I am trying to model the unsteady flow in the vaneless diffuser of a centrifugal compressor, by solving the linearized 2D compressible Euler equations in r-theta for perturbed quantities (represented by tilde). The mean background flow is assumed known(in terms of flow rate Q and circulation G condtants, since the mean background flow is non-uniforn and assumed to behave like a logarithmic spiral and hence the free vortex assumption) and the solution is assumed to be periodic in time and to be a Fourier sum of n spatial harmonics in theta, leading to the reduction of the set of PDEs to a set of linear ODEs as shown below.
Doing the same thing in the axial ducts (solving in x-theta) gave explicit analytical solutions for each of the perturbed quantities, but attempting the same thing for the r-theta domain I run into problems. My code is shown below
syms r Vt(r) s Q G J n RHO P(r) Vr(r) a rho(r)
ode1 = diff(rho,r) + (s*r/Q + G*J*n/(Q*r))*rho + (RHO*J*n)/r * Vt + RHO*diff(Vr,r) == 0;
ode2 = diff(Vr,r) + (s*r/Q + G*J*n/(Q*r))*Vr == -r/(RHO*Q)*diff(P,r);
ode3 = diff(Vt,r) + (s*r/Q + G*J*n/(Q*r))*Vt == -1/(RHO*Q)*J*n*P;
ode4 = diff(P,r) + (s*r/Q + G*J*n/(Q*r))*P == a^2*( diff(rho,r) + (s*r/Q + G*J*n/(Q*r))*rho);
odes = [ode1; ode2; ode3; ode4];
S = dsolve(odes);
And MATLAB returns
Warning: Unable to find explicit solution.
What could be the reason? Could it be that there is no explicit solution for this set of ODEs?