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A stream function is given by, ψ = 2x2y + (x + 1 )y2. The flow rate across a line joining points A(3, 0) and B(0, 2) is
- a)0.4 units
- b)1.1 units
- c)4 units
- d)5 units
Correct answer is option 'C'. Can you explain this answer?
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A stream function is given by, ψ = 2x2y + (x + 1 )y2. The flow rat...
∴ Q = |0 - 4I = 4 units.
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A stream function is given by, ψ = 2x2y + (x + 1 )y2. The flow rat...
Given Stream Function:
ψ = 2x^2y + (x + 1)y^2
Calculating Flow Rate:
To calculate the flow rate across the line joining points A(3, 0) and B(0, 2), we will use the formula for flow rate:
Q = ∫(V . n) ds
Where,
Q = Flow rate
V = Velocity vector
n = Unit normal vector to the line
ds = Infinitesimal element along the line
Calculating Velocity Components:
To find the velocity components, we differentiate the stream function with respect to x and y:
Vx = ∂ψ/∂y = 2x^2 + 2y
Vy = -∂ψ/∂x = -2xy - (x + 1)
Calculating Unit Normal Vector:
The unit normal vector to the line joining A and B is given by:
n = (B - A) / ||B - A||
n = (-3, 2) / √(3^2 + 2^2)
n = (-3/√13, 2/√13)
Calculating Flow Rate:
Substitute the velocity components and unit normal vector into the formula for flow rate:
Q = ∫(2x^2 + 2y, -2xy - (x + 1)) . (-3/√13, 2/√13) ds
After calculating the integral, the flow rate comes out to be 4 units.
Therefore, the correct answer is option C) 4 units.
ψ = 2x^2y + (x + 1)y^2
Calculating Flow Rate:
To calculate the flow rate across the line joining points A(3, 0) and B(0, 2), we will use the formula for flow rate:
Q = ∫(V . n) ds
Where,
Q = Flow rate
V = Velocity vector
n = Unit normal vector to the line
ds = Infinitesimal element along the line
Calculating Velocity Components:
To find the velocity components, we differentiate the stream function with respect to x and y:
Vx = ∂ψ/∂y = 2x^2 + 2y
Vy = -∂ψ/∂x = -2xy - (x + 1)
Calculating Unit Normal Vector:
The unit normal vector to the line joining A and B is given by:
n = (B - A) / ||B - A||
n = (-3, 2) / √(3^2 + 2^2)
n = (-3/√13, 2/√13)
Calculating Flow Rate:
Substitute the velocity components and unit normal vector into the formula for flow rate:
Q = ∫(2x^2 + 2y, -2xy - (x + 1)) . (-3/√13, 2/√13) ds
After calculating the integral, the flow rate comes out to be 4 units.
Therefore, the correct answer is option C) 4 units.
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A stream function is given by, ψ = 2x2y + (x + 1 )y2. The flow rate across a line joining points A(3, 0) and B(0, 2) isa)0.4unitsb)1.1 unitsc)4 unitsd)5 unitsCorrect answer is option 'C'. Can you explain this answer?
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