Velocity Potential Function

Q. For a two-dimensional irrotational flow, the velocity potential is defined as $\varphi =lo{g}_e(x^2+y^2)$. Which of the following is a possible stream function, $\psi$, for this flow?

1. $\frac{1}{2}ta{n}^{-1}\left(y/x\right)$
2. $ta{n}^{-1}\left(y/x\right)$
3. $2ta{n}^{-1}\left(y/x\right)$
4. $2ta{n}^{-1}\left(x/y\right)$

Q. The velocity potential $\varphi$ at any point for a two dimensional steady irrotational flow in polar coordinates, is given by (with usual notations)
$\varphi =\frac{m\cos \theta }{r}$
This equation represents a

1. vortex
2. sink
3. source
4. doublet

Q. In a two-dimensional flow of fluid, if a velocity potential function $\varphi$ exists which satisfies the relation $\frac{{\partial }^2\varphi }{\partial x^2}+\frac{{\partial }^2\varphi }{\partial y^2}=0$, then the flow is :

1. Steady incompressible.
2. Steady laminar incompressible.
3. Irrotational and incompressible.
4. Turbulent incompressible.

Q. Consider the following statements.

1. Velocity potential exists for rotational and irrotational flow both.

2. Stream function exists only for irrotational flow.

Which of the above statement(s) is/are correct ?

1. 1 only
2. 2 only
3. 1 and 2
4. None

Q. Existence of velocity potential implies that

1. Fluid is in continuum
2. fluid is irrotational
3. Fluid is ideal
4. Fluid is compressible

Q. If the fluid velocity for a potential flow is given by V(x, y) = u(x, y) i + v(x, y)j with usual notations, then the slope of the potential line at (x, y) is

1. $\frac{v}{u}$
2. $-\frac{u}{v}$
3. $-\frac{v^2}{u_2}$
4. $\frac{u}{v}$

Q. Statement(I): $▽\times u=0$ represents a fluid motion is irrotational velocity potential, where u = flow velocity of fluid.
Statement (II): $u=▽\varphi$ represents the velocity u is gradient of a scalar function of velocity potential $\left(\varphi \right)$

1. Statement - I is only correct
2. Statement - II is only correct
3. Statement - I and statement - II both are correct
4. Statement -I and statement - II both are not correct

Q. Match List -I (Format of representation) with List -II (Context/Relevant to) and select the correct answer using the codes given below the lists:

Lists -I
A. $\frac{\partial u}{\partial x}+\frac{\partial V}{\partial y}$
B. $\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}$
C. $u\frac{\partial v}{\partial x}+v\frac{\partial u}{\partial y}$
D. $\frac{q\theta }{2\pi }$

List -II
1. Relevant to a velocity potential
2. Rate of rotation about a relevant axis
3. Pressure gradient in a relevant direction
4. Continuity of flow

Codes:

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