Internal Travelling Waves in the Limit of a Discontinuously Stratified Fluid

We consider internal travelling waves in a perfect stratified fluid, in the singular limit case when smooth stratifications approach a discontinuous two-layer profile. Our analysis concerns two-dimensional waves of small amplitude, propagating in an infinite horizontal strip of finite depth. The problems with smooth or discontinuous stratification are formulated as a unifying spatial evolution problem, where the stratification ρ plays the role of a functional parameter. The vector field is not smooth with respect to ρ, but has some weak continuity. When the Froude number is close to a critical value, we reduce the problem to one on a center manifold in a neighborhood of the trivial state independent of ρ (for the usual topology). Considering a weaker topology, we prove the continuity in ρ of the center manifold. Then the small solutions are described by an ordinary differential equation in ?², which depends continuously on ρ in the C ^k norm.     

Osgood’s Lemma and Some Results for the Slightly Supercritical 2D Euler Equations for Incompressible Flow

We investigate the (slightly) super-critical two-dimensional Euler equations. The paper consists of two parts. In the first part we prove well-posedness in C ^s spaces for all s > 0. We also give growth estimates for the C ^s norms of the vorticity for ${0 < s \leqq 1}$ . In the second part we prove global regularity for the vortex patch problem in the super-critical regime. This paper extends the results of Chae et al. where they prove well-posedness for the so-called LogLog-Euler equation. We also extend the classical results of Chemin and Bertozzi–Constantin on the vortex patch problem to the slightly supercritical case. Both problems we study in the setting of the whole space.     

Stress concentration in a uniformly rotating circular plate weakened by an elliptic hole and two rectilinear cuts

We consider a uniformly rotating isotropic plate whose cross-section is a two-connected domain bounded from the outside by a circle of radius R and from inside by an ellipse with semiaxes a and b. The rectilinear cuts coming to the ellipse are located symmetrically on the real axis OX. On the imaginary axis OY, two point masses m are located at the distances ±id from the center of the plate. Such a plate can be considered as a plate under the action of bulk forces (forces of inertia), X, Y, and under the action of two lumped tensile forces P = mdw ² applied at the points ±id.     

Quasi-Periodic Stability of Subfamilies of an Unfolded Skew Hopf Bifurcation

In the skew Hopf bifurcation a quasi-periodic attractor with nontrivial normal linear dynamics loses hyperbolicity. The simplest setting concerns rotationally symmetric diffeomorphisms of S ¹×R ². Their dynamics involve periodicity, quasi-periodicity and chaos, including mixed spectrum. The present paper deals with the persistence under symmetry-breaking of quasi-periodic invariant circles in this bifurcation. It turns out that, when adding sufficiently many unfolding parameters, the invariant circle persists for a large Hausdorff measure subset of a submanifold in parameter space. Accepted: September 3, 1999     

Poisson–Nernst–Planck Systems for Ion Flow with Density Functional Theory for Hard-Sphere Potential: I–V Relations and Critical Potentials. Part II: Numerics

We consider a one-dimensional steady-state Poisson–Nernst–Planck type model for ionic flow through membrane channels. Improving the classical Poisson–Nernst–Planck models where ion species are treated as point charges, this model includes ionic interaction due to finite sizes of ion species modeled by hard sphere potential from the Density Functional Theory. The resulting problem is a singularly perturbed boundary value problem of an integro-differential system. We examine the problem and investigate the ion size effect on the current–voltage (I–V) relations numerically, focusing on the case where two oppositely charged ion species are involved and only the hard sphere components of the excess chemical potentials are included. Two numerical tasks are conducted. The first one is a numerical approach of solving the boundary value problem and obtaining I–V curves. This is accomplished through a numerical implementation of the analytical strategy introduced by Ji and Liu in [Poisson–Nernst–Planck systems for ion flow with density functional theory for hard-sphere potential: I–V relations and critical potentials. Part I: Analysis, J. Dyn. Differ. Equ. (to appear)]. The second task is to numerically detect two critical potential values V _c and V ^c .The existence of these two critical values is first realized for a relatively simple setting and analytical approximations of V _c and V ^c are obtained in the above mentioned reference. We propose an algorithm for numerical detection of V _c and V ^c without using any analytical formulas but based on the defining properties and numerical I–V curves directly. For the setting in the above mentioned reference, our numerical values for V _c and V ^c agree well with the analytical predictions. For a setting including a nonzero permanent charge in which case no analytic formula for the I–V relation is available now, our algorithms can still be applied to find V _c and V ^c numerically.     

Thermal waves induced by a rapidly moving line source

For the cases involving a fast moving heat source or extremely short pulses emitted by lasers or short time after the start of transients, the classical theory of heat conduction breaks down since the wave nature of heat transport dominates. In this study, the temperature field due to a fast moving line source was determined analytically using the wave concept. The results are given for different values of thermal Mach number (M=V/C). For M>1 the heat affected zone is confined in a wedge shape region behind the source. The wedge half angle is equal to sin^?1 (1/M). It was confirmed that the difference between the results of diffusion and wave models depends on the corresponding time scale and the relaxation time.     

Periodic Motions of Stokes and Navier–Stokes Flows Around a Rotating Obstacle

We prove the existence and uniqueness of periodic motions to Stokes and Navier–Stokes flows around a rotating obstacle \({D \subset \mathbb{R}^3}\) with the complement \({\Omega = \mathbb{R}^3 \backslash D}\) being an exterior domain. In our strategy, we show the C _b -regularity in time for the mild solutions to linearized equations in the Lorentz space \({L^{3,\infty}(\Omega)}\) (known as weak-L ³ spaces) and prove a Massera-typed Theorem on the existence and uniqueness of periodic mild solutions to the linearized equations in weak-L ³ spaces. We then use the obtained results for such equations and the fixed point argument to prove such results for Navier–Stokes equations around a rotating obstacle. We also show the stability of such periodic solutions.     

Effect of surfactant concentration on saturated flow boiling in vertical narrow annular channels

Saturated flow boiling of environmentally acceptable nonionic surfactant solutions of Alkyl (8–16) was compared to that of pure water. The concentration of surfactant solutions was in the range of 100–1000 ppm. The liquid flowed in an annular gap of 2.5 and 4.4 mm between two vertical tubes. The heat was transferred from the inner heated tube to two-phase flow in the range of mass flux from 5 to 18 kg/m² s and heat flux from 40 to 200 kW/m². Boiling curves of water were found to be heat flux and channel gap size dependent but essentially mass flux independent. An addition of surfactant to the water produced a large number of bubbles of small diameter, which, at high heat fluxes, tend to cover the entire heater surface with a vapor blanket. It was found that the heat transfer increased at low values of relative surfactant concentration C/C₀, reaches a maximum close to the value of C/C₀ = 1 (where C₀ = 300 ppm is the critical micelle concentration) and decreased with further increase in the amount of additive. The dependence of the maximal values of the relative heat transfer enhancement, obtained at the value of relative concentration of C/C₀ = 1, on the boiling number Bo may be presented as single curve for both gap sizes and the whole range of considered concentrations.     

Differential Systems with Feedback: Time Discretizations and Lyapunov Functions

In this paper we examine a class of Eulerian time discretizations for a monotone cyclic feedback system with a time delay; see Mallet-Paret and Sell (1996a, 1996b) for background information. We construct an integer-valued function V for the discrete-time problem. The Main Theorem shows that V is a Lyapunov function, that is, V(x ⁿ⁺¹)≤V(x ⁿ ) along a solution {x ⁿ }^∞ _n=0, where the time steps can be relatively large.     

Poisson–Nernst–Planck Systems for Ion Flow with Density Functional Theory for Hard-Sphere Potential: I–V Relations and Critical Potentials. Part I: Analysis

In this work, we analyze a one-dimensional steady-state Poisson–Nernst–Planck type model for ionic flow through a membrane channel including ionic interactions modeled from the Density Functional Theory in a simple setting: Two oppositely charged ion species are involved with electroneutrality boundary conditions and with zero permanent charge, and only the hard sphere component of the excess (beyond the ideal) electrochemical potential is included. The model can be viewed as a singularly perturbed integro-differential system with a parameter resulting from a dimensionless scaling of the problem as the singular parameter. Our analysis is a combination of geometric singular perturbation theory and functional analysis. The existence of a solution of the model problem for small ion sizes is established and, treating the sizes as small parameters, we also derive an approximation of the I–V (current–voltage) relation. For this relatively simple situation, it is found that the ion size effect on the I–V relation can go either way—enhance or reduce the current. More precisely, there is a critical potential value V _c so that, if V > V _c , then the ion size enhances the current; if V < V _c , it reduces the current. There is another critical potential value V ^c so that, if V > V ^c , the current is increasing with respect to λ =? r ₂/r ₁ where r ₁ and r ₂ are, respectively, the radii of the positively and negatively charged ions; if V < V ^c , the current is decreasing in λ. To our knowledge, the existence of these two critical values for the potential was not previously identified.     

Isoparametric shear spring element applied to crack patching and instability

In this work the isoparametric shear spring element is applied to the stress and energy analysis of a center-crack panel reinforced by a rectangular patch. In this model, only transverse shears are assumed to prevail in the adhesive layer. The stresses and crack-tip stress intensity factors are obtained for reinforcement on both sides and one side of the panel, and are found to be in agreement with those obtained by previous authors using the triangular shear spring element.Crack stability that tends to vary with patch thickness is determined from the local and global maximum of the minimum strain energy density function denoted, respectively, as [(dW/dV)_min^max]_L at point L and [(dW/dV)_min^max]_G at point G. The distance l between L and G gives the prospective path of subcritical crack growth and its magnitude provides a measure of the degree of crack stability. A patched panel with small l tends to be more stable than that with large l. By increasing the patch thickness beyond a certain value, l can be contained within the patch such that failure, if initiated, will be highly localized. Such a behavior is exhibited. Numerical results are provided to support the foregoing conclusion.     

Extending Circle Mappings to the Annulus

We show that any monotone degree one C mapping of the circle can be realized as the induced mapping on the boundary by a C area preserving diffeomorphism of the open annulus. The circle mapping may have critical points, allowing the possibility of Denjoy behavior on the boundary.     

Asymptotic Behavior of Global Classical Solutions to the Mixed Initial-Boundary Value Problem for Quasilinear Hyperbolic Systems with Small BV Data

In this paper, we investigate the asymptotic behavior of global classical solutions to the mixed initial-boundary value problem with small BV data for linearly degenerate quasilinear hyperbolic systems with general nonlinear boundary conditions in the half space {(t,x)|t≥0,x≥0}. Based on the existence result on the global classical solution, we prove that when t tends to the infinity, the solution approaches a combination of C ¹ traveling wave solutions, provided that the C ¹ norm of the initial and boundary data is bounded and the BV norm of the initial and boundary data is sufficiently small. Applications to quasilinear hyperbolic systems arising in physics and mechanics, particularly to the system describing the motion of the relativistic string in the Minkowski space-time R ¹⁺ⁿ , are also given.     

Bifurcation of traveling waves resulting from resonant mode crossings in oceanic currents

We consider parallel currents in a rotating system. The current has a finite width and is bounded by density fronts on both sides. It has been shown that such flows have instabilities associated with a 1:1 resonance, i.e. the merging of two real wave speeds which then become complex. In this paper, we analyze bifurcations resulting from these instabilities. Two types of bifurcations arise; one is a reversible Hopf bifurcation, and the other is a reversible Hopf bifurcation with O(2)-symmetry.     

Universal Moduli of Continuity for Solutions to Fully Nonlinear Elliptic Equations

This paper provides universal, optimal moduli of continuity for viscosity solutions to fully nonlinear elliptic equations F(X, D ² u) =  f(X), based on the weakest and borderline integrability properties of the source function f in different scenarios. The primary result established in this work is a sharp Log-Lipschitz estimate on u based on the L ⁿ norm of f, which corresponds to optimal regularity bounds for the critical threshold case. Optimal C ^1,α regularity estimates are also delivered when ${f\in L^{n+\varepsilon}}$ . The limiting upper borderline case, ${f \in L^\infty}$ , also has transcendental importance to elliptic regularity theory and its applications. In this paper we show, under the convexity assumption on F, that ${u \in C^{1,{\rm Log-Lip}}}$ , provided f has bounded mean oscillation. Once more, such an estimate is optimal. For the lower borderline integrability condition allowed by the theory, we establish interior a priori estimates on the ${C^{0,\frac{n-2\varepsilon}{n-\varepsilon}}}$ norm of u based on the L ^n-ε norm of f, where ? is the Escauriaza universal constant. The exponent ${\frac{n-2\varepsilon}{n-\varepsilon}}$ is optimal. When the source function f lies in L ^q , n > q > n?ε, we also obtain the exact, improved sharp Hölder exponent of continuity.     

Hydrodynamic forces on a rotating sphere

The wake dynamics of a rotating sphere with prescribed rotation axis angles are quantitatively analysed by carrying out numerical simulations at Reynolds numbers of Re = 100, 250 and 300, non-dimensional rotational rates Ω¹ = 0–1 and rotation axis angles α = 0, π/6, π/3 and π/2 measured from the free stream axis. These parameters are the same as those in an earlier study (Poon et al., 2010, Int. J. Heat Fluid Flow) where the instantaneous flow structures were discussed qualitatively. This study extends the findings of the earlier study by employing phase diagrams (C_Lx, C_Ly) and (C_D, C_L) to provide a quantitative analysis of the time-dependent behaviour of the flow structures. At Re = 300 and Ω¹ = 0.05, the phase diagrams (C_Lx, C_Ly) show ‘saw tooth’ patterns for both α = 0 and π/6. The ‘saw tooth’ pattern indicates that the flow structures comprise a higher frequency oscillation component at a Reynolds number of 300 which is not observed until Re ≈ 800 for a stationary sphere. This ‘saw tooth’ pattern disappears as Ω¹ increases. The employment of the phase diagrams also reveals that different flow structures induce different oscillation amplitudes on both lateral force coefficients. With the exception of the vortices formed from a shear layer instability, all other flow regimes show larger fluctuations in C_L than C_D.     

Periodic solutions to some problems of n-body type

We prove the existence of at least one T-periodic solution to a dynamical system of the type $$ - m_i \ddot u_i = \sum\limits_{j = 1,j \ne i}^n {\triangledown V_{ij} (u_i - u_j ,{\text{ }}t)}$$ (1) where the potentials V _ij are T-periodic in t and singular at the origin, u _i ε R ^k i=1, ..., n, and k≧3. We also provide estimates on the H ¹ norm of this solution. The proofs are based on a variant of the Ljusternik-Schnirelman method. The results here generalize to the n-body problem some results obtained by Bahri & Rabinowitz on the 3-body problem in [6].     

Heteroclinic Cycles and Homoclinic Closures for Generic Diffeomorphisms

We prove some C ¹ generic results about orbit-connecting, in particular about heteroclinic cycles and homoclinic closures. As a consequence we obtain a three-ways C ¹ density theorem: Diffeomorphisms with either infinitely many weakly transitive components or a heterodimensional cycle are C ¹ dense in the complement of the C ¹ closure of Axiom A and no-cycle diffeomorphisms.     

Alternate pacing of border-collision period-doubling bifurcations

Unlike classical bifurcations, border-collision bifurcations occur when, for example, a fixed point of a continuous, piecewise C ¹ map crosses a boundary in state space. Although classical bifurcations have been much studied, border-collision bifurcations are not well understood. This paper considers a particular class of border-collision bifurcations, i.e., border-collision period-doubling bifurcations. We apply a subharmonic perturbation to the bifurcation parameter, which is also known as alternate pacing, and we investigate the response under such pacing near the original bifurcation point. The resulting behavior is characterized quantitatively by a gain, which is the ratio of the response amplitude to the applied perturbation amplitude. The gain in a border-collision period-doubling bifurcation has a qualitatively different dependence on parameters from that of a classical period-doubling bifurcation. Perhaps surprisingly, the differences are more readily apparent if the gain is plotted versus the perturbation amplitude (with the bifurcation parameter fixed) than if plotted versus the bifurcation parameter (with the perturbation amplitude fixed). When this observation is exploited, the gain under alternate pacing provides a useful experimental tool to identify a border-collision period-doubling bifurcation.     

Multiple Timescales Analysis for 1:2 and 1:3 Resonant Hopf Bifurcations

The postcritical behavior of a generaln-dimensional system around a resonant double Hopf bifurcation isanalyzed. Both cases in which the critical eigenvalues are in ratios of1:2 and 1:3 are investigated. The Multiple Scale Method is employedto derive the bifurcation equations systematically in terms of thederivatives of the original vector field evaluated at the criticalstate. Expansions of the n-dimensional vector of state variables andof a three-dimensional vector of control parameters are performed interms of a unique perturbation parameter ε, of the order ofthe amplitude of motion. However, while resonant terms only appear atthe ε³-order in the 1:3 case, they already arise at theε²-order in the 1:2 case. Thus, by truncating theanalysis at the ε³-order in both cases, first orsecond-order bifurcation equations are respectively drawn, the latterrequiring resort to the reconstitution principle. A two-degrees-of-freedom system undergoing resonant double Hopf bifurcations isstudied. The complete postcritical scenario is analyzed in terms of thethree control parameters and the asymptotic results are compared withexact numerical integrations for both resonances. Branches of periodicas well as periodically modulated solutions are found and theirstability analyzed.