Search results for "Euler equations"
showing 5 items of 35 documents
Zero viscosity limit of the Oseen equations in a channel
2001
Oseen equations in the channel are considered. We give an explicit solution formula in terms of the inverse heat operators and of projection operators. This solution formula is used for the analysis of the behavior of the Oseen equations in the zero viscosity limit. We prove that the solution of Oseen equations converges in W1,2 to the solution of the linearized Euler equations outside the boundary layer and to the solution of the linearized Prandtl equations inside the boundary layer. © 2001 Society for Industrial and Applied Mathematics.
Hydrodynamics and Stochastic Differential Equation with Sobolev Coefficients
2013
In this chapter, we will explain how the Brenier’s relaxed variational principle for Euler equation makes involved the ordinary differential equations with Sobolev coefficients and how the investigation on stochastic differential equations (SDE) with Sobolev coefficients is useful to establish variational principles for Navier–Stokes equations. We will survey recent results on this topic.
Modelling and simulation of gas-liquid hydrodynamics in mechnically stirred tanks
2007
Abstract Computational fuid dynamics (CFD) is an increasingly important tool for carrying out realistic simulations of process equipment. In the case of multiphase systems the development of CFD models is less advanced than for single-phase systems. In the present work CFD simulations of gas–liquid stirred tanks are reported. An Eulerian–Eulerian multi-fluid approach is used in conjunction with the simplest two-phase extension of the k–ɛ turbulence model. All bubbles are assumed to share the same size. The effect of inter-phase forces on simulation results is separately considered. As concerns drag, it is shown that the sole parameter needed to characterize the dispersed phase behaviour is …
Computation of conjugate times in smooth optimal control: the COTCOT algorithm
2006
Conjugate point type second order optimality conditions for extremals associated to smooth Hamiltonians are evaluated by means of a new algorithm. Two kinds of standard control problems fit in this setting: the so-called regular ones, and the minimum time singular single-input affine systems. Conjugate point theory is recalled in these two cases, and two applications are presented: the minimum time control of the Kepler and Euler equations.
$$\mathscr {K}$$-Convergence of Finite Volume Solutions of the Euler Equations
2020
We review our recent results on the convergence of invariant domain-preserving finite volume solutions to the Euler equations of gas dynamics. If the classical solution exists we obtain strong convergence of numerical solutions to the classical one applying the weak-strong uniqueness principle. On the other hand, if the classical solution does not exist we adapt the well-known Prokhorov compactness theorem to space-time probability measures that are generated by the sequences of finite volume solutions and show how to obtain the strong convergence in space and time of observable quantities. This can be achieved even in the case of ill-posed Euler equations having possibly many oscillatory s…