GATE Past Year Questions: Fluid Kinematics | Fluid Mechanics for Mechanical Engineering PDF Download

Q1: The velocity field of a two-dimensional, incompressible flow is given by

where and denote the unit vectors in x and y directions, respectively. If v (x, 0) = cos x, then v (0, -1) is [GATE ME 2024]
(a) 1
(b) 2

(c) 3
(d) 4
Ans: (c)
For an incompressible flow, ∇⋅ = 0
Given; the velocity field of a two-dimensional, incompressible flow,

2 sinh  & v (x, 0) = cosh x

Now, ∇. = 0 (for an incompressible flow)
i. e. ∂v/∂x + ∂v/∂y = 0
2 cosh x + ∂v/∂y = 0
⇒ ∂v = -2 cos h . ∂y
Integrate both sides,

v = -2y cosh x + C.... (1)
Given; v (x, 0) = cosh⁡ x
⇒ C = cosh x
Now, from equation (1)
v (x, y)=−2y cosh x + cosh x
= (1−2y) cosh x
At (0,−1)
v (0,−1) = [1−2(−1)] cosh (0) = 3

Q1: Consider a unidirectional fluid flow with the velocity field given by

where u ( 0 , t ) = 1. If the spatially homogeneous density field varies with time t as
ρ (t)=1 + 0.2e ^−t the value of u (2 , 1) is _______. (Rounded off to two decimal places)
Assume all quantities to be dimensionless.  [GATE  ME 2023]
Ans: (1.1 to 1.2)
Continuity equation for unsteady flow
Here V (x, y, z, t)=u (x, t)
So v = 0

Given ρ(t)=1 + 0.2 e^−t

Since

u (2, 1) = 1.137 m/s ≈ 1.14 m/sec

Q2: The velocity field of a certain two-dimensional flow is given by

where k = 2s^{− 1} . The coordinates x  and y  are in meters. Assume gravitational effects to be negligible. If the density of the fluid is 1000 kg / m³ and the pressure at the origin is 100 kPa, the pressure at the location (2 m, 2 m) is _____________ kPa. (Answer in integer) [GATE ME 2023]
Ans: (83.999 to 84.001
To find the pressure at location (2m,2m) we apply  Bernoulli's equation

We will apply this equation between two points Origin(0, 0) and location  (2 m, 2 m) 
At Origin (0, 0)

V₁  =2 (0 − 0) = 0 
P₁ =100kPa 
 At Iocation   (2, 2)

magnitude of velocity

Applying Bernoulli's theorem
100,000 + 1/2 × 1000 × 0 = P₂ + 1/2 × 1000 ×32
So P₂ + 16,000 = 100,000

P₂ = 100,000 −16,000 
P₂ = 84 , 000 Pa = 84 k Pa

Q3: Air (density = 1.2kg/m³ , kinematic viscosity = 1. 5 × 1 0 ^{− 5} m²/ s ) flows over a flat plate with a free-stream velocity of 2 m/s . The wall shear stress at a location 15mm from the leading edge is τ w  . What is the wall shear stress at a location 30mm from the leading edge?  [GATE ME 2023]

(a) T_w/2
(b)

(c) 2/T_w
(d)

Ans: (d)
Step-1: First check type of flow by Reynold No
 Re ∝ u ∞ L/v 
Re = 2 x 0.03/1.5 x 10^-5 = 4000
As Reynold no. is less than 5 × 1 0⁵  , it is laminar flow

Step-2: Wall shear stress in laminar flow -

Q1: The steady velocity field in an inviscid fluid of density 1.5 is given to be Neglecting body forces, the pressure gradient at (x = 1, y = 1) is  [GATE ME 2022, SET-2]
(a)

(b)

(c)

 (d)
Ans: (c) 
By Euler's equation of motion,

Neglecting body forces (i.e. g _x = g _y = 0 )

= - 1.5 x (2 x 1 x 1² + 2 x 1³)
= - 6 pa/m
Similarly,

= - 1.5 x (2 x 1 x 1² + 2 x 1³)
= - 6 pa/m
The pressure gradient vector is given by

Q2:The velocity field in a fluid is given to be

Which of the following statement(s) is/are correct?  [GATE ME 2022 SET-2]
(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).
Ans: (b, c, d)
For given flow,
u = 4 xy, v = 2 (x² − y²)
As velocity field is function of two space variables, flow is two dimensional.

Therefore, flow is incompressible.
ω z = 1/2  = 1/2 (4x - 4x) = 0
Therefore, flow is irrotational.

= 4 xy (4y) + 2 (x² − y²) (4x)

= 16 xy2 + 8x ³  - 8xy²
= 16 x 0 x 0² + 8 x 0³ - 8 x 0 x 0²
= 0 

= 4 xy (4x) + 2 (x² − y²) ( - 4y)
= 16 x²y  - 8x ² + 8xy³
= 16 x 0 x 0² x 8 x 0² x 0 + 8 x 0³

= 0

0

Q3: A steady two-dimensional flow field is specified by the stream function
ψ = kx³y
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/s² (round off to 2 decimal places). [GATE ME 2022, SET- 1]
Ans: (4.2 to 4.28)
Given,
Stream function,

At (1, 1)

Q1: A two dimensional flow has velocities in x x and y y directions given by u = 2xyt and v = − y ²t , where t denotes time. The equation for streamline passing through x = 1 , y = 1 is [GATE ME 2021, SET-2]
(a) x²y = 1
(b) xy² = 1
(c) x²y² = 1
(d) x/y² = 1
Ans: (b) 
u = 2xyt
v = -y²t
dx/u = dy/v = dz/w
-ydx = 2xdy
In xy²= c
xy² = 1

Q2: For a two-dimensional, incompressible flow having velocity components u  and v  in the x  and y  directions, respectively, the expression

can be simplified to  [GATE ME 2021, SET-2]

(a) u
(b) 2u

(c) 2u
(d) u

 Ans: (d) 

By differentiating:

According to Continuity  eq. :   = 0
So, u

Q1: Consider a flow through a nozzle, as shown in the figure below:  [GATE ME 2020,  SET-2]

Ans: (1.5 to 1.55)

A₁ V₁ = A₂ V₂
0.2 x V₁ = 0.02 x 50
V₁ = 1/10 x 50 = 5m/s
Applying BE

= 1522.125Pa = 1.52kPa

Q2: The velocity field of an incompressible flow in a Cartesian system is represented by

Which one of the following expressions for v is valid?  [GATE ME 2020, SET-1]
(a) - 4 xz + 6xy
(b)  - 4 xy + 6xz
(c) 4 xz + 6xy
(d) 4 xy + 6xz
Ans: (b) 

For Incompressible flow

v = -4xy  + f (x, z)
f (x, z) is an arbitary function of x and z
Hence the most suitable answer is option (B)

Question for GATE Past Year Questions: Fluid Kinematics

Try yourself:For a two-dimensional flow, the velocity field is 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,

4. x-component of acceleration,

[2016, Set-3]

• A.

  2 and 3
• B.

  1 and 3
• C.

  1 and 2
• D.

  3 and 4

Explanation

The velocity components are not functions of time, so flow is steady according to continuity equation,

Since it satisfies the above continuity equation for 2D and incompressible flow.
∴ The flow is in compressible.

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Question for GATE Past Year Questions: Fluid Kinematics

Try yourself:A two-dimensional in compressible friction less flow field is given by  . If ρ is the density of the fluid, the expression for pressure gradient vector at any point in the flow field is given as

[2019, Set -2]

• A.

  
• B.

  
• C.

  
• D.

  

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Question for GATE Past Year Questions: Fluid Kinematics

Try yourself:For a steady flow, the velocity field is . The magnitude of the acceleration of a particle at (1, - 1) is

[2017 Set-1]

• A.

  2
• B.

  1
• C.

  2√5
• D.

  0

Explanation

Now magnitude of particleat (1, – 1)

 

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Question for GATE Past Year Questions: Fluid Kinematics

Try yourself:For a certain two-dimensional incompressible flow, velocity field is given by . The streamlines for this flow are given by the family of curves

[2016,Set-3]

• A.

  x2y2 = constant
• B.

  xy2 = constant
• C.

  2xy – y2 = constant
• D.

  xy – constant

Explanation

on integrating

ψ = xy² + f (x)
= y¹ + f'(x)
f'(x) = 0
⇒ f (x)= constant
so y = xy² + constant

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Question for GATE Past Year Questions: Fluid Kinematics

Try yourself:Consider a velocity field where K is a constant. The vorticity, Ω Z, is

[2014 Set-4]

• A.

  –K
• B.

  K
• C.

  –K/2
• D.

  K/2

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Question for GATE Past Year Questions: Fluid Kinematics

Try yourself:A fluid flow is represented by the velocity field ​, where a is a constant. The equation of stream line passing through a point (1, 2) is

[2004]

• A.

  x – 2y = 0
• B.

  2x + y = 0
• C.

  2x – y = 0
• D.

  x + 2y = 0

Explanation

Given: u_x = ax and u_y = ay
Equation of steam line is,

Integrating both sides, we have
log(ax) = log(ay) + log c
or ax = c×ay
or x = cy
Since the steam line is passed through point (1, 2), therefore
1 = 2c
⇒ c = 1/2
∴ x = y/2
Hence equation of steam line is
2x – y = 0.

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Question for GATE Past Year Questions: Fluid Kinematics

Try yourself:The Newtonian fluid has the following velocity field:

The rate shear deformation ∈_yz at the point x  = -2, y = -1 and z = 2 for the given flow is

[1988]

• A.

  -6
• B.

  -2
• C.

  -12
• D.

  4

Explanation

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Question for GATE Past Year Questions: Fluid Kinematics

Try yourself:For a two-dimensional incompressible flow field given by , where A > 0, which one of the following statements is FALSE?
A. It satisfies continuity equation B. It is unidirectional when x → 0 and y → ∞.
C. Its streamlines are given by x = y.
D. It is irrotational

[2018, Set-1]

• A.

  A
• B.

  B
• C.

  C
• D.

  D

Explanation

C is the false statement 2D incompressible flow continuity equation.

A – A = 0 it satisfies continuity equation.

As y → ∞.velocity vector field will not be defined along y axis.
So flow will be along x-axis i.e. 1-D flow
⇒ Stream line equation for 2D


In x = – ln y + ln c
ln xy = ln c
xy = c → stream line equation.

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Question for GATE Past Year Questions: Fluid Kinematics

Try yourself:Match the following pairs:

[2015: Set-1]

• A.

  P-IV, Q-I, R-II, S-III
• B.

  P-IV, Q-III, FRS-II
• C.

  P-III, Q-I, R-IV, S-II
• D.

  P-III, Q-I, R-II, S-IV

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Question for GATE Past Year Questions: Fluid Kinematics

Try yourself:For the continuity equation given by to be valid, where is the velocity vector,which one of the following is a necessary condition?

[2008]

• A.

  Steady flow
• B.

  Irrotational flow
• C.

  Inviscid flow
• D.

  Incompressible flow

Explanation

The basic equation of continuity for fluid flow is given by

Now if ρ remains constant, then only we can write

hence incompressible flow.

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Question for GATE Past Year Questions: Fluid Kinematics

Try yourself:Consider the two-dimensional velocity field given by = (5 + a₁x + +b₁y) + (4 + a₂x + b₂y) , where a₁, b₁, a₂ and b₂ are constants.

Which one of the following conditions needs to be satisfied for the flow to be incompressible?

[2017: Set-1]

• A.

  a₁ + b₁ = 0
• B.

  a₁ + b₂ = 0
• C.

  a₂ + b₂ = 0
• D.

  a₂ + b₁ = 0

Explanation

For continuous and in compressible flow
u_x + u_y = 0
a₁ + b₂ = 0

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Question for GATE Past Year Questions: Fluid Kinematics

Try yourself:A leaf is caught in a whirlpool. At a given instant, the leaf is at a distance of 120 m from the centre of the whirlpool. The whirlpool can be described by the following velocity distribution: 
 m/s and m/s, where r (in meters) is thedistance from the centre of the whirlpool. What will be the distance of the leaf from the centre when it has moved through half a revolution?

[2005]

• A.

  48 m
• B.

  64 m
• C.

  120 m
• D.

  142 m

Explanation

Radial distance = 120 m


By equating (i) & (ii), we get

By solving above, we get
r = 64 m

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