Periodic solutions for one-dimensional nonlinear nonlocal problem with drift including singular nonlinearities

In this paper, we prove existence results of a one-dimensional periodic solution to equations with the fractional Laplacian of order $s\in (1/2,1)$ , singular nonlinearity and gradient term under various situations, including nonlocal contra-part of classical Lienard vector equations, as well other nonlocal versions of classical results know only in the context of second-order ODE. Our proofs are based on degree theory and Perron's method, so before that we need to establish a variety of priori estimates under different assumptions on the nonlinearities appearing in the equations. Besides, we obtain also multiplicity results in a regime where a priori bounds are lost and bifurcation from infinity occurs.

Fujita type global solvability to double nonlinear parabolic equation with nonlinear gradient term for the salt transfer process

In this paper we investigated the qualitative properties of non-negative and bounded continuous solutions to problem Cauchy for a degenerate parabolic equation with nonlinear gradient term. We based on splitting algorithm suggest estimate of weak solution for slowly, fast diffusion and critical cases, Fujita type global solvability of the Cauchy problem to degenerate type parabolic equation with nonlinear gradient term established. The theorems proven with comparison principle.

Effects of Pressure Gradient on Convective Heat Transfer in a Boundary Layer Flow of a Maxwell Fluid Past a Stretching Sheet

The pressure gradient term plays a vital role in convective heat transfer in the boundary layer flow of a Maxwell fluid over a stretching sheet. The importance of the effects of the term can be monitored by developing Maxwell’s equation of momentum and energy with the pressure gradient term. To achieve this goal, an approximation technique, i.e. Homotopy Perturbation Method (HPM) is employed with an application of algorithms of Adams Method (AM) and Gear Method (GM). With this approximation method we can study the effects of the pressure gradient (m), Deborah number (β), the ratio of the free stream velocity parameter to the stretching sheet parameter (ɛ) and Prandtl number (Pr) on both the momentum and thermal boundary layer thicknesses. The results have been compared in the absence and presence of the pressure gradient term m . It has an impact of thinning of the momentum and boundary layer thickness for non-zero values of the pressure gradient. The convergence of the system has been taken into account for the stretching sheet parameter ɛ. The result of the system indicates the significant thinning of the momentum and thermal boundary layer thickness in velocity and temperature profiles. On the other hand, some results show negative values of f '(η) and θ (η) which indicates the case of fluid cooling.

A unified description of gravity- and kinematics-induced segregation forces in dense granular flows

Particle segregation is common in natural and industrial processes involving flowing granular materials. Complex, and seemingly contradictory, segregation phenomena have been observed for different boundary conditions and forcing. Using discrete element method simulations, we show that segregation of a single particle intruder can be described in a unified manner across different flow configurations. A scaling relation for the net segregation force is obtained by measuring forces on an intruder particle in controlled-velocity flows where gravity and flow kinematics are varied independently. The scaling law consists of two additive terms: a buoyancy-like gravity-induced pressure gradient term and a shear rate gradient term, both of which depend on the particle size ratio. The shear rate gradient term reflects a kinematics-driven mechanism whereby larger (smaller) intruders are pushed toward higher (lower) shear rate regions. The scaling is validated, without refitting, in wall-driven flows, inclined wall-driven flows, vertical silo flows, and free-surface flows down inclines. Comparing the segregation force with the intruder weight results in predictions of the segregation direction that match experimental and computational results for various flow configurations.

Evolution of Strain Gradient Formulation in Capturing Localization Phenomena in Granular Materials

This paper highlights the use of incorporating strain gradient into flow stress to study localization behavior in materials. Pioneered by Zbib and Aifantis in the late 1980s, the formulation enabled incorporation of length scales into continuum formulations naturally. The formulation has also evolved into being able to study the effects of microstructure and heterogeneity on localization in granular materials. A multi-slip Mohr-Coulomb type plasticity model with the flow stress in the constitutive equation modified with a higher order gradient term of the effective plastic strain is used for this purpose. The possibility of abrupt changes of mobilized friction caused by intense shearing rate often leads to particle breakage. Its effects on localization is accounted for by modifying the material properties such as mobilized friction using a scaling parameter averaged over a representative elementary area. The change of shearing rate in the integration points was monitored through quasi-statistically measure parameter called inertia number. The inertia number was set to be all the time to consider quasi static less than l.0E-3. The formulation was implemented into a finite element code and used to simulate plane strain compression tests on dry sand. The model highlights effects of confining pressure, anisotropic microstructure, the non-coaxial angle between the direction of principal stress and principal plastic strain rate directions on shear band characteristics.

Instantons in quantum mechanics (QM)

Perturbative expansion can be generated by calculating Euclidean functional integrals by the steepest descent method always looking, in the absence of external sources, for saddle points in the form of constant solutions to the classical field equations. However, classical field equations may have non-constant solutions. In Euclidean stable field theories, non-constant solutions have always a larger action than minimal constant solutions, because the gradient term gives an additional positive contribution. The non-constant solutions whose action is finite, are called instanton solutions and are the saddle points relevant for a calculation, by the steepest descent method, of barrier penetration effects. This chapter is devoted to simple examples of non-relativistic quantum mechanics (QM), where instanton calculus is an alternative to the semi-classical Wentzel–Kramers–Brillouin (WKB) method. The role of instantons in some metastable systems in QM is explained. In particular, instantons determine the decay rate of metastable states in the semi-classical limit initially confined in a relative minimum of a potential and decaying through barrier penetration. The contributions of instantons at leading order for the quartic anharmonic oscillator with negative coupling are calculate explicitly. The method is generalized to a large class of analytic potentials, and explicit expressions, at leading order, for one-dimensional systems are obtained.

Inherent dissipation of upwind-biased potential temperature advection and its feedback on model dynamics

<p>Higher order upwind biased advection schemes are often used for potential temperature advection in dynamical cores of atmospheric models. The inherent diffusive and anti-diffusive fluxes are interpreted here as the effect of irreversible sub-gridscale dynamics. For those, total energy conservation and positive internal entropy production must be guaranteed. As a consequence of energy conservation, the pressure gradient term should be formulated in Exner pressure form. The presence of local antidiffusive fluxes in potential temperature advection schemes foils the validity of the second law of thermodynamics. Due to this failure, a spurious wind acceleration into the wrong direction is locally induced via the pressure gradient term. When correcting the advection scheme to be more entropically consistent, the spurious acceleration is avoided, but two side effects come to the fore: (i) the overall accuracy of the advection scheme decreases and (ii) the now purely diffusive fluxes become more discontinuous compared to the original ones, which leads to more sudden body forces in the momentum equation. Therefore the amplitudes of excited gravity waves from jets and fronts increase compared to the original formulation with inherent local antidiffusive fluxes.</p><p>The means used for supporting the argumentation line are theoretical arguments concerning total energy conservation and internal entropy production, pure advection tests, one-dimensional advection-dynamics interaction tests and evaluation of runs with a global atmospheric dry dynamical core.</p>

Efficient and accurate modeling of wave-driven flooding on coral reef-lined coasts: Case Study of Majuro Atoll, Republic of the Marshall Islands

<p>Many coral reef islands are low-lying, which in combination with population growth, sea level rise and possibly more frequent extreme weather events is likely to result in increased coastal risk (e.g. Storlazzi et al., 2015). On smaller scales of O(10 km) wave-driven coastal inundation can be accurately predicted with advanced models such as XBeach (Roelvink et al., 2009), at already high computational costs. For larger scales, larger number of islands, for scenario modelling, and for implementation in early warning systems, computationally faster methods are needed. Reduced physics models, which neglect some of the processes (e.g. non-hydrostatic pressure gradient term and viscosity), are a potential solution. However, their accuracy and the best method to force them has not been established.</p><p>In this research we propose a new methodology to model wave-driven flooding on coral reef-lined coasts. A look-up-table (LUT), composed of XBeach model runs, is combined with a reduced-physics model, SFINCS (Leijnse et al., 2021), to achieve high accuracy predictions at limited computational expense. The LUT consists of pre-run 1D XBeach simulations for several reef profiles from Scott et al. (2020), forced with different offshore wave and water level conditions. Wave conditions close to the shore as predicted by the LUT are used to force SFINCS which then simulates the wave runup, overtopping and flooding. These are forced in SFINCS using random wave timeseries from an interpolated parameterized wave spectrum following Athif (2020).</p><p>The accuracy of the method is investigated for 6 distinctive cross-shore profiles from Scott et al. (2020), for two wave scenarios (gentle swell and stormy conditions). Results of complete XBeach simulations are compared to LUT-SFINCS simulations with different boundary forcing locations. The sensitivity analysis shows that the preferred boundary location to initialize the SFINCS model is at a water depth between 0.5 m and 2.5 m, preferably shoreward of the reef edge. Errors introduced by the generated parameterized spectra lead to runup estimation errors of up to around 40% depending on reef geometry. The developed methodology will be applied to a case study of Majuro Island, the Republic of Marshall Islands, as proof of concept.</p><p> </p><p><strong>References</strong></p><p>Athif, A. A. (2020). Computationally efficient modelling of wave driven flooding in Atoll Islands: Investigation on the use of a reduced-physics model solver SFINCS. Master’s thesis, IHE, the Netherlands.</p><p>Leijnse, T., van Ormondt, M., Nederhoff, K., and van Dongeren, A. (2021). Modeling compound flooding in coastal systems using a computationally efficient reduced-physics solver: Including fluvial, pluvial, tidal, wind-and wave- driven processes. <em>Coastal Engineering</em>, 163:103796.</p><p>Roelvink, D., Reniers, A., Van Dongeren, A. P., De Vries, J. V. T., McCall, R., and Lescinski, J. (2009). Modelling storm impacts on beaches, dunes and barrier islands. <em>Coastal engineering</em>, 56(11-12), 1133-1152.</p><p>Scott, F., Antolinez, J. A. A., Mccall, R., Storlazzi, C., Reniers, A., and Pearson, S. (2020). Hydro-Morphological Characterization of Coral Reefs for Wave Runup Prediction. <em>Frontiers in Marine Science</em>, 7(May):1–20.</p><p>Storlazzi, C. D., Elias, E. P., and Berkowitz, P. (2015). Many atolls may be uninhabitable within decades due to climate change. <em>Scientific reports</em>, 5:14546.</p>