Fluid Mechanics and Hydraulic Machinery Miscellaneous Easy Questions and Answers | Page - 8

Fluid Mechanics and Hydraulic Machinery Miscellaneous

In a two-dimensional velocity field with velocities u and v along the x and y directions respectively, the convective acceleration along the x-direction is given by

1. u ∂V + v ∂u
  ∂x ∂y
1. u ∂u + v ∂v
  ∂x ∂y
1. u ∂u + v ∂u
  ∂x ∂y
1. u ∂u + u ∂u
  ∂x ∂y

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Two dimensional velocity field with velocities u, v and along x and y direction.
∴ Acceleration along x direction, a_x = a_convective + a_{temporal or local}
= u ∂u + v ∂u + w ∂u + ∂u
∂x ∂y ∂z ∂t

Since, (∂u/∂z) = 0 for 2-dimensional field, therefore
Convective acceleration u ∂u + v ∂u
∂x ∂y

Correct Option: C

Two dimensional velocity field with velocities u, v and along x and y direction.
∴ Acceleration along x direction, a_x = a_convective + a_{temporal or local}
= u ∂u + v ∂u + w ∂u + ∂u
∂x ∂y ∂z ∂t

Since, (∂u/∂z) = 0 for 2-dimensional field, therefore
Convective acceleration u ∂u + v ∂u
∂x ∂y

A flow field which has only convective acceleration is

1. a steady uniform flow
1. an unsteady uniform flow
1. a steady non-uniform flow
1. an unsteady non-uniform flow

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Convective acceleration is the effect of time independent acceleration of fluid with respect to space that means flow is steady non-uniform flow.

Correct Option: C

Convective acceleration is the effect of time independent acceleration of fluid with respect to space that means flow is steady non-uniform flow.

Consider the two-dimensional velocity field given by V = (5 + a₁x + +b₁y) ˆ i + (4 + a₂x + b₂y) ˆ j, where a₁, b₁, a₂ and b₂ are constants. Which one of the following conditions needs to be satisfied for the flow to be incompressible?

1. a₁ + b₁ = 0
1. a₁ + b₂ = 0
1. a₂ + b₂ = 0
1. a₂ + b₁ = 0

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For continuous and incompressible flow
u_x + u_y = 0
a₁ + b₂ = 0

Correct Option: B

For continuous and incompressible flow
u_x + u_y = 0
a₁ + b₂ = 0

Water flows though a pipe with a velocity given by V^→ = 4 + x + y ĵ m/s
t

m/s, where ĵ is the unit vector in the y direction, t(>0) is in seconds, and x and y are in meters. The magnitude of total acceleration at the point (x, y) = (1, 1) at t = 2 s is _____ m/s².

1. 3 m/s²
1. 29 m/s²
1. 10 m/s²
1. 30 m/s²

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Given:
V = 4 + x + y ĵ m/s
t

V = uî + vĵ + wk̂
u = 0, w = 0
V = 4 + x + y ĵ
t

a = a_xî + a_yĵ + a_zk̂
a_x = u ∂u + v ∂u + w ∂u + ∂u = 0
∂x ∂y ∂z ∂t

a_z = u ∂w + v ∂w + w ∂w + ∂w
∂x ∂y ∂z ∂t

a_y = u ∂v + v ∂v + w ∂v + ∂v
∂x ∂y ∂z ∂t

= u ∂v + v ∂v
∂y ∂t

= 4 + x + y × 1 + 4
t t²

a_y = 4 + x + y − 4 ĵ
t t²

Now, At (x,y) = 1,1 and t = 2 sec.
Total acceleration is given by,
a = a_y = 4 + 1 + 1 − 4 = 3 m/s²
2 4

Correct Option: A

Given:
V = 4 + x + y ĵ m/s
t

V = uî + vĵ + wk̂
u = 0, w = 0
V = 4 + x + y ĵ
t

a = a_xî + a_yĵ + a_zk̂
a_x = u ∂u + v ∂u + w ∂u + ∂u = 0
∂x ∂y ∂z ∂t

a_z = u ∂w + v ∂w + w ∂w + ∂w
∂x ∂y ∂z ∂t

a_y = u ∂v + v ∂v + w ∂v + ∂v
∂x ∂y ∂z ∂t

= u ∂v + v ∂v
∂y ∂t

= 4 + x + y × 1 + 4
t t²

a_y = 4 + x + y − 4 ĵ
t t²

Now, At (x,y) = 1,1 and t = 2 sec.
Total acceleration is given by,
a = a_y = 4 + 1 + 1 − 4 = 3 m/s²
2 4

A velocity field is given as
VV^→ 3x²yî − 6xyzk̂
where x, y, z are in m and V in m/s. Determine if
(i) It represents an incompressible flow
(ii) The flow is irrotational
(iii) The flow is steady

1. (ii)
1. (i)
1. (i) and (ii)
1. (iii)

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V = 3x²î − 6xyzk̂
∂u + ∂v + ∂w = 6xy + 0 − 6xy = 0
∂x ∂y ∂z

Flow is incompressible
w_y = 1 ∂u − ∂w
2 ∂z ∂x

= 1 [0 − (−6yz)]
2

w_y = 3yz
Flow is rotational. Flow field is independent of time since terms does not include time ‘t’.

Correct Option: A

V = 3x²î − 6xyzk̂
∂u + ∂v + ∂w = 6xy + 0 − 6xy = 0
∂x ∂y ∂z

Flow is incompressible
w_y = 1 ∂u − ∂w
2 ∂z ∂x

= 1 [0 − (−6yz)]
2

w_y = 3yz
Flow is rotational. Flow field is independent of time since terms does not include time ‘t’.