Numerical Solution of the Falkner-Skan 
Equation for various Wedge Angles
Author's Data: Housam BINOUS
Department of Chemical Engineering
National Institute of Applied Sciences and Technology
Tunis, TUNISIA
Email: binoushousam@yahoo.com

Acknowledgement: The governing equation is taken from Analysis of Transport Phenomena, William M. Deen, OUP, 1998.

case 1 : β=0. The Blasius equation is recovered.

sol[ξ_] := 
      NDSolve[{D[f[η], {η, 3}] + 
              f[η] f''[η] + β (1 - (f'[η])^2) == 0, 
          f[0] ==  0, f'[0] ==  
            0, f''[0] ==  ξ}, {f[η]}, {η, 0, 
          5}]

β = 0;

DFEnd[ξ_?NumericQ] := 
      D[f[η] /. sol[ξ], {η, 1}] /. η -> 5

SOL1 = FindRoot[DFEnd[ξ] ==  1, {ξ, 0.5}]

plt1 = 
      Plot[Evaluate[
          D[f[η] /. sol[ξ /. SOL1], {η, 1}]], {η, 0, 5}, 
        PlotStyle -> Hue[1/5], PlotRange -> All]

case 2 : β=0.4

β = 0.4;

DFEnd[ξ_?NumericQ] := 
      D[f[η] /. sol[ξ], {η, 1}] /. η -> 5

SOL2 = FindRoot[DFEnd[ξ] ==  1, {ξ, 0.5}]

plt2 = 
      Plot[Evaluate[
          D[f[η] /. sol[ξ /. SOL2], {η, 1}]], {η, 0, 5}, 
        PlotStyle -> Hue[2/5], PlotRange -> All]

case 3 : β=-0.199

β = -0.199;

sol[ξ_] := 
      NDSolve[{D[f[η], {η, 3}] + 
              f[η] f''[η] + β (1 - (f'[η])^2) == 0, 
          f[0] ==  0, f'[0] ==  
            0, f''[0] ==  ξ}, {f[η]}, {η, 0, 
          5}]

DFEnd[ξ_?NumericQ] := 
      D[f[η] /. sol[ξ], {η, 1}] /. η -> 5

SOL3 = FindRoot[DFEnd[ξ] ==  1, {ξ, 0.5}]

plt3 = 
      Plot[Evaluate[
          D[f[η] /. sol[ξ /. SOL3], {η, 1}]], {η, 0, 5}, 
        PlotStyle -> Hue[3/5], PlotRange -> All]

case 4 : β=1.0

β = 1;

sol[ξ_] := 
      NDSolve[{D[f[η], {η, 3}] + 
              f[η] f''[η] + β (1 - (f'[η])^2) == 0, 
          f[0] ==  0, f'[0] ==  
            0, f''[0] ==  ξ}, {f[η]}, {η, 0, 
          3}]

DFEnd[ξ_?NumericQ] := 
      D[f[η] /. sol[ξ], {η, 1}] /. η -> 3

SOL4 = FindRoot[DFEnd[ξ] ==  1, {ξ, 0.5}]

plt4 = 
      Plot[Evaluate[
          D[f[η] /. sol[ξ /. SOL4], {η, 1}]], {η, 0, 3}, 
        PlotStyle -> Hue[4/5], PlotRange -> All]

case 5 : β=1.6

β = 1.6;

sol[ξ_] := 
      NDSolve[{D[f[η], {η, 3}] + 
              f[η] f''[η] + β (1 - (f'[η])^2) == 0, 
          f[0] ==  0, f'[0] ==  
            0, f''[0] ==  ξ}, {f[η]}, {η, 0, 
          2.2}]

DFEnd[ξ_?NumericQ] := 
      D[f[η] /. sol[ξ], {η, 1}] /. η -> 
          2.2

SOL5 = FindRoot[DFEnd[ξ] ==  1, {ξ, 0.5}]

plt5 = 
      Plot[Evaluate[
          D[f[η] /. sol[ξ /. SOL5], {η, 1}]], {η, 0, 2.2}, 
        PlotStyle -> Hue[5/5], PlotRange -> All]

We obtain the same result as figure 8-10 in page 352 of Deen's book

Show[plt1, plt2, plt3, plt4, plt5, Frame -> True]