Adding boundary condition to dsolve argument

I've used the dsolve function to get a solution for , but I want matlab to solve for such that . That is, I'm trying to add the boundary condition that when r = 0, α = . How can I input this boundary condition into the dsolve argument? I looked through the documentation but couldn't figure it out.

My code is below...

syms F_lift(r) C_lift(r) rho v_rel(r) A_blade r omega alpha(r) v_air P_in T_in

    % r..........effective radius of turbine blade's rotation at a given distance along its length
    % alpha......turbine blade's angle of attack at a given distance along its length
    % F_lift.....lift force generated by turbine blade at a given distance along its length
    % C_lift.....coefficient of lift for turbine airfoil at a given angle of attack
    % v_rel......relative velocity between incoming air and turbine blade at a given distance along its length 
    % rho........mass density of the air after passing through the stator
    % A_blade....surface area of turbine blade coming into contact with the airflow
    % omega......angular velocity of the turbine and shaft
    
  %%%%%%% major assumptions made
    
% Give the generic equation for lift of an airfoil
F_lift = 1/2*C_lift*rho*v_rel^2*A_blade

F_lift(r) = 

% Give C_lift as a function of alpha
C_lift = .12*alpha + .2;

% Give v_rel as a function of v_air and omega
omega = 10471.975511; % rad/s
v_rel = sqrt(v_air^2 + omega^2*r^2);

% Give area of each blade
A_blade = 5.8064e-5; % m^2

% Determine the air density at the turbine inlet using the ideal gas law
T_in = 1023.15; %%%%%%% 750 deg C
P_in = 1100*10^3; %%%%%%% ~10 atm = ~1100 kPa
R = 287.058; %%%%%%% R_air
rho = P_in/(R*T_in);

% Give v_air
v_rel = subs(v_rel,v_air,150); %%%%%%% assumed to be 150 m/s

% Give the new equation
F_lift = 1/2*C_lift*rho*v_rel^2*A_blade

F_lift(r) = 

% Let F_lift be constant along the length of the blade
liftDSTRBTN = diff(F_lift,r) 

liftDSTRBTN(r) = 

alpha = dsolve(liftDSTRBTN == 0) % NEED HELP HERE!

alpha = 

Answer 1

[Cris LaPierre]

I'm no expert here, so take this for what it is worth.

You mention needing to set a condition. Does the Solve Differential Equation with Conditions example help? You can then call dsolve with the conditions you have set.

• S = dsolve(eqn,cond)

If so, your code might look like this. One change I did make was to use alpha(r) everywhere to keep alpha a symfun. It may not matter, as I get the same result either way.

syms F_lift(r) C_lift(r) rho v_rel(r) A_blade r omega alpha(r) v_air P_in T_in
    
% Give the generic equation for lift of an airfoil
F_lift = 1/2*C_lift*rho*v_rel^2*A_blade

F_lift(r) = 

% Give C_lift as a function of alpha
C_lift = .12*alpha(r) + .2;
%            ^^^^^^^^      -----------> used alpha(r) to preserver symfun

% Give v_rel as a function of v_air and omega
omega = 10471.975511; % rad/s
v_rel = sqrt(v_air^2 + omega^2*r^2);

% Give area of each blade
A_blade = 5.8064e-5; % m^2

% Determine the air density at the turbine inlet using the ideal gas law
T_in = 1023.15; %%%%%%% 750 deg C
P_in = 1100*10^3; %%%%%%% ~10 atm = ~1100 kPa
R = 287.058; %%%%%%% R_air
rho = P_in/(R*T_in);

% Give v_air
v_rel = subs(v_rel,v_air,150); %%%%%%% assumed to be 150 m/s

% Give the new equation
F_lift = 1/2*C_lift*rho*v_rel^2*A_blade

F_lift = 

% Let F_lift be constant along the length of the blade
liftDSTRBTN = diff(F_lift,r) 

liftDSTRBTN = 

% Set condition
cond = alpha(0) == pi/2;
%    vvv   -----------> used alpha(r) to preserver symfun
alpha(r) = dsolve(liftDSTRBTN == 0,cond)

alpha(r) = 

You can test the solution at r=0 using subs. It does equal

subs(alpha,r,0)

ans(r) = 

[Caleb Rudick]

This is fantastic, thank you so much!