Ordinary differential equation - pratical problem

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mortain 2016-7-9

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https://ww2.mathworks.cn/matlabcentral/answers/294607-ordinary-differential-equation-pratical-problem

评论： Walter Roberson 2016-7-13

Hello,

I have a single equation of the form: u'=a*u^2+b*u^1.5+c*u+d. I am trying to solve it in Matlab but it does not find the solution. The problem is a body falling down. The body is under the effect of four forces: 1) weight (m*g) 2) flow in opposite direction of the weight (0.5*da*A*U0^2) 3) drag force (0.5*da*C_d*A*u^2), which would change verse depending on flow direction 4) acceleration (m*u') u(t=0)=0;

The C_d changes with velocity (relationship from the sphere in free flow C_d = 21.12/Reynolds+6.3/sqrt(Reynolds)+.25; % drag coefficient)

I coded this problem, but I can't get it to give a result, unless I fix C_d to a value. Would you be able to give a help, please?

da = 1.161; %density of air
g = 9.81; % Gravitational acceleration
d =1E-6*100; % characteristic length of the body
m = 1000*.957251*d.^3.09275; %Mass of the the body
A = 3.3108*d.^2.21672; %Surface area of the body
u_flow = .5; % Velocity air flow
syms u(t) c_D Re_ver 
Re_ver = da*u(t)*d/1.8E-5; % Reynolds number
c_D = 21.12/Re_ver+6.3/sqrt(Re_ver)+.25; % drag coefficient;
a = m;
b = -.5*da*A*c_D;
c = m*g-.5*da*u_flow^2*A;
if c<0
  c = -c;
end
u(t) = dsolve(a*diff(u,t) == b*u^2+c,u(0)==0);
t = linspace(0,10,500);
figure, plot(t,double(u(t)))

I am using Matlab R2014a

Thanks Antonio

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Karan Gill 2016-7-13

在 MATLAB Online 中打开

In the ODE a*u' = b*u^2+c , b has a term 1/sqrt(u). My guess is that this term complicates the calculation. The ODE is of the form:

C*u' = u^2(C + C/u + C/sqrt(C*u)) + C
           |----'b' term-------|

where C represents the constants. I'm afraid I don't see how b = constant/u. Could you point out where I misunderstand?

Walter Roberson 2016-7-13

At your initial condition, u(0) = 0, your Reynold's number is 0. That is giving you a division by 0 for your drag coefficient, which is giving a singularity right from the start, which is preventing any resolution of the system.

I am not familiar with fluid dynamics, but when I look at various material it appears to me that the equation you are using might perhaps not apply at very low velocities.

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