Plannar Jet
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 equations and analytical solution are taken from Analysis of Transport Phenomena, William M. Deen, OUP, 1998.

Off[General::"spell1"]

Numerical solution

sol[ξ_] := 
      NDSolve[{D[f[η], {η, 3}] + 
              2 (f[η] D[
                        f[η], {η, 2}] + (D[
                          f[η], {η, 1}])^2) == 
            0, 
f[0] ==  
            0, f'[0] ==  ξ, f''[0] ==  
            0}, {f[η]}, {η, 0, 10}]

DFEnd[ξ_?NumericQ] := 
      D[f[η] /. sol[ξ], {η, 1}] /. η -> 10

SOL1 = FindRoot[DFEnd[ξ] ==  0, {ξ, 0.5}]

plt1 = 
      Plot[Evaluate[
          D[f[η] /. sol[ξ /. SOL1], {η, 1}]], {η, 0, 10}, 
        PlotStyle -> RGBColor[1, 0, 0], 
        PlotRange -> All]

Determination of parameter b that appears in the analytical solution (in Deen's book) of the velocity (Subscript[v, x])

solt = Solve[b^2 ==  ξ /. SOL1, b]

Analytical solution

plt2 = 
      Plot[(ξ /. SOL1) (1 - 
              Tanh[(Sqrt[ξ/.SOL1 ]) η]^2), {η, 0, 10}, PlotStyle -> RGBColor[0, 0, 1], 
        PlotRange -> All]

Comparing analytical and numerical solutions

Show[plt1, plt2]

Determination of parameter K that appears in the analytical solution (in Deen's book) of the velocity (Subscript[v, x])

K = 
      1/(3  ν) First[
          NIntegrate[
            Evaluate[
              D[f[η] /. sol[ξ /. SOL1], {η, 1}]^2], {η, 0, 
              10}]]

2 (b /. b -> solt[[2, 1, 2]])^3/(9 ν)

2D, contour and density plots of the velocity (Subscript[v, x])(x, y)

vx = 
      First[x^1/
              3/(6 ν x^2/3) (D[
                f[η] /. sol[ξ /. SOL1], {η, 
                  1}] /. η -> 
                Abs[y]/(6  ν x^2/3))]

ν = 0.1;

Plot3D[vx, {x, 1, 2}, {y, -1, 1}]

ContourPlot[vx, {x, 1, 2}, {y, -1, 1}, Contours -> 30, 
      ColorFunction -> ((Hue[
                ArcTan[1.35 - 1.3 #]] &))]

DensityPlot[vx, {x, 1, 2}, {y, -1, 1}, PlotPoints -> 200, 
      Mesh -> False, 
      ColorFunction -> ((Hue[
                ArcTan[1.35 - 1.3  #]] &))]