How can i solve these systems of ODE

(d^3 f)/(dη^3 )=-1/2.f(η).(d^2 f)/(dη^2 )-Grθ(η)…….. (1)

(d^2 θ)/(dη^2 )=-Pr.f(η). dθ/dη…………………….. (2)

(d^2 ϕ)/(dη^2 )=-1/2 Sc.f(η). dϕ/dη……….………… (3)

Gr, Sc and Pr are constants.

Domain is η from zero to infinity and I want the iteration to stop the moment the difference is 〖 10〗^(-6)

Boundary conditions f'(0)=0,θ(0)=1 and ϕ(0)=1 f'(∞)=1,θ(∞)=0 and ϕ(∞)=0

Answer 1

[Zoltán Csáti]

These are typical boundary layer equations. There are several strategies to tackle it. The two main solution methods: truncate it to some finite [0 L] interval or solve it on the semi-implicit domain. I recommend you the chebfun library because it is very easy to use. You can also use the built-in bvp4c function which is based on this article. There is also an example for the Falkner-Skan problem in it on page 16.

[samuel okeleye]

I have never used matlab, are you sure i can do all these things as a first time user? I appreciate your answer but i am sorry, it is not detailed enough for a first time user that i am.

[Zoltán Csáti]

Well, I recommend you the following.

1. Download chebfun
2. Install it
3. Open it's GUI
4. Type the equations
5. Solve it

If you need any help, feel free to write.

[samuel okeleye]

i had to read a lot to be able to solve the problem with bvp4c but i have been able to solve it now. Thank God and thank you. My problem now is to be able to have a plot of y(4) and y(6) at various values of Gr_nf,Sc_nf and Pr_nf in just one plot. Below is my code

function [yprime]=myblayerode1(eta,y)
%Gr_nf,Pr_nf and Sc_nf are the grashof number, Prantl number and Schmidt
%number of the concerned fluid respectively.
%yprime is a vector of all the first order ODEs on the left hand side
%having turned all 2nd and 3rd order ODEs to 1st order. y(1)to y(7) are the
%variables on the right hanside of the resulting 1st order
%ODE.f=y(1),f'=y(2),f''=y(3),theta=y(4),theta'=y(5),phi=y(6),phi'=y(7) and
%yprime is a column vector
%df/deta,df'/deta,df''/deta,dtheta/deta,dtheta'/deta,dphi/deta and
%dphi'/deta.
Gr_nf=11.297;
Pr_nf=17.716;
Sc_nf=100;
yprime=[y(2,:);y(3,:);-0.5.*y(1,:).*y(3,:)-Gr_nf.*y(4,:);y(5,:);-0.5.*Pr_nf*y(1,:).*y(5,:);y(7,:);-0.5.*Sc_nf.*y(1,:).*y(7,:)];

function res= bvpbc(ya,yb)
Bi=1;
%BC: Evaluates the residue of the boundary condition
res1=[ya(1,:);ya(2,:);yb(2,:)-1];
res2=[(ya(5,:)+Bi.*(1-ya(4,:)));yb(4,:)];
res3=[ya(6,:)-1;yb(6,:)];
res=[res1;res2;res3;];

solinit=bvpinit(linspace(0,10),[ 0 0 0 1 0 1 0]);
options =bvpset('RelTol',1e-06,'AbsTol',1e-06,'stats','on');
sol = bvp4c(@myblayerode1,@bvpbc,solinit,options);
eta=0:0.5:10;
y=deval(sol,eta);
sol.eta=eta;
sol.y=y;
hold off
plot(eta,y(7,:),'r:s');
print plot;

so instead of having one number for Gr_nf,Pr_nf and Sc_nf i want to have like 5 numbers and i want the result to be displayed as numbers and on a graph, just like the velocity profile for power law fluid in your work with gabriella bognar where you varied n from 1.0 to 1.5. Thanks in anticipation of your assistance.

Answer 2

[Zoltán Csáti]

Well, you have to solve the system of BVPs several times for each different values of Gr, Pr and Sc. Then you can plot the data in one figure (see plot command) and can also add a legend. You may also put an arrow representing the effect of the different parameter values on the boundary layer. These can either be done programatically or by using the interactive tools.

Sorry for not answering earlier, but the comments - in contrast to the answers - are not sent to my e-mail address.

[samuel okeleye]

Thanks Zoltan i appreciate your assistance, i had to solve the problem with one value at a time since i did not hear from you, the work has been completed and accepted by my university. Once again i appreciate your response. Zoltan i do not know if i could get a Phd position at your school in Hungary, i am specialized in thermo-fluid area of mechanical engineering.

[Zoltán Csáti]

I recommend you the Department of Fluid and Heat Engineering , because of your specialization. I suggest you to gather information about the university and the research area here and if you take it seriously, I can ask the department if they need a researcher or not.