Solved Numericals: Fluid Acceleration | Fluid Mechanics for Mechanical Engineering PDF Download

Solved Numericals

Q1: Velocity along the centreline of the nozzle of length L is given by 

v = velocity (m/s), t = time (s), x = distance (m)
Find the magnitude of total acceleration at a distance of 0.5 m and at 3 sec and use L = 0.8 m. (Round off to two decimal places,neglecting sign of acceleration)
Ans: 13.65 - 13.70
Acceleration =

=

Putting x = 0.5 m, t = 3 sec, L = 0.8 m, We get

= (-14.623) + 0.945
= -13.68 m/s²

Q2: A velocity field is given by the equation v = (3x)i + (2y)j. The y-component of the acceleration, 'a_y' is
Ans: For a velocity Vector:

where u, v, and w are velocity components along x, y, and z-direction respectively.
Then its acceleration
Along x-direction:

Along y-direction:

Along z-direction:

Calculation:
V = (3x)i + (2y)j
u = 3x and v = 2y
Along y-direction:

Q3: Calculate the normal component of acceleration when 8 m³/s of water passes over the bucket of a spillway of radius 4 m. Consider the thickness of sheet of water of ver the bucket as 0.5 m and take unit width.
Ans: Discharge, Q = A × V
where Q = Discharge, A = Cross-sectional area, V = velocity of flow
Normal component of acceleration,

Calculation:
Q = 8 m3/s, A = 1 × 0.5, R = 4 m
Velocity,
Normal component of acceleration,

Q4: If a nozzle is so shaped that the velocity of flow among center line changes from 1.5 m/sec to 15 m/sec in a distance of 0.375 m, what will be the magnitude of convective acceleration at the beginning?
Ans: Convective acceleration is given by,

Convective acceleration is defined as the rate of change of velocity due to the change in the position of fluid particles in a fluid flow.
Convective acceleration in one direction is given as:

Where, u = Initial velocity, du = Change in velocity
Given,
u = 1.5 m/s and du = 15 - 1.5 = 13.5 m/s
dx = 0.375 m

a_x = 54 m/s²

Q5: The velocity components in a flow field are given as u = xy² + 5t and v = 3 + 4t² + 2xy
where t (> 0) is in second, and x and y are in meters. The magnitude of total acceleration vector and direction acceleration vector at location (1, 0, 1) m and time t = 0.5 s  will be______ m/s²
Ans: 13
For a velocity field,
Total acceleration is the vector sum of the convective acceleration vector (directional acceleration vector) and local acceleration vector.

For 2 D flow field, the above equation can be reduced as,

The magnitude of total acceleration,

Calculation:
Given:
u = xy² + 5t, v = 3 + 4t² + 2xy, w = 0;
Using equation (3) and (4),

At location (1, 0, 1) m and time t = 0.5 s  
⇒ a_x = 5 m/s²: a_y = 12 m/s²: a_z = 0 m/s²
Using equation (5),
The magnitude of total accelaration,

Q6: In a stream line steady flow, two points A and B on a stream line are 1 m apart and the flow velocity varies uniformly from 2 m/s to 5 m/s. What is the acceleration of fluid at B? 
Ans: For flow along a stream line acceleration is given as
If V = f(s, t)

For steady flow dv/dt$=0$

Then

Since V = f(s) only for steady flow therefore dv/ds$=dv/ds$

Therefore

Calculation:
Given, V_A = 2 m/s, V_B = 5 m/s, and distance s = 1 m
$dV/ds\; =\frac{(}{5}=3$
So acceleration of fluid at B is

Q7: If the total acceleration of fluid flow is always zero, then it is:
Ans: For a velocity field,
where u, v and w are velocity components along x, y and z-direction respectively
Then its acceleration

Then the magnitude of total acceleration
There are two components of the acceleration

•  this group is called convective or advective acceleration and
• du/dt this group is called temporal acceleration.

Various condition and the respective type of flow is described in the table below.

∴ Total acceleration will be zero if both temporal and convective acceleration will be zero which is the case of steady and uniform flow.

Q8: If a nozzle is so shaped that the velocity of flow among center line changes from 1.5 m/sec to 15 m/sec in a distance of 0.375 m, what will be the magnitude of convective acceleration at the beginning?
Ans: Convective acceleration is given by,

Convective acceleration is defined as the rate of change of velocity due to the change in the position of fluid particles in a fluid flow.
Convective acceleration in one direction is given as:

Where, u = Initial velocity, du = Change in velocity
Calculation:
Given,
u = 1.5 m/s and du = 15 - 1.5 = 13.5 m/s
dx = 0.375 m

Q9: Calculate the normal component of acceleration when 8 m³/s of water passes over the bucket of a spillway of radius 4 m. Consider the thickness of sheet of water of ver the bucket as 0.5 m and take unit width.
Ans: Discharge, Q = A × V
where Q = Discharge, A = Cross-sectional area, V = velocity of flow
Normal component of acceleration,
Calculation:
Given:
Q = 8 m³/s, A = 1 × 0.5, R = 4 m
Velocity,
Normal component of acceleration,

Q10: For a steady flow, the velocity field is . The magnitude of the acceleration of a particle at (1, -1) is 
Ans:

a_x = [-(1)² + 3 (-1)] (-2(1))+6(1)(-1)
a₁ = 8 – 6
a_x = 2