On the evolution of two-dimensional patterns in an inviscid liquid sheet

We perform an analysis of the pattern formation for a moving sheet of inviscid fluid. The sheet, which is assumed to have an infinite horizontal extent, moves at some prescribed velocity into a passive surrounding gas. The sheet’s thickness is assumed much smaller than the horizontal scale of the fluid motion. By considering a system that is symmetric with respect to the horizontal planes, long scale asymptotics are used to reduce the full governing equations in three dimensions to a set of three coupled nonlinear partial differential equations for the horizontal components of the velocity field and the height of the interface profile. The interfacial conditions consisting of the kinematic and normal stress balance are incorporated into these evolution equations. Investigations are carried out as function of the sole dimensionless parameter, namely the Weber number. A small amplitude stability analysis around the planar gas–liquid interface reveals that wave patterns in the form of traveling plane waves occur subcritically, and are therefore unstable. The reduced evolution equations are solved numerically for fixed values of the Weber number. Since the reduced system of equations is homogeneous, the wave motion is generated by initial conditions. Five initial conditions have been imposed: one-dimensional rolls, two-dimensional squares, two-dimensional hexagons, two-dimensional ridges, and smooth peaks. The ensuing evolution of the liquid sheet’s shape and corresponding flow fields are described by illustrations of the changes in the sheet’s morphology with time.     

Stability of a bubble expanding and translating through an inviscid liquid

A bubble expands adiabatically and translates in an incompressible and inviscid liquid. We investigate the number of equilibrium points of the bubble and the nature of stability of the bubble at these points. We find that there is only one equilibrium point and the bubble is stable there.     

The inviscid limit for an inflow problem of compressible viscous gas in presence of both shocks and boundary layers

In this paper, we study the inviscid limit problem for the Navier-Stokes equations of one-dimensional compressible viscous gas on half plane. We prove that if the solution of the inviscid Euler system on half plane is piecewise smooth with a single shock satisfying the entropy condition, then there exist solutions to Navier-Stokes equations which converge to the inviscid solution away from the shock discontinuity and the boundary at an optimal rate of ε¹ as the viscosity ε tends to zero.     

Existence theorem in the variational problem for compressible inviscid fluids

Existence of minima of hemibounded functionals with nonconvex integrand is considered. For this purpose the concept of generalized solutions is introduced. The approach is based on the Young's method; the integrand is imbedded in a suitable topological space, and the dual problem is formulated. The results are obtained for the particular Hamilton action functional which arises in the theory of compressible inviscid fluids.     

Free boundary problem for a two-layer inviscid incompressible fluid

The existence of a local (in time) classical solution of a free boundary problem for a two-layer inviscid incompressible fluid is shown. The method of successive approximations and the novel approach to Lagrangian coordinates of Solonnikov are used.     

The propagation problem and far-field patterns in a stratified finite-depth ocean

In this paper we investigate the direct problem associated with the scattering of ‘plane waves’ from an object submerged in an ocean of finite depth. An integral representation for the Dirichlet problem is found, from which a formula for the far-field pattern evolves. A density theorem is established concerning the set of all far-field patterns. This theorem is essential for the reconstruction of the submerged object, the ‘inverse’ problem [2], [4], [5].     

Sobolev problem on a stratified manifold

We study Sobolev problems for the case in which the boundary condition is posed on a submanifold X that is a stratified manifold, more precisely, a union of several transversally intersecting smooth manifolds. Using the theory of elliptic morphisms, we state conditions for a Sobolev problem to be Fredholm and construct the solution.     

Remarks on the minimizing geodesic problem in inviscid incompressible fluid mechanics

We consider L2 minimizing geodesics along the group of volume preserving maps SDiff(D) of a given 3-dimensional domain D. The corresponding curves describe the motion of an ideal incompressible fluid inside D and are (formally) solutions of the Euler equations. It is known that there is a unique possible pressure gradient for these curves whenever their end points are fixed. In addition, this pressure field has a limited but unconditional (internal) regularity. The present paper completes these results by showing: (1) the uniqueness property can be viewed as an infinite dimensional phenomenon (related to the possibility of relaxing the corresponding minimization problem by convex optimization), which is false for finite dimensional configuration spaces such as O(3) for the motion of rigid bodies; (2) the unconditional partial regularity is necessarily limited.     

The inverse problem of electromagnetic frequency sounding of stratified media

A solution method is proposed for the inverse problem of electromagnetic sounding of a stratified medium with a field excited by an electrical source. The numerical solution of the problem is analyzed using a one-dimensional model of the medium with piecewise-constant variation of the geoelectrical parameters.Translated from Vychislitel'naya Matematika i Matematicheskoe Obespechenie EVM, pp. 159–165, 1985.     

The contact angle in inviscid fluid mechanics

We show that in general, the specification of a contact angle condition at the contact line in inviscid fluid motions is incompatible with the classical field equations and boundary conditions generally applicable to them. The limited conditions under which such a specification is permissible are derived; however, these include cases where the static meniscus is not flat. In view of this situation, the status of the many ‘solutions’ in the literature which prescribe a contact angle in potential flows comes into question. We suggest that these solutions which attempt to incorporate a phenomenological, but incompatible, condition are in some, imprecise sense ‘weak-type solutions’; they satisfy or are likely to satisfy, at least in the limit, the governing equations and boundary conditions everywhere except in the neighbourhood of the contact line. We discuss the implications of the result for the analysis of inviscid flows with free surfaces.     

On small oscillations of a compressible stratified liquid

We study the structure of the spectrum and the completeness and basis property of a system of eigenvectors. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 12, pp. 1614–1623, December, 2006.     

Concentrated force acting in an inviscid and incompressible parallel flow

The paper discusses the steady disturbance velocity field induced by a concentrated force acting at a fixed point of space in a parallel flow of an inviscid and incompressible fluid. Force and parallel flow have the same direction. The induced velocity field, its pressure field and its acceleration field are described by means of generalized functions (distributions). Copyright © 2000 John Wiley & Sons, Ltd.