Streamlines for flow past a solid cylinder
Author's Data: Housam BINOUS
Department of Chemical Engineering
National Institute of Applied Sciences and Technology
Tunis, TUNISIA
Email: binoushousam@yahoo.com

Non-rotating solid cylinder : Γ = 0. There are two stagnation points located at r=R and θ=π and 0.

U = 10; R = 1;

r = (Sqrt[x^2 + y^2])

θ = ArcSin[y/r]

ContourPlot[
      U R Sin[θ] ((r/R) - (R/r)), {x, -5, 
        5}, {y, - 5,  5}, 
      Contours -> 10  Table[i, {i, -10, 10, 0.25}], 
      ColorFunction -> ((Hue[ArcTan[1.45 - 1.45 #]] &)), 
      DisplayFunction -> Identity]

Show[%, Graphics[Disk[{0, 0}, 1]], 
      DisplayFunction -> $DisplayFunction]

Rotating solid cylinder : Γ = 4 π R U. There is only one stagnation point located at r=R and θ=-(π/2).

Γ = 4 Pi R U

ContourPlot[
      U R Sin[θ] ((r/R) - (R/
                  r)) + Γ/(2 Pi) Log[r], {x, -5, 
        5}, {y, - 5,  5}, 
      Contours -> 10  Table[i, {i, -10, 10, 0.25}], 
      ColorFunction -> ((Hue[ArcTan[1.45 - 1.45 #]] &)), 
      DisplayFunction -> Identity]

Show[%, Graphics[Disk[{0, 0}, 1]], 
      DisplayFunction -> $DisplayFunction]

Rotating solid cylinder : Γ < 4 π R U. There are two stagnation points located at r=R and θ=((7  π)/6)and -(π/6).

Γ = 2 Pi R U

ContourPlot[
      U R Sin[θ] ((r/R) - (R/
                  r)) + Γ/(2 Pi) Log[r], {x, -5, 
        5}, {y, - 5,  5}, 
      Contours -> 10  Table[i, {i, -10, 10, 0.25}], 
      ColorFunction -> ((Hue[ArcTan[1.45 - 1.45 #]] &)), 
      DisplayFunction -> Identity]

Show[%, Graphics[Disk[{0, 0}, 1]], 
      DisplayFunction -> $DisplayFunction]

Rotating solid cylinder : Γ > 4 π R U. There is only one stagnation located point below the cylinder.

Γ = 6 Pi R U

ContourPlot[
      U R Sin[θ] ((r/R) - (R/
                  r)) + Γ/(2 Pi) Log[r], {x, -5, 
        5}, {y, - 5,  5}, 
      Contours -> 10  Table[i, {i, -10, 10, 0.25}], 
      ColorFunction -> ((Hue[ArcTan[1.45 - 1.45 #]] &)), 
      DisplayFunction -> Identity]

Show[%, Graphics[Disk[{0, 0}, 1]], 
      DisplayFunction -> $DisplayFunction]