{\partial v}{\partial y}\\ &= 4xy(4x)+2(x^2-y^2)(-4y)\\ &= 16x^2y-8x^2y+8y^3 \\ &=16 \times 0^2 \times 0-8 \times 0^2 \times 0+8 \times 0^3 \\ &= 0\\ |\vec{a}|&=\sqrt{a_x^2+a_y^2}=0 \end{aligned}
 Question 5
Consider a steady flow through a horizontal divergent channel, as shown in the figure, with supersonic flow at the inlet. The direction of flow is from left to right. Pressure at location B is observed to be higher than that at an upstream location A. Which among the following options can be the reason?

MSQ
 A Since volume flow rate is constant, velocity at B is lower than velocity at A B Normal shock C Viscous effect D Boundary layer separation
GATE ME 2022 SET-2      Viscous, Turbulent Flow and Boundary Layer Theory
Question 5 Explanation:
If the supersonic flow enters to the diverging duct, it will act as nozzle the pressure of the flow is decreases in the nozzle. But it is given the pressure of the flow at B is more than that A It will happen only with the development of normal shoot in the diverging flow (the normal shock wave is the characteristic of only supersonic flow) Because of normal shock wave in the diverging nozzle pressure increases and velocity decreases. But mass flow rate remains unchanged.
 Question 6
A steady two-dimensional flow field is specified by the stream function
$\psi =kx^3y$
where $x$ and $y$ are in meter and the constant $k = 1 \; m^{-2}s^{-1}$. The magnitude of acceleration at a point $(x,y) = (1 m, 1 m)$ is ________ $m/s^2$ (round off to 2 decimal places).
 A 2.42 B 1.25 C 3.62 D 4.24
GATE ME 2022 SET-1      Fluid Kinematics
Question 6 Explanation:
Given,
Stream function,
\begin{aligned} \psi &=kx^3y; \; k=1m^{-2}s^{-1}\\ u&=-\frac{\partial \psi }{\partial y}\Rightarrow u=-x^3\\ v&=-\frac{\partial \psi }{\partial x}\Rightarrow v=3x^2y\\ \vec{V}&=-x^3\hat{i}+3x^2y\hat{j}\\ a_x&=\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z}\\ a_x&=-x^3(-3x^2)\Rightarrow a_x=3x^5\\ At\; &(1,1), \; a_x=3m/sec^2\\ a_y&=\frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}+w\frac{\partial v}{\partial z}\\ a_y&=-x^3(6xy)+3(x^2y)(3x^2)\\ &=-6x^4y+9x^4y=3x^4y\\ \vec{a}&=3x^5\hat{i}+3x^4y\hat{j}\\ At& \;\;(1,1)\\ \vec{a}&=3\hat{i}+3\hat{j}\\ \Rightarrow |a|=\sqrt{3^2+3^2}&=4.24m/s^2 \end{aligned}
 Question 7
Consider steady, one-dimensional compressible flow of a gas in a pipe of diameter 1 m. At one location in the pipe, the density and velocity are 1 $kg/m^3$ and 100 m/s, respectively. At a downstream location in the pipe, the velocity is 170 m/s. If the pressure drop between these two locations is 10 kPa, the force exerted by the gas on the pipe between these two locations is ____________ N.
 A $350 \pi ^2$ B $750 \pi$ C $1000 \pi$ D $3000$
GATE ME 2022 SET-1      Fluid Dynamics
Question 7 Explanation: Applying momentum equation to the pipe flow,
$\Sigma \vec{F}=(\dot{m}\vec{V})_{out}-(\dot{m}\vec{V})_{in}+\frac{\partial }{\partial t}(m\vec{V})_{e,v}$
$P_1A-P_2A+F=\dot{m}(V_2)-\dot{m}(V_1)$
\begin{aligned} \therefore \; F&=\dot{m}(V_2-V_1)-(P_1-P_2)A\\ &=\rho _1AV_1(V_2-V_1)-(P_1-P_2)A\\ &=\frac{ \pi d^2}{4}\left [ \rho (V_1V_2-V_1^2)-(P_1-P_2) \right ]\\ &=\frac{\pi \times 1^2}{4}\left [ 1 \times (100 \times 170 -100^2)-(10 \times 10^3) \right ]\\ &=-750 \pi N \end{aligned}
Negative sign shows that assumed direction of force is opposite of actual direction.
 Question 8
A solid spherical bead of lead (uniform density $=11000\; kg/m^3$) of diameter $d = 0.1 mm$ sinks with a constant velocity $V$ in a large stagnant pool of a liquid (dynamic viscosity $=1.1 \times 10^{-3} kg.m^{-1}.s^{-1}$). The coefficient of drag is given by $C_D=\frac{24}{Re}$, where the Reynolds number ($Re$) is defined on the basis of the diameter of the bead. The drag force acting on the bead is expressed as $D=(C_D)(0.5 \rho V^2)\left ( \frac{\pi d^2}{4} \right )$, where $\rho$ is the density of the liquid. Neglect the buoyancy force. Using $g = 10 m/s^2$, the velocity $V$ is __________ m/s.
 A $\frac{1}{24}$ B $\frac{1}{6}$ C $\frac{1}{18}$ D $\frac{1}{12}$
GATE ME 2022 SET-1      Viscous, Turbulent Flow and Boundary Layer Theory
Question 8 Explanation:
As buoyancy force is neglected we can neglect density of fluid () in the following equation.
\begin{aligned} V&=\frac{1}{18\mu _f}(\rho _b-\rho _f)gd^2\\ &=\frac{11000 \times 10 \times (10^{-4})^2}{18 \times 1.1 \times 10^{-3}}\\ &=\frac{1}{18}m/s \end{aligned}
 Question 9
The figure shows a purely convergent nozzle with a steady, inviscid compressible flow of an ideal gas with constant thermophysical properties operating under choked condition. The exit plane shown in the figure is located within the nozzle. If the inlet pressure $(P_0)$ is increased while keeping the back pressure $(P_{back})$ unchanged, which of the following statements is/are true? MSQ
 A Mass flow rate through the nozzle will remain unchanged. B Mach number at the exit plane of the nozzle will remain unchanged at unity C Mass flow rate through the nozzle will increase. D Mach number at the exit plane of the nozzle will become more than unity.
GATE ME 2022 SET-1      Viscous, Turbulent Flow and Boundary Layer Theory
Question 9 Explanation:
\begin{aligned} \frac{\dot{m}}{A}&=\sqrt{\frac{\gamma }{R}}\frac{P_0}{\sqrt{T_0}}\frac{1}{\left ( \frac{\gamma +1}{2} \right )^{\frac{\gamma +1}{2(\gamma -1)}}}\\ \frac{\dot{m}}{A}&=0.685 \times \sqrt{P_0\rho _0} \;\;\;...\text{ for Air}\\ \frac{T_c}{T_0}&=\frac{T^*}{T_0}=\frac{2}{\gamma +1}=0.8333\\ &(\text{critical temp ratio})\\ \frac{P_c}{P_0}&=\frac{P^*}{P_0}=\left ( \frac{2}{\gamma +1} \right )^{\frac{\gamma }{\gamma -1}}=0.5282\\ &(\text{critical pressure ratio}) \end{aligned}
where $\gamma=$ Adiabatic index = 1.4
If $P_0$ increases , mass flow rate increases. Mach number at the exit plane of the nozzle will remain unchanged at unity.
Since the maximum expected velocity from the converging nozzle is sound velocity.


 Question 10
In the following two-dimensional momentum equation for natural convection over a surface immersed in a quiescent fluid at temperature $T_\infty ( g$ is the gravitational acceleration, $\beta$ is the volumetric thermal expansion coefficient, $v$ is the kinematic viscosity, $u$ and $v$ are the velocities in $x$ and $y$ directions, respectively, and $T$ is the temperature)
$u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=g\beta (T-T_\infty )+v\frac{\partial u^2}{\partial y^2}$
the term $g\beta (T-T_\infty )$ represents
 A Ratio of inertial force to viscous force. B Ratio of buoyancy force to viscous force. C Viscous force per unit mass. D Buoyancy force per unit mass.
GATE ME 2022 SET-1      Viscous, Turbulent Flow and Boundary Layer Theory
Question 10 Explanation:
$u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=g\beta (T-T_\infty )+v\frac{\partial^2 u}{\partial y^2}$
This is the equation that governs the fluid motion in the boundary layer due to effect of Buoyancy
$g\beta (T-T_\infty )=\frac{m}{s^2} \times \frac{1}{K} \times K=m/s^2$
Unit of $g\beta (T-T_\infty )= m/s^2$
Buoyancy force is due to density difference and gravitational effect
$g\beta (T-T_\infty )$ represents Buoyancy force per unit mass.

There are 10 questions to complete.

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