= Exact incompressible shearing wave
For background shear $\mathbf u_0=-2Ax\mathbf e_y$, take $\mathbf u'=\operatorname{Re}[\mathbf v(t)e^{i\mathbf k(t)\cdot\mathbf r}]$ with real $\mathbf k$ and $\mathbf k\cdot\mathbf v=0$. Then $\mathbf u'\cdot\nabla\mathbf u'=0$ pointwise, including conjugate terms. Choosing $\dot k_x=2Ak_y$, $\dot k_y=\dot k_z=0$ cancels background phase <advection>. The resulting linear <amplitude> equations are therefore exact finite-amplitude solutions of the incompressible <Navier-Stokes equation> in this unbounded or compatible shearing-periodic setting.
// Target: gravitational-instability-of-an-astrophysical-disk.bigb
Back to article page
