# ContourPlot of a polar flow field (complex function)

I am currently trying to contourplot a dimensionless velocity-field given as:

W =  1/\[Alpha] Im[Sec[Cos[-(1/2) q I Log[10, (2 \[Alpha] \[CapitalTheta])/q]]/
Cos[\[Alpha]]] + 1/2 I q Log[10, (2 \[Alpha] \[CapitalTheta])/q]]


with

\[Alpha] -> \[Pi]/8, q -> 1, a -> 10/10^6


and the complex polar coordinate \[CapitalTheta] -> a R E^(I \[CurlyPhi]) with {R, 0, 4/\[Pi]}, {\[CurlyPhi], 0, 2 \[Pi]} My problem now is to generate the ContourPlot of the function. I know how the result should look like (at least similar)

I have already tried a coordinate transformation and much more, but unfortunately that did not work either.

I´d be happy for every bit of support. Thanks and best regards, Seb

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With the proper replacements, I get something different from your solution:

W = 1/\[Alpha] Im[
Sec[Cos[-(1/2) q I Log[10, (2 \[Alpha] \[CapitalTheta])/q]]/
Cos[\[Alpha]]] +
1/2 I q Log[
10, (2 \[Alpha] \[CapitalTheta])/q]] /. {\[Alpha] -> \[Pi]/8,
q -> 1, \[CapitalTheta] -> a R E^(I \[CurlyPhi])} /. {a ->
10/10^6};

ContourPlot[W, {R, 0, 4/\[Pi]}, {\[CurlyPhi], 0, 2 \[Pi]},
FrameLabel -> {"X", "Y"}]


• Hey, without transforming the polar coordinates into cartesian coordinates, I get the same result as you.
– SebZ
Jan 26, 2022 at 14:16