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You are asked to evaluate assorted fluid flows for their suitability in a given laboratory application. The following three flow choices, expressed in terms of 2-D velocity field in the xy plane, are made available
P. u = 2y,v = -3x
Q. u = 3xy, v = 0
R. u = -2x , v = 2 y
P. u = 2y,v = -3x
Q. u = 3xy, v = 0
R. u = -2x , v = 2 y
Which flow(s) should be recommended when the application requires the flow to be incompressible and irrotational
- a)P and R
- b)Q
- c)Q and R
- d)R
Correct answer is option 'D'. Can you explain this answer?
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You are asked to evaluate assorted fluid flows for their suitability i...
Answer:
To determine which flow(s) should be recommended for an incompressible and irrotational fluid flow, we need to analyze the given velocity fields and check if they satisfy the conditions for incompressibility and irrotationality.
Incompressible Flow:
An incompressible flow is characterized by the condition of constant density throughout the flow field. Mathematically, this condition is expressed as:
∇ · V = ∂u/∂x + ∂v/∂y + ∂w/∂z = 0
Where ∇ · V represents the divergence of the velocity field V.
Irrotational Flow:
An irrotational flow is characterized by the condition that the fluid particles do not rotate as they move along their paths. Mathematically, this condition is expressed as:
∇ x V = curl(V) = 0
Where ∇ x V represents the curl of the velocity field V.
Now let's analyze each given flow choice:
P. u = 2y, v = -3x
- ∇ · V = ∂u/∂x + ∂v/∂y = 0 + 0 = 0 (satisfies incompressibility condition)
- ∇ x V = ∂v/∂x - ∂u/∂y = 0 + 0 = 0 (satisfies irrotational condition)
Q. u = 3xy, v = 0
- ∇ · V = ∂u/∂x + ∂v/∂y = 3y + 0 = 3y ≠ 0 (does not satisfy incompressibility condition)
- ∇ x V = ∂v/∂x - ∂u/∂y = 0 - 3x = -3x ≠ 0 (does not satisfy irrotational condition)
R. u = -2x, v = 2y
- ∇ · V = ∂u/∂x + ∂v/∂y = -2 + 2 = 0 (satisfies incompressibility condition)
- ∇ x V = ∂v/∂x - ∂u/∂y = 0 - (-2) = 2 (does not satisfy irrotational condition)
Conclusion:
From the analysis, we can see that only flow choice P satisfies both the conditions of incompressibility and irrotationality. Therefore, the recommended flow for the given laboratory application is flow choice P.
To determine which flow(s) should be recommended for an incompressible and irrotational fluid flow, we need to analyze the given velocity fields and check if they satisfy the conditions for incompressibility and irrotationality.
Incompressible Flow:
An incompressible flow is characterized by the condition of constant density throughout the flow field. Mathematically, this condition is expressed as:
∇ · V = ∂u/∂x + ∂v/∂y + ∂w/∂z = 0
Where ∇ · V represents the divergence of the velocity field V.
Irrotational Flow:
An irrotational flow is characterized by the condition that the fluid particles do not rotate as they move along their paths. Mathematically, this condition is expressed as:
∇ x V = curl(V) = 0
Where ∇ x V represents the curl of the velocity field V.
Now let's analyze each given flow choice:
P. u = 2y, v = -3x
- ∇ · V = ∂u/∂x + ∂v/∂y = 0 + 0 = 0 (satisfies incompressibility condition)
- ∇ x V = ∂v/∂x - ∂u/∂y = 0 + 0 = 0 (satisfies irrotational condition)
Q. u = 3xy, v = 0
- ∇ · V = ∂u/∂x + ∂v/∂y = 3y + 0 = 3y ≠ 0 (does not satisfy incompressibility condition)
- ∇ x V = ∂v/∂x - ∂u/∂y = 0 - 3x = -3x ≠ 0 (does not satisfy irrotational condition)
R. u = -2x, v = 2y
- ∇ · V = ∂u/∂x + ∂v/∂y = -2 + 2 = 0 (satisfies incompressibility condition)
- ∇ x V = ∂v/∂x - ∂u/∂y = 0 - (-2) = 2 (does not satisfy irrotational condition)
Conclusion:
From the analysis, we can see that only flow choice P satisfies both the conditions of incompressibility and irrotationality. Therefore, the recommended flow for the given laboratory application is flow choice P.
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You are asked to evaluate assorted fluid flows for their suitability in a given laboratory application. The following three flow choices, expressed in terms of 2-D velocity field in the xy plane, are made availableP. u = 2y,v = -3xQ. u = 3xy, v = 0R. u = -2x , v = 2 yWhich flow(s) should be recommended when the application requires the flow to be incompressible and irrotationala)P and Rb)Qc)Q and Rd)RCorrect answer is option 'D'. Can you explain this answer?
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