Transfer model and kinetic characteristics of $${\rm NH}_{4}^{+}$$ –K+ ion exchange on K-zeolite

Based on the mass transfer theory, a new mass transfer model of ion-exchange process on zeolite under liquid film diffusion control is established, and the kinetic curves and the mass transfer coefficients of –K⁺ ion-exchange under different conditions were systemically determined using the shallow-bed experimental method. The results showed that the –K⁺ ion-exchange rates and transfer coefficients are directly proportional to solution flow rate and temperature, and inversely proportional to solution viscosity and the size of zeolite granules. It also showed that the transfer coefficient is not influenced by the ion concentrations. For a large ranges of operational conditions including temperatures (10 − 75°C), flow rates (0.031 m s⁻¹ −0.26 m s⁻¹), liquid viscosities (1.002 × 10⁻³ N s m⁻² − 4.44 × 10⁻³ N s m⁻²), and zeolite granular sizes (0.2 − 1.45 mm), the average mass transfer coefficients calculated by the model agree with the experimental results very well.     

Transient elongational viscosities of aqueous polyacrylamide solutions measured with an optical rheometer

An anionic polyacrylamide solution was characterized in elongational flow by combining laser-Doppler velocimetry to determine the strain rate in the flow direction and the two-color flow-induced birefringence method to measure the first normal stress difference along the axial centerline of a hyperbolic die. The elongational rate was constant along the axial centerline of the planar hyperbolic die as long as vortices at the die entrance did not occur. The transient elongational viscosity μ ⁺ was determined as a function of the elongational rate. The parameters varied are the Hencky strain rate and the polymer concentration. μ ⁺ showed a pronounced increase over the linear viscoelastic behavior above critical Hencky strains of 1.2 to 1.5; that is, a significant strain hardening could be observed for polyacrylamide solutions. This strain hardening is stronger the higher the elongational rate. A slight enhancement of strain hardening was found by increasing the concentration from 0.5 to 1 g/l. The stress optical coefficient was determined as 1.8 × 10⁻⁷ Pa⁻¹ (0.5 g/l) and 1.2 × 10⁻⁷ Pa⁻¹ (1 g/l).

A New Approach to the Fundamental Theorem of Surface Theory

The fundamental theorem of surface theory classically asserts that, if a field of positive-definite symmetric matrices (a _αβ ) of order two and a field of symmetric matrices (b _αβ ) of order two together satisfy the Gauss and Codazzi-Mainardi equations in a simply connected open subset ω of , then there exists an immersion such that these fields are the first and second fundamental forms of the surface , and this surface is unique up to proper isometries in . The main purpose of this paper is to identify new compatibility conditions, expressed again in terms of the functions a _αβ and b _αβ , that likewise lead to a similar existence and uniqueness theorem. These conditions take the form of the matrix equation

Non-autonomous 2D Navier–Stokes System with Singularly Oscillating External Force and its Global Attractor

We study the global attractor of the non-autonomous 2D Navier–Stokes (N.–S.) system with singularly oscillating external force of the form . If the functions g ₀(x, t) and g ₁ (z, t) are translation bounded in the corresponding spaces, then it is known that the global attractor is bounded in the space H, however, its norm may be unbounded as since the magnitude of the external force is growing. Assuming that the function g ₁ (z, t) has a divergence representation of the form where the functions (see Section 3), we prove that the global attractors of the N.–S. equations are uniformly bounded with respect to for all . We also consider the “limiting” 2D N.–S. system with external force g ₀(x, t). We have found an estimate for the deviation of a solution of the original N.–S. system from a solution u ⁰(x, t) of the “limiting” N.–S. system with the same initial data. If the function g ₁ (z, t) admits the divergence representation, the functions g ₀(x, t) and g ₁ (z, t) are translation compact in the corresponding spaces, and , then we prove that the global attractors converges to the global attractor of the “limiting” system as in the norm of H. In the last section, we present an estimate for the Hausdorff deviation of from of the form: in the case, when the global attractor is exponential (the Grashof number of the “limiting” 2D N.–S. system is small).      

On exact solutions of a class of fractional Euler–Lagrange equations

In this paper, first a class of fractional differential equations are obtained by using the fractional variational principles. We find a fractional Lagrangian L(x(t), where _a ^c D _t ^α x(t)) and 0<α<1, such that the following is the corresponding Euler–Lagrange

The Semigeostrophic Equations Discretized in Reference and Dual Variables

We study the evolution of a system of n particles in . That system is a conservative system with a Hamiltonian of the form , where W ₂ is the Wasserstein distance and μ is a discrete measure concentrated on the set . Typically, μ(0) is a discrete measure approximating an initial L ^∞ density and can be chosen randomly. When d  =  1, our results prove convergence of the discrete system to a variant of the semigeostrophic equations. We obtain that the limiting densities are absolutely continuous with respect to the Lebesgue measure. When converges to a measure concentrated on a special d–dimensional set, we obtain the Vlasov–Monge–Ampère (VMA) system. When, d = 1 the VMA system coincides with the standard Vlasov–Poisson system.     

An Example of Finite-time Singularities in the 3d Euler Equations

Let be the exterior of the closed unit ball. Consider the self-similar Euler system

The Asymptotic Finite-dimensional Character of a Spectrally-hyperviscous Model of 3D Turbulent Flow

We obtain attractor and inertial-manifold results for a class of 3D turbulent flow models on a periodic spatial domain in which hyperviscous terms are added spectrally to the standard incompressible Navier–Stokes equations (NSE). Let P _m be the projection onto the first m eigenspaces of A =−Δ, let μ and α be positive constants with α ≥3/2, and let Q _m =I − P _m , then we add to the NSE operators μ A _φ in a general family such that A _φ≥Q _m A ^α in the sense of quadratic forms. The models are motivated by characteristics of spectral eddy-viscosity (SEV) and spectral vanishing viscosity (SVV) models. A distinguished class of our models adds extra hyperviscosity terms only to high wavenumbers past a cutoff λ _m0 where m ₀ ≤ m, so that for large enough m ₀ the inertial-range wavenumbers see only standard NSE viscosity. We first obtain estimates on the Hausdorff and fractal dimensions of the attractor (respectively and ). For a constant K _α on the order of unity we show if μ ≥ ν that and if μ ≤ ν that where ν is the standard viscosity coefficient, l ₀ = λ₁^−1/2 represents characteristic macroscopic length, and is the Kolmogorov length scale, i.e. where is Kolmogorov’s mean rate of dissipation of energy in turbulent flow. All bracketed constants and K _α are dimensionless and scale-invariant. The estimate grows in m due to the term λ _m /λ₁ but at a rate lower than m ^3/5, and the estimate grows in μ as the relative size of ν to μ. The exponent on is significantly less than the Landau–Lifschitz predicted value of 3. If we impose the condition , the estimates become for μ ≥ ν and for μ ≤ ν. This result holds independently of α, with K _α and c _α independent of m. In an SVV example μ ≥ ν, and for μ ≤ ν aspects of SEV theory and observation suggest setting for 1/c within α orders of magnitude of unity, giving the estimate where c _α is within an order of magnitude of unity. These choices give straight-up or nearly straight-up agreement with the Landau–Lifschitz predictions for the number of degrees of freedom in 3D turbulent flow with m so large that (e.g. in the distinguished-class case for m ₀ large enough) we would expect our solutions to be very good if not virtually indistinguishable approximants to standard NSE solutions. We would expect lower choices of λ _m (e.g. with a > 1) to still give good NSE approximation with lower powers on l ₀/l _ε, showing the potential of the model to reduce the number of degrees of freedom needed in practical simulations. For the choice , motivated by the Chapman–Enskog expansion in the case m = 0, the condition becomes , giving agreement with Landau–Lifschitz for smaller values of λ _m then as above but still large enough to suggest good NSE approximation. Our final results establish the existence of a inertial manifold for reasonably wide classes of the above models using the Foias/Sell/Temam theory. The first of these results obtains such an of dimension N > m for the general class of operators A _φ if α > 5/2. The special class of A _φ such that P _m A _φ = 0 and Q _m A _φ ≥ Q _m A ^α has a unique spectral-gap property which we can use whenever α ≥ 3/2 to show that we have an inertial manifold of dimension m if m is large enough. As a corollary, for most of the cases of the operators A _φ in the distinguished-class case that we expect will be typically used in practice we also obtain an , now of dimension m ₀ for m ₀ large enough, though under conditions requiring generally larger m ₀ than the m in the special class. In both cases, for large enough m (respectively m ₀), we have an inertial manifold for a system in which the inertial range essentially behaves according to standard NSE physics, and in particular trajectories on are controlled by essentially NSE dynamics.      

Radial Solutions and Phase Separation in a System of Two Coupled Schrödinger Equations

We consider the nonlinear elliptic system

Rotating Navier-Stokes Equations in $${\mathbb R}^{3}_{+}$$ with Initial Data Nondecreasing at Infinity: The Ekman Boundary Layer Problem

We prove time local existence and uniqueness of solutions to a boundary layer problem in a rotating frame around the stationary solution called the Ekman spiral. We choose initial data in the vector-valued homogeneous Besov space for 2 <  p <  ∞. Here the L ^p -integrability is imposed in the normal direction, while we may have no decay in tangential components, since the Besov space contains nondecaying functions such as almost periodic functions. A crucial ingredient is theory for vector-valued homogeneous Besov spaces. For instance we provide and apply an operator-valued bounded H ^∞-calculus for the Laplacian in for a general Banach space .     

Countable Compacta Admitting Homeomorphisms with Positive Sequence Entropy

Let (X, T) be a topological dynamical system (TDS), Ω(T) be the set of non-wondering points and be the topological sequence entropy. In this paper, an example on a countable compactum X with is given. Then for TDSs on countable compacta X, it is proved that when d(X) ≤ 1, ; and when d(X) ≥ 2, there exists a homeomorphism T on X such that X ^d is the sequence entropy set of (X, T), where d(X) and X ^d are the derived degree of X and the set of all accumulation points of X respectively. Dedicated to Professor Zhifen Zhang on the occasion of her 80th birthday     

Heat Flow of Harmonic Maps Whose Gradients Belong to L^{n}_{x}L^{\infty}_{t}

For any compact n-dimensional Riemannian manifold (M, g) without boundary, a compact Riemannian manifold without boundary, and 0 < T ≦ +∞, we prove that for n ≧ 4, if u : M × (0, T] → N is a weak solution to the heat flow of harmonic maps such that , then u ∈C ^∞(M × (0, T], N). As a consequence, we show that for n ≧3, if 0 < T < +∞ is the maximal time interval for the unique smooth solution u ∈C ^∞(M × [0, T), N) of (1.1), then blows up as t ↑ T.     

Solutions of the Dirac–Fock Equations and the Energy of the Electron-Positron Field

We consider atoms with closed shells, i.e. the electron number N is 2, 8, 10,..., and weak electron-electron interaction. Then there exists a unique solution γ of the Dirac–Fock equations with the additional property that γ is the orthogonal projector onto the first N positive eigenvalues of the Dirac–Fock operator . Moreover, γ minimizes the energy of the relativistic electron-positron field in Hartree–Fock approximation, if the splitting of into electron and positron subspace is chosen self-consistently, i.e. the projection onto the electron-subspace is given by the positive spectral projection of. For fixed electron-nucleus coupling constant g:=α Z we give quantitative estimates on the maximal value of the fine structure constant α for which the existence can be guaranteed.     

Resolvent Estimates and Maximal Regularity in Weighted L q -spaces of the Stokes Operator in an Infinite Cylinder

Let be an infinite cylinder of , n ≥ 3, with a bounded cross-section of C ^1,1-class. We study resolvent estimates and maximal regularity of the Stokes operator in for 1 < q, r < ∞ and for arbitrary Muckenhoupt weights ω ∈ A _r with respect to x′ ∈ Σ. The proofs use an operator-valued Fourier multiplier theorem and techniques of unconditional Schauder decompositions based on the -boundedness of the family of solution operators for a system in Σ parametrized by the phase variable of the one-dimensional partial Fourier transform. Supported by the Gottlieb Daimler- und Karl Benz-Stiftung, grant no. S025/02-10/03.     

Momentum and heat transfer from an asymmetrically confined circular cylinder in a plane channel

Unsteady momentum and heat transfer from an asymmetrically confined circular cylinder in a plane channel is numerically investigated using FLUENT for the ranges of Reynolds numbers as 10≤Re≤500, of the blockage ratio as 0.1≤β≤0.4, and of the gap ratio as 0.125≤γ≤1 for a constant value of the Prandtl number of 0.744. The transition of the flow from steady to unsteady (characterized by critical Re) is determined as a function of γ and β. The effect of γ on the mean drag and lift coefficients, Strouhal number (St), and Nusselt number (Nu _w ) is studied. Critical Re was found to increase with decreasing γ for all values of β. and St were found to increase with decreasing values of γ for fixed β and Re. The effect of decrease in γ on was found to be negligible for all blockage ratios investigated.     

High Reynolds Number Flow past a Plate Mounted at a Small Angle of Attack

A plane incompressible fluid flow past a plate mounted in a homogeneous stream at a small angle of attack ^* is investigated on the basis of an asymptotic analysis of the Navier-Stokes equations at high Reynolds numbers (Re). In the neighborhood of the leading edge the flow structure is studied in detail. It is found that separation is initiated in a small vicinity of the leading edge at and the length of the slow reverse stream zone is of the order O(1) at . The nonuniqueness of the solution is detected at and the hysteresis phenomenon is explained. It is shown that under certain conditions the solutions obtained also hold for flows past bodies of small thickness.     

Least Supersolution Approach to Regularizing Free Boundary Problems

In this paper, we study a free boundary problem obtained as a limit as ε → 0 to the following regularizing family of semilinear equations , where β _ε approximates the Dirac delta in the origin and F is a Lipschitz function bounded away from 0 and infinity. The least supersolution approach is used to construct solutions satisfying geometric properties of the level surfaces that are uniform in ε. This allows to prove that the free boundary of a limit has the “right” weak geometry, in the measure theoretical sense. By the construction of some barriers with curvature, the classification of global profiles of the blow-up analysis is carried out and the limit functions are proven to be viscosity and pointwise solution ( almost everywhere) to a free boundary problem. Finally, the free boundary is proven to be a C ^1,α surface around almost everywhere point. An erratum to this article can be found at     

Positive Solutions of Boundary Value Problems for Second-Order Singular Nonlinear Differential Equations

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