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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 + cy2 and v = cxy. What should be the condition for the flow field to be continuous?
- a)a + c = 0
- b)b + c = 0
- c)2a + c = 0
- d)2b + c = 0
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
In a two dimensional flow, the component of the velocity along the X-a...
Explanation: According to the condition for continuity,
2ax + cx = 0
2a + c = 0.
2ax + cx = 0
2a + c = 0.
Most Upvoted Answer
In a two dimensional flow, the component of the velocity along the X-a...
To determine the condition for the flow field to be continuous, we need to analyze the velocity components along the X-axis and the Y-axis.
Given:
u = ax^2 + bxy + cy^2
v = cxy
To have a continuous flow field, the velocity components must satisfy the following condition:
1. Continuity Equation:
The continuity equation states that the divergence of the velocity field must be zero, i.e., ∇·V = 0. In two dimensions, this equation can be written as:
∂u/∂x + ∂v/∂y = 0
Substituting the given velocity components into the continuity equation:
∂(ax^2 + bxy + cy^2)/∂x + ∂(cxy)/∂y = 0
Differentiating with respect to x and y:
2ax + by + 0 = 0
bx + 2cy + 0 = 0
2. Condition for Continuous Flow:
For the flow field to be continuous, the coefficients of x and y in both equations must be zero simultaneously. This is because if either of the coefficients is non-zero, the flow will have a jump or discontinuity at that point.
From the first equation, we have:
2a = 0 ----(1)
From the second equation, we have:
b = 0 ----(2)
Multiplying equation (2) by 2, we get:
2b = 0 ----(3)
Comparing equations (1) and (3), we can see that the condition for a continuous flow field is:
2a = 2b = 0
Therefore, the correct option is (c) 2a * c = 0.
This means that the coefficient of x in the velocity component u and the coefficient of y in the velocity component v should both be zero for the flow field to be continuous.
Given:
u = ax^2 + bxy + cy^2
v = cxy
To have a continuous flow field, the velocity components must satisfy the following condition:
1. Continuity Equation:
The continuity equation states that the divergence of the velocity field must be zero, i.e., ∇·V = 0. In two dimensions, this equation can be written as:
∂u/∂x + ∂v/∂y = 0
Substituting the given velocity components into the continuity equation:
∂(ax^2 + bxy + cy^2)/∂x + ∂(cxy)/∂y = 0
Differentiating with respect to x and y:
2ax + by + 0 = 0
bx + 2cy + 0 = 0
2. Condition for Continuous Flow:
For the flow field to be continuous, the coefficients of x and y in both equations must be zero simultaneously. This is because if either of the coefficients is non-zero, the flow will have a jump or discontinuity at that point.
From the first equation, we have:
2a = 0 ----(1)
From the second equation, we have:
b = 0 ----(2)
Multiplying equation (2) by 2, we get:
2b = 0 ----(3)
Comparing equations (1) and (3), we can see that the condition for a continuous flow field is:
2a = 2b = 0
Therefore, the correct option is (c) 2a * c = 0.
This means that the coefficient of x in the velocity component u and the coefficient of y in the velocity component v should both be zero for the flow field to be continuous.
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