Fluid Mechanics and Hydraulic Machinery Miscellaneous
 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

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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 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 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.
 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?

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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

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 = 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
 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
(iii) The flow is steady

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V = 3x²î − 6xyzk̂
∂u + ∂v + ∂w = 6xy + 0 − 6xy = 0 ∂x ∂y ∂z
Flow is incompressiblew_{y} = 1 ∂u − ∂w 2 ∂z ∂x = 1 [0 − (−6yz)] 2
w_{y} = 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 incompressiblew_{y} = 1 ∂u − ∂w 2 ∂z ∂x = 1 [0 − (−6yz)] 2
w_{y} = 3yz
Flow is rotational. Flow field is independent of time since terms does not include time ‘t’.