I tried to solve the ODE system. But I have a initial conditions are variables. So I want to solve this problem using shooting technique and find the initial values...
(*Define constants*)
phi = 0.5; (*example value,replace with actual values*)
rhoF = 1.0; (*example value*)
rhoS = 1.5; (*example value*)
alpha = 30 Degree; (*example value*)
M = 0.1; (*example value*)
kNF = 1.0; (*example value*)
kF = 0.5; (*example value*)
Pr = 0.7; (*example value*)
CpF = 1.0; (*example value*)
CpS = 1.2; (*example value*)
(*Define the equations*)
eq1 = D[f[eta], {eta,
3}]/((1 - phi)^2.5*((1 - phi)*rhoF + phi*rhoS)) + (1/2)*
D[f[eta], {eta, 2}]*f[eta]*Sin[alpha] + (1/2)*
D[f[eta], {eta, 2}]*eta*Cos[alpha] -
M*(D[f[eta], eta] - 1)/((1 - phi)*rhoF + phi*rhoS) == 0;
eq2 = kNF*D[theta[eta], {eta, 2}]/kF + (1/2)*
Pr*((1 - phi)*rhoF*CpF + phi*rhoS*CpS)*(f[eta]*Sin[alpha] +
eta*Cos[alpha])*D[theta[eta], eta] == 0;
(*Convert to first-order system*)
sys = {x1'[eta] == x2[eta], x2'[eta] == x3[eta],
x3'[eta] == -((1 - phi)^2.5*((1 - phi)*rhoF +
phi*rhoS)*(1/2*
x3[eta]*(x1[eta]*Sin[alpha] + eta*Cos[alpha]) -
M*(x2[eta] - 1))), x4'[eta] == x5[eta],
x5'[eta] == -(kF*
Pr*((1 - phi)*rhoF*CpF +
phi*rhoS*CpS)*(1/2*(x1[eta]*Sin[alpha] + eta*Cos[alpha])*
x5[eta]/kNF))};
(*Initial conditions for the first-order system*)
ics = {x1[0] == 0, x2[0] == 0,
x3[0] == a,(*a should be defined with an appropriate value*)
x4[0] == 1,
x5[0] == b (*b should be defined with an appropriate value*)};