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A velocity potential is not unique. If ϕ is a velocity potential, then ϕ + f(t) is also a velocity potential for u, where f(t) is a scalar function of time and can be constant. Velocity potentials are unique up to a constant, or a function solely of the temporal variable. The Laplacian of a velocity potential is equal to the divergence of the ...
Potential flow describes the velocity field as the gradient of a scalar function: the velocity potential. As a result, a potential flow is characterized by an irrotational velocity field, which is a valid approximation for several applications.
In mathematics, potential flow around a circular cylinder is a classical solution for the flow of an inviscid, incompressible fluid around a cylinder that is transverse to the flow. Far from the cylinder, the flow is unidirectional and uniform. The flow has no vorticity and thus the velocity field is irrotational and can be modeled as a ...
The velocity at all points at a given distance from the source is the same. Fig 2 - Streamlines and potential lines for source flow. The velocity of fluid flow can be given as - ¯ = ^. We can derive the relation between flow rate and velocity of the flow. Consider a cylinder of unit height, coaxial with the source.
Note that the relation of the potential to the flow velocity is unaffected by this transformation: ∇Φ = ∇φ. The Bernoulli equation for unsteady potential flow also appears to play a central role in Luke's variational principle, a variational description of free-surface flows using the Lagrangian mechanics.
A practical example of this type of flow is a bridge pier or a strut placed in a uniform stream. The resulting stream function ( ψ {\displaystyle \psi } ) and velocity potential ( ϕ {\displaystyle \phi } ) are obtained by simply adding the stream function and velocity potential for each individual flow.
so the flow velocity components in relation to the stream function must be =, =. Notice that the stream function is linear in the velocity. Consequently if two incompressible flow fields are superimposed, then the stream function of the resultant flow field is the algebraic sum of the stream functions of the two original fields.
Examples of common boundary conditions include the velocity of the fluid, determined by =, being 0 on the boundaries of the system. There is a great amount of overlap with electromagnetism when solving this equation in general, as the Laplace equation also models the electrostatic potential in a vacuum.