Fluid Mechanics and Hydraulic Machinery Miscellaneous
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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.
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Given:
V = 4 + x + y ĵ m/s t
V = uî + vĵ + wk̂
u = 0, w = 0V = 4 + x + y ĵ t
a = axî + ayĵ + azk̂ax = 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 = 0V = 4 + x + y ĵ t
a = axî + ayĵ + azk̂ax = 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
- 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?
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For continuous and incompressible flow
ux + uy = 0
a1 + b2 = 0Correct Option: B
For continuous and incompressible flow
ux + uy = 0
a1 + b2 = 0
- A flow field which has only convective acceleration is
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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.
- 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
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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, 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, ax = aconvective + atemporal or local= u ∂u + v ∂u + w ∂u + ∂u ∂x ∂y ∂z ∂t
Since, (∂u/∂z) = 0 for 2-dimensional 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).
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By mass conservation:- 2(s × uavg × db) = (s. u × Lb)
⇒ 2 × vavg × 0.1 = 0.01 × 1
uavg = 0.05 m/sCorrect Option: C
By mass conservation:- 2(s × uavg × db) = (s. u × Lb)
⇒ 2 × vavg × 0.1 = 0.01 × 1
uavg = 0.05 m/s