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In a two dimensional flow, the component of the velocity along the X-axis and the Y-axis are u = axy and v = bx2 + cy2. What should be the condition for the flow field to be continuous?
- a)a + b = 0
- b)a + c = 0
- c)a + 2b = 0
- d)a + 2c = 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:
ay + 2cy = 0
a + 2c = 0.
ay + 2cy = 0
a + 2c = 0.
Most Upvoted Answer
In a two dimensional flow, the component of the velocity along the X-a...
Solution:
To determine the continuity of the flow field, we need to check whether the continuity equation is satisfied or not.
Continuity Equation:
∂ρ/∂t + ∂(ρu)/∂x + ∂(ρv)/∂y = 0
where ρ is the density of the fluid, t is time, u and v are the velocity components along x and y directions respectively.
From the given velocity components:
u = axy
v = bx² - cy²
Density (ρ) is not given, but we can assume that it is constant.
Now, let's substitute the velocity components in the continuity equation:
∂ρ/∂t + ∂(ρaxy)/∂x + ∂(ρ(bx²-cy²))/∂y = 0
Simplifying this equation, we get:
ay + 2bxρx - 2cyρy = 0
where ρx and ρy are the partial derivatives of density with respect to x and y respectively.
For the flow to be continuous, the above equation should be satisfied at all points in the flow field. This means that the coefficients of all the variables (a, b, c, ρx, and ρy) should be equal to zero.
Now, let's compare the coefficients:
a = 0 (for continuity, a should not be zero as it is the x and y component of the velocity)
2bxρx = 0 (for continuity, either b or ρx should be zero)
2cyρy = 0 (for continuity, either c or ρy should be zero)
Therefore, the condition for the flow field to be continuous is:
2c = 0 (as c cannot be zero, ρy should be zero)
Hence, the correct answer is option 'D': a = 2c = 0.
To determine the continuity of the flow field, we need to check whether the continuity equation is satisfied or not.
Continuity Equation:
∂ρ/∂t + ∂(ρu)/∂x + ∂(ρv)/∂y = 0
where ρ is the density of the fluid, t is time, u and v are the velocity components along x and y directions respectively.
From the given velocity components:
u = axy
v = bx² - cy²
Density (ρ) is not given, but we can assume that it is constant.
Now, let's substitute the velocity components in the continuity equation:
∂ρ/∂t + ∂(ρaxy)/∂x + ∂(ρ(bx²-cy²))/∂y = 0
Simplifying this equation, we get:
ay + 2bxρx - 2cyρy = 0
where ρx and ρy are the partial derivatives of density with respect to x and y respectively.
For the flow to be continuous, the above equation should be satisfied at all points in the flow field. This means that the coefficients of all the variables (a, b, c, ρx, and ρy) should be equal to zero.
Now, let's compare the coefficients:
a = 0 (for continuity, a should not be zero as it is the x and y component of the velocity)
2bxρx = 0 (for continuity, either b or ρx should be zero)
2cyρy = 0 (for continuity, either c or ρy should be zero)
Therefore, the condition for the flow field to be continuous is:
2c = 0 (as c cannot be zero, ρy should be zero)
Hence, the correct answer is option 'D': a = 2c = 0.
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In a two dimensional flow, the component of the velocity along the X-axis and the Y-axis are u = axy and v = bx2+ cy2. What should be the condition for the flow field to be continuous?a)a + b = 0b)a + c = 0c)a + 2b = 0d)a + 2c = 0Correct answer is option 'D'. Can you explain this answer?
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