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What will be the circulation around rectangle defined by x = 0, y = 0, x = 1, y = 1 for a velocity field u = x and v = x + y ?
- a)Infinity
- b)0
- c)1
- d)4
Correct answer is option 'C'. Can you explain this answer?
Most Upvoted Answer
What will be the circulation around rectangle defined by x = 0, y = 0,...
Concept:
Circulation = Vorticity × Area
where,
Vorticity is defined as the value twice of the rotation.
ζ = 2ω
Rotation (ω) =
Now,
∴ Vorticity = ζ =
Calculation:
Given,
Velocity field, u = x, v = x + y
∴ ∂u/∂y = 0, ∂v/∂x = 1
Rectangular Field x = 0, y = 0, x = 1, y = 1 ( i.e Dimension of rectangular are 1 × 1 )
Area of Recatangle = 1 × 1 = 1
Circulation = Vorticity × Area
Circulation= (1 – 0) × 1
∴ Circulation= 1 units
Circulation = Vorticity × Area
where,
Vorticity is defined as the value twice of the rotation.
ζ = 2ω
Rotation (ω) =
Now,
∴ Vorticity = ζ =
Calculation:
Given,
Velocity field, u = x, v = x + y
∴ ∂u/∂y = 0, ∂v/∂x = 1
Rectangular Field x = 0, y = 0, x = 1, y = 1 ( i.e Dimension of rectangular are 1 × 1 )
Area of Recatangle = 1 × 1 = 1
Circulation = Vorticity × Area
Circulation= (1 – 0) × 1
∴ Circulation= 1 units
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What will be the circulation around rectangle defined by x = 0, y = 0,...
Understanding Circulation
Circulation in fluid dynamics is defined as the line integral of the velocity field around a closed curve. For this problem, we need to compute the circulation around the rectangle defined by the vertices (0,0), (1,0), (1,1), and (0,1) for the velocity field given by u = x and v = x + y.
Velocity Field Components
- u = x
- v = x + y
Defining the Path
1. Segment 1 (0,0) to (1,0):
- y = 0, dy = 0
- Integral: ∫(u dx + v dy) = ∫(x dx) from 0 to 1 = [0.5 * x^2] from 0 to 1 = 0.5
2. Segment 2 (1,0) to (1,1):
- x = 1, dx = 0
- Integral: ∫(u dx + v dy) = ∫(1 + y) dy from 0 to 1 = [y + 0.5 * y^2] from 0 to 1 = 1.5
3. Segment 3 (1,1) to (0,1):
- y = 1, dy = 0
- Integral: ∫(u dx + v dy) = ∫(x + 1) dx from 1 to 0 = [-0.5 * x^2 - x] from 1 to 0 = -1.5
4. Segment 4 (0,1) to (0,0):
- x = 0, dx = 0
- Integral: ∫(u dx + v dy) = ∫(0 + y) dy from 1 to 0 = [-0.5 * y^2] from 1 to 0 = -0.5
Calculating Total Circulation
- Total circulation = 0.5 + 1.5 - 1.5 - 0.5 = 1
Thus, the circulation around the rectangle is 1.
Conclusion
The correct answer is option 'C' (1). This illustrates how to calculate circulation using the line integral of a velocity field around a defined path.
Circulation in fluid dynamics is defined as the line integral of the velocity field around a closed curve. For this problem, we need to compute the circulation around the rectangle defined by the vertices (0,0), (1,0), (1,1), and (0,1) for the velocity field given by u = x and v = x + y.
Velocity Field Components
- u = x
- v = x + y
Defining the Path
1. Segment 1 (0,0) to (1,0):
- y = 0, dy = 0
- Integral: ∫(u dx + v dy) = ∫(x dx) from 0 to 1 = [0.5 * x^2] from 0 to 1 = 0.5
2. Segment 2 (1,0) to (1,1):
- x = 1, dx = 0
- Integral: ∫(u dx + v dy) = ∫(1 + y) dy from 0 to 1 = [y + 0.5 * y^2] from 0 to 1 = 1.5
3. Segment 3 (1,1) to (0,1):
- y = 1, dy = 0
- Integral: ∫(u dx + v dy) = ∫(x + 1) dx from 1 to 0 = [-0.5 * x^2 - x] from 1 to 0 = -1.5
4. Segment 4 (0,1) to (0,0):
- x = 0, dx = 0
- Integral: ∫(u dx + v dy) = ∫(0 + y) dy from 1 to 0 = [-0.5 * y^2] from 1 to 0 = -0.5
Calculating Total Circulation
- Total circulation = 0.5 + 1.5 - 1.5 - 0.5 = 1
Thus, the circulation around the rectangle is 1.
Conclusion
The correct answer is option 'C' (1). This illustrates how to calculate circulation using the line integral of a velocity field around a defined path.
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What will be the circulation around rectangle defined by x = 0, y = 0, x = 1, y = 1 for a velocity field u = x and v = x + y ?a)Infinityb)0c)1d)4Correct answer is option 'C'. Can you explain this answer?
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