Question

In a motion velocity potential is single valued,the write the nature of motion​

Answer 1

Answer:

Irrotational

Step-by-step explanation:

A velocity potential is a scalar potential utilized in potential flow theory. It become added with the aid of using Joseph-Louis Lagrange in 1788.

It is utilized in continuum mechanics, while a continuum occupies a simply-related region and is irrotational. In any such case,

                         ${\displaystyle \nabla \times \mathbf {u} =0\,,} {\displaystyle \nabla \times \mathbf {u} =0\,,}$

in which u denotes the flow velocity. As a result, u may be represented because the gradient of a scalar function Φ:

         ${\displaystyle \mathbf {u} =\nabla \Phi \ ={\frac {\partial \Phi }{\partial x}}\mathbf {i} +{\frac {\partial \Phi }{\partial y}}\mathbf {j} +{\frac {\partial \Phi }{\partial z}}\mathbf {k} \,.} {\displaystyle \mathbf {u} =\nabla \Phi \ ={\frac {\partial \Phi }{\partial x}}\mathbf {i} +{\frac {\partial \Phi }{\partial y}}\mathbf {j} +{\frac {\partial \Phi }{\partial z}}\mathbf {k} \,.}$

Φ is known as a velocity potential for u.

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Answer 2

Answer:

The Laplacian of the velocity potential is equal to the divergence of the corresponding flux. Thus, if the velocity potential satisfies Laplace's equation, the flow is incompressible.

Step-by-step explanation:

The velocity potential is a scalar potential used in potential flow theory. It was introduceds by Joseph-Louis Lagrange in 1788.[1]

It is used in continuum mechanics where the continuum occupies a simply connected region and is irrotational. In that case,

{\displaystyle \nabla \times \mathbf {u} =0\,,}where u denotes the flow velocity. Consequently, u can be represented as the gradient of a scalar function Φ:

{\displaystyle \mathbf {u} =\nabla \Phi \ ={\frac {\partial \Phi }{\partial x}}\mathbf {i} +{\frac {\partial \Phi }{\partial y} }\mathbf {j} +{\frac {\partial \Phi }{\partial z}}\mathbf {k} \,.}

{\displaystyle \mathbf {u} =\nabla \Phi \ ={\frac {\partial \Phi }{\partial x}}\mathbf {i} +{\frac {\partial \Phi }{\partial y} }\mathbf {j} +{\frac {\partial \Phi }{\partial z}}\mathbf {k} \,.}

Φ is known as the velocity potential for u.

The speed potential is not unique. If Φ is the rate potential, then Φ + a(t) is also the rate potential for u, where a(t) is a scalar function of time and can be constant. In other words, the rate potentials are unique up to a constant or function of only the time variable.

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