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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 = bxy + ay2. The condition for the flow field to be continuous is
- a)independent of the constants (a; b) but dependent on the variables (x; y)
- b)independent of the variables (x; y) but dependent on the constants (a; b)
- c)independent of both the constants (a; b) and the variables (x; y)
- d)dependent on both the constants (a; b) and the variables (x; y)
Correct answer is option 'A'. 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 + by + 2ay + bx = 0
x + y = 0
Hence, the condition for the flow field to be continuous is independent of the constants (a; b) and dependent only on the variables (x; y).
2ax + by + 2ay + bx = 0
x + y = 0
Hence, the condition for the flow field to be continuous is independent of the constants (a; b) and dependent only on the variables (x; y).
Most Upvoted Answer
In a two dimensional flow, the component of the velocity along the X-a...
Explanation:
To determine if the flow field is continuous, we need to check if the velocity components have the same value at any given point (x, y) in the flow field.
The given velocity components are:
u = ax^2 - bxy
v = bxy - ay^2
To check for continuity, we need to compare the velocity components at any given point (x, y) with the velocity components at a neighboring point (x+dx, y+dy).
Step 1: Compare the u component:
u(x, y) = ax^2 - bxy
u(x+dx, y+dy) = a(x+dx)^2 - b(x+dx)(y+dy)
To check for continuity, we need to compare u(x, y) with u(x+dx, y+dy):
ax^2 - bxy = a(x+dx)^2 - b(x+dx)(y+dy)
Expanding and simplifying the equation:
ax^2 - bxy = ax^2 + 2axdx + adx^2 - bxy - bdyx - bdydx
Canceling out the common terms:
0 = 2axdx + adx^2 - bdyx - bdydx
Step 2: Compare the v component:
v(x, y) = bxy - ay^2
v(x+dx, y+dy) = b(x+dx)(y+dy) - a(y+dy)^2
To check for continuity, we need to compare v(x, y) with v(x+dx, y+dy):
bxy - ay^2 = b(x+dx)(y+dy) - a(y+dy)^2
Expanding and simplifying the equation:
bxy - ay^2 = bxy + bxdy + bydx + bdydy - ay^2 - 2adydy - ady^2
Canceling out the common terms:
0 = bxdy + bydx + bdydy - 2adydy - ady^2
Step 3: Conclusion:
From the equations obtained in Step 1 and Step 2, we can observe that the terms involving the constants (a, b) are not canceled out. This implies that the continuity condition is dependent on the constants (a, b). Therefore, the flow field will be continuous only if the constants (a, b) have specific values.
Hence, the correct answer is option A: The condition for the flow field to be continuous is independent of the constants (a, b) but dependent on the variables (x, y).
To determine if the flow field is continuous, we need to check if the velocity components have the same value at any given point (x, y) in the flow field.
The given velocity components are:
u = ax^2 - bxy
v = bxy - ay^2
To check for continuity, we need to compare the velocity components at any given point (x, y) with the velocity components at a neighboring point (x+dx, y+dy).
Step 1: Compare the u component:
u(x, y) = ax^2 - bxy
u(x+dx, y+dy) = a(x+dx)^2 - b(x+dx)(y+dy)
To check for continuity, we need to compare u(x, y) with u(x+dx, y+dy):
ax^2 - bxy = a(x+dx)^2 - b(x+dx)(y+dy)
Expanding and simplifying the equation:
ax^2 - bxy = ax^2 + 2axdx + adx^2 - bxy - bdyx - bdydx
Canceling out the common terms:
0 = 2axdx + adx^2 - bdyx - bdydx
Step 2: Compare the v component:
v(x, y) = bxy - ay^2
v(x+dx, y+dy) = b(x+dx)(y+dy) - a(y+dy)^2
To check for continuity, we need to compare v(x, y) with v(x+dx, y+dy):
bxy - ay^2 = b(x+dx)(y+dy) - a(y+dy)^2
Expanding and simplifying the equation:
bxy - ay^2 = bxy + bxdy + bydx + bdydy - ay^2 - 2adydy - ady^2
Canceling out the common terms:
0 = bxdy + bydx + bdydy - 2adydy - ady^2
Step 3: Conclusion:
From the equations obtained in Step 1 and Step 2, we can observe that the terms involving the constants (a, b) are not canceled out. This implies that the continuity condition is dependent on the constants (a, b). Therefore, the flow field will be continuous only if the constants (a, b) have specific values.
Hence, the correct answer is option A: The condition for the flow field to be continuous is independent of the constants (a, b) but dependent on the variables (x, y).
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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 = bxy + ay2. The condition for the flow field to be continuous isa)independent of the constants (a; b) but dependent on the variables (x; y)b)independent of the variables (x; y) but dependent on the constants (a; b)c)independent of both the constants (a; b) and the variables (x; y)d)dependent on both the constants (a; b) and the variables (x; y)Correct answer is option 'A'. Can you explain this answer?
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