Hello everybody in Mathematica SE. Although my question is related to flow stability analysis, this should be a general application of MMA to solve a system of ODEs. Thank you for your suggestion!
For the special case of zero streamwise wave number ($\alpha=0$), the dispersion relation (frequency $\omega$ as a function of spanwise wave number $\beta$ and other control parameter) and the eigenfunctions could be determined analytically because the Orr-Sommerfeld and Squire equations (OSS eqn, see Fig.1) reduce to constant coefficient ODEs.
Fig.1 snapshot of OSS eqns (from a highly cited textbook in the flow stability field): $\omega$ is a frequency, $\alpha$ and $\beta$ are streamwise and spanwise wave number, respectively, and $k^2=\alpha^2+\beta^2$, $U(y)$ is a mean flow, for plan Pliseuille flow $U=1-y^2$, $D=\frac{d}{d y}$, and a prime stands for differential w.r.t $y$, $\tilde{v}(y)$ and $\tilde{\eta}(y)$ are normal velocity and normal vorticity, and $Re$ is Reynolds number (just a constant parameter).
My question arises from the following snapshot from this book when I have been trying to find the solution to the OSS equation and the dispersion relation in the special case with Mathematica.
Fig.2 
My traces:
- For plan Poiseuille flow: $U(y)=1-y^2$; with $\alpha=0$, it follows that $k=\beta$, then the OSS eqn reduces to
$$\left[-\mathrm{i} \omega (D^2-\beta^2)-\frac{1}{Re}(D^2-\beta^2)^2 \right] \tilde{v} =0, \tag{1}$$
$$\left[-\mathrm{i} \omega -\frac{1}{Re}(D^2-\beta^2) \right] \tilde{\eta} = -\mathrm{i} \beta U^{\prime} \tilde{v}. \tag{2}$$
With the following code,
DSolvereturns the equations unsolved.DSolve[{-I \[Omega]*(v''[y] - \[Beta]^2 v[y]) - 1/Rey (v''''[y] - 2 \[Beta]^2 v''[y] + \[Beta]^4 v[y]) == 0, -I*\[Omega] \[Eta][y] - 1/Rey*(\[Eta]''[y] - \[Beta]^2*\[Eta][y]) == -I \[Beta] (-2 y) v[y], v[1] == 0, v[-1] == 0, v'[1] == 0, v'[-1] == 0, \[Eta][-1] == 0, \[Eta][1] == 0}, {v[y], \[Eta][y]}, y]If I solve the two equations separately
DSolve[{-I \[Omega]*(v''[y] - \[Beta]^2 v[y]) - 1/Rey (v''''[y] - 2 \[Beta]^2 v''[y] + \[Beta]^4 v[y]) == 0, v[1] == 0, v[-1] == 0, v'[1] == 0, v'[-1] == 0}, v[y], y] (*{{v[y] -> 0}}*)
With $v(y)=0$, the second equation becomes homogeneous. The code gives {{\[Eta][y] -> 0}}:
DSolve[{-I*\[Omega] \[Eta][y] -
1/Rey*(\[Eta]''[y] - \[Beta]^2*\[Eta][y]) == 0, \[Eta][-1] ==
0, \[Eta][1] == 0}, \[Eta][y], y]
My questions are:
(1) How to modify the code to solve the OSS equation in this special case analytically?
(2) How to introduce an intermediate variable $\mu$ in the solution of $\omega$ (eigenvalue), see eqs.(3.61)--(3.63) in Fig.2, which is related to $\beta$ through a transcendental equation. Similarly, how to introduce another intermediate variabel $n$ in (3.64), which is the solution of Eq.(2). But, I found that $n$ was not defined there in that book.
(3) How could I account for odd and even modes in MMA's DSolve?
Thank you in advance!
DSolveshould be able to give the answer, which isv = η = 0, but it cannot without some human assistance. The appropriate function, I would expect, isDEigensystem, but it returns unevaluated, which is bizarre, because the problem is so simple. So, I suggest you useDSolveto solve the first equation without boundary conditions, after which you can do the same for the second equation, Then, apply the boundary conditions, yielding six linear algebraic equations for the six constants of integration. Construct the determinant of the equations to obtain a dispersion relation. $\endgroup$DEigensystemwas unevaluated, could you pls post it that all of us can debug together? However, it depends on you of course :) $\endgroup$