Fluid Mechanics and Hydraulic Machinery Miscellaneous
 For a steady flow, the velocity field is V = (–x² + 3y) i + (2xy)j. The magnitude of the acceleration of a particle at (1,  1) is

View Hint View Answer Discuss in Forum
NA
Correct Option: C
NA
 For a twodimensional flow, the velocity field is
u = x i + y j x² + y² x² + y²
where i and j are the basis vectors in the x – y Cartesian coordinate system. Identify the CORRECT statements from below.
1. The flow is incompressible
2. The flow is unsteady
3. ycomponent of acceleration,a_{y} =  y (x² + y²)²
4. xcomponent of acceleration,a_{x} = (x + y) (x² + y²)²

View Hint View Answer Discuss in Forum
a_{x}u δu + v δu δx δy = x(x² + y² 2x²)2xy² =  x^{3}  xy² (x² + y²)(x² + y²) (x² + y²) ∴ a_{x} = x (x² + y²)² a_{y} = u δv + v δv δx δy = x² + yx²  y^{3} =  y (x² + y²)^{3} (x² + y²)²
The velocity components are not functions of time, so flow is steady according to continuity equation,
Since it satisfies the above continuity equation for 2D and incompressible flow.
∴ The flow is incompressible.Correct Option: B
a_{x}u δu + v δu δx δy = x(x² + y² 2x²)2xy² =  x^{3}  xy² (x² + y²)(x² + y²) (x² + y²) ∴ a_{x} = x (x² + y²)² a_{y} = u δv + v δv δx δy = x² + yx²  y^{3} =  y (x² + y²)^{3} (x² + y²)²
The velocity components are not functions of time, so flow is steady according to continuity equation,
Since it satisfies the above continuity equation for 2D and incompressible flow.
∴ The flow is incompressible.
 For a certain twodimensional incompressible flow, velocity field is given by 2xyi – y² j. The streamlines for this flow are given by the family of curves

View Hint View Answer Discuss in Forum
v = 2xy i – y²j
u = δx , v =  δx δy δx 2xy = δx = 2 δy
2xyδy = δψ
on integrating
ψ = xy² + f(x)
= y^{1} + f'(x)
f'(x) = 0
⇒ f(x)= constant
so ψ= xy² + constantCorrect Option: B
v = 2xy i – y²j
u = δx , v =  δx δy δx 2xy = δx = 2 δy
2xyδy = δψ
on integrating
ψ = xy² + f(x)
= y^{1} + f'(x)
f'(x) = 0
⇒ f(x)= constant
so ψ= xy² + constant
 The volumetric flow rate (per unit depth) between two streamlines having stream function ψ1 and ψ2 is

View Hint View Answer Discuss in Forum
NA
Correct Option: D
NA
 The velocity field of an incompressible flow is given by V = (a_{1}x + a_{2}y + a_{3} z)i + (b_{1}x + b_{2}y + b_{3} z)j + (c_{1}x + c_{2}y + c_{3} z)k, where a_{1} = 2 and c_{3} = –4. The value of b_{2} is _________.

View Hint View Answer Discuss in Forum
δu + δv + δw = 0 δx δy δz
a_{1} + b_{2} + c_{3} = 0
2 – 4 + b_{2} = 0
b_{2} = 2
Correct Option: A
δu + δv + δw = 0 δx δy δz
a_{1} + b_{2} + c_{3} = 0
2 – 4 + b_{2} = 0
b_{2} = 2