Fluid Kinematics

Question 1
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 Mechanics
Question 1 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 2
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 Mechanics
Question 2 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 3
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 Mechanics
Question 3 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 4
A two dimensional flow has velocities in x and y directions given by u=2xyt and v=-y^2 t, where t denotes time. The equation for streamline passing through x=1, \; y=1 is
A
x^2y=1
B
xy^2=1
C
x^2y^2=1
D
x/y^2=1
GATE ME 2021 SET-2   Fluid Mechanics
Question 4 Explanation: 
\begin{aligned} u &=2 x y t \\ v &=-y^{2} t \\ \frac{d x}{u} &=\frac{d y}{v}=\frac{d z}{w} \\ \frac{d x}{2 x y t} &=\frac{d y}{-y^{2} t} \\ -y d x &=2 x d y \\ \ln x y^{2} &=c \\ x y^{2} &=1 \end{aligned}
Question 5
For a two-dimensional, incompressible flow having velocity components u and v in the x and y directions, respectively, the expression

\frac{\partial (u^2)}{\partial x}+\frac{\partial (uv)}{\partial y}
can be simplified to
A
u\frac{\partial u}{\partial x}+u\frac{\partial v}{\partial y}
B
2u\frac{\partial u}{\partial x}+u\frac{\partial v}{\partial y}
C
2u\frac{\partial u}{\partial x}+v\frac{\partial v}{\partial y}
D
u\frac{\partial u}{\partial x}+v\frac{\partial v}{\partial y}
GATE ME 2021 SET-2   Fluid Mechanics
Question 5 Explanation: 
\frac{\partial\left(u^{2}\right)}{\partial x}+\frac{\partial(u v)}{\partial y}
By differentiating:
\Rightarrow 2 u\left[\frac{\partial u}{\partial x}\right]+u \frac{\partial v}{\partial y}+v \frac{\partial u}{\partial y}
\Rightarrow u \frac{\partial u}{\partial x}+\left[u \frac{\partial u}{\partial x}+u \frac{\partial v}{\partial y}\right]+v \frac{\partial u}{\partial y}
According to continity eq. : \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0
So, u\frac{\partial u}{\partial x}+v \frac{\partial u}{\partial y}
Question 6
Consider a flow through a nozzle, as shown in the figure below:

The air flow is steady, incompressible and inviscid. The density of air is 1.23 kg/m^3. The pressure difference (p_1 - p_{atm}) is _______ kPa (round off to the nearest integer)
A
1522.12
B
1.52
C
321.32
D
3.21
GATE ME 2020 SET-2   Fluid Mechanics
Question 6 Explanation: 


\begin{aligned} A_{1} V_{1} &=A_{2} V_{2} \\ 0.2 \times V_{1} &=0.02 \times 50 \\ V_{1} &=\frac{1}{10} \times 50=5 \mathrm{m} / \mathrm{s} \end{aligned}
Applying BE
\begin{aligned} \frac{P_{1}}{w}+\frac{V_{1}^{2}}{2 g}+z_{1} &=\frac{P_{2}}{w}+\frac{V_{2}^{2}}{2 g}+z_{2} \; \; \; (\because z_{1}=z_{2})\\ \frac{P_{1}-P_{2}}{\rho_{\text {air }} g} &=\frac{V_{2}^{2}-V_{1}^{2}}{2 g} \\ P_{1}-P_{2} &=\left(\frac{50^{2}-5^{2}}{2}\right) \times 1.23\\ &=1522.125 \mathrm{Pa}=1.52 \mathrm{kPa} \end{aligned}
Question 7
The velocity field of an incompressible flow in a Cartesian system is represented by

\vec{V}=2(x^2-y^2)\hat{i}+v\hat{j}+3\hat{k}

Which one of the following expressions for v is valid?
A
-4xz+6xy
B
-4xy-4xz
C
4xz-6xy
D
4xy+4xz
GATE ME 2020 SET-1   Fluid Mechanics
Question 7 Explanation: 
\vec{V}=\left(2 x^{2}-2 y^{2}\right) \hat{i}+v \hat{j}+3 \hat{k}
For Incompressible flow
\begin{aligned} \frac{\partial u}{\partial x}+\frac{\partial V}{\partial y}+\frac{\partial W}{\partial z} &=0 \\ 4 x+\frac{\partial V}{\partial y} &=0 \\ \frac{\partial V}{\partial y} &=-4 x \\ v &=-4 x y+f(x, z) \end{aligned}
f(x,z) is an arbitary function of x and z
Hence the most suitable answer is option (B)
Question 8
For a steady flow, the velocity field is \overrightarrow{\textrm{v}}=(-x^{2}+3y)\hat{i}+(2xy)\hat{j} , The magnitude of the acceleration of a particle at (1, -1) is
A
2
B
1
C
2\sqrt{5}
D
0
GATE ME 2017 SET-1   Fluid Mechanics
Question 8 Explanation: 
Given velocity field,
\vec{V}=\left(-x^{2}+3 y\right) \hat{i}+(2 x y) \hat{j}
where \quad u=-x^{2}+3 y \text { and } v=2 x y
The acceleration components along x and y -axis.
\begin{aligned} a_{x}&=u \frac{\partial u}{\partial x}+v \frac{\partial u}{\partial y}\\ \text{and}\quad a_{y} &=u \frac{\partial v}{\partial x}+v \frac{\partial v}{\partial y} \\ &=\left(-x^{2}+3 y\right) \times(-2 x)+2 x y \times 3 \\ &=2 x^{3}-6 x y+6 x y=2 x^{3}\\ \text{and}\quad a_{y} &=\left(-x^{2}+3 y\right) \times 2 y+2 x y \times 2 x \\ &=-2 y x^{2}+6 y^{2}+4 x^{2} y=2 y x^{2}+6 y^{2} \end{aligned}
At point (1,-1)
\begin{aligned} a_{x}&=2\\ \text{and}\quad a_{y} &=2 \times(-1) \times 1+6 \times(-1)^{2} \\ &=-2+6=4 \\ \vec{a} &=a_{x} \hat{i}+a_{y} \hat{j}=2 i+4 \hat{j} \end{aligned}
Resultant acceleration:
\begin{aligned} a &=\sqrt{4+16}=\sqrt{20} \\ &=\sqrt{4 \times 5}=2 \sqrt{5} \end{aligned}
Question 9
Consider the two-dimensional velocity field given by \overrightarrow{\mathrm{v}}=(5+a_{1}x+b_{1}y)\hat{i}\: + \: (4+a_{2}x+b_{2}y)\hat{j} ,where a_1,b_1,a_2 text{ and }b_2 are constants. Which one of the following conditions needs to be satisfied for the flow to be incompressible?
A
a_{1}+\, b_{1}=0
B
a_{1}+\, b_{2}=0
C
a_{2}+\, b_{2}=0
D
a_{2}+\, b_{1}=0
GATE ME 2017 SET-1   Fluid Mechanics
Question 9 Explanation: 
Given that the flow is incompressible.
\therefore \quad \operatorname{div} \bar{V}=0
\frac{\partial}{\partial x}\left(5+a_{1} x+b_{1} y\right)+\frac{\partial}{\partial y}\left(8+a_{2} x+b_{2} y\right)=0
a_{1}+b_{2}=0
Question 10
For a two-dimensional flow, the velocity field is \vec{u}=\frac{x}{x^{2}+y^{2}}\hat{i}+\frac{y}{x^{2}+y^{2}}\hat{j}, where \hat{i} and \hat{j} are the basis vectors in the x-y Cartesian coordinate system. Identify the CORRECT statements from below.
(1)The flow is incompressible
(2)The flow is unsteady.
(3) y-component of acceleration, a_{y}=\frac{-y}{(x^{2}+y^{2})^{2}}
(4) x-component of acceleration,a_{x}=\frac{-(x+y)}{(x^{2}+y^{2})^{2}}
A
(2) and (3)
B
(1) and (3)
C
(1) and (2)
D
(3) and (4)
GATE ME 2016 SET-3   Fluid Mechanics
Question 10 Explanation: 
For 2 \mathrm{D} -flow velocity field is:
\begin{aligned} \vec{V}&=\frac{x}{x^{2}+y^{2}} \hat{i}+\frac{y}{x^{2}+y^{2}} \hat{j}\\ so \quad u&=\frac{x}{x^{2}+y^{2}} and v=\frac{y}{x^{2}+y^{2}} \\ a_{x} &=u \frac{\partial u}{\partial x}+v \frac{\partial u}{\partial y} \\ &=\frac{x}{x^{2}+y^{2}} \frac{\partial}{\partial x}\left[\frac{x}{x^{2}+y^{2}}\right] \\ &+\frac{y}{x^{2}+y^{2}} \frac{\partial}{\partial y}\left[\frac{x}{x^{2}+y^{2}}\right] \\ &= \frac{x}{x^{2}+y^{2}}\left[\frac{x(-2 x)}{\left(x^{2}+y^{2}\right)^{2}}+\frac{1}{x^{2}+y^{2}}\right]\\ &+\frac{y}{x^{2}+y^{2}} \left[\frac{x(2 y)}{\left(x^{2}+y^{2}\right)^{2}}\right]\\ &=\frac{-2 x^{3}}{\left(x^{2}+y^{2}\right)^{3}}+\frac{x}{\left(x^{2}+y^{2}\right)^{2}}\\ &+ \frac{2xy^{2}}{\left(x^{2}+y^{2}\right)^{3}}\\ &=\frac{-2 x^{3}+x\left(x^{2}+y^{2}\right)-2 x y^{2}}{\left(x^{2}+y^{2}\right)^{3}} \\ &=\frac{-2 x^{3}+x^{3}+x y^{2}-2 x y^{2}}{\left(x^{2}+y^{2}\right)^{3}}\\ &=\frac{-x^{3}-x y^{2}}{\left(x^{2}+y^{2}\right)^{3}} \\ &=\frac{-x\left(x^{2}+y^{2}\right)}{\left(x^{2}+y^{2}\right)^{3}} \end{aligned}
\begin{aligned} a_{x} &=\frac{x}{\left(x^{2}+y^{2}\right)^{2}} \\ a_{y} &=u \frac{\partial v}{\partial x}+u \frac{\partial v}{\partial y} \\ &=\frac{x}{x^{2}+y^{2}} \frac{\partial}{\partial x}\left[\frac{y}{x^{2}+y^{2}}\right] \\& +\frac{y}{x^{2}+y^{2}} \frac{\partial}{\partial y}\left[\frac{y}{x^{2}+y^{2}}\right] \\ &=\frac{x}{x^{2}+y^{2}} \frac{(-2 x y)}{\left(x^{2}+y^{2}\right)^{2}}\\& +\frac{y}{x^{2}+y^{2}} \times\left[\frac{-2 y^{2}}{\left(x^{2}+y^{2}\right)^{2}}+\frac{1}{x^{2}+y^{2}}\right]\\ &=\frac{-2 x^{2} y}{\left(x^{2}+y^{2}\right)^{3}}+\frac{2 y^{3}}{\left(x^{2}+y^{2}\right)^{3}}\\&+\frac{y}{\left(x^{2}+y^{2}\right)^{2}} \\ &=\frac{-2 x^{2} y-2 y^{3}+y\left(x^{2}+y^{2}\right)}{\left(x^{2}+y^{2}\right)^{3}} \\ &=\frac{-x^{2} y-y^{3}}{\left(x^{2}+y^{2}\right)^{3}}=\frac{y\left(x^{2}+y^{2}\right)}{\left(x^{2}+y^{2}\right)^{3}} \\ a_{y}&=\frac{-y}{\left(x^{2}+y^{2}\right)^{2}} \\ \end{aligned}
There are 10 questions to complete.

7 thoughts on “Fluid Kinematics”

  1. In question 18, the vorticity vector is to be found, but its written ‘velocity vector’. Please rectify that.

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