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Complex Velocity

Equations (5.17), (5.20), and (6.15) imply that

 (6.35)

Consequently, is termed the complex velocity. It follows that

 (6.36)

where is the flow speed.

A stagnation point is defined as a point in a flow pattern at which the flow speed, , falls to zero. (See Section 5.8.) According to the previous expression,

 (6.37)

at a stagnation point. For instance, the stagnation points of the flow pattern produced when a cylindrical obstacle of radius , centered on the origin, is placed in a uniform flow of speed , directed parallel to the -axis, and the circulation of the flow around is cylinder is , are found by setting the derivative of the complex potential (6.32) to zero. It follows that the stagnation points satisfy the quadratic equation

 (6.38)

The solutions are

 (6.39)

where , with the proviso that , because the region is occupied by the cylinder. Thus, if then there are two stagnation points on the surface of the cylinder at and . On the other hand, if then there is a single stagnation point below the cylinder at and .

According to Section 4.15, Bernoulli's theorem in an steady, irrotational, incompressible fluid takes the form

 (6.40)

where is a uniform constant. Here, gravity (and any other body force) has been neglected. Thus, the pressure distribution in such a fluid can be written

 (6.41)

Next: Method of Images Up: Two-Dimensional Potential Flow Previous: Complex Velocity Potential
Richard Fitzpatrick 2016-03-31