Which of the following satisfies the Laplace Equation?a)Stream function and Velocity Potential Functionb)Only Velocity Potential functionc)Only Stream functiond)Neither Stream function nor Velocity Potential functionCorrect answer is option 'B'. Can you explain this answer?

Question

Answer

The Laplace equation is a partial differential equation that describes the behavior of many physical phenomena, such as heat flow, electrostatics, and fluid dynamics. In the context of fluid dynamics, the Laplace equation is used to describe the behavior of a fluid flow that is both steady and irrotational.

Velocity Potential Function:

A velocity potential function is a scalar field that describes the velocity of a fluid flow as a function of position. It satisfies the Laplace equation, which means that it describes a fluid flow that is both steady and irrotational.

Stream Function:

A stream function is a scalar field that describes the flow of a fluid in terms of its paths or streamlines. It also satisfies the Laplace equation, which means that it describes a fluid flow that is both steady and irrotational.

Answer:

Option 'B' Only Velocity Potential function satisfies the Laplace Equation. The reason is that the velocity potential function is a scalar field that describes the velocity of a fluid flow as a function of position. It satisfies the Laplace equation, which means that it describes a fluid flow that is both steady and irrotational. The stream function also satisfies the Laplace equation, but it describes the flow of a fluid in terms of its paths or streamlines, not its velocity. Therefore, the correct answer is option 'B'.

Answer

Velocity potential (ϕ) satisfied the Laplace equation only if it represents the flow field. This can be proved in the following way:
By definition of ϕ:
u = - δϕ/δx and v = - δϕ/δy
If ϕ represents the flow field, then it must satisfy the continuity equation i.e.
δu/δx + δv/δy = 0
or
Substitute the values of u and v in the above continuity equation, we get the Laplace Equation.

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Which one of the following is NOT a correct statement?

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