Fluid mechanics problem concerning flow over a wedge

Question

Below is the code I am using to generate a velocity profile and temperature profile for flow over a wedge.

sol[ξ_] := 
  NDSolve[{
   D[f[η], {η, 3}] + (m + 1)/2 f[η] f''[η] + m* (1 - (f'[η])^2) == 0,
   D[θ[η], {η, 2}] + f[η] θ'[η]*(m + 1)*0.5*0.7 == 0,
   θ[0] ==  1, θ[5] == 0, f[0] == -2/(m + 1)*bsp, 
   f'[0] == 0, 
   f''[0] == ξ}, 
   {f[η], θ[η], θ'[η]}, {η, 0, 9}] 

m = β/(2 π - β);
m = 0;
DFEnd[ξ_?NumericQ] := D[f[η] /. sol[ξ], {η, 1}] /. η -> 9
SOL1[bsp_] := FindRoot[DFEnd[ξ] == 1, {ξ, 0.5}]

plt1 = 
  Plot[Evaluate[
        Flatten[Table[D[f[η] /. sol[ξ /. SOL1[bsp]], {η, 1}], {bsp, 0, -1, -0.25}]] /. 
          η -> x], 
    {x, 0, 9},  
    PlotLegends -> Range[BSP = 0, BSP = -1, BSP = -0.25], 
    PlotRange -> All]
plt2 = 
  Plot[Evaluate[
         Flatten[Table[D[θ[η] /. sol[ξ /. SOL1[bsp]], {η,  1}], {bsp, 0, -1, -0.25}]] /. 
           η -> x], 
    {x, 0, 5}, 
    PlotLegends -> Range[0, 1, .25], 
    PlotRange -> All]

Here is how I am generating values of f''[0], varying the bsp parameter for various values of m.

I wish to generate a table having values of η for which f'[η] is equal to 0.99 for different values of bsp for a given m, and then I wish to vary m as well.

Let's say, for example, I first want a table for m = 0, where bsp ranges over {-2.5,2.5,0.5}; then m = 1, etc.

Is there a way to generate such a table for f'[η] == 0.99 and similarly θ'[0] for different values of m and bsp.

Answer 1

You mention two parameters, m and bsp, but not the third ξ. I will concentrate on solving f'[η] == 0.99 for ξ == 0.2. Use ParametricNDSolveValue to set up the differential equations, and use WhenEvent to solve any ancillary equations that you wish, for example f'[η] == 0.99. If you use Sow as I did, you need to turn the caching off, or the root will be sown only on the first call for the given parameters.

Clear[m, bsp, ξ, f, η];
sol = ParametricNDSolveValue[
  {D[f[η], {η, 3}] + (m + 1)/2 f[η] f''[η] + m*(1 - (f'[η])^2) == 0,
   D[θ[η], {η, 2}] + f[η] θ'[η]*(m + 1)*0.5*0.7 == 0,
   θ[0] == 1, θ[5] == 0,
   f[0] == -2/(m + 1)*bsp, f'[0] == 0, f''[0] == ξ,
   WhenEvent[f'[η] == 0.99, Sow[η]]},
  {f, θ, Derivative[1][θ]},
  {η, 0, 9},
  {m, bsp, ξ},
  Method -> {"ParametricCaching" -> None}]

Flatten@Table[Last@Reap[sol[0, bsp, 0.2]] /. {} -> None, {bsp, -2.5, 2.5, 0.5}]
(*
  {None, None, None, None, None, None, 2.70527, 1.83437, 1.44146, 1.20493, 1.04352}
*)

If you don't want to use Sow, the equation may be solved after getting the solution to the differential equation like this:

With[{f = First[sol[0, 1., 0.2]]},
 FindRoot[
  f'[η] == 0.99, {η, 1}]
 ]
(*
  {η -> 1.83437}
*)