- #1
stedwards
- 416
- 46
I would like to obtain the conformal map from a uniform rectilinear fluid flowing in the x-direction, where the field is bounded below by the x-axis, to the flow in the w-plane.
In the w-plane the flow is correspondingly bounded from below by a trochoid. (A trochoid is a continuous waveform shaped something a sine wave but with pointier tops.)
With
the trochoid boundary is given parametrically as
where a is greater than b.
But how do I map the x-axis to the trochoid? There seem to be an infinity of maps. How do I select the correct one?
This problem seems to be isomorphic to finding the electric field and equipotentials of a charged trochoid shaped conductor with the return conductor at [itex]y=\inf[/itex].
In the w-plane the flow is correspondingly bounded from below by a trochoid. (A trochoid is a continuous waveform shaped something a sine wave but with pointier tops.)
With
[itex]z=x+iy[/itex]
[itex]w=u+iv[/itex],
[itex]w=u+iv[/itex],
the trochoid boundary is given parametrically as
[itex]u=a \theta – b sin \theta[/itex]
[itex]v= -a + b cos \theta[/itex]
[itex]v= -a + b cos \theta[/itex]
where a is greater than b.
But how do I map the x-axis to the trochoid? There seem to be an infinity of maps. How do I select the correct one?
This problem seems to be isomorphic to finding the electric field and equipotentials of a charged trochoid shaped conductor with the return conductor at [itex]y=\inf[/itex].