Lets say we have a system of partial differential equation $\Delta(x,y,t,u^{(n)})=0$ (Navier-Stokes Equations) with a given stationary solution $u_s(x,y)$ for a inviscid flow. Note that $u$ is a 2D-vector.
Imagine we have a plate with infinite length along the x-axis and the inviscid flow is at the upper portion of the x plane.
If we introduce a linear perturbation $u=u_s+\varepsilon\hat{u}(x,y,t)$ and linearize the PDE neglecting quadratic terms. What are the boundary conditions and initial conditions for the perturbations?
These are my thoughts:
- I would assume that the initial conditions for all velocity components of the perturbations to be zero. As the perturbation starts for $t=0$ to deviate from the stationary solution.
- The Boundary conditions for the $y$-component of the perturbations is $0$ at the plate $(y=0)$. As we have a inviscid flow the no slip condition ($x$-component of perturbations is 0) at the wall cannot be applied.
- Also I would assume that the energy of the pertrubations over the whole region $(x,y)\in D=\mathbb{R}\times\mathbb{R}_0^+$ is finite. I would calculate this by the integral over D with the sum of squares of the velocities.
Are my boundary conditions fine or am I missing something.
Typically all the boundary conditions will be zero and once substitued into the ,Navier Stokes equations the perturbed quantity will be the solution of an ODE. Typically the perturbation will take the form
$\hat{u}=e^{i(kx -\omega t)}U(y)$
Then either $k$ or omega is specified and the $\omega$ found or vice versa whether spatial or temporal stability is sort. This leads to the dispersion relation giving a functional form $\omega=\omega(k)$ say. Do a search for the Orr-Sommerfeld equation for more details