Unsteady Prandtl equation ($u_t+u{u}_x+v{u}_y={\mathcal{U}}_t+\mathcal{U}{\mathcal{U}}_x+\nu u_{yy},u_x+v_y=0$)

The unsteady form of the Prandtl boundary-layer equation evolves the tangential boundary-layer velocity under advection, outer pressure forcing and normal viscous diffusion. Incompressibility supplies the normal velocity. A stationary wall imposes the no-slip boundary condition, and tangential velocity matches the prescribed outer slip velocity far from the wall. The normal far-field disturbance need not vanish.

For a fixed outer pressure and slip velocity, retain terms linear in a disturbance of a base flow $(U,V)$. The tangential equation has the displayed form and continuity is ${\tilde{u}}_x+{\tilde{v}}_y=0$. Disturbances vanish at the wall and the tangential disturbance vanishes in the far field. A local parallel-profile approximation may freeze the coefficients even when the profile is not an exact global steady solution.

For a frozen real parallel profile and modes proportional to $e^{i\alpha (x-ct)}$, continuity eliminates tangential velocity and gives this third-order equation. Wall conditions are $v=v^{\prime }=0$; outer tangential matching gives $v^{\prime }\to 0$, with bounded $v$. Real coefficients give the spectral symmetry $(\alpha ,c)\mapsto (-\alpha ,c^{\ast })$. For positive real wavenumber, growth means $\mathrm{Im}c>0$.