Fluid Mechanics and Hydraulic Machinery Miscellaneous

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.

View Hint View Answer Discuss in Forum
Given:
V = 4 + x + y ĵ m/s t
V = uî + vĵ + wk̂
u = 0, w = 0V = 4 + x + y ĵ t
a = a_{x}î + a_{y}ĵ + a_{z}k̂a_{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 = 0V = 4 + x + y ĵ t
a = a_{x}î + a_{y}ĵ + a_{z}k̂a_{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
 Consider the twodimensional velocity field given by V = (5 + a_{1}x + +b_{1}y) ˆ i + (4 + a_{2}x + b_{2}y) ˆ j, where a_{1}, b_{1}, a_{2} and b_{2} are constants. Which one of the following conditions needs to be satisfied for the flow to be incompressible?

View Hint View Answer Discuss in Forum
For continuous and incompressible flow
u_{x} + u_{y} = 0
a_{1} + b_{2} = 0Correct Option: B
For continuous and incompressible flow
u_{x} + u_{y} = 0
a_{1} + b_{2} = 0
 A flow field which has only convective acceleration is

View Hint View Answer Discuss in Forum
Convective acceleration is the effect of time independent acceleration of fluid with respect to space that means flow is steady nonuniform flow.
Correct Option: C
Convective acceleration is the effect of time independent acceleration of fluid with respect to space that means flow is steady nonuniform flow.
 In a twodimensional velocity field with velocities u and v along the x and y directions respectively, the convective acceleration along the xdirection is given by

View Hint View Answer Discuss in Forum
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 2dimensional field, thereforeConvective 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 2dimensional field, thereforeConvective acceleration u ∂u + v ∂u ∂x ∂y
 A flat plate of width L = 1 m is pushed down with a velocity U = 0.01 m/s towards a wall resulting in the drainage of the fluid between the plate and the wall as shown in the figure. Assume two dimensional incompressible flow and that the plate remains parallel to the wall. The average velocity, uavg of the fluid (in m/s) draining out at the instant shown in the figure is _____ (correct to three decimal places).

View Hint View Answer Discuss in Forum
By mass conservation: 2(s × u_{avg} × db) = (s. u × Lb)
⇒ 2 × v_{avg} × 0.1 = 0.01 × 1
u_{avg} = 0.05 m/sCorrect Option: C
By mass conservation: 2(s × u_{avg} × db) = (s. u × Lb)
⇒ 2 × v_{avg} × 0.1 = 0.01 × 1
u_{avg} = 0.05 m/s