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Which of the following velocity relations satisfy the requirements for incompressible irrotational flow,
- u = x + y, v = x - y
- u = xt2 + 2y, v = x2 - yt2
- u = xt2, v = xyt + y2
- a)1 only
- b)2 only
- c)3 only
- d)None of the above
Correct answer is option 'A'. Can you explain this answer?
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Which of the following velocity relations satisfy the requirements for...
Incompressible Irrotational Flow:
Incompressible irrotational flow is a type of fluid flow where the fluid is assumed to be incompressible (i.e., its density remains constant) and irrotational (i.e., the fluid particles rotate about their own axis). In this type of flow, the velocity of the fluid is described by two components, u and v, which are functions of the position coordinates x and y.
Velocity Relations:
Three velocity relations are given in the question, and we need to determine which of them satisfy the requirements for incompressible irrotational flow. Let's analyze each of them.
1. u = x y, v = x - y
To check if this velocity relation satisfies the requirements for incompressible irrotational flow, we need to calculate the partial derivatives of u and v with respect to x and y and check if they are equal. If they are equal, the flow is irrotational.
∂u/∂x = y, ∂v/∂y = -1
∂u/∂y = x, ∂v/∂x = 1
As we can see, ∂u/∂y = ∂v/∂x and ∂u/∂x = -∂v/∂y, which means the flow is irrotational. Also, the divergence of the velocity field is zero, which means the flow is incompressible. Therefore, this velocity relation satisfies the requirements for incompressible irrotational flow.
2. u = xt^2/2y, v = x^2 - yt^2
To check if this velocity relation satisfies the requirements for incompressible irrotational flow, we need to calculate the partial derivatives of u and v with respect to x and y and check if they are equal. If they are equal, the flow is irrotational.
∂u/∂x = t^2/2y, ∂v/∂y = -t^2
∂u/∂y = -xt^2/2y^2, ∂v/∂x = 2x
As we can see, ∂u/∂y ≠ ∂v/∂x and ∂u/∂x ≠ -∂v/∂y, which means the flow is not irrotational. Therefore, this velocity relation does not satisfy the requirements for incompressible irrotational flow.
3. u = xt^2, v = xyt - y^2
To check if this velocity relation satisfies the requirements for incompressible irrotational flow, we need to calculate the partial derivatives of u and v with respect to x and y and check if they are equal. If they are equal, the flow is irrotational.
∂u/∂x = t^2, ∂v/∂y = xt - 2y
∂u/∂y = 0, ∂v/∂x = yt
As we can see, ∂u/∂y ≠ ∂v/∂x and ∂u/∂x ≠ -∂v/∂y, which means the flow is not irrotational. Therefore, this velocity relation does not satisfy the requirements for incompress
Incompressible irrotational flow is a type of fluid flow where the fluid is assumed to be incompressible (i.e., its density remains constant) and irrotational (i.e., the fluid particles rotate about their own axis). In this type of flow, the velocity of the fluid is described by two components, u and v, which are functions of the position coordinates x and y.
Velocity Relations:
Three velocity relations are given in the question, and we need to determine which of them satisfy the requirements for incompressible irrotational flow. Let's analyze each of them.
1. u = x y, v = x - y
To check if this velocity relation satisfies the requirements for incompressible irrotational flow, we need to calculate the partial derivatives of u and v with respect to x and y and check if they are equal. If they are equal, the flow is irrotational.
∂u/∂x = y, ∂v/∂y = -1
∂u/∂y = x, ∂v/∂x = 1
As we can see, ∂u/∂y = ∂v/∂x and ∂u/∂x = -∂v/∂y, which means the flow is irrotational. Also, the divergence of the velocity field is zero, which means the flow is incompressible. Therefore, this velocity relation satisfies the requirements for incompressible irrotational flow.
2. u = xt^2/2y, v = x^2 - yt^2
To check if this velocity relation satisfies the requirements for incompressible irrotational flow, we need to calculate the partial derivatives of u and v with respect to x and y and check if they are equal. If they are equal, the flow is irrotational.
∂u/∂x = t^2/2y, ∂v/∂y = -t^2
∂u/∂y = -xt^2/2y^2, ∂v/∂x = 2x
As we can see, ∂u/∂y ≠ ∂v/∂x and ∂u/∂x ≠ -∂v/∂y, which means the flow is not irrotational. Therefore, this velocity relation does not satisfy the requirements for incompressible irrotational flow.
3. u = xt^2, v = xyt - y^2
To check if this velocity relation satisfies the requirements for incompressible irrotational flow, we need to calculate the partial derivatives of u and v with respect to x and y and check if they are equal. If they are equal, the flow is irrotational.
∂u/∂x = t^2, ∂v/∂y = xt - 2y
∂u/∂y = 0, ∂v/∂x = yt
As we can see, ∂u/∂y ≠ ∂v/∂x and ∂u/∂x ≠ -∂v/∂y, which means the flow is not irrotational. Therefore, this velocity relation does not satisfy the requirements for incompress
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Which of the following velocity relations satisfy the requirements for...
To check for incompressible flow use continuity equation,
1) ∂u/∂x = 1, ∂v/∂y = -1
∴ ∂u/∂x + ∂v/∂y = 0 ,the flow is incompressible
2) ∂u/∂x = t2,∂v/∂y = -t2
∴ ∂u/∂x + ∂v/∂y = 0, the flow is incompressible
3) ∂u/∂x = t2, ∂v/∂y = xt + 2y
∴ ∂u/∂x + ∂v/∂y ≠ 0 ,the flow is not incompressible
To check for irrotational flow,∂u/∂y - ∂v/∂x = 0
1) ∂u/∂y = 1,∂v/∂x = 1, flow is irrotational.
2) ∂u/∂y = 2,∂v/∂x = 2x, flow is not irrotational
3) ∂u/∂y =0,∂v/∂x = yt, flow is not irrotational
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Which of the following velocity relations satisfy the requirements for incompressible irrotational flow, u = x + y, v = x - y u = xt2 + 2y, v = x2 - yt2 u = xt2, v = xyt + y2a)1 onlyb)2 onlyc)3 onlyd)None of the aboveCorrect answer is option 'A'. Can you explain this answer?
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