In other words, the stream function accounts for the solenoidal part of a two-dimensional Helmholtz decomposition, while the velocity potential accounts for the irrotational part.
22.
Consequently, assuming incompressible flow, the velocity field "'u "'is divergence-free and the velocity potential ? satisfies Laplace's equation
23.
The calculation of the velocity potential associated to constant sink / source panels and vortex rings ( constant doublets ) at a given point in space is quite complex.
24.
Observe that for finite depth the velocity potential ? contains a linear drift in time, independent of position ( " x " and " z " ).
25.
While other formulations converge much more rapidly, e . g . in the Eulerian frame of reference ( with the velocity potential or stream function as a function of the spatial coordinates ).
26.
So the velocity in each layer is given by the gradient of a velocity potential, { \ vec { u } = \ nabla \ phi, } and the potential itself satisfies Laplace's equation:
27.
For instance, an inverse formulation of the periodic wave problem as used by Stokes with the spatial coordinates as a function of velocity potential and stream function does not converge for high-amplitude waves.
28.
A flow in a simply-connected domain which is irrotational can be described as a potential flow, through the use of a velocity potential \ Phi, with \ mathbf { u } = \ nabla \ Phi.
29.
According to contemporary scientific theory regarding comparative velocity measurement, all moving objects can be classified into two distinct categories vis-a-vis their velocity potential relative to that of a dog'( Canis lupus familiaris ) '.
30.
Again the oscillatory part of the velocity vector is related to the velocity potential by " " ? " } }, while as before is the Laplace operator, and " " is the average speed of sound in the homogeneous medium.
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