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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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Most Upvoted Answer
In a two dimensional flow, the component of the velocity along the X-a...
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).
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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 consider the continuity equation, which states that the rate of mass flow into a control volume must be equal to the rate of mass flow out of the control volume. In two-dimensional flow, this equation can be expressed as:
∂(ρu)/∂x + ∂(ρv)/∂y = 0
where ρ is the density of the fluid.
Now let's substitute the given expressions for u and v into the continuity equation and see if we can simplify it.
1. Substitute u = ax^2 + bxy and v = bxy + ay^2 into the continuity equation:
∂(ρ(ax^2 + bxy))/∂x + ∂(ρ(bxy + ay^2))/∂y = 0
2. Expand the derivatives:
2ax(ρa + ρbx) + 2by(ρb + ρay) = 0
3. Factor out the common terms:
2(ρa + ρbx)(ax + by) = 0
4. Equate each factor to zero:
ρa + ρbx = 0 (1)
ax + by = 0 (2)
Now let's analyze each factor separately.
Condition for (ρa + ρbx) = 0:
The factor ρa + ρbx represents a linear combination of constants (a and b) and variables (x and y). For this factor to be equal to zero, it means that the coefficients of the constants and variables must sum up to zero. Therefore, this condition is independent of the constants (a and b) but dependent on the variables (x and y).
Condition for (ax + by) = 0:
The factor ax + by represents a linear combination of variables (x and y) only. For this factor to be equal to zero, it means that the coefficients of x and y must sum up to zero. Therefore, this condition is independent of the constants (a and b) but dependent on the variables (x and y).
Conclusion:
Since both factors in the continuity equation have conditions that are independent of the constants (a and b) but dependent on the variables (x and y), the condition for the flow field to be continuous is independent of the constants (a and b) but dependent on the variables (x and y). Therefore, the correct answer is option 'A'.
∂(ρu)/∂x + ∂(ρv)/∂y = 0
where ρ is the density of the fluid.
Now let's substitute the given expressions for u and v into the continuity equation and see if we can simplify it.
1. Substitute u = ax^2 + bxy and v = bxy + ay^2 into the continuity equation:
∂(ρ(ax^2 + bxy))/∂x + ∂(ρ(bxy + ay^2))/∂y = 0
2. Expand the derivatives:
2ax(ρa + ρbx) + 2by(ρb + ρay) = 0
3. Factor out the common terms:
2(ρa + ρbx)(ax + by) = 0
4. Equate each factor to zero:
ρa + ρbx = 0 (1)
ax + by = 0 (2)
Now let's analyze each factor separately.
Condition for (ρa + ρbx) = 0:
The factor ρa + ρbx represents a linear combination of constants (a and b) and variables (x and y). For this factor to be equal to zero, it means that the coefficients of the constants and variables must sum up to zero. Therefore, this condition is independent of the constants (a and b) but dependent on the variables (x and y).
Condition for (ax + by) = 0:
The factor ax + by represents a linear combination of variables (x and y) only. For this factor to be equal to zero, it means that the coefficients of x and y must sum up to zero. Therefore, this condition is independent of the constants (a and b) but dependent on the variables (x and y).
Conclusion:
Since both factors in the continuity equation have conditions that are independent of the constants (a and b) but dependent on the variables (x and y), the condition for the flow field to be continuous is independent of the constants (a and b) but dependent on the variables (x and y). Therefore, the correct answer is option 'A'.
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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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