Displacements by Successive rotations for Vehicles Subject to Given Constraints

The displacements of a vehicle on a plane can be subject to constraints depending on the nature of the vehicle. One can, for instance, think of the existence of a smallest turning circle for a car. In this paper our purpose is to show, on a simple example, how such constraints can be handled. We, in fact, consider the case of a vehicle the motions of which consist of a finite sequence of rotations, each rotation being subject to the following constraints. 1) The radius of the circles along which the displacements of the vehicle take place are larger than a critical radius. 2) The centers of the successive rotations are located along a straight line defined by the geometry of the vehicle. The mathematical analysis of this problem relies on a suitable choice of frames of reference in which the expression of the constraints is particularly simple. It is then shown that, under the above constraints, an arbitrary displacement can always be achieved by three approriate rotation

Impact of Magnetohydrodynamic and Bubbles Driving Forces on the Alumina Concentration in the Bath of an Hall-Heroult Cell

The alumina concentration in the bath plays a fundamental role on cell operation. Local depletion may lead to an anode effect when using carbon anodes. A mathematical model describing the alumina convection-diffusion process in the bath coupled to the cell magneto-hydrodynamic (MHD) in the presence of small bubbles is presented. Small bubbles may be assumed when slotted anodes are used. The relative importance of the velocity felds generated by the magnetic effects and/or the small bubbles on the alumina concentration in the bath is discussed.Концентрация глинозёма в алюминиевом электролизере играет ключевую роль в технологических операциях. Локальное истощение может привести к анодному эффекту при использовании углеродных анодов. Представлена математическая модель, описывающая конвективно диффузионный перенос глинозёма, связанный с магнитогидродинамикой (МГД) в ванне при малых размерах пузырьков. Маленькие пузырьки можно получить при использовании анодов с пазами. Обсуждена относительная значимость полей скоростей, создаваемых магнитными эффектами и/или маленькими пузырями, на концентрацию глинозёма в электролизере

Motion of gas bubbles, considered as massless bodies, affording deformations within a prescribed family of shapes, in an incompressible fluid under the action of gravitation and surface tension

A model allowing to describe motion and coalescence of gas bubbles in a liquid under the action of gravitation and surface tension is proposed. The shape of the bubbles is described by a pre-defined family of mappings, for example ellipsoids with a fixed volume and the effects of the gas motions inside the bubbles are neglected. The motion of a bubble is obtained in a Lagrangian form using the D'Alembert principle of virtual works. The set of equations is numerically solved with the help of the fictitious domain technique in which the Navier-Stokes equations in the domain formed by both fluid and gas are considered. The equations governing the bubbles motion are imposed by introducing Lagrange multipliers on the bubbles boundaries. Numerical results in 2D and 3D are presented

Impact of Magnetohydrodynamic and Bubbles Driving Forces on the Alumina Concentration in the Bath of an Hall-Heroult Cell

The alumina concentration in the bath plays a fundamental role on cell operation. Local depletion may lead to an anode effect when using carbon anodes. A mathematical model describing the alumina convection-diffusion process in the bath coupled to the cell magneto-hydrodynamic (MHD) in the presence of small bubbles is presented. Small bubbles may be assumed when slotted anodes are used. The relative importance of the velocity felds generated by the magnetic effects and/or the small bubbles on the alumina concentration in the bath is discussed.Концентрация глинозёма в алюминиевом электролизере играет ключевую роль в технологических операциях. Локальное истощение может привести к анодному эффекту при использовании углеродных анодов. Представлена математическая модель, описывающая конвективно диффузионный перенос глинозёма, связанный с магнитогидродинамикой (МГД) в ванне при малых размерах пузырьков. Маленькие пузырьки можно получить при использовании анодов с пазами. Обсуждена относительная значимость полей скоростей, создаваемых магнитными эффектами и/или маленькими пузырями, на концентрацию глинозёма в электролизере