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Scalar velocity potential functional form

The boundary condition at large r is employed to calculate the radial and tangential velocity components, as well as the functional form of the scalar velocity potential. Since the velocity vector far from the bubble is... [Pg.211]

Hence, two-dimensional axisymmetric potential flow in spherical coordinates is described by = 0 for the scalar velocity potential and = 0 for the stream function. Recall that two-dimensional axisymmetric creeping viscous flow in spherical coordinates is described by E E ir) = 0. This implies that potential flow solutions represent a subset of creeping viscous flow solutions for two-dimensional axisymmetric problems in spherical coordinates. Also, recall from the boundary condition far from submerged objects that sin 0 is the appropriate Legendre polynomial for the E operator in spherical coordinates. The methodology presented on pages 186 through 188 is employed to postulate the functional form for xlr. [Pg.216]

In summary, Laplace s equation must be satisfied by the scalar velocity potential and the stream function for all two-dimensional planar flows that lack an axis of symmetry. The Laplacian operator is replaced by the operator to calculate the stream function for two-dimensional axisymmetric flows. For potential flow transverse to a long cylinder, vector algebra is required to determine the functional form of the stream function far from the submerged object. This is accomplished from a consideration of Vr and vg via equation (8-255) ... [Pg.220]

Except for the difference between sin 6 and cos0, notice the similarity between this form of Laplace s equation and (8-262) for the scalar velocity potential 4>. In fact, the general solution for the radial part of the stream function is exactly the same as that for from the preceding section. This is expected because and f satisfy the same equation for two-dimensional ideal flows that lack an axis of symmetry. The general solution for is... [Pg.221]

The key feature of the theory of QED—whether it is cast in nonrelativis-tic or fully covariant forms is that the electromagnetic field obeys quantum mechanical laws. A frequent first step in the construction of either version of the theory is the writing of the classical Lagrangian function for the interaction of a charged particle with a radiation field. For a particle of mass m, electronic charge —e, located at position vector q, and moving with velocity d /df c in a position-dependent potential V( ) subject to electromagnetic radiation described by scalar and vector potentials cp0) and a(r), at field point... [Pg.4]


See other pages where Scalar velocity potential functional form is mentioned: [Pg.212]    [Pg.186]    [Pg.379]    [Pg.258]    [Pg.192]   
See also in sourсe #XX -- [ Pg.211 ]




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Scalar

Scalar velocity potential

Velocity form

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