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In a two dimensional flow in x-y plane, if ∂u / ∂y = ∂v / ∂x , then the fluid element will undergo
- a)Translation only
- b)Translation and rotation
- c)Translation and deformation
- d)Rotation and deformation
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
In a two dimensional flow in x-y plane, if ∂u / ∂y = ∂v / ∂x , then t...
For a 2D planar flow the deformation rate is given as ε̇ xy = ∂v / ∂x + ∂u / ∂y And rotation rate is given as
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ωz = 1/[∂v / ∂x − ∂u / ∂y ]
Therefore in the given case ε̇ xy = 2 ∂u / ∂y and ωz = 0
Thus along with translation, elements will deform but shall not rotate.
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In a two dimensional flow in x-y plane, if ∂u / ∂y = ∂v / ∂x , then t...
Two dimensional flow in x-y plane
- In a two-dimensional flow in the x-y plane, the velocity of the fluid can be described by two components: u in the x-direction and v in the y-direction.
- The flow can be considered as a collection of fluid elements or particles that move in the x-y plane.
Relationship between ∂u/∂y and ∂v/∂x
- ∂u/∂y and ∂v/∂x are partial derivatives of the velocity components u and v with respect to the y and x coordinates, respectively.
- If ∂u/∂y = ∂v/∂x, then the flow is said to be irrotational, which means that there is no net rotation of the fluid particles around their own axes.
- This condition is also known as the condition of zero vorticity.
Behavior of fluid element
- If a fluid element undergoes translation, it means that it moves without any deformation or change in shape.
- If a fluid element undergoes rotation, it means that it moves in a circular path around its own axis.
- If a fluid element undergoes deformation, it means that its shape changes while it moves.
Conclusion
- If ∂u/∂y = ∂v/∂x, then the fluid element will undergo translation and deformation.
- This is because the condition of zero vorticity implies that there is no net rotation of the fluid particles, but it does not imply that there is no deformation.
- Therefore, the fluid element will move without any rotation but may undergo deformation due to the non-uniformity of the flow.
- In a two-dimensional flow in the x-y plane, the velocity of the fluid can be described by two components: u in the x-direction and v in the y-direction.
- The flow can be considered as a collection of fluid elements or particles that move in the x-y plane.
Relationship between ∂u/∂y and ∂v/∂x
- ∂u/∂y and ∂v/∂x are partial derivatives of the velocity components u and v with respect to the y and x coordinates, respectively.
- If ∂u/∂y = ∂v/∂x, then the flow is said to be irrotational, which means that there is no net rotation of the fluid particles around their own axes.
- This condition is also known as the condition of zero vorticity.
Behavior of fluid element
- If a fluid element undergoes translation, it means that it moves without any deformation or change in shape.
- If a fluid element undergoes rotation, it means that it moves in a circular path around its own axis.
- If a fluid element undergoes deformation, it means that its shape changes while it moves.
Conclusion
- If ∂u/∂y = ∂v/∂x, then the fluid element will undergo translation and deformation.
- This is because the condition of zero vorticity implies that there is no net rotation of the fluid particles, but it does not imply that there is no deformation.
- Therefore, the fluid element will move without any rotation but may undergo deformation due to the non-uniformity of the flow.
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In a two dimensional flow in x-y plane, if ∂u / ∂y = ∂v / ∂x , then the fluid element will undergoa) Translation onlyb) Translation and rotationc) Translation and deformationd) Rotation and deformationCorrect answer is option 'C'. Can you explain this answer?
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