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Steady two-dimensional fluid flow over an obstacle is solved using complex variable methods. We consider the cases of rectangular obstacles, such as large boulders, submerged in a potential flow. These may arise in geophysics, marine and civil engineering. Our models are applicable to initiation of motion that may result in subsequent transport. The local flow depends on the obstacle shape, slowing down in confining corners and speeding up in expanding corners. The flow generates hydrodynamic forces, drag and lift, and their associated moments, which differ around each face. Our model replaces the need for ill-defined drag and lift coefficients with geometry-dependent functions. We predict smaller flow velocities to initiate motion. We show how a joint-bound boulder can be transported against gravity, and analyse the influence of a wake region behind an isolated boulder. [ABSTRACT FROM AUTHOR]

We formulate a fully discrete finite difference numerical method to approximate the incompressible Euler equations and prove that the sequence generated by the scheme converges to an admissible measure valued solution. The scheme combines an energy conservative flux with a velocity-projection temporal splitting in order to efficiently decouple the advection from the pressure gradient. With the use of robust Monte Carlo approximations, statistical quantities of the approximate solution can be computed. We present numerical results that agree with the theoretical findings obtained for the scheme. [ABSTRACT FROM AUTHOR]

This paper treats the large deformation of closed nonlinearly elastic axisymmetric shells under an external pressure field generated by the steady, irrotational, axisymmetric flow of an incompressible, inviscid fluid. The flow is assumed to have a prescribed velocity U and pressure Ρ at infinity. The deformation of the shells is described by a geometrically exact theory. The parameters U and Ρ and the deformed shape of the shell uniquely determine the velocity field of the steady flow. The most difficult part of the analysis is to show that the velocity and pressure of the flow on the shell depend continuously and compactly on the function describing the shape. The pressure field on the shell is substituted into the equilibrium equations for the shell, yielding a system of ordinary functional-differential equations. These are converted into a fixed-point form, which is analyzed by a global implicit function theorem. The problem has technical difficulties that do not arise in problems with rigid obstacles. [ABSTRACT FROM AUTHOR]

Abstract: Starting from the property that the velocity is a divergence free filtered field we construct the equations of motion for an ideal fluid on a space–time-scale. Methods of approximation are briefly examined by which the macroscopic equations can be solved on individual scale slices. Validation of such approximations based on general residual models are discussed. [Copyright &y& Elsevier]

We construct a feedback law which allows us to asymptotically stabilize the Euler system for incompressible inviscid fluids in two dimensions, in the case of a multiconnected bounded domain, by means of a control localized on a part of the boundary that meets every connected component of the boundary. This generalizes a result of Coron [{SIAM J. Control Optim., 37 (1999), pp. 1874--1896] concerning simply connected domains. [ABSTRACT FROM AUTHOR]

Abstract: The solutions of time-dependent PDEs may show a bewildering variety of behaviors. The example of the Kuramoto–Sivashinsky (KS) equation illustrates how simple an equation can be with extremely complex solutions. The equations for incompressible fluids are also notorious for the extremely complex behavior of their solutions. But in the latter case, the situation is in a sense far less satisfactory than for the KS equations, since one still ignores if their solution remains smooth as time goes on, with smooth initial data in 3D. This leaves in a quite uncertain state all theories of 3D turbulence, which may be either a way of coping with our ignorance of the true dynamics of real fluids, or a way of analyzing the behavior of the smooth solutions of the Navier–Stokes equations. Below, in this homage to Professor Kuramoto, we present some remarks on this question, limited to the 3D Euler equations without viscosity. The existence of solutions of the 3D incompressible Euler equations for fluids blowing-up in finite time remains an open question if the initial energy is finite. The consensus seems to be that the singularity, if there is any, is local in space and time, although it is difficult to find a straight statement supporting such a claim in the literature. Based on the properties of self-similar equations, we claim that this is not possible at least if one assumes such a self-similar blow-up, including some refinements in the time dependence that extend the class of self-similar solutions. Actually, the possible blow-up solutions are very much constrained by the conservation of energy and of circulation. Based on scaling transformations, we argue that the singularity cannot be a simple self-similar blow-up, but requires some sort of oscillations on a logarithmic time-scale. The space dependence cannot be simple either. The various constraints lead to a self-similar collapse towards a line (at least for an axisymmetric flow with swirl), with a wavelength along the axis decreasing by steps. Our final result is a set of explicit equations such that, if they have a smooth solution satisfying certain conditions, the original Euler equations have a finite-time singularity. [Copyright &y& Elsevier]