Planar two-dimensional impinging streams

For planar two-dimensional impinging streams, the velocity vector, u, can be represented in the well-known way, as 

The equations above can be represented in terms of the velocity potential to yield 

From the definitions for the velocity components, Eqs. (1.9) and (1.10), using Eq. (1.13) and considering that d if/ = nxdv - uydy, one can obtain the expressions for the velocity components below 

The streamlines in two-dimensional impinging streams are shown in Fig. 1.5, the slope of which demonstrates the direction of flow and is equal to the local velocity vector meeting the relationship below  

This relationship indicates that the velocity profile is flat for large x or y while when x and y are of the same order of magnitude, the velocity is minimal at the centerline and increases with x or y increasing. This is indicated in Fig. 1.5. For the case of neglecting viscosity, these results are reasonable, because no shearing force is imposed on the jets. Integrating Eqs. (1.18) and (1.9) leads to the streamlines equation  

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Champion and Libby analyzed both the planar two-dimensional impinging streams and the impingement of two co-axial-cylindrical jets in which the flow is axis symmetrical. Actually, the results they obtained are applicable for both the two cases, provided the two-dimensional coordinates in planar impinging streams are replaced by the cylindrical coordinates. 

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