# 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.

There are 5 questions to complete.