The effect of the nose radius of a body on boundary-layer receptivity is analyzed for the case of a symmetric mean flow past a body with a parabolic leading edge. Asymptotic methods based on large Reynolds number are used, supplemented by numerical results. The Mach number is assumed small, and acoustic free-stream disturbances are considered. The case of free-stream acoustic waves, propagating obliquely to the symmetric mean flow is considered. The body nose radius, r n , enters the theory through a Strouhal number, S = ωr n /U , where ω is the frequency of the acoustic wave and U is the mean flow speed. The finite nose radius dramatically reduces the receptivity level compared to that for a flat plate, the amplitude of the instability waves in the boundary layer being decreased by an order of magnitude when S = 0.3. Oblique acoustic waves produce much higher receptivity levels than acoustic waves propagating parallel to the body chord.

We consider waves on a layer of finite depth governed by the Euler equations in the presence of gravity, surface tension, and vertical electric fields. We use perturbation theory to identify canonical scalings and to derive a Korteweg-de Vries Benjamin-Ono equation arising in interfacial electrohydrodynamics. When the Bond number is equal to 1 / 3, dispersion disappears and the equation reduces to the Benjamin-Ono equation. In the additional limit of vanishing electric fields, we show how to obtain a new evolution equation that contains third-and fifth-order dispersion as well as a nonlocal electric field term.Classical nonlinear water wave theories leading to evolution equations such as that of Korteweg-de Vries ͑KdV͒ 1 are important in uncovering nonlinear mechanisms and establishing an analytical foundation for further studies, e.g., direct computations. Our main interest here is the novel derivation of such nonlinear models including the influence of a vertical electric field. The importance of interfacial electrohydrodynamics phenomena has been reviewed by Melcher and Taylor, 2 and recent applications can be found in processes such as the electrostatic liquid film radiator ͑Griffing et al. 3 and references therein͒ and lithographically induced self-assembly using liquid films on microfluidic scales ͑Wu and Russel 4 and references therein͒. In some of our recent studies, 5-8 we considered the effect of horizontal and vertical electric fields on nonlinear features ͑such as traveling waves or film rupture͒ of inviscid liquid sheets and layers, by direct computations of the Euler equations and reduced long wave models; comparison between the two indicates that long wave models do quite well in capturing flow quantities such as wave speeds, for example.We consider an inviscid, incompressible, and irrotational flow in a liquid layer ͑region 1͒, bounded below by a wall electrode at y = 0 and above by a free surface at y = h 0 + ͑x , t͒, where h 0 is the mean layer thickness. The fluid motion is described by a velocity potential ͑x , y , t͒ satisfying Laplace's equation in region 1. Surface tension with coefficient and gravity, g, are included. We denote the voltage potential by V and choose V =0 at y = 0. A vertical electric field is imposed by requiring V ϳ E 0 y as y → ϱ, with E 0 a constant. We assume that the fluid in region 1 is a perfect conductor so that V = 0 everywhere in it. The region y Ͼ h 0 + , denoted by region 2, is occupied by a hydrodynamically passive dielectric having permittivity . The voltage potential V͑x , y , t͒ satisfies Laplace's equation there.Given a typical velocity c 0 = ͱ gh 0 , typical free surface amplitude a, and a typical horizontal length scale l, we define dimensionless variables as follows:Here, y ͑1,2͒ denote the vertical coordinates in regions 1 and 2, respectively. We also definewhich represent an amplitude and depth parameters, respectively ͑see Keller 9 and Whitham 10 ͒. In terms of the variables ͑1͒ and ͑2͒, the governing equations and boundary conditions ͑dropping the pr...

We consider the interaction of free-stream disturbances with the leading edge of a body and its effect on the transition point. We present a method which combines an asymptotic receptivity approach, and a numerical method which marches through the Orr-Sommerfeld region. The asymptotic receptivity analysis produces a three deck eigensolution which in its far downstream limiting form, produces an upstream boundary condition for our numerical Parabolized Stability Equation (PSE). We discuss the advantages of this method against existing numerical and asymptotic analysis and present results which justifies this method for the case of a semi-infinite flat plate, where asymptotic results exist in the Orr-Sommerfeld region. We also discuss the limitations of the PSE and comment on the validity of the upstream boundary conditions. Good agreement is found between the present results and the numerical results of Haddad & Corke (1998).

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