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In a two dimensional flow, the component of the velocity along the X-axis and the Y-axis are u = ax2 + bxy and v = cxy +dy2. What should be the condition for the flow field to be continuous?
- a)(a + b)x + (c + d)y = 0
- b)(a + c)x + (b + d)y = 0
- c)(2a + b)x + (c + 2d)y = 0
- d)(2a + c)x + (b + 2d)y = 0
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
In a two dimensional flow, the component of the velocity along the X-a...
Explanation: The condition for the flow field to be continuous is:
2ax + cx + by + 2dy = 0
(2a + c)x + (b + 2d)y = 0.
2ax + cx + by + 2dy = 0
(2a + c)x + (b + 2d)y = 0.
Most Upvoted Answer
In a two dimensional flow, the component of the velocity along the X-a...
Understanding Continuity in Fluid Flow
In fluid mechanics, the continuity equation ensures the conservation of mass in a flow field. For a two-dimensional flow defined by velocity components u and v, the flow field is continuous if the divergence of the velocity field is zero.
Velocity Components
- The velocity components are given as:
- u = ax^2 + bxy
- v = cxy + dy^2
Divergence of Velocity Field
- The divergence of the velocity field (∇·V) in two-dimensional flow is calculated as follows:
- ∇·V = (∂u/∂x) + (∂v/∂y)
Calculating Partial Derivatives
- For u:
- ∂u/∂x = 2ax + by
- For v:
- ∂v/∂y = cx + 2dy
Setting Up the Continuity Condition
- Therefore, the total divergence is:
- ∇·V = (2ax + by) + (cx + 2dy)
- For continuity, we set this equal to zero:
- (2a + c)x + (b + 2d)y = 0
Identifying the Correct Condition
- The condition for the flow field to be continuous simplifies to:
- (2a + c)x + (b + 2d)y = 0
- After examining the options provided, we see that option 'D' corresponds to this condition:
- (2a + c)x + (b + 2d)y = 0
Conclusion
- The continuity condition ensures that mass is conserved in the flow, and option 'D' correctly represents this requirement. Thus, ensuring that the fluid flow is continuous in the given context.
In fluid mechanics, the continuity equation ensures the conservation of mass in a flow field. For a two-dimensional flow defined by velocity components u and v, the flow field is continuous if the divergence of the velocity field is zero.
Velocity Components
- The velocity components are given as:
- u = ax^2 + bxy
- v = cxy + dy^2
Divergence of Velocity Field
- The divergence of the velocity field (∇·V) in two-dimensional flow is calculated as follows:
- ∇·V = (∂u/∂x) + (∂v/∂y)
Calculating Partial Derivatives
- For u:
- ∂u/∂x = 2ax + by
- For v:
- ∂v/∂y = cx + 2dy
Setting Up the Continuity Condition
- Therefore, the total divergence is:
- ∇·V = (2ax + by) + (cx + 2dy)
- For continuity, we set this equal to zero:
- (2a + c)x + (b + 2d)y = 0
Identifying the Correct Condition
- The condition for the flow field to be continuous simplifies to:
- (2a + c)x + (b + 2d)y = 0
- After examining the options provided, we see that option 'D' corresponds to this condition:
- (2a + c)x + (b + 2d)y = 0
Conclusion
- The continuity condition ensures that mass is conserved in the flow, and option 'D' correctly represents this requirement. Thus, ensuring that the fluid flow is continuous in the given context.
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In a two dimensional flow, the component of the velocity along the X-axis and the Y-axis are u = ax2+ bxy and v = cxy +dy2. What should be the condition for the flow field to be continuous?a)(a + b)x + (c + d)y = 0b)(a + c)x + (b + d)y = 0c)(2a + b)x + (c + 2d)y = 0d)(2a + c)x + (b + 2d)y = 0Correct answer is option 'D'. Can you explain this answer?
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In a two dimensional flow, the component of the velocity along the X-axis and the Y-axis are u = ax2+ bxy and v = cxy +dy2. What should be the condition for the flow field to be continuous?a)(a + b)x + (c + d)y = 0b)(a + c)x + (b + d)y = 0c)(2a + b)x + (c + 2d)y = 0d)(2a + c)x + (b + 2d)y = 0Correct answer is option 'D'. Can you explain this answer? for Civil Engineering (CE) 2024 is part of Civil Engineering (CE) preparation. The Question and answers have been prepared according to the Civil Engineering (CE) exam syllabus. Information about In a two dimensional flow, the component of the velocity along the X-axis and the Y-axis are u = ax2+ bxy and v = cxy +dy2. What should be the condition for the flow field to be continuous?a)(a + b)x + (c + d)y = 0b)(a + c)x + (b + d)y = 0c)(2a + b)x + (c + 2d)y = 0d)(2a + c)x + (b + 2d)y = 0Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for Civil Engineering (CE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In a two dimensional flow, the component of the velocity along the X-axis and the Y-axis are u = ax2+ bxy and v = cxy +dy2. What should be the condition for the flow field to be continuous?a)(a + b)x + (c + d)y = 0b)(a + c)x + (b + d)y = 0c)(2a + b)x + (c + 2d)y = 0d)(2a + c)x + (b + 2d)y = 0Correct answer is option 'D'. Can you explain this answer?.
In a two dimensional flow, the component of the velocity along the X-axis and the Y-axis are u = ax2+ bxy and v = cxy +dy2. What should be the condition for the flow field to be continuous?a)(a + b)x + (c + d)y = 0b)(a + c)x + (b + d)y = 0c)(2a + b)x + (c + 2d)y = 0d)(2a + c)x + (b + 2d)y = 0Correct answer is option 'D'. Can you explain this answer? for Civil Engineering (CE) 2024 is part of Civil Engineering (CE) preparation. The Question and answers have been prepared according to the Civil Engineering (CE) exam syllabus. Information about In a two dimensional flow, the component of the velocity along the X-axis and the Y-axis are u = ax2+ bxy and v = cxy +dy2. What should be the condition for the flow field to be continuous?a)(a + b)x + (c + d)y = 0b)(a + c)x + (b + d)y = 0c)(2a + b)x + (c + 2d)y = 0d)(2a + c)x + (b + 2d)y = 0Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for Civil Engineering (CE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In a two dimensional flow, the component of the velocity along the X-axis and the Y-axis are u = ax2+ bxy and v = cxy +dy2. What should be the condition for the flow field to be continuous?a)(a + b)x + (c + d)y = 0b)(a + c)x + (b + d)y = 0c)(2a + b)x + (c + 2d)y = 0d)(2a + c)x + (b + 2d)y = 0Correct answer is option 'D'. Can you explain this answer?.
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