Long-time Solution Research Articles

Time-dependent Hermitian Yang–Mills flow on surfaces

We investigate a generalisation of Hermitian Yang–Mills flow in which the base metric itself is allowed to depend on the time-parameter. For technical reasons we restrict our attention to the case of a holomorphic vector bundles E over compact Riemann surfaces which is assumed to be slope stable with respect to a fixed Kahler class \(\tau \) on the base. We show that if \(\omega (t)\) is a family of Kahler metrics in \(\tau \) converging to a limit metric \(\omega _\infty \) at an exponential rate, then starting at a smooth initial Hermitian metric \(h_0\) on E, the Hermitian Yang–Mills flow with respect to the time-dependent metric \(\omega (t)\) admits a unique long-time solution h(t) converging exponentially to a \(\omega _\infty \)-Hermite–Einstein metric. The proof utilises an extension of Donadson’s techniques used to treat the case where the base metrics does not depend on the time-parameter.

The asymptotic behavior of symplectic mean curvature flow with pinched curvatures in [formula omitted

It was proved by Han–Li–Yang [4] that the mean curvature flow of symplectic surfaces with suitably pinched curvatures in CP2 has a longtime solution and converges to a holomorphic curve as the time approaches infinity. In this note, we give a refinement of this theorem. We prove that the evolving surface becomes more and more umbilical and the limit surface is in fact CP1. As a consequence, the initial surface is symplectically isotopic to CP1.

Unsteady free convection from a heated sphere in the presence of internal heat generation or absorption

This paper is concerned with an unsteady, laminar, free convective flow over a heated sphere with the effect of internal heat generation/absorption. The dimensionless governing equations have been solved employing the finite difference method as well as a perturbation method for short time and an asymptotic method for long time. We examine the effects of the physical parameters, such as, the Prandtl number, Pr, and the heat generation/absorption parameter, γ, on the friction factor and heat transfer rate as well as the velocity and temperature profiles. It is observed that when the Prandtl number, Pr, is increased, the friction factor decreases while the heat transfer rate increases. In the presence of internal heat generation, the friction factor increases while the heat transfer rate reduces. The reverse pattern is found with the heat absorption parameter. The momentum and thermal boundary layers become thicker with an increase of the heat generation parameter. A comparison among the numerical solutions, the perturbation solutions for short time and the asymptotic solutions for long time has been presented which provides a good agreement among the solutions.

On the nonexistence of non-constant exact periodic solutions in a class of the Caputo fractional-order dynamical systems

Using Fourier analysis as tool, we prove that the Caputo fractional-order derivative of the non-constant periodic function cannot be periodic, and thus we confirm the nonexistence of the non-constant exact periodic solutions in a class of the Caputo fractional-order initial-valued dynamical systems (including the autonomous case) from a different viewpoint. We also demonstrate with theoretical discussion and numerical examples that the non-constant long-time periodic solution might exist in the Caputo fractional-order systems. The investigation clarifies the difference and similarity of the periodic solutions between the Caputo fractional-order dynamical systems and the integer-order dynamical systems in a systematic way.

Evaluation of a Heuristic Model for Tropical Cyclone Resilience

Abstract This work examines the applicability of a previously postulated heuristic model for the temporal evolution of the small-amplitude tilt of a tropical cyclone–like vortex under vertical shear forcing for both a dry and cloudy atmosphere. The heuristic model hinges on the existence of a quasi-discrete vortex Rossby wave and its ability to represent the coherent precession and tilt decay of a stable vortex in the free-alignment problem. Linearized numerical solutions for a dry and cloudy vortex confirm the model predictions that an increase in the magnitude of the radial potential vorticity (PV) gradient within the vortex skirt surrounding the core yields a more rapid evolution of a sheared vortex toward the equilibrium, left-of-shear tilt configuration. However, in the moist-neutral limit, in which the effective static stability vanishes in rising and sinking regions, the heuristic model yields a poor approximation to the simulated vortex core evolution, but a left-of-shear tilt of the near-core vortex, radially beyond the heating region, remains the preferred long-time solution. Within the near-core skirt, the PV perturbation generated by vertical shearing exhibits continuous-spectrum-type vortex Rossby waves, features that are not captured by the heuristic model. Nevertheless, the heuristic model continues to predict the rapid vertical alignment and equilibrium, left-of-shear tilt configuration of the simulated near-core vortex in the moist-neutral limit.

Ebola-challenge and revival of theoretical epidemiology: Why Extrapolations from early phases of epidemics are problematic.

Ebola-challenge and revival of theoretical epidemiology: Why Extrapolations from early phases of epidemics are problematic.

On the J-flow in Sasakian manifolds

We study the space of Sasaki metrics on a compact manifold \(M\) by introducing an odd-dimensional analogue of the \(J\)-flow. That leads to the notion of critical metric in the Sasakian context. In analogy to the Kähler case, on a polarised Sasakian manifold, there exists at most one normalised critical metric. The flow is a tool for texting the existence of such a metric. We show that some results proved by Chen (Commun. Anal. Geom. 12: 837–852, 2004) can be generalised to the Sasakian case. In particular, the Sasaki \(J\) -flow is a gradient flow which has always a long-time solution minimising the distance on the space of Sasakian potentials of a polarised Sasakian manifold. The flow minimises an energy functional whose definition depends on the choice of a background transverse Kähler form \(\chi \). When \(\chi \) has nonnegative transverse holomorphic bisectional curvature, the flow converges to a critical Sasakian structure.

Time-convolutionless mode-coupling theory near the glass transition—A recursion formula and its asymptotic solutions

The time-convolutionless mode-coupling theory (TMCT) equation for the intermediate scattering function fα(q,t) derived recently by the present author is analyzed mathematically and numerically, where α=c stands for a collective case and α=s for a self case. All the mathematical formulations discussed by Götze for the MCT equation are then shown to be directly applicable to the TMCT equation. Firstly, it is shown that similarly to MCT, there exists an ergodic to non-ergodic transition at a critical point, above which the long-time solution fα(q,t=∞), that is, the so-called Debye–Waller factor fα(q), reduces to a non-zero value. The critical point is then shown to be definitely different from that of MCT. Secondly, it is also shown that there is a two-step relaxation process in a β stage near the critical point, which is described by the same two different power-law decays as those obtained in MCT. In order to discuss the asymptotic solutions, the TMCT equation is then transformed into a recursion formula for a cumulant function Kα(q,t)(=−ln[fα(q,t)]). By employing the same simplified model as that proposed by MCT, the simplified asymptotic recursion formula is then numerically solved for different temperatures under the initial conditions obtained from the simulations. Thus, it is discussed how the TMCT equation can describe the simulation results within the simplified model.

Time-stepping techniques to enable the simulation of bursting behavior in a physiologically realistic computational islet

Physiologically realistic simulations of computational islets of beta cells require the long-time solution of several thousands of coupled ordinary differential equations (ODEs), resulting from the combination of several ODEs in each cell and realistic numbers of several hundreds of cells in an islet. For a reliable and accurate solution of complex nonlinear models up to the desired final times on the scale of several bursting periods, an appropriate ODE solver designed for stiff problems is eventually a necessity, since other solvers may not be able to handle the problem or are exceedingly inefficient. But stiff solvers are potentially significantly harder to use, since their algorithms require at least an approximation of the Jacobian matrix. For sophisticated models, systems of several complex ODEs in each cell, it is practically unworkable to differentiate these intricate nonlinear systems analytically and to manually program the resulting Jacobian matrix in computer code. This paper demonstrates that automatic differentiation can be used to obtain code for the Jacobian directly from code for the ODE system, which allows a full accounting for the sophisticated model equations. This technique is also feasible in source-code languages Fortran and C, and the conclusions apply to a wide range of systems of coupled, nonlinear reaction equations. However, when we combine an appropriately supplied Jacobian with slightly modified memory management in the ODE solver, simulations on the realistic scale of one thousand cells in the islet become possible that are several orders of magnitude faster than the original solver in the software Matlab, a language that is particularly user friendly for programming complicated model equations. We use the efficient simulator to analyze electrical bursting and show non-monotonic average burst period between fast and slow cells for increasing coupling strengths. We also find that interestingly, the arrangement of the connected fast and slow heterogeneous cells impacts the peak bursting period monotonically.

Buoyant convection from a discrete source in a leaky porous medium

AbstractThe application of turbulent plume theory in describing the dynamics of emptying filling boxes, control volumes connected to an infinite exterior through a series of openings along the upper and lower boundaries, has yielded novel strategies for the natural ventilation of buildings. Making the plume laminar and having it fall through a porous medium yields a problem of fundamental significance in its own right, insights from which may be applied, for example, in minimizing the contamination of drinking water by geologically sequestered $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}{\mathrm{CO}}_2$ or the chemicals leached from waste piles. After reviewing the theory appropriate to rectilinear and axisymmetric plumes in porous media, we demonstrate how the model equations may be adapted to the case of an emptying filling box. In this circumstance, the long-time solution consists of two ambient layers, each of which has a uniform density. The lower and upper layers comprise fluid that is respectively discharged by the plume and advected into the box through the upper opening. Our theory provides an estimate for both the height and thickness of the associated interface in terms of, for example, the source volume and buoyancy fluxes, the outlet area and permeability, and the depth-average solute dispersion coefficient, which is itself a function of the far-field horizontal flow speed. Complementary laboratory experiments are provided for the case of a line source plume and show very good agreement with model predictions. Our measurements also indicate that the permeability, $k_f$, of the lower opening (or fissure) decreases with the density of the fluid being discharged, a fact that has been overlooked in some previous studies, wherein $k_f$ is assumed to depend only on the fissure geometry.

Analytical modeling of non-Fickian wave-diffusion of gas in heterogeneous media

An isothermal transient-state non-Fickian diffusion model is developed and analytically solved for description of gas dissolution in locally heterogeneous media suddenly exposed to a high pressure gas. The full-, short-, and long-time analytical solutions are used to establish the significance of the non-Fickian gas dissolution in heterogeneous media compared to the Fickian diffusion assumption. Parametric studies are carried out by means of the special analytical-solutions obtained for gas transport in the semi-infinite and finite-thickness heterogeneous media involving a delay time. The profiles of concentration and diffusion flux obtained for the non-Fickian wave-diffusion case are compared with the Fickian pure-diffusion case. The initial propagation of a right-running wave and its reflection from the wall are illustrated for the concentrations and diffusion fluxes. The small-time behavior is shown to be inherently wave-like and the discontinuity wave front propagates into the medium with the speed decreasing with time. For small times, the differences between the wave- and pure-diffusion cases are found to be significant depending on the magnitude of the delay time. For sufficiently large times, the wave behavior dies out and the wave solutions approach the equilibrium pure-diffusion solutions, except very near the decaying wave front. The formulations presented in this paper are of practical importance because they can be instrumental in determination of the diffusivity, interface surface mass-transfer coefficient, and rate of dissolution of gases in heterogeneous medium. A parameter estimation method is also proposed and elaborated for estimation of the diffusion and interface surface mass-transfer coefficients from measured pressure decay data.

Analysis of a model for foam improved oil recovery

AbstractDuring improved oil recovery (IOR), gas may be introduced into a porous reservoir filled with surfactant solution in order to form foam. A model for the evolution of the resulting foam front known as ‘pressure-driven growth’ is analysed. An asymptotic solution of this model for long times is derived that shows that foam can propagate indefinitely into the reservoir without gravity override. Moreover, ‘pressure-driven growth’ is shown to correspond to a special case of the more general ‘viscous froth’ model. In particular, it is a singular limit of the viscous froth, corresponding to the elimination of a surface tension term, permitting sharp corners and kinks in the predicted shape of the front. Sharp corners tend to develop from concave regions of the front. The principal solution of interest has a convex front, however, so that although this solution itself has no sharp corners (except for some kinks that develop spuriously owing to errors in a numerical scheme), it is found nevertheless to exhibit milder singularities in front curvature, as the long-time asymptotic analytical solution makes clear. Numerical schemes for the evolving front shape which perform robustly (avoiding the development of spurious kinks) are also developed. Generalisations of this solution to geologically heterogeneous reservoirs should exhibit concavities and/or sharp corner singularities as an inherent part of their evolution: propagation of fronts containing such ‘inherent’ singularities can be readily incorporated into these numerical schemes.

Analysis of capillary drainage from a flat solid strip

A long and narrow solid strip coated with a thin liquid layer is used as a model of a generic fluid mass probe in a spacecraft propellant tank just after a small thruster firing. The drainage dynamics of the initial coating layer into the settled bulk fluid affects the interpretation of probe measurements as the sensors' signal depends strongly on whether a sensor is in contact with vapor or with liquid. We analyze the drainage under various conditions of zero-gravity (i.e., capillary drainage) and with gravity aligned with the strip length, corresponding to the thruster acceleration. Long-time analytical solutions are found for zero and non-zero gravity. In the case with gravity, an approximate solution is found using matched asymptotics. Estimates show that a thrust of 10−3g0 significantly reduces drainage times.

Theory of electromagnetic drying: An asymptotic solution of an initial-boundary value problem for a cylinder

A long-time asymptotic analytical solution to a system of equations and boundary and initial conditions modeling electromagnetic drying has been constructed for a cylindrical region. The solution determines the temperature and moisture content fields in the during period in which the drying rate is constant. Calculations have been carried out for the maximum drying intensity regime with allowance made for the overeating of the material or its strain-induced breakup.

Development of doubled haploid mapping population for drought tolerant high chapati quality Indian wheat variety C306 using wheat x maize system

Doubled haploid (DH) technology allows production of pure homozygous lines in a single generation. Integration of DH technology in wheat breeding programmes and genetic studies has the potential to reduce the breeding time for new varieties and enhance our understanding of the genetics of important agronomic traits. A protocol was standardized for the production of DH wheat plants using wheat into maize hybridization system and a mapping population of sixty DH wheat lines was produced involving drought tolerant high chapati quality Indian wheat variety C306 and a poor quality variety WH157. Factors were identified for efficient production of DH plants, including early excision of the haploid embryos after pollination, low temperature treatment of the excised embryos and use of lower concentration of colchicine solution for longer time for the chromosome doubling of the haploid plantlets.

Effective Duration of Gas Nitriding Process on AISI 316L for the Formation of a Desired Thickness of Surface Nitrided Layer

High temperature gas nitriding performed on AISI 316L at the temperature of 1200°C. The microstructure of treated AISI 316L samples were observed to identify the formation of the microstructure of nitrided surface layer. The grain size of austenite tends to be enlarged when the nitriding time increases, but the austenite single phase structure is maintained even after the long-time solution nitriding. Using microhardness testing, the hardness values drop to the center of the samples. The increase in surface hardness is due to the high nitrogen concentration at or near the surface. At 245HV, the graph of the effective duration of nitriding process was plotted to achieve the maximum depth of nitrogen diffuse under the surface. Using Sigma Plot software best fit lines of the experimental result found and plotted to find out effective duration of nitriding equation as Y=1.9491(1-0.7947 x ), where Y is the thickness of nitrided layer below the surface and X is duration of nitriding process. Based on this equation, the duration of gas nitriding process can be estimated to produce desired thickness of nitrided layer.

Rayleigh–Taylor instability in a finite cylinder: linear stability analysis and long-time fingering solutions

AbstractWe consider the Rayleigh–Taylor instability problem of two initially stationary immiscible viscous fluids positioned with the denser above the less dense in a finite circular cylinder, such that their starting fluid–fluid interface is the horizontal midplane of the cylinder. The ensuing linear instability problem has a five-dimensional parameter space – defined by the density ratio, the viscosity ratio, the cylinder aspect ratio, the surface tension between the fluids and the ratio of viscous to gravitational time scales – of which we explore only part, motivated by recent experiments where viscous fluids exchange in vertical tubes (Beckett et al., J. Fluid Mech., 2011, vol. 682, pp. 652–670). We find that for these experiments, the instability is invariably ‘side-by-side’ (of azimuthal wavenumber 1 type) but we also uncover parameter regions where the preferred instability is axisymmetric. The fact that both ‘core-annular’ (axisymmetric) and ‘side-by-side’ (asymmetric) long-time flows are seen experimentally highlights the fact that the initial Rayleigh–Taylor instability of the interface does not determine the long-time flow configuration in these situations. Finally, long-time flow solutions are presented on the basis that they will be slowly varying fingering solutions.

Entraining gravity currents

AbstractEntrainment of ambient fluid into a gravity current, while often negligible in laboratory-scale flows, may become increasingly significant in large-scale natural flows. We present a theoretical study of the effect of this entrainment by augmenting a shallow water model for gravity currents under a deep ambient with a simple empirical model for entrainment, based on experimental measurements of the fluid entrainment rate as a function of the bulk Richardson number. By analysing long-time similarity solutions of the model, we find that the decrease in entrainment coefficient at large Richardson number, due to the suppression of turbulent mixing by stable stratification, qualitatively affects the structure and growth rate of the solutions, compared to currents in which the entrainment is taken to be constant or negligible. In particular, mixing is most significant close to the front of the currents, leading to flows that are more dilute, deeper and slower than their non-entraining counterparts. The long-time solution of an inviscid entraining gravity current generated by a lock-release of dense fluid is a similarity solution of the second kind, in which the current grows as a power of time that is dependent on the form of the entrainment law. With an entrainment law that fits the experimental measurements well, the length of currents in this entraining inviscid regime grows with time approximately as ${t}^{0. 447} $. For currents instigated by a constant buoyancy flux, a different solution structure exists in which the current length grows as ${t}^{4/ 5} $. In both cases, entrainment is most significant close to the current front.

Correlated two-photon scattering in cavity optomechanics

We present an exact analytical solution of the two-photon scattering in a cavity optomechanical system. This is achieved by solving the quantum dynamics of the total system, including the optomechanical cavity and the cavity-field environment, with the Laplace transform method. The long-time solution reveals detailed physical processes involved as well as the corresponding resonant photon frequencies. We characterize the photon correlation induced in the scattering process by calculating the two-photon joint spectrum of the long-time state. Clear evidence for photon frequency anti-correlation can be observed in the joint spectrum. In addition, we calculate the equal-time second-order correlation function of the cavity photons. The results show that the radiation pressure coupling can induce photon blockade effect, which is strongly modulated by the phonon sideband resonance. In particular, we obtain an explicit expression of optomechanical coupling strength determining these sideband modulation peaks based on the two-photon resonance condition.

Theory of reversible diffusion-influenced reactions with non-Markovian dissociation in two space dimensions.

We investigate the reversible diffusion-influenced reaction of an isolated pair in the presence of a non-Markovian generalization of the backreaction boundary condition in two space dimensions. Following earlier work by Agmon and Weiss, we consider residence time probability densities that decay slower than an exponential and that are characterized by a single parameter 0 < σ ≤ 1. We calculate an exact expression for a Green's function of the two-dimensional diffusion equation subject to a non-Markovian backreaction boundary condition that is valid for arbitrary σ and for all times. We use the obtained expression to derive the survival probability for the initially unbound pair and we calculate an exact expression for the probability S(t[line]*) that the initially bound particle is unbound. Finally, we obtain an approximate solution for long times. In particular, we show that the ultimate fate of the bound state is complete dissociation, as in the Markovian case. However, the limiting value is approached quite differently: Instead of a ~t(-1) decay, we obtain 1 - S(t[line]*) ~ t(-σ)ln t. The derived expressions should be relevant for a better understanding of reversible membrane-bound reactions in cell biology.