Parallel shear flows problem is a classical hydrodynamic instability problem and continues to attract attention of researchers [1,2,3,5]. Kuo [8] considered slightly general case of a homogeneous, inviscid, parallel shear flow problem in the β -plane. He obtained the linearized perturbation equation for this problem and derived an extension of the celebrated Rayleigh’s [9] inflexion point crite...

In this work we consider the linear stability of a sub-to-super inviscid transonic stationary wave of a one-dimensional model of isentropic compressible flows through a nozzle of varying area. This sub-to-super inviscid transonic stationary wave is newly founded by the authors using the geometric singular perturbation theory. The main result of this work is to show that the sub-to-super invisci...

We study the large time asymptotic shock front speed in an inviscid Burgers equation with a spatially random flux function. This equation is a prototype for a class of scalar conservation laws with spatial random coefficents such as the wellknown Buckley-Leverett equation for two-phase flows, and the contaminant transport equation in groundwater flows. The initial condition is a shock located a...

The computation of boundary layer properties and laminar-to-turbulent transition location is a very complex problem, generally not undertaken in the context of aircraft design. Yet if an aircraft is to exploit the advantages of laminar flow and proper trade-offs between inviscid drag, structural weight, and skin friction are to be made, this is just what must be done. This paper summarizes a de...

We consider the linear, viscous stability of the boundary layer induced by an unbounded vortex whose outer inviscid structure coincides near the axis with Long's vortex. The viscous boundary layer induced by the interaction of such a vortex with a solid plane perpendicular to the axis has a known self-similar structure. The spatial stability of this self-similar solution is analyzed here for ax...

In this paper it is shown that the inviscid Burgers equation is nonlinearly self-adjoint. Then, from Ibragimov’s theorem on conservation laws, local conserved quantities are obtained. Mathematical subject classification: Primary: 76M60; Secondary: 58J70.

We prove an existence and uniqueness theorem for stationary solutions of the inviscid Burgers equation on a segment with random boundary conditions. We also prove exponential convergence to the stationary distribution.

We introduce a class of multi-scale systems with discrete time, motivated by the problem inviscid limit in fluid dynamics presence small-scale noise. These are infinite-dimensional and defined on scale-invariant space-time lattice. propose qualitative theory describing vanishing regularization (inviscid) as an attractor renormalization group operator acting space flow maps or respective probabi...