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A 2-D flow having velocity V = (x + 2y + 2)i + (4 - y)j will be
- a)compressible and irrotational
- b)compressible and not irrotational
- c)in compressible and irrotational
- d)in compressible and not irrotational
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
A 2-D flow having velocity V = (x + 2y + 2)i + (4 - y)jwill bea)compre...
u - x + 2y + 2
v = 4 - y
As if satisfy the continuity equation for incompressible flow so this is incompressible,
and since rotational component is not zero so flow is not irrotational.
v = 4 - y
As if satisfy the continuity equation for incompressible flow so this is incompressible,
and since rotational component is not zero so flow is not irrotational.
Most Upvoted Answer
A 2-D flow having velocity V = (x + 2y + 2)i + (4 - y)jwill bea)compre...
Given: Velocity field, V = (x 2y 2)i (4 - y)j
To determine: Whether the flow is compressible and irrotational or not.
Solution:
Compressibility of the flow:
The flow is compressible if the density of the fluid changes with respect to time and position.
The continuity equation for incompressible flow is given as:
∇·V = 0
where, ∇·V is the divergence of the velocity field.
Taking the divergence of the given velocity field:
∇·V = ∂(x) / ∂x + ∂(2y) / ∂y + ∂(2) / ∂z
+ ∂(4 - y) / ∂y
= 1 + 2 + 0 + (-1)
= 2
Since the divergence of the velocity field is non-zero, the flow is compressible.
Irrotationality of the flow:
The flow is irrotational if the curl of the velocity field is zero.
The curl of the given velocity field is given as:
∇ x V = (∂(4 - y) / ∂x - ∂(2) / ∂y) k
= 0i - 0j + (2 - 0)k
= 2k ≠ 0
Since the curl of the velocity field is non-zero, the flow is not irrotational.
Therefore, the flow is compressible and not irrotational, and the correct option is (D).
Note: The continuity equation is based on the principle of conservation of mass, which states that the mass of the fluid remains constant within a control volume. The curl of the velocity field is based on the principle of conservation of angular momentum, which states that the angular momentum of the fluid remains constant within a control volume.
To determine: Whether the flow is compressible and irrotational or not.
Solution:
Compressibility of the flow:
The flow is compressible if the density of the fluid changes with respect to time and position.
The continuity equation for incompressible flow is given as:
∇·V = 0
where, ∇·V is the divergence of the velocity field.
Taking the divergence of the given velocity field:
∇·V = ∂(x) / ∂x + ∂(2y) / ∂y + ∂(2) / ∂z
+ ∂(4 - y) / ∂y
= 1 + 2 + 0 + (-1)
= 2
Since the divergence of the velocity field is non-zero, the flow is compressible.
Irrotationality of the flow:
The flow is irrotational if the curl of the velocity field is zero.
The curl of the given velocity field is given as:
∇ x V = (∂(4 - y) / ∂x - ∂(2) / ∂y) k
= 0i - 0j + (2 - 0)k
= 2k ≠ 0
Since the curl of the velocity field is non-zero, the flow is not irrotational.
Therefore, the flow is compressible and not irrotational, and the correct option is (D).
Note: The continuity equation is based on the principle of conservation of mass, which states that the mass of the fluid remains constant within a control volume. The curl of the velocity field is based on the principle of conservation of angular momentum, which states that the angular momentum of the fluid remains constant within a control volume.
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A 2-D flow having velocity V = (x + 2y + 2)i + (4 - y)jwill bea)compressible and irrotationalb)compressible and not irrotationalc)in compressible and irrotationald)in compressible and not irrotationalCorrect answer is option 'D'. Can you explain this answer?
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