## Fluid Mechanics and Hydraulic Machinery Miscellaneous

#### Fluid Mechanics and Hydraulic Machinery

1. 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. Two dimensional velocity field with velocities u, v and along x and y direction.
∴  Acceleration along x direction, ax = aconvective + atemporal 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, ax = aconvective + atemporal 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

1. A flow field which has only convective acceleration is

1. 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.

1. 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. For continuous and incompressible flow
ux + uy = 0
a1 + b2 = 0

##### Correct Option: B

For continuous and incompressible flow
ux + uy = 0
a1 + b2 = 0

1.  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. Given:

 V = 4 + x + y ĵ m/s t

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

a = axî + ayĵ + az
 ax = u ∂u + v ∂u + w ∂u + ∂u = 0 ∂x ∂y ∂z ∂t

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

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

 = u ∂v + v ∂v ∂y ∂t

 = 4 + x + y × 1 + 4 t t²

 ay = 4 + x + y − 4 ĵ t t²

Now, At (x,y) = 1,1 and t = 2 sec.
Total acceleration is given by,
 a = ay = 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 = axî + ayĵ + az
 ax = u ∂u + v ∂u + w ∂u + ∂u = 0 ∂x ∂y ∂z ∂t

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

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

 = u ∂v + v ∂v ∂y ∂t

 = 4 + x + y × 1 + 4 t t²

 ay = 4 + x + y − 4 ĵ t t²

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

1. 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

1. V = 3x²î − 6xyzk̂

 ∂u + ∂v + ∂w = 6xy + 0 − 6xy = 0 ∂x ∂y ∂z

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

 = 1 [0 − (−6yz)] 2

wy = 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
 wy = 1 ∂u − ∂w 2 ∂z ∂x

 = 1 [0 − (−6yz)] 2

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