On the Controllability of the Hydrostatic Stokes Equations

This paper is devoted to present some results on the controllability of the hydrostatic Stokes equations. The first main result of this paper states that the approximate null controllability of this system holds. This is proved whatever the boundary conditions are. Then, we extend this result to an exact null controllability result when the boundary conditions are ∂ _z u = 0 (the vertical derivative of the horizontal velocity) at the bottom.      

Bubble finite elements for the primitive equations of the ocean

We introduce in this paper two original Mixed methods for the numerical resolution of the (stationary) Primitive Equations (PE) of the Ocean. The PE govern the behavior of oceanic flows in shallow domains for large time scales. We use a reduced formulation (Lions et al. [28]) involving horizontal velocities and surface pressures. By using bubble functions constructed ad-hoc, we are able to define two stable Mixed Methods requiring a low number of degrees of freedom. The first one is based on the addition of bubbles of reduced support to velocities elementwise. The second one makes use of conic bubbles of extended support along the vertical coordinate. The latter constitutes a genuine mini-element for the PE, e.g., it requires the least number of extra degrees of freedom to stabilize piecewise linear hydrostatic pressures. Both methods verify a specific inf-sup condition and provide stability and convergence. Finally, we compare several numerical features of the proposed pairs in the context of other FE methods found in the literature.     

Conformational study of galphimines A and B

A conformational study has been performed for galphimines A and B, which differ from each other only in an acetate moiety on ring B of galphimine A. Mechanical molecular calculations showed that the predominant conformers in a Boltzman distribution are those which establish an intramolecular hydrogen bond between the hydroxyls on rings A and B, keeping a similar conformation on the rest of the molecule. The existence of these conformers was confirmed by NMR spectroscopy in (D6) DMSO solution. Furthermore, an unbound hydrogen conformation was found. These types of conformations very probably coexist in solution, for both types of galphimines A and B. Additional experiments suggest that the acetate group on galphimine A does not distort rings B and A, neither does it disturb the intramolecular hydrogen bond formation that also shows galphimine B. Finally, it does not present a steric effect on ring A to avoid any type of interaction of the functional groups on this ring with the biological receptor. The acetate group, which is responsible for the lost of activity of galphimine A very probably prevent that the hydroxyls OH4 and OH7 from interacting, either in a hydrogen bounded or free form, with the receptor, indicating the importance that these hydroxyls play in the biological activity of the molecule.     

Error estimates of a linear decoupled Euler–FEM scheme for a mass diffusion model

We present error estimates of a linear fully discrete scheme for a three-dimensional mass diffusion model for incompressible fluids (also called Kazhikhov–Smagulov model). All unknowns of the model (velocity, pressure and density) are approximated in space by C ⁰-finite elements and in time an Euler type scheme is used decoupling the density from the velocity–pressure pair. If we assume that the velocity and pressure finite-element spaces satisfy the inf–sup condition and the density finite-element space contains the products of any two discrete velocities, we first obtain point-wise stability estimates for the density, under the constraint lim_(h,k)→0 h/k = 0 (h and k being the space and time discrete parameters, respectively), and error estimates for the velocity and density in energy type norms, at the same time. Afterwards, error estimates for the density in stronger norms are deduced. All these error estimates will be optimal (of order O(h+k){\mathcal{O}(h+k)}) for regular enough solutions without imposing nonlocal compatibility conditions at the initial time. Finally, we also study two convergent iterative methods for the two problems to solve at each time step, which hold constant matrices (independent of iterations).     

CO2 and CH4 mole fraction measurements during hydrate growth in a semi-batch stirred tank reactor and its significance to kinetic modeling

A new experimental technique has been developed to measure the mole fraction of the gas hydrate former in the bulk liquid phase, at the onset of hydrate growth and thereafter, in a semi-batch stirred tank reactor. The mole fraction of carbon dioxide and methane in the bulk liquid phase was obtained for the first 11 and 13 min of the growth stage, for the carbon dioxide–water and methane–water systems respectively. Experiments were conducted at temperatures ranging from 275.3 K to 281.4 K and at pressures ranging from 2017 kPa to 4000 kPa for the carbon dioxide–water system, while temperatures ranging from 275.1 K to 279.1 K and pressures ranging from 3858 kPa to 6992 kPa were investigated for the methane–water system. The mole fraction of carbon dioxide in the bulk liquid phase was found to be constant during the growth period, varying on average by 0.6% and 0.3% at 275.4 K and 279.5 K. Similarly, the mole fraction of methane in the bulk liquid phase was found to remain constant during the growth stage, varying on average by 2.0%, 0.8% and 0.2% at 275.1 K, 277.1 K and 279.1 K respectively. The mole fraction of the gas hydrate former in the bulk liquid phase was also found to increase with pressure and decrease with temperature, while remaining greater than its hydrate-liquid water equilibrium value. As a result, an alternate formulation of a hydrate growth model is proposed.     

A modified soft-core fluid model for the direct correlation function of the square-shoulder and square-well fluids

We propose a simple analytical expression of the direct correlation function for the square-shoulder and square-well fluids. Our approximation is based on an ansatz for the direct correlation function of a modified soft-core fluid, whose parameters are adjusted by fitting the data obtained from Monte Carlo computer simulations. Moreover, it is complemented with a Wertheim-like parametrization to reproduce correctly the direct correlation inside the hard-core. We demonstrate that this approach is in quantitative agreement with the numerical solution of the Ornstein–Zernike equation within the Percus–Yevick approximation. We also show that our results are accurate in a large regime of densities for different interaction ranges and potential strengths. Therefore, this opens up the possibility of introducing the square-shoulder or the square-well potentials as new reference systems in advanced theoretical approximations.     

Effect of temperature and pressure on the solubility of carbon dioxide in water in the presence of gas hydrate

The solubility of carbon dioxide in pure water in the presence of CO₂ gas hydrate has been measured at temperatures between 273 and 284 K and pressures ranging from 20 to 60 bar. It was found that the solubility decreases with decreasing temperature in the hydrate formation region. In the absence of gas hydrate, the gas solubility increases with decreasing temperature, but the hydrate formation process changes this trend. This confirms theoretical calculations as well as removes previously reported ambiguities. It was also observed that pressure was not a strong factor on the solubility.     

Localized SVEP, Property (b) and Property (ab)

Property (b) for a bounded linear operator ${T \in L(X)}$ on a Banach space X means that the points λ of the approximate point spectrum for which λI – T is upper semi-Weyl are exactly the spectral points λ for which λI – T is Browder. In this paper we shall give several characterizations of operators T for which T, or its dual T*, has property (b). We also investigate the property (ab) which is closely related to property (b).     

Property (R) under Perturbations

Property (R) holds for a bounded linear operator ${T \in L(X)}$ , defined on a complex infinite dimensional Banach space X, if the isolated points of the spectrum of T which are eigenvalues of finite multiplicity are exactly those points λ of the approximate point spectrum for which λI ? T is upper semi-Browder. In this paper we consider the permanence of this property under quasi nilpotent, Riesz, or algebraic perturbations commuting with T.