Vorticity and Circulation

Q. The Newtonian fluid has the following velocity field:
$\overrightarrow{V}=x^{2}y\hat{i}+2x{y}^{2}z\hat{j}-y{z}^3\hat{k}$

The rate shear deformation ${\epsilon }_{yz}$ at the point x = -2, y = -1 and z = 2 for given flow is

1. -6
2. -2
3. -12
4. 4

Q. Velocity vector of a flow field is given as
$\overrightarrow{V}=2xy\hat{i}-x^{2}z\hat{j}$. The vorticity vector at (1, 1, 1) is

1. $4\hat{i}-\hat{j}$
2. $4\hat{i}-\hat{k}$
3. $\hat{i}-4\hat{j}$
4. $\hat{i}-4\hat{k}$

Q. For a fluid element in a two dimensional flow field (x-y plane), if it will undergo

1. Translation only
2. Translation and rotation
3. Translation and deformation
4. Deformation only

Q. The circulation ${}^{\prime }{\Gamma }^{\prime }$ around a circle of radius 2 units for the velocity field $u=2x+3yandv=-2yis$

1. $-6\pi$ units
2. $-12\pi$ units
3. $-18\pi$ units
4. $-24\pi$ units

Q. Which one of the following gives the lift force $F_L$ produced on a rotating circular cylinder per unit length in a uniform flow ?
Where,
u = free stream velocity
$\Gamma =circulation$
$\rho =densityofthefluid$

1. $F_L=\frac{u\Gamma }{\rho }$
2. $F_L=\rho u\Gamma$
3. $F_L=\frac{\rho \Gamma }{u}$
4. $F_L=\frac{\rho u}{\Gamma }$

Q. Consider the following parameters related to fluid
1. Vorticity
2. Velocity
3. Stream function
Which of these parameters exist both in rotational and irrotational flows?

1. 1 and 2
2. 2 and 3
3. 1 and 3
4. 1, 2 and 3

Q. For a two-dimensional incompressible flow field given by $\overrightarrow{U}=A(x\hat{i}-y\hat{j})$, where A>0, which one of the following statements is FALSE?
A. It satisfies continuity equation
B. It is unidirectional when x$\to$0 and y$\to \infty$.
C. Its streamlines are given by x = y.
D. It is irrotational

1. A
2. B
3. C
4. D

Q. The velocity of water at the outer edge of a 60 cm diameter whirlpool, where the water level is horizontal is 2.5 m/s. The velocity of water at a level where the diameter of the whirlpool is 15 cm, is

1. 1 m/s
2. 5 m/s
3. 8 m/.s
4. 10 m/s

Q. Consider a velocity field $\overrightarrow{V}=K(y\hat{i}+x\hat{k}$), where K is constant. The vorticity , ${\Omega }_z$, is

1. -K
2. K
3. -K/2
4. K/2

Q. Which of the following statements is / are correct?

1. "Vorticity" is a vector field, which gives a micro measure of the rotation at any point in the fluid.

2. "Circulation" is a scalar integral quantity, which gives a macro scopic measure of rotation for a finite area of the field

1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2

Q. Match the following pairs:
$\begin{array}{ccc}& \text{Equation}& \text{Physical Interpretation}\\ \text{P}& \nabla \times \overrightarrow{V}=0& \text{I. Incompressible}\\ & & \text{continuity equation}\\ \text{Q}& \nabla .\overrightarrow{V}=0& \text{II. Steady flow}\\ \text{R}& \frac{\Delta \overrightarrow{V}}{\Delta t}=0& \text{III. Irrotational flow}\\ \text{S}& \frac{\partial \overrightarrow{V}}{\partial t}=0& \text{IV. Zero acceleration}\\ & & \text{of fluid particles}\end{array}$

1. P-IV, Q-I, R-II, S-III
2. P-IV, Q-III, R-I, S-II
3. P-III, Q-I, R-IV, S-II
4. P-III, Q-I, R-II, S-IV