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The Bernoulli’s constant for points lying on the same streamline and those which lie on other streamlines will have the same value if the flow is
- a)incompressible
- b)Steady
- c)Irrotational
- d)Uniform
Correct answer is option 'C'. Can you explain this answer?
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The Bernoulli’s constant for points lying on the same streamline and ...
Bernoulli’s equation states that the,
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P + 1/2ρV2 + ρgh = c
P is the pressure, r is the density, v is the velocity, h is the elevation difference.
The value of C in the above equation is constant only along a streamline and should essentially vary from streamline to streamline.
The equation can be used to define the relation between flow variables at point B on the streamline and at point A, along the same streamline. So, to apply this equation, one should have knowledge of the velocity field beforehand. This is one of the limitations of the application of Bernoulli's equation.
Therefore, the total mechanical energy remains constant everywhere in inviscid and irrotational flow, while it is constant only along a streamline for inviscid but rotational flow.
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The Bernoulli’s constant for points lying on the same streamline and ...
Understanding Bernoulli’s Principle
Bernoulli's equation states that for an incompressible, non-viscous fluid in steady flow, the sum of pressure energy, kinetic energy per unit volume, and potential energy per unit volume remains constant along a streamline.
Key Concepts
- Incompressible Flow:
- Fluid density remains constant.
- Applies to liquids and gases at low velocities.
- Steady Flow:
- Fluid properties (velocity, pressure) at any given point do not change over time.
- Ensures that Bernoulli’s equation can be applied consistently.
- Irrotational Flow:
- Flow where fluid particles do not rotate about their own axes.
- Allows for the application of Bernoulli’s equation across different streamlines.
- Uniform Flow:
- Velocity is constant across any cross-section of the flow.
- Not necessary for Bernoulli's principle but simplifies analysis.
Why Irrotational Flow Matters
In irrotational flow, the velocity field can be expressed as the gradient of a scalar potential function (known as the velocity potential). This allows for the application of Bernoulli’s equation across different streamlines.
Conclusion
Therefore, the Bernoulli’s constant remains the same for points on separate streamlines if the flow is irrotational. In contrast, for incompressible, steady, or uniform flows, Bernoulli's constant is consistent along a single streamline but may differ between different streamlines. Thus, the correct answer is option 'C' - Irrotational.
Bernoulli's equation states that for an incompressible, non-viscous fluid in steady flow, the sum of pressure energy, kinetic energy per unit volume, and potential energy per unit volume remains constant along a streamline.
Key Concepts
- Incompressible Flow:
- Fluid density remains constant.
- Applies to liquids and gases at low velocities.
- Steady Flow:
- Fluid properties (velocity, pressure) at any given point do not change over time.
- Ensures that Bernoulli’s equation can be applied consistently.
- Irrotational Flow:
- Flow where fluid particles do not rotate about their own axes.
- Allows for the application of Bernoulli’s equation across different streamlines.
- Uniform Flow:
- Velocity is constant across any cross-section of the flow.
- Not necessary for Bernoulli's principle but simplifies analysis.
Why Irrotational Flow Matters
In irrotational flow, the velocity field can be expressed as the gradient of a scalar potential function (known as the velocity potential). This allows for the application of Bernoulli’s equation across different streamlines.
Conclusion
Therefore, the Bernoulli’s constant remains the same for points on separate streamlines if the flow is irrotational. In contrast, for incompressible, steady, or uniform flows, Bernoulli's constant is consistent along a single streamline but may differ between different streamlines. Thus, the correct answer is option 'C' - Irrotational.
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The Bernoulli’s constant for points lying on the same streamline and those which lie on other streamlines will have the same value if the flow isa)incompressibleb)Steadyc)Irrotationald)UniformCorrect answer is option 'C'. Can you explain this answer?
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