in bernoulli’s theorem which of the following is constant
In Bernoulli’s Theorem, Which of the Following is Constant?
Answer: In Bernoulli’s Theorem, the Bernoulli’s Equation describes the conservation of mechanical energy in a fluid flow and is typically expressed as follows:
P + \frac{1}{2} \rho v^2 + \rho gh = \text{constant}
In this equation:
- P represents the pressure energy per unit volume.
- \rho is the density of the fluid.
- v is the velocity of the fluid flow.
- h denotes the height above a reference point.
- g is the acceleration due to gravity.
Explanation of the Constant Components
Let’s break down the components of the Bernoulli’s Equation:
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Pressure Energy (P): This is the energy due to the pressure exerted by the fluid. It is often referred to as static pressure energy.
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Kinetic Energy Term (\frac{1}{2} \rho v^2): This represents the energy due to the fluid’s motion, known as dynamic pressure or kinetic energy per unit volume.
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Potential Energy Term (\rho gh): This is the potential energy due to the fluid’s elevation. It takes into account the gravitational influence on the fluid.
The Constant in Bernoulli’s Theorem
In a streamline flow (steady, incompressible, and non-viscous fluid flow), the sum of the pressure energy, kinetic energy, and potential energy per unit volume remains constant along a streamline.
Thus, in the context of Bernoulli’s Theorem, the total mechanical energy per unit volume (i.e., the combination of pressure energy, kinetic energy, and potential energy) remains constant:
- Total Mechanical Energy = P + \frac{1}{2} \rho v^2 + \rho gh
Important Assumptions in Bernoulli’s Equation
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Steady Flow: The velocity of fluid particles at any point does not change with time.
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Incompressible Flow: The density of the fluid is constant.
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Non-Viscous Fluid: The fluid has no viscosity (frictionless flow); hence internal resistance does not dissipate mechanical energy.
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Along a Streamline: The theorem holds along a streamline, which is a path traced by a fluid particle following the flow. In parallel streamlines, each streamline would have its own constant value.
Applications of Bernoulli’s Theorem
- Venturi Effect: Used in devices like carburetors and Venturi meters, where a fluid’s pressure is reduced as it flows through a constricted section of pipe.
- Aerofoils: Airplane wings use Bernoulli’s principle, where faster air movement over the wing generates lift.
- Hydraulic Structures: Used in the design of hydraulic structures and predicting the flow of water in pipes and channels.
In summary, the sum of the pressure energy, kinetic energy, and potential energy per unit volume is constant in Bernoulli’s theorem for fluid flow along a streamline, provided the flow adheres to the theorem’s assumptions. This conservation principle underpins many practical engineering applications in fluid dynamics. If you have a specific list of options from which the constant is to be identified, please provide more details, and I would be happy to help further! @username