clear
clc
close all
% Define global variable
global Pr
% Set the Prandtl number and solve for different cases
Pr = 1;
options = optimset('TolFun', 1e-6, 'TolX', 1e-6);
a = fsolve(@big_Pr, [0.5 -0.6], options);
x0 = [0 0 a(1) 1 a(2)];
[t, Y] = ode15s(@governing_equation, [0 4], x0);
plot_results(t, Y, 'r', 'Pr = 1');
Pr = 100;
a = fsolve(@big_Pr, [0.25 -0.7], options);
x0 = [0 0 a(1) 1 a(2)];
[t, Y] = ode15s(@governing_equation, [0 4], x0);
plot_results(t, Y, 'b', 'Pr = 100');
Pr = 1000;
a = fsolve(@big_Pr, [0.1 -0.65], options);
x0 = [0 0 a(1) 1 a(2)];
[t, Y] = ode15s(@governing_equation, [0 4], x0);
plot_results(t, Y, 'k', 'Pr = 1000');
Pr = 10;
a = fsolve(@big_Pr, [0.5 -0.52], options);
x0 = [0 0 a(1) 1 a(2)];
[t, Y] = ode15s(@governing_equation, [0 4], x0);
plot_results(t, Y, 'g', 'Pr = 10');
Pr = 0.01;
a = fsolve(@small_Pr, [0.15 -0.17], options);
x0 = [0 0 a(1) 1 a(2)];
[t, Y] = ode15s(@governing_equation, [0 4], x0);
plot_results(t, Y, 'y', 'Pr = 0.01');
% Functions
function f = big_Pr(a)
x0 = [0 0 a(1) 1 a(2)];
[t, Y] = ode15s(@governing_equation, [0 7], x0);
f(1) = Y(end, 2);
f(2) = Y(end, 4)^2 + Y(end, 5)^2;
end
function f = small_Pr(a)
x0 = [0 0 a(1) 1 a(2)];
[t, Y] = ode15s(@governing_equation, [0 30], x0);
f(1) = Y(end, 2);
f(2) = Y(end, 4)^2 + Y(end, 5)^2;
end
function xdot = governing_equation(t, x)
global Pr
xdot(1) = x(2);
xdot(2) = x(3);
xdot(3) = -(1/Pr)*(0.5*x(2)^2 - (3*x(1)*x(3))/4) + x(4);
xdot(4) = x(5);
xdot(5) = 3*x(1)*x(5)/4;
xdot = xdot';
end
function plot_results(t, Y, color, label)
figure(1);
plot(t, -Y(:, 2), color);
hold on;
title('Vertical Velocity Profiles');
xlabel('η = (x/y) Ra_y^(^0^.^2^5^)');
ylabel('G = v(y/α) Ra_y^(^0^.^2^5^)');
legend(label);
figure(2);
plot(t, Y(:, 4), color);
hold on;
title('Temperature Profiles');
xlabel('η = (x/y) Ra_y^(^0^.^2^5^)');
ylabel('ϴ');
legend(label);
end
- For Cases Where a Solution is Found: The solutions seem fine, but you should still verify them against expected results or additional criteria to ensure they are accurate and meaningful.
- For the Case with Inaccuracy Possible: You might want to adjust the solver options like increasing the maximum number of iterations (MaxIter) or changing the function tolerance (TolFun). This could help the solver take more steps to find a better solution.
- For the No Solution Found Case: This is a bit trickier. You might need to provide a better initial guess or adjust solver settings. In some cases, the problem formulation itself might need to be revisited, especially if it's a particularly challenging nonlinear problem.
Remember, solving nonlinear equations, especially those from complex systems, can be quite sensitive to initial conditions and solver settings. Fine-tuning these parameters is often a key part of the process.
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