Fluid Mechanics and Hydraulic Machinery Miscellaneous


Fluid Mechanics and Hydraulic Machinery Miscellaneous

Fluid Mechanics and Hydraulic Machinery

  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.









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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
    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

    ay =
    4
    + x + y −
    4
    t

    Now, At (x,y) = 1,1 and t = 2 sec.
    Total acceleration is given by,
    a = ay =
    4
    + 1 + 1 −
    4
    = 3 m/s²
    24

    Correct Option: A

    Given:

    V =
    4
    + x + y ĵ m/s
    t

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

    ay =
    4
    + x + y −
    4
    t

    Now, At (x,y) = 1,1 and t = 2 sec.
    Total acceleration is given by,
    a = ay =
    4
    + 1 + 1 −
    4
    = 3 m/s²
    24


  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?









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    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. 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.


  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









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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, 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 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/s

    Correct Option: C


    By mass conservation:- 2(s × uavg × db) = (s. u × Lb)
    ⇒  2 × vavg × 0.1 = 0.01 × 1
    uavg = 0.05 m/s