Starting from the Euler equations governing the flow of two immiscible incompressible fluids in a horizontal channel, allowing gravity and surface tension, and imposing an electric field across the channel, a nonlinear long-wave analysis is used to derive a $$2\times 2$$ 2 × 2 system of evolution equations describing the interface position and a modified tangential velocity jump across it. Travelling waves of permanent form are shown to exist and are constructed in the periodic case producing wave trains and the infinite case yielding novel gravity electro-capillary solitary waves. Various regimes are analysed including a hydrodynamically passive but electrically active upper layer, pairs of perfect dielectric fluids and a perfectly conducting lower fluid. In all cases, the presence of the field produces both depression and elevation waves travelling at the same speed, for given sets of parameters. The stability of the non-uniform travelling waves is investigated by numerically solving appropriate linearised eigenvalue problems. It is found that depression waves are neutrally stable whereas elevation ones are unstable unless the surface tension is large. Stability or instability is shown to be linked mathematically to the type of local eigenvalues of the nonlinear flux matrix used to obtain travelling and solitary waves; if these are real (hyperbolic flux matrix), the system is stable, and if they are complex (elliptic), the system is unstable. The latter is a manifestation of Kelvin–Helmholtz instability in electrified flows.

This paper studies the generation of Tollmien–Schlichting waves by free-stream turbulence in transonic flow over a half-infinite flat plate with a roughness element using an asymptotic approach. It is assumed that the Reynolds number (denoted Re) is large, and that the free-stream turbulence is uniform so it can be modelled as vorticity waves. Close to the plate, a Blasius boundary layer forms at a thickness of $$O(\mathrm{{Re}}^{-{1}/{2}})$$ O ( Re - 1 / 2 ) , and a vorticity deformation layer is also present with thickness $$O(\mathrm{{Re}}^{-{1}/{4}})$$ O ( Re - 1 / 4 ) . The report shows that there is no mechanism by which the vorticity waves can penetrate from the vorticity deformation layer into the classical boundary layer; therefore, a transitional layer is introduced between them in order to prevent a discontinuity in vorticity. The flow in the interaction region in the vicinity of the roughness element is then analysed using the triple-deck model for transonic flow. A novel asymptotic expansion is used to analyse the upper deck, which enables a viscous–inviscid interaction problem to be derived. In order to make analytical progress, the height of the roughness element is assumed to be small, and from this, we find an explicit formula for the receptivity coefficient of the Tollmien–Schlichting wave far downstream of the roughness.

We consider high Reynolds number supersonic flow over a compression ramp in the triple-deck formulation. Previous studies of compression-ramp stability have shown rapid growth of high-frequency disturbances in initial-value computations; however, no physical or numerical origin has yet been identified robustly. By considering linear perturbations to steady compression-ramp solutions, we show that instabilities observed in previous studies do not have a growth rate that is described by the integral eigenrelation of Tutty & Cowley (J. Fluid Mech., vol. 168, 1986, pp. 431–456) for a (long-wave) Rayleigh instability. We solve both the temporal and spatial instability problems in the limit of asymptotically large wavenumber $K$ (or equivalently frequency) and show that the growth rate of the instability remains $o(K)$ , being dominated by higher-order terms in the expansion at moderate ramp angles.