I am investigating the stability/instability of the plane Couette flow. Looking at a flow, with at solid wall and a moving wall in x1 direction, while x2 being the wall-normal direction and x3 the spanwise direction, one gets following Navier Stokes equation:
$\frac{\partial u_1'}{\partial t}+u_1'\frac{\partial u_1'}{\partial x_1}+u_2'\frac{\partial u_1'}{\partial x_2}+u_3'\frac{\partial u_1'}{\partial x_3}+A x_2 \frac{\partial u_1'}{\partial x_1}+u_2'A+\frac{\partial p}{\partial x_1}=\nu(\frac{\partial^2 u_1'}{\partial x_1^2}+\frac{\partial^2 u_1'}{\partial x_2^2}+\frac{\partial^2 u_1'}{\partial x_3^2})$
$\frac{\partial u_2'}{\partial t}+u_1'\frac{\partial u_2'}{\partial x_1}+u_2'\frac{\partial u_2'}{\partial x_2}+u_3'\frac{\partial u_2'}{\partial x_3}+A x_2 \frac{\partial u_2'}{\partial x_1}+\frac{\partial p}{\partial x_2}=\nu(\frac{\partial^2 u_2'}{\partial x_1^2}+\frac{\partial^2 u_2'}{\partial x_2^2}+\frac{\partial^2 u_2'}{\partial x_3^2})$
$\frac{\partial u_3'}{\partial t}+u_1'\frac{\partial u_3'}{\partial x_1}+u_2'\frac{\partial u_3'}{\partial x_2}+u_3'\frac{\partial u_3'}{\partial x_3}+A x_2 \frac{\partial u_3'}{\partial x_1}+\frac{\partial p}{\partial x_3}=\nu(\frac{\partial^2 u_3'}{\partial x_1^2}+\frac{\partial^2 u_3'}{\partial x_2^2}+\frac{\partial^2 u_3'}{\partial x_3^2})$
The Squires theorem states that of all the perturbations that may be applied to a shear flow (i.e. a velocity field of the form, the perturbations which are least stable are two-dimensional. Thus its seems sufficient to look at $u_3'=0$.
This leads to:
$\frac{\partial u_1'}{\partial t}+u_1'\frac{\partial u_1'}{\partial x_1}+u_2'\frac{\partial u_1'}{\partial x_2}+A x_2 \frac{\partial u_1'}{\partial x_1}+u_2'A+\frac{\partial p}{\partial x_1}=\nu(\frac{\partial^2 u_1'}{\partial x_1^2}+\frac{\partial^2 u_1'}{\partial x_2^2}+\frac{\partial^2 u_1'}{\partial x_3^2})$
$\frac{\partial u_2'}{\partial t}+u_1'\frac{\partial u_2'}{\partial x_1}+u_2'\frac{\partial u_2'}{\partial x_2}+A x_2 \frac{\partial u_2'}{\partial x_1}+\frac{\partial p}{\partial x_2}=\nu(\frac{\partial^2 u_2'}{\partial x_1^2}+\frac{\partial^2 u_2'}{\partial x_2^2}+\frac{\partial^2 u_2'}{\partial x_3^2})$
Taking the derivate $\frac{\partial}{\partial x_1}$ of the second equation and $\frac{\partial}{\partial x_2}$ of the first equation, leads to the vorticity trtansport equation for $\omega_3'=\frac{\partial u_2'}{\partial x_1}-\frac{\partial u_1'}{\partial x_3}$, you get:
$\frac{\partial \omega_3'}{\partial t}+u_1' \frac{\partial \omega_3'}{\partial x_1}+u_2' \frac{\partial \omega_3'}{\partial x_2}+A x_2 \frac{\partial \omega_3'}{\partial x_1}=\nu (\frac{\partial^2 \omega_3'}{\partial x_1^2}+\frac{\partial^2 \omega_3'}{\partial x_2^2}+\frac{\partial^2 \omega_3'}{\partial x_3^2})$
Introducing the stream function with:
$u_1'=\frac{\partial \Psi'}{\partial x_2}$ and $u_2'=-\frac{\partial \Psi'}{\partial x_1}$, one gets:
$-\frac{\partial \Delta \Psi'}{\partial t}-\frac{\partial \Psi'}{\partial x_2} \frac{\partial \Delta \Psi'}{\partial x_1}+\frac{\partial \Psi'}{\partial x_1} \frac{\partial \Delta \Psi'}{\partial x_2}-A x_2 \frac{\partial \Delta \Psi'}{\partial x_1}=-\nu \Delta^2 \Psi'$
Ist it reasonable assume a similar approach as in the Rayleigh equation?
$\Psi'=\Phi(x_2) e^{i(x_1-c t)}$
This would leat to an ODE for $\Phi$, with $\Phi'$ and $\Phi''$, etc. being the derivatives of$\Phi$:
$ikc\Phi''+ik^3 c \Phi+i k \Phi'\Phi''-ik\Phi \Phi'''+ikA x_2(\Phi''+k^2\Phi)=\nu(-\Phi''''+k^4 \Phi)$
And how can one solve this equation?