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... 

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. 

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) ... 

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... 

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... 

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