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Consider a steady incompressible flow through a channel as shown below. The velocity profile is uniform with a value of uo at the inlet section A. The velocity profile at section B downstream is u=⎧⎩⎨⎪⎪⎪⎪Vmyδ,Vm,VmH−yδ,0≤y≤δδ≤y≤H−δH−δ≤y≤H The ratio pA−pB12ρuo2(where pA and pB are the pressures at section A and B, respectively, and ρ is the density of the fluid) is?
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Consider a steady incompressible flow through a channel as shown below...
**Steady Incompressible Flow through a Channel**
- In this problem, we are considering a steady incompressible flow through a channel. The flow is shown in the diagram below.
**Velocity Profile**
- The velocity profile in the channel is uniform at the inlet section A with a value of uo. However, downstream at section B, the velocity profile varies according to the following conditions:
1. For 0 ≤ y ≤ δ: The velocity is Vm.
2. For δ ≤ y ≤ H-δ: The velocity is Vm(H-y)/δ.
3. For H-δ ≤ y ≤ H: The velocity is 0.
**Pressure Difference between Section A and B**
- We are required to find the ratio pA - pB / 1/2 ρ uo^2, where pA and pB are the pressures at section A and B, respectively, and ρ is the density of the fluid.
- To find this ratio, we need to consider the Bernoulli's equation for incompressible flow:
p + 1/2 ρ u^2 + ρgh = constant
- We can apply this equation at sections A and B to find the pressure difference.
**Applying Bernoulli's Equation at Section A**
- At section A, the velocity is uniform with a value of uo. Therefore, the term 1/2 ρ u^2 can be written as 1/2 ρ uo^2.
- Since the flow is horizontal and there is no change in height, the term ρgh can be neglected.
- Therefore, the Bernoulli's equation at section A becomes:
pA + 1/2 ρ uo^2 = constant
**Applying Bernoulli's Equation at Section B**
- At section B, the velocity varies according to the given velocity profile.
- For 0 ≤ y ≤ δ, the velocity is Vm.
- For δ ≤ y ≤ H-δ, the velocity is Vm(H-y)/δ.
- For H-δ ≤ y ≤ H, the velocity is 0.
- Therefore, the Bernoulli's equation at section B becomes:
pB + 1/2 ρ Vm^2 = constant
**Pressure Difference Calculation**
- By comparing the Bernoulli's equations at section A and B, we can see that the constant terms are the same.
- Therefore, we can subtract the two equations to find the pressure difference:
pA - pB = 1/2 ρ uo^2 - 1/2 ρ Vm^2
- Simplifying the equation further, we get:
pA - pB = 1/2 ρ (uo^2 - Vm^2)
- Finally, the ratio pA - pB / 1/2 ρ uo^2 can be calculated as:
(pA - pB) / (1/2 ρ uo^2) = (uo^2 - Vm^2) / uo^2
**Conclusion**
- The ratio pA - pB / 1/2 ρ uo^2 is (uo^2 - Vm^2) / uo^2, where uo is the velocity at section A and Vm is the velocity at section B according to the given velocity profile
- In this problem, we are considering a steady incompressible flow through a channel. The flow is shown in the diagram below.
**Velocity Profile**
- The velocity profile in the channel is uniform at the inlet section A with a value of uo. However, downstream at section B, the velocity profile varies according to the following conditions:
1. For 0 ≤ y ≤ δ: The velocity is Vm.
2. For δ ≤ y ≤ H-δ: The velocity is Vm(H-y)/δ.
3. For H-δ ≤ y ≤ H: The velocity is 0.
**Pressure Difference between Section A and B**
- We are required to find the ratio pA - pB / 1/2 ρ uo^2, where pA and pB are the pressures at section A and B, respectively, and ρ is the density of the fluid.
- To find this ratio, we need to consider the Bernoulli's equation for incompressible flow:
p + 1/2 ρ u^2 + ρgh = constant
- We can apply this equation at sections A and B to find the pressure difference.
**Applying Bernoulli's Equation at Section A**
- At section A, the velocity is uniform with a value of uo. Therefore, the term 1/2 ρ u^2 can be written as 1/2 ρ uo^2.
- Since the flow is horizontal and there is no change in height, the term ρgh can be neglected.
- Therefore, the Bernoulli's equation at section A becomes:
pA + 1/2 ρ uo^2 = constant
**Applying Bernoulli's Equation at Section B**
- At section B, the velocity varies according to the given velocity profile.
- For 0 ≤ y ≤ δ, the velocity is Vm.
- For δ ≤ y ≤ H-δ, the velocity is Vm(H-y)/δ.
- For H-δ ≤ y ≤ H, the velocity is 0.
- Therefore, the Bernoulli's equation at section B becomes:
pB + 1/2 ρ Vm^2 = constant
**Pressure Difference Calculation**
- By comparing the Bernoulli's equations at section A and B, we can see that the constant terms are the same.
- Therefore, we can subtract the two equations to find the pressure difference:
pA - pB = 1/2 ρ uo^2 - 1/2 ρ Vm^2
- Simplifying the equation further, we get:
pA - pB = 1/2 ρ (uo^2 - Vm^2)
- Finally, the ratio pA - pB / 1/2 ρ uo^2 can be calculated as:
(pA - pB) / (1/2 ρ uo^2) = (uo^2 - Vm^2) / uo^2
**Conclusion**
- The ratio pA - pB / 1/2 ρ uo^2 is (uo^2 - Vm^2) / uo^2, where uo is the velocity at section A and Vm is the velocity at section B according to the given velocity profile
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Consider a steady incompressible flow through a channel as shown below. The velocity profile is uniform with a value of uo at the inlet section A. The velocity profile at section B downstream is u=⎧⎩⎨⎪⎪⎪⎪Vmyδ,Vm,VmH−yδ,0≤y≤δδ≤y≤H−δH−δ≤y≤H The ratio pA−pB12ρuo2(where pA and pB are the pressures at section A and B, respectively, and ρ is the density of the fluid) is?
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Consider a steady incompressible flow through a channel as shown below. The velocity profile is uniform with a value of uo at the inlet section A. The velocity profile at section B downstream is u=⎧⎩⎨⎪⎪⎪⎪Vmyδ,Vm,VmH−yδ,0≤y≤δδ≤y≤H−δH−δ≤y≤H The ratio pA−pB12ρuo2(where pA and pB are the pressures at section A and B, respectively, and ρ is the density of the fluid) is? for Mechanical Engineering 2024 is part of Mechanical Engineering preparation. The Question and answers have been prepared according to the Mechanical Engineering exam syllabus. Information about Consider a steady incompressible flow through a channel as shown below. The velocity profile is uniform with a value of uo at the inlet section A. The velocity profile at section B downstream is u=⎧⎩⎨⎪⎪⎪⎪Vmyδ,Vm,VmH−yδ,0≤y≤δδ≤y≤H−δH−δ≤y≤H The ratio pA−pB12ρuo2(where pA and pB are the pressures at section A and B, respectively, and ρ is the density of the fluid) is? covers all topics & solutions for Mechanical Engineering 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider a steady incompressible flow through a channel as shown below. The velocity profile is uniform with a value of uo at the inlet section A. The velocity profile at section B downstream is u=⎧⎩⎨⎪⎪⎪⎪Vmyδ,Vm,VmH−yδ,0≤y≤δδ≤y≤H−δH−δ≤y≤H The ratio pA−pB12ρuo2(where pA and pB are the pressures at section A and B, respectively, and ρ is the density of the fluid) is?.
Consider a steady incompressible flow through a channel as shown below. The velocity profile is uniform with a value of uo at the inlet section A. The velocity profile at section B downstream is u=⎧⎩⎨⎪⎪⎪⎪Vmyδ,Vm,VmH−yδ,0≤y≤δδ≤y≤H−δH−δ≤y≤H The ratio pA−pB12ρuo2(where pA and pB are the pressures at section A and B, respectively, and ρ is the density of the fluid) is? for Mechanical Engineering 2024 is part of Mechanical Engineering preparation. The Question and answers have been prepared according to the Mechanical Engineering exam syllabus. Information about Consider a steady incompressible flow through a channel as shown below. The velocity profile is uniform with a value of uo at the inlet section A. The velocity profile at section B downstream is u=⎧⎩⎨⎪⎪⎪⎪Vmyδ,Vm,VmH−yδ,0≤y≤δδ≤y≤H−δH−δ≤y≤H The ratio pA−pB12ρuo2(where pA and pB are the pressures at section A and B, respectively, and ρ is the density of the fluid) is? covers all topics & solutions for Mechanical Engineering 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider a steady incompressible flow through a channel as shown below. The velocity profile is uniform with a value of uo at the inlet section A. The velocity profile at section B downstream is u=⎧⎩⎨⎪⎪⎪⎪Vmyδ,Vm,VmH−yδ,0≤y≤δδ≤y≤H−δH−δ≤y≤H The ratio pA−pB12ρuo2(where pA and pB are the pressures at section A and B, respectively, and ρ is the density of the fluid) is?.
Solutions for Consider a steady incompressible flow through a channel as shown below. The velocity profile is uniform with a value of uo at the inlet section A. The velocity profile at section B downstream is u=⎧⎩⎨⎪⎪⎪⎪Vmyδ,Vm,VmH−yδ,0≤y≤δδ≤y≤H−δH−δ≤y≤H The ratio pA−pB12ρuo2(where pA and pB are the pressures at section A and B, respectively, and ρ is the density of the fluid) is? in English & in Hindi are available as part of our courses for Mechanical Engineering.
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