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A 2-D flow defined by velocity field ⃗V⃗ = (x + 2y + 2)î + (4 − y)ĵ is
- a)compressible and irrotational
- b)compressible and not irrotational
- c)incompressible and irrotational
- d)incompressible and not irrotational
Correct answer is option 'D'. Can you explain this answer?
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A 2-D flow defined by velocity field ⃗V⃗ = (x + 2y + 2)î + (4 − y)ĵ...
Given u = (x + 2y + 2) ; v = (4 − y) For the flow to be incompressible, the flow must satisfy the following continuity equation i.e ∂u /∂x + ∂v /∂y = 0
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Substituting the given velocity components in LHS.
∂u /∂x + ∂v / ∂y = 1 − 1 = 0
Hence flow is incompressible.
For the flow to be irrotational, Ωz must be zero. i. e ∂v /∂x − ∂u /∂y = 0
Substituting velocity components in LHS
∂v /∂x − ∂u /∂y = 0 − (2) = −2
Hence flow is rotational or not irrotational because vorticity is non-zero.
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A 2-D flow defined by velocity field ⃗V⃗ = (x + 2y + 2)î + (4 − y)ĵ...
To determine whether the given 2-D flow is compressible and irrotational, we need to examine the properties of the velocity field ⃗V⃗ = (x 2y 2)î (4 − y)ĵ.
1. Compressible Flow:
- Compressible flow refers to a flow where the density of the fluid changes throughout the flow field.
- In 2-D flow, compressibility is determined by the divergence of the velocity field.
- The divergence of ⃗V⃗ is given by div(⃗V⃗) = ∂Vx/∂x + ∂Vy/∂y.
- In this case, ∂Vx/∂x = 1 and ∂Vy/∂y = -1, as there are no other terms involving x and y in the given velocity field.
- Therefore, div(⃗V⃗) = 1 + (-1) = 0.
- Since the divergence is zero, the flow is incompressible.
2. Irrotational Flow:
- Irrotational flow refers to a flow where the fluid particles rotate about their own axes without any net rotation.
- In 2-D flow, irrotationality is determined by the curl of the velocity field.
- The curl of ⃗V⃗ is given by curl(⃗V⃗) = (∂Vy/∂x - ∂Vx/∂y)k̂.
- In this case, ∂Vy/∂x = 0 and ∂Vx/∂y = -2, as there are no other terms involving x and y in the given velocity field.
- Therefore, curl(⃗V⃗) = (0 - (-2))k̂ = 2k̂.
- Since the curl is not zero, the flow is not irrotational.
Based on the above analysis, we can conclude that the given 2-D flow is incompressible and not irrotational. Therefore, option D is the correct answer.
1. Compressible Flow:
- Compressible flow refers to a flow where the density of the fluid changes throughout the flow field.
- In 2-D flow, compressibility is determined by the divergence of the velocity field.
- The divergence of ⃗V⃗ is given by div(⃗V⃗) = ∂Vx/∂x + ∂Vy/∂y.
- In this case, ∂Vx/∂x = 1 and ∂Vy/∂y = -1, as there are no other terms involving x and y in the given velocity field.
- Therefore, div(⃗V⃗) = 1 + (-1) = 0.
- Since the divergence is zero, the flow is incompressible.
2. Irrotational Flow:
- Irrotational flow refers to a flow where the fluid particles rotate about their own axes without any net rotation.
- In 2-D flow, irrotationality is determined by the curl of the velocity field.
- The curl of ⃗V⃗ is given by curl(⃗V⃗) = (∂Vy/∂x - ∂Vx/∂y)k̂.
- In this case, ∂Vy/∂x = 0 and ∂Vx/∂y = -2, as there are no other terms involving x and y in the given velocity field.
- Therefore, curl(⃗V⃗) = (0 - (-2))k̂ = 2k̂.
- Since the curl is not zero, the flow is not irrotational.
Based on the above analysis, we can conclude that the given 2-D flow is incompressible and not irrotational. Therefore, option D is the correct answer.
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A 2-D flow defined by velocity field ⃗V⃗ = (x + 2y + 2)î + (4 − y)ĵ isa) compressible and irrotationalb) compressible and not irrotationalc) incompressible and irrotationald) incompressible and not irrotationalCorrect answer is option 'D'. Can you explain this answer?
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