# Fluid Kinematics

 Question 1
The velocity components in the x and y directions for an incompressible flow are given as u=(-5+6x) and v=-(9+6y), respectively. The equation of the streamline is
 A (-5+6x)-(9+6y)= constant B $\frac{-5+6x}{9+6y}=constant$ C (-5+6x)(9+6y)= constant D $\frac{9+6y}{-5+6x}=constant$
GATE CE 2020 SET-2   Fluid Mechanics and Hydraulics
Question 1 Explanation:
\begin{aligned} \text{Given } \; \; \; & u=-5+6x \\ & v=-(9+6y) \end{aligned}
Equation of streamline
\begin{aligned} \frac{dx}{u}&=\frac{dy}{u} \\ \frac{dx}{-5+6x}&=\frac{dy}{-(9+6y)} \\ \text{Integrating it,} \\ \ln(-5+6x)^{1/6}&= -\ln(9+6y)^{1/6}+ \ln C^{1/6}\\ \frac{1}{6} \ln (-5+6x)(9+6y)&=\frac{1}{6} \ln C \\ \text{Take antilog,} \\ (-5 + 6x)(9 + 6y) &= \text{constant} \\ \text{u.v} &= \text{constant} \end{aligned}
 Question 2
Uniform flow with velocity U makes an angle $\theta$ with the y-axis, as shown in the figure

The velocity potential ($\Phi$), is
 A $\pm U(x \sin \theta +y \cos \theta )$ B $\pm U(y \sin \theta -x \cos \theta )$ C $\pm U(x \sin \theta -y \cos \theta )$ D $\pm U(y \sin \theta +x \cos \theta )$
GATE CE 2020 SET-1   Fluid Mechanics and Hydraulics
Question 2 Explanation:
Velocity in x-depth, $u_x=u\sin \theta$
Velocity in y-depth, $u_y=u\cos \theta$
\begin{aligned} -\frac{\partial \phi }{\partial x}&=u_x \\ &{Integrating it} \\ \phi &= -u_xx+f(y)+c\\ &= -(u \sin \theta )x+f(y)+c...(i)\\ -\frac{\partial \phi }{\partial y}&=u_y \\ &\text{Integrating it} \\ \phi &= -u_y y+f(x)+c\\ &= -(u \cos \theta )y+f(x)+c...(ii)\\ & \text{By equation (i) and (ii),}\\ \phi &= -u(x \sin \theta+y \cos \theta )\\ &\text{If we take,} \\ \frac{\partial \phi }{\partial x}&= u_x \; and \; \frac{\partial \phi }{\partial y}= u_y\\ \text{then}, \; \phi &= u(x \sin \theta +y \cos \theta )\\ \text{So}, \; \phi &= \pm u(x \sin \theta +y \cos \theta ) \end{aligned}
 Question 3
The velocity field in a flow system is given by $v=2i+(x+y)j+(xyz)k$. The acceleration of the fluid at (1,1,2) is
 A 2i + 10k B 4i + 12k C l + k D 4j + 10k
GATE CE 2019 SET-2   Fluid Mechanics and Hydraulics
Question 3 Explanation:
\begin{aligned} \vec{v} &=2\hat{i}(x+y)\hat{j}+xyz\hat{k} \\ u&=2 \\ v&=x+y \\ w&=xyz \\ a_x&=u\frac{\partial u}{\partial x} +v\frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z}+\frac{\partial u}{\partial t}=0\\ a_y&=x+y+2 \\ a_z&= u\frac{\partial w}{\partial x} +v\frac{\partial w}{\partial y}+w\frac{\partial w}{\partial z}+\frac{\partial w}{\partial t}\\ &= 2(yz)+(x+y)(xz)+xyz(xy)\\ a_z&=2yz+x^2z+xyz+x^2y^2z \\ &At\;(1,1,2) \\ a_y&=11+2=4\\ a_z&=2(1)(2)+(1)^2(2)+(1)(1)(2)+(1)^2(1)^2(2)\\ &=4+2+2+2=10\\ \vec{a}&=4\hat{j}+10\hat{k} \end{aligned}
 Question 4
A solid sphere of radius, r, and made of material with density, $\rho _s$, is moving through the atmosphere (constant pressure, p)with a velocity, v. The net force ONLY due to atmospheric pressure ($F_p$) acting on the sphere at any time, t, is
 A $\pi r^2p$ B $4 \pi r^2p$ C $\frac{4}{3}\pi r^3 \rho _s\frac{dv}{dt}$ D zero
GATE CE 2019 SET-2   Fluid Mechanics and Hydraulics
Question 4 Explanation:

When object of any shape ( not only sphere) is subjected to uniform pressure over entire surface, the net force is zero because pressure force at any point is balanced by an equal and opposite force on opposite point.
 Question 5
A flow field is given by $u=y^{2} ,v=-xy,w=0.$ Value of the component of the angular velocity(in radians per unit time,up to two decimal places)at the point (0,-1,1) is_____
 A 0.5 B 1 C 1.5 D 2
GATE CE 2018 SET-1   Fluid Mechanics and Hydraulics
Question 5 Explanation:
\begin{aligned} \omega_{z} &=\frac{1}{2}\left[\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}\right] \\ &=\frac{1}{2}\left[\frac{\partial}{\partial x}(-x y)-\frac{\partial}{\partial y}(y)^{2}\right] \\ &=\frac{1}{2}[-y-2 y] \\ &=-\frac{3 y}{2} \\ \text { At point } &(0,-1,1) \\ \omega_{z} &=-\frac{3}{2} \times-1=1.50 \mathrm{rad} / \mathrm{s} \end{aligned}
Note: Since the density variation is not given continuity equation of incompressible flow can not be applied directly to check the possibility of flow.
 Question 6
The velocity components of a two dimensional plane motion of a fluid are: $u=\frac{y^{3}}{3}+2x-x^{2}y$ and $v=xy^{2}-2y-\frac{x^{3}}{3}$.
The correct statement is:
 A Fluid is incompressible and flow is irrotational B Fluid is incompressible and flow is rotational C Fluid is compressible and flow is irrotational D Fluid is compressible and flow is rotational
GATE CE 2015 SET-2   Fluid Mechanics and Hydraulics
Question 6 Explanation:
Continuity equation
\begin{aligned} \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y} &=0 \\ \Rightarrow \quad(2-2 x y)+(2 x y-2) &=0 \\ 0 &=0 \end{aligned}
Hence, incompressible fluid.
\begin{aligned} \omega_{z} &=\frac{1}{2}\left[\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}\right] \\ &=\frac{1}{2}\left[y^{2}-\frac{3 x^{2}}{3}-\left(\frac{3 y^{2}}{3}-x^{2}\right)\right] \\ &=\frac{1}{2}\left[y^{2}-x^{2}-y^{2}+x^{2}\right] \\ &=0 \end{aligned}
Hence, flow is irrotational.
 Question 7
A nozzle is so shaped that the average flow velocity changes linearly from 1.5 m/s at the beginning to 15 m/s at its end in a distance of 0.375 m. The magnitude of the convective acceleration (in m/$s^{2}$) at the end of the nozzle is _________.
 A 74 B 54 C 540 D 740
GATE CE 2015 SET-2   Fluid Mechanics and Hydraulics
Question 7 Explanation:
Convective acceleration $=\frac{u \partial u}{\partial x}$
$=15 \times \frac{15-1.5}{0.375}=540 \mathrm{m} / \mathrm{s}^{2}$
 Question 8
In a two-dimensional steady flow field, in a certain region of the x-y plane, the velocity component in the x-direction is given by $v_{x}=x^{2}$ and the density varies as $\rho =\frac{1}{x}$. Which of the following is a valid expression for the velocity component in the y- direction, $v_{y}$ ?
 A $v_{y}=-x/y$ B $v_{y}=x/y$ C $v_{y}=-xy$ D $v_{y}=xy$
GATE CE 2015 SET-1   Fluid Mechanics and Hydraulics
Question 8 Explanation:
\begin{aligned} \frac{\partial(\rho u)}{\partial x}+\frac{\partial(\rho v)}{\partial y} &=0 \\ \frac{\partial}{\partial x}\left[\frac{1}{x} \cdot x^{2}\right]+\frac{\partial}{\partial y}\left(\frac{1}{x} \cdot v_{y}\right) &=0 \\ \frac{\partial}{\partial x}(x)+\frac{1}{x} \frac{\partial v_{y}}{x \partial y} &=0 \\ 1+\frac{1}{x} \frac{\partial v_{y}}{\partial y} &=0 \\ -x &=\frac{\partial v_{y}}{\partial y} \end{aligned}
On integrating both sides
$v_{y}=-x y$
 Question 9
A plane flow has velocity components $u=\frac{x}{T_{1}}, \; v=-\frac{y}{T_{2}}$ and w=0 along x, y and z directions respectively, where$T_{1}(\neq 0)$ and $T_{2}(\neq 0)$ are constants having the dimensions of time. The given flow is incompressible if
 A $T_{1}=-T_{2}$ B $T_{1}=-\frac{T_{2}}{2}$ C $T_{1}=\frac{T_{2}}{2}$ D $T_{1}=T_{2}$
GATE CE 2014 SET-2   Fluid Mechanics and Hydraulics
Question 9 Explanation:
For a flow to exist
\begin{aligned} \Rightarrow \quad \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}&=0 \\ \Rightarrow \quad \frac{1}{T_{1}}-\frac{1}{T_{2}}&=0 \\ \Rightarrow \quad T_{1}&=T_{2} \end{aligned}
 Question 10
A particle moves along a curve whose parametric equations are: $x=t^{3}+2t$, $y=-3e^{-2t}$ and $z=2 \sin (5t)$ , where x, y and z show variations of the distance covered by the particle (in cm)with time t (in s). The magnitude of the acceleration of the particle (in $cm/s^{2}$) at t = 0 is ________
 A 8 B 10 C 12 D 16
GATE CE 2014 SET-1   Fluid Mechanics and Hydraulics
Question 10 Explanation:
\begin{aligned} x &=t^{3}+2 t \\ y &=-3 e^{-2 t} \\ z &=2 \sin (5 t) \\ \frac{d x}{d t} &=3 e^{2}+2 \\ \Rightarrow\quad a_{x} &=\frac{d^{2} x}{d t^{2}}=6 t \\ \frac{d y}{d t} &=-3 e^{-2} \times(-2)=6 e^{-2 t} \\ \Rightarrow\quad a_{y} &=\frac{d^{2} y}{d t^{3}}=-12 e^{-2 t}\\ \Rightarrow\quad \frac{d z}{d t} &=2 \times 5 \cos (5 t) \\ &=10 \cos (5 t) \\ \Rightarrow\quad a_{z} &=\frac{d^{2} z}{d t^{2}}=-50 \sin 5 t \\ \vec{a} &=a_{x} \hat{i}+a_{y} \hat{j}+a_{z} \hat{k} \\ \vec{a} \text { at } t=0 &=0 \hat{i}-12 \hat{j}+0 \hat{k} \\ \vec{a} &=-12 \hat{j} \end{aligned}
$\Rightarrow\quad$ Magnitude of acceleration at t=0
$=12 \mathrm{cm} / \mathrm{s}^{2}$
There are 10 questions to complete.