Centralized and decentralized management of groundwater with multiple users

In this work, we investigate two groundwater inventory management schemes with multiple users in a dynamic game-theoretic structure: (i) under the centralized management scheme, users are allowed to pump water from a common aquifer with the supervision of a social planner, and (ii) under the decentralized management scheme, each user is allowed to pump water from a common aquifer making usage decisions individually in a non-cooperative fashion. This work is motivated by the work of Saak and Peterson [14], which considers a model with two identical users sharing a common aquifer over a two-period planning horizon. In our work, the model and results of Saak and Peterson [14] are generalized in several directions. We first build on and extend their work to the case of n non-identical users distributed over a common aquifer region. Furthermore, we consider two different geometric configurations overlying the aquifer, namely, the strip and the ring configurations. In each configuration, general analytical results of the optimal groundwater usage are obtained and numerical examples are discussed for both centralized and decentralized problems.     

Three-dimensional non-Darcy flow analysis using a hybrid numerical model

A hybrid numerical model is developed for the simulation of three-dimensional, unsteady non-Darcy flow through an unconfined aquifer. The major problem in analysing flow through unconfined aquifers is that they involve two boundaries, namely a surface of seepage and a free surface, the location of which is not known beforehand. The model that is presented here determines these boundaries via a two stage modelling technique. In the first stage a one-dimensional finite difference model is used to estimate the surface of seepage height whereas in the second stage a vertically integrated finite element model determines the free surface solution within the flow domain. A comparison between numerical and experimental results is included which indicates the sensitivity of the numerical solution to the selected aquifer parameters, particularly to those associated with the determination of the height of the surface of seepage.     

Steady Groundwater Flow with Steep Gradients: An Analytic Approach to the 2D–Problem with a Vertical Pressure Boundary Condition and Steady Recharge/Drainage

Steady groundwater flow with steep gradients in a vertical plane due to superficial recharge/drainage, inner sources/sinks and a one‐sided pressure boundary condition can be described by a 2D Poisson equation with a nonlinear free surface boundary condition. By means of conformal mapping techniques Schmitz and Edenhofer [1] derived the exact explicit solution of this problem in a horizontally infinite aquifer. Their results are extended to problems with a one‐sided vertical pressure boundary condition, modelling f. ex. the boundary between an aquifer and an adjacent free water body. According to ist simple parametrization, this approach can be applied on one hand to model various real world phenomena like river–aquifer–systems. It may on the other hand serve as a tool for investigating the exactness of numerical solutions and the range of validity of simplifying assumptions. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the regularity of a free boundary near contact points with a fixed boundary

We investigate the regularity of a free boundary near contact points with a fixed boundary, with C1,1 boundary data, for an obstacle-like free boundary problem. We will show that under certain assumptions on the solution, and the boundary function, the free boundary is uniformly C¹ up to the fixed boundary. We will also construct some examples of irregular free boundaries.     

A kinetic formulation for a model coupling free surface and pressurised flows in closed pipes

The aim of this paper is to present a kinetic formulation of a model for the coupling of transient free surface and pressurised flows. Firstly, we revisit the system of Saint-Venant equations for free surface flow: we state some properties of Saint-Venant equations, we propose a kinetic formulation and we verify that this kinetic formulation leads to a Gibbs equilibrium that minimises (in some general case) an energy and preserves the still water steady state. Secondly, we propose a model for pressurised flows in a Saint-Venant-like conservative formulation. We then propose a kinetic formulation and we verify that this kinetic formulation leads to a Gibbs equilibrium that minimises in any case an energy and preserves the still water steady state. Finally, we propose a dual model that couples these two types of flow.     

On the microcanonical solution of a system of fully coupled particles

We study the Hamiltonian mean field (HMF) model, a system of N fully coupled particles, in the microcanonical ensemble. We use the previously obtained free energy in the canonical ensemble to derive entropy as a function of energy, using Legendre transform techniques. The temperature–energy relation is found to coincide with the one obtained in the canonical ensemble and includes a metastable branch which represents spatially homogeneous states below the critical energy. “Water bag” states, with removed tails momentum distribution, lying on this branch, are shown to relax to equilibrium on a time which diverges linearly with N in an energy region just below the phase transition.     

Bifurcation for a free boundary problem modeling the growth of a tumor with a necrotic core

We consider a free boundary problem for a system of partial differential equations, which arises in a model of tumor growth with a necrotic core. For any positive numbers ρ<R, there exists a radially symmetric stationary solution with tumor boundary r=R and necrotic core boundary r=ρ. The system depends on a positive parameter μ, which describes the tumor aggressiveness. There also exists a sequence of values μ₂<μ₃<? for which branches of symmetry-breaking stationary solutions bifurcate from the radially symmetric solution branch.     

Application of Hankel transforms to boundary value problems of water flow due to a circular source

With the aid of Hankel transform technique, we obtain close-form solutions for discontinuous boundary-condition problems of water flow due to a circular source, which located on the upper surface of a confined aquifer. Owing to difficult evaluations of the original solutions that are in a form of an infinite range integral with a singular point and Bessel functions in integrands, we adopt two numerical algorisms to transform the original solutions as a series form for convenient practical applications. We apply the solutions in series form to numerical examples to analyze the characteristics of the flow in the confined aquifers subjected to pumping or recharge. By numerical examples, it indicates that: the drawdown will reduce with the increase of the layer thickness and the distance from the center of a circular source when pumping in a region with a finite thickness and a finite width; two algorisms for closed-form solutions of an infinite range integral have almost the same results, but the second algorism is superior for a faster convergence; in a semi-infinite confined aquifer, the drawdown due to a constant pumping rate Q and uplift due to recharge by a given hydraulic head s₀ will both decrease with the increase of K_r/K_v; however, the radius r₀ of the circular source has a reverse influence on the drawdown and the uplift, i.e., the drawdown decrease with the increase of r₀, while the uplift increase with r₀.     

Global existence of weak solutions to a nonlocal Cahn–Hilliard–Navier–Stokes system

A well-known diffuse interface model consists of the Navier–Stokes equations nonlinearly coupled with a convective Cahn–Hilliard type equation. This system describes the evolution of an incompressible isothermal mixture of binary fluids and it has been investigated by many authors. Here we consider a variant of this model where the standard Cahn–Hilliard equation is replaced by its nonlocal version. More precisely, the gradient term in the free energy functional is replaced by a spatial convolution operator acting on the order parameter φ, while the potential F may have any polynomial growth. Therefore the coupling with the Navier–Stokes equations is difficult to handle even in two spatial dimensions because of the lack of regularity of φ. We establish the global existence of a weak solution. In the two-dimensional case we also prove that such a solution satisfies the energy identity and a dissipative estimate, provided that F fulfills a suitable coercivity condition.     

Asymptotic stability for a free boundary problem arising in a tumor model

We consider a tumor model in which all cells are proliferating at a rate μ and their density is proportional to the nutrient concentration. The model consists of a coupled system of an elliptic equation and a parabolic equation, with the tumor boundary as a free boundary. It is known that for an appropriate choice of parameters, there exists a unique spherically symmetric stationary solution with radius RS which is independent of μ. It was recently proved that there is a function μ_∗(RS) such that the spherical stationary solution is linearly stable if μ<μ_∗(RS) and linearly unstable if μ>μ_∗(RS). In this paper we prove that the spherical stationary solution is nonlinearly stable (or, asymptotically stable) if μ<μ_∗(RS).     

Finite element approximation of a free boundary plasma problem

In this article, we study a finite element approximation for a model free boundary plasma problem. Using a mixed approach (which resembles an optimal control problem with control constraints), we formulate a weak formulation and study the existence and uniqueness of a solution to the continuous model problem. Using the same setting, we formulate and analyze the discrete problem. We derive optimal order energy norm a priori error estimates proving the convergence of the method. Further, we derive a reliable and efficient a posteriori error estimator for the adaptive mesh refinement algorithm. Finally, we illustrate the theoretical results by some numerical examples.     

A Point Model for the Free Cyclic Submodules over Ternions

We show that the set of all (unimodular and non-unimodular) free cyclic submodules of T ², where T is the ring of ternions over a commutative field, admits a point model in terms of a smooth algebraic variety.     

Streamwise oscillations of a cylinder beneath a free surface: Free surface effects on vortex formation modes

A computational study of a viscous incompressible two-fluid model with an oscillating cylinder is investigated at a Reynolds number of 200 and at a dimensionless displacement amplitude of A=0.13 and for the dimensionless forcing cylinder oscillation frequency-to-natural vortex shedding frequency ratios, f/f₀=1.5,2.5,3.5. Specifically, two-dimensional flow past a circular cylinder subject to forced in-line oscillations beneath a free surface is considered. The method is based on a finite volume discretization of the two-dimensional continuity and unsteady Navier-Stokes equations (when a solid body is present) on a fixed Cartesian grid. Two-fluid model based on improved volume-of-fluid method is used to discretize the free surface interface. The study focuses on the laminar asymmetric flow structure in the near wake region and lock-on phenomena at a Froude number of 0.2 and for the dimensionless cylinder submergence depths, h=0.25, 0.5 and 0.75. The equivorticity patterns and pressure distribution contours are used for the numerical flow visualization. The code validations in special cases show good comparisons with previous numerical results.     

Qualitative analysis of a delayed free boundary problem for tumor growth under the effect of inhibitors

In this paper we study a delayed free boundary problem for the growth of tumors under the effect of inhibitors. The establishing of the model is based on the diffusion of nutrient and inhibitors, and mass conservation for the two processes proliferation and apoptosis. It is assumed that the process of proliferation is delayed compared to apoptosis. We mainly study the asymptotic behavior of the solution, and prove that under some assumptions, in the case where c₁ and c₂ are sufficiently small, the volume of the tumor cannot expand without limit; it will either disappear or evolve to a dormant state as t→∞.     

Vector fields and a family of linear type modules related to free divisors

This paper has three main goals. We start describing a method for computing the polynomial vector fields tangent to a given algebraic variety; this is of interest, for instance, in view of (effective) foliation theory. We then pass to furnishing a family of modules of linear type (that is, the Rees algebra equals the symmetric algebra), formed with vector fields related to suitable pairs of algebraic varieties, one of them being a free divisor in the sense of K. Saito. Finally, we derive freeness criteria for modules retaining a certain tangency feature, so that, in particular, well-known criteria for free divisors are recovered.     

Smooth interface in a two-component Stokes flow

The problem considered is that of evolution of the free boundary Γ(t) separating two immiscible viscous fluids with different constant densities and viscosities. The motion is described by the Stokes equations driven by the gravity force. We prove the existence of classical solutions for small timet and establish that the free boundary Γ(t)∈C ^l+2 (l>0 is an arbitrary non-integer number)     

On the classification problem of the quasi-isomorphism classes of free chain algebras

The present paper is devoted to the classification problem of the quasi-isomorphism classes of free differential graded algebras (dgas) over a (P.I.D) R. We introduce the notion of coherent homomorphisms, perfect and quasi-perfect dgas (the Adams-Hilton model of simply connected CW-complex such that H_∗(X,R) is free is a such a dga) and our first main result asserts that two perfect (quasi-perfect) dgas are quasi-isomorphic if and only if their Whitehead exact sequences are coherently isomorphic. Moreover we define the notion of a strong isomorphism between the Whitehead exact sequences and we show that two free R-dgas, of which their Whitehead exact sequences are strongly isomorphic, are quasi-isomorphic.     

The diffusive logistic model with a free boundary in heterogeneous environment

In this paper, we study the population dynamics of an invasive species in heterogeneous environment which is modeled by a diffusive logistic equation with free boundary condition. To understand the effect of the dispersal rate D and the parameter μ (the ratio of the expansion speed of the free boundary and the population gradient at the expanding front) on the dynamics of this model, we divide the heterogeneous environment into two cases: strong heterogeneous environment and weak heterogeneous environment. By choosing D and μ as variable parameters, we derive sufficient conditions for species spreading (resp. vanishing) in the strong heterogeneous environment; while in the weak heterogeneous environment, we obtain sharp criteria for the spreading and vanishing. Moreover, when spreading happens, we give an estimate for the asymptotic spreading speed of the free boundary. These theoretical results may have important implications for prediction and prevention of biological invasions.     

The vibrations of a free and loaded tyre

Using the model of a wheel with a reinforced tyre [Vil’ke VG, Kozhevnikov IF. A model of a wheel with a reinforced tyre. Vestnik MGU. Ser.1. Matematika Mekhanika 2004;4:37–45], the natural frequencies and natural forms of vibrations of a free or loaded tyre in the neighbourhood of the equilibrium position are determined. The spectrum of natural frequencies and natural forms of vibration are found analytically for a free (unloaded) tyre with a fixed disc. A similar problem is solved numerically in the case of a loaded tyre. The results of this analysis can be used to estimate the level of noise which occurs when a vehicle moves on an uneven surface.     

On a class of partial planes related to biplanes

In this note we consider partial planes in which for each element x (point or line) there exists a unique opposite element or antipode x* which cannot be joined to x or has no intersection with x. We also require the existence of a triangle. Such partial planes will be called antipodal planes. We are mainly interested in the subclass of regular antipodal planes satisfying: p I L implies p* I L* for all points p and lines L. We shall provide a free construction of infinite regular antipodal planes. The objects thus constructed are not free objects in the usual sense since between antipodal planes there do not exist proper homomorphisms. On the other hand, regular antipodal planes do have a canonical homomorphic image which is a biplane (cf. Payne, J Comb Theory A 12:268–282, 1972). Regular antipodal planes can be coordinatized by certain algebraic systems in a similar way as projective planes are coordinatized by ternary rings. Again by a free construction, we shall provide examples satisfying a configuration theorem comparable to the Fano condition with fixed line at infinity.