Potentials coupled three-dimensional problems

Potential functions for two coupled three-dimensional problems... 

In the 1960s, the start of application of computers to the practice of marine research gave a pulse to the development of numerical diagnostic hydrodynamic models [33]. In them, the SLE (or the integral stream function) field is calculated from the three-dimensional density field in the equation of potential vorticity balance over the entire water column from the surface to the bottom. The iterative computational procedure is repeated until a stationary condition of the SLE (or the integral stream function) is reached at the specified fixed density field. Then, from equations of momentum balance, horizontal components of the current vector are obtained, while the continuity equation provides the calculations of the vertical component. The advantage of this approach is related to the absence of the problem of the choice of the zero surface and to the account for the coupled effect of the baroclinicity of... 

Hyperspherical coordinates were introduced by Delves [52] and the formalism of hyperspherical expansion was further developed by many authors [40,53,54] for three-body or more complicated bound states. The usefulness of this method for baryon spectroscopy was shown by several groups [55]. The basic idea is rather simple the two relative coordinates are merged into a single six-dimensional vector. The three-body problem in ordinary space becomes equivalent to a two-body problem in six dimensions, with a noncentral potential. A generalized partial wave expansion leads to an infinite set of coupled radial equations. In practice, however, a very good convergence is achieved with a few partial waves only. 

The HF equations are a system of simultaneous three-dimensional (3D) partial integrodifferential equations since each equation is coupled to the others through the potentials and multipliers. Solution of such a complex system obviously presents formidable problems. However, equation (1) can be... 

One concludes from (12-17a) and (12-17c) that neither 4> nor Vp is a function of the Reynolds number because Re does not appear in either equation. Consequently, dynamic pressure and its gradient in the x direction are not functions of the Reynolds number because Re does not appear in the dimensionless potential flow equation of motion, given by (12-16), from which /dx is calculated. In summary, two-dimensional momentum boundary layer problems in the laminar flow regime (1) focus on the component of the equation of motion in the primary flow direction, (2) use the equation of continuity to calculate the other velocity component transverse to the primary flow direction, (3) use potential flow theory far from a fluid-solid interface to calculate the important component of the dynamic pressure gradient, and (4) impose this pressme gradient across the momentum boundary layer. The following set of dimensionless equations must be solved for Vp, IP, u, and v in sequential order. The first three equations below are solved separately, but the last two equations are coupled ... 

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