Young Laplace equation with shooting method

Question

I am attempting to follow the method outlined in this paper on page 5, where the first step requires solving a 2 point boundary value problem. I have tried using the Shooting Method, but to no avail.

ClearAll["Global`*"]
Subscript[\[Rho], s] = 8000; Subscript[\[Rho], w] = 1000; g = 9.81; \[Gamma] = 72.8/10^3; R = 50/10^3; Bo = R^2/\[Lambda]^2; Q = Bo*X[\[Psi]] - Sin[\[Psi]]/\[CapitalSigma][\[Psi]]; 
  \[Lambda] = Sqrt[\[Gamma]/(Subscript[\[Rho], w]*g)]; 
zero = 10^(-6); 
\[Omega] = 40*Degree; 
\[Theta] = 100*Degree; 
\[Beta] = \[Theta] - \[Omega]; 
inf = 1; 
H0 = -R/7; sol = NDSolve[{Derivative[1][X][\[Psi]] == Sin[\[Psi]]/Q, Derivative[1][\[CapitalSigma]][\[Psi]] == Cos[\[Psi]]/Q, \[CapitalSigma][zero] == inf, X[zero] == zero}, {X, \[CapitalSigma]}, 
    {\[Psi], \[Beta], 0}, Method -> {"Shooting", "StartingInitialConditions" -> {X[\[Beta]] == H0, \[CapitalSigma][\[Beta]] == Sin[\[Theta]]}}]

It seems that I have specified the boundary condition wrongly. Thank you for any help.

Answer 1

We can use your code as it is with small modification

ClearAll["Global`*"]
Subscript[\[Rho], s] = 8000; 
Subscript[\[Rho], w] = 1000; g = 9.81; \[Gamma] = 72.8/10^3; R = 
 50/10^3; Bo = R^2/\[Lambda]^2; Q = 
 Bo*X[\[Psi]] - Sin[\[Psi]]/\[CapitalSigma][\[Psi]];
\[Lambda] = Sqrt[\[Gamma]/(Subscript[\[Rho], w]*g)];
zero = 10^(-6);
\[Omega] = 40/360 (2  Pi);
\[Theta] = 100/360  2  Pi;
\[Beta] = \[Theta] - \[Omega];
inf = 1;
H0 = -R/7; sol = 
 NDSolve[{Derivative[1][X][\[Psi]] == Sin[\[Psi]]/Q, 
   Derivative[1][\[CapitalSigma]][\[Psi]] == 
    Cos[\[Psi]]/Q, \[CapitalSigma][zero] == inf, 
   X[zero] == zero}, {X, \[CapitalSigma]}, {\[Psi], \[Beta], 0}]

Visualization

{Plot[X[t] /. sol[[1]], {t, 0, 1.05}, AxesLabel -> {"\[Psi]", "X"}], 
 Plot[\[CapitalSigma][t] /. sol[[1]], {t, 0, 1.05}, 
  AxesLabel -> {"\[Psi]", "\[CapitalSigma]"}]}