Non-optimality of conical parts for Newton’s problem of minimal resistance in the class of convex bodies and the limiting case of infinite height

We consider Newton’s problem of minimal resistance, in particular we address the problem arising in the limit if the height goes to infinity. We establish existence of solutions and lack radial symmetry of solutions. Moreover, we show that certain conical parts contained in the boundary of a convex body inhibit the optimality in the classical Newton’s problem with finite height. This result is applied to certain bodies considered in the literature, which are conjectured to be optimal for the classical Newton’s problem, and we show that they are not.

Stability of a Buoyant Oldroyd-B Flow Saturating a Vertical Porous Layer with Open Boundaries

The performance of several engineering applications are strictly connected to the rheology of the working fluids and the Oldroyd-B model is widely employed to describe a linear viscoelastic behaviour. In the present paper, a buoyant Oldroyd-B flow in a vertical porous layer with permeable and isothermal boundaries is investigated. Seepage flow is modelled through an extended version of Darcy’s law which accounts for the Oldroyd-B rheology. The basic stationary flow is parallel to the vertical axis and describes a single-cell pattern where the cell has an infinite height. A linear stability analysis of such a basic flow is carried out to determine the onset conditions for a multicellular pattern. This analysis is performed numerically by employing the shooting method. The neutral stability curves and the values of the critical Rayleigh number are evaluated for different retardation time and relaxation time characteristics of the fluid. The study highlights the extent to which the viscoelasticity has a destabilising effect on the buoyant flow. For the limiting case of a Newtonian fluid, the known results available in the literature are recovered, namely a critical value of the Darcy–Rayleigh number equal to 197.081 and a corresponding critical wavenumber of 1.05950.

A Simple Way for Implementing Extraction Columns of Infinite Height in Flowsheet Simulators

A surrogate model to implement an extraction column of in?nite height in pro- cess simulation software is presented. The model consists of three decanters and a few speci?cations which are easy to implement in commercial process simula- tors. Using the model, the minimum solvent ow rate and the limiting product compositions are determined. In an example, the surrogate model for the ex- traction column is combined with standard unit operation models for describing an extraction process with a distillative solvent recycle.

Exact solutions for the 2d-strip packing problem using the positions-and-covering methodology

We use the Positions and Covering methodology to obtain exact solutions for the two-dimensional, non-guillotine restricted, strip packing problem. In this classical NP-hard problem, a given set of rectangular items has to be packed into a strip of fixed weight and infinite height. The objective consists in determining the minimum height of the strip. The Positions and Covering methodology is based on a two-stage procedure. First, it is generated, in a pseudo-polynomial way, a set of valid positions in which an item can be packed into the strip. Then, by using a set-covering formulation, the best configuration of items into the strip is selected. Based on the literature benchmark, experimental results validate the quality of the solutions and method’s effectiveness for small and medium-size instances. To the best of our knowledge, this is the first approach that generates optimal solutions for some literature instances for which the optimal solution was unknown before this study.

Towards automated reasoning in Herbrand structures

Herbrand structures have the advantage, computationally speaking, of being guided by the definability of all elements in them. A salient feature of the logics induced by them is that they internally exhibit the induction scheme, thus providing a congenial, computationally oriented framework for formal inductive reasoning. Nonetheless, their enhanced expressivity renders any effective proof system for them incomplete. Furthermore, the fact that they are not compact poses yet another proof-theoretic challenge. This paper offers several layers for coping with the inherent incompleteness and non-compactness of these logics. First, two types of infinitary proof system are introduced—one of infinite width and one of infinite height—which manipulate infinite sequents and are sound and complete for the intended semantics. The restriction of these systems to finite sequents induces a completeness result for finite entailments. Then, in search of effectiveness, two finite approximations of these systems are presented and explored. Interestingly, the approximation of the infinite-width system via an explicit induction scheme turns out to be weaker than the effective cyclic fragment of the infinite-height system.

Godunov-Type Numerical Methods for a Model of Granular Flow on Open Tables with Walls

We propose and analyse finite volume Godunov type methods based on discontinuous flux for a 2×2 system of non-linear partial differential equations proposed by Hadeler and Kuttler to model the dynamics of growing sandpiles generated by a vertical source on a flat bounded rectangular table. The problem considered here is the so-called partially open table problem where sand is blocked by a wall (of infinite height) on some part of the boundary of the table. The novelty here is the corresponding modification of boundary conditions for the standing and the rolling layers and generalization of the techniques of the well-balancedness proposed in [1]. Presence of walls may lead to unbounded or discontinuous surface flow density at equilibrium resulting in solutions with singularities propagating from the extreme points of the walls. A scheme has been proposed to approximate efficiently the Hamiltonians with the coefficients which can be unbounded and discontinuous. Numerical experiments are presented to illustrate that the proposed schemes detect these singularities in the equilibrium solutions efficiently and comparisons are made with the previously studied finite difference and Semi-Lagrangian approaches by Finzi Vita et al.

Implementation of an Infinite-Height Levee in CaFunwave Using an Immersed-Boundary Method

Numerical simulations of storm-surge–wave actions on coastal highways and levees are very important research topics for coastal engineering. In a large-scale region hydrodynamic model, highways and levees are often complicated in geometry and much smaller in size compared to the grid spacing. The immersed-boundary method (IBM) allows for those complicated geometries to be modeled in a less expensive way. It can allow very small geometries to be modeled in a large-scale simulation, without requiring them to be explicitly on the grid. It can also allow for complicated geometries not collocated on the grid points. CaFunwave is a project that uses the Cactus Framework for modeling a solitary coastal wave impinging on a coastline and is the wave solver in this research. The IBM allows for a levee with different geometries to be implemented on a simple Cartesian grid in the CaFunwave package. The IBM has not been often used previously for these types of applications. Implementing an infinite-height levee using the IBM into the Cactus project CaFunwave involves introducing immersed-boundary (IB) forcing terms into the standard two-dimensional (2D) depth-averaged shallow water equation set. These forcing terms cause the 2D solitary wave to experience a virtual force at the grid points surrounding the IB levee. In this paper, the levee was implemented and tested using two different IBMs. The first method was a feedback-forcing method, which proved to be more effective at modeling the levee than the second method, the direct-forcing method. In this study, the results of the two methods are presented and discussed. The effect of levee shape on the flow is also investigated and discussed in this paper.