Reflection and Transmission of Plane Waves at a Water–Porous Sediment Interface with a Double-Porosity Substrate

This paper investigates the wave propagation at the interface between the ocean and the ocean floor. The ocean floor is assumed to be composed of covered porous sediment with an underlying double-porosity substrate. For this purpose, plane wave reflection and transmission in the coupled water–porous sediment–double-porosity substrate system are analytically solved in terms of displacement potentials. Using numerical examples, the effects of the material properties of the underlying double-porosity substrate on the reflection coefficients are discussed in detail. Variations in pore and fracture fluid, fracture volume fraction, and permeability coefficients are considered. In addition, two cases of boundary conditions at the porous sediment–double-porosity substrate interface, i.e., sealed-pore boundary and open-pore boundary, are compared in the numerical calculations. Results show that material property variations in the double-porosity substrate may significantly affect the reflected wave in the overlying water if the sandwiched sediment depth is less than the critical value.     

Joule–Thomson Effects on the Flow of Liquid Water

We present a revised form of the energy balance for the coupled thermodynamics of liquid water flowing in porous media and give examples of situations where a commonly used formulation based on transport of enthalpy leads to erroneous results. Assuming negligible contribution from kinetic energy as well as sources and sinks such as energy from radioactive decay, total energy conservation is reduced to a balance between changes in internal energy, enthalpy, conductive heat flux, and gravitational potential energy. The Joule–Thomson coefficient is defined as the change in temperature with respect to an increase in pressure at constant enthalpy. Because liquid water has a negative Joule–Thomson coefficient at low temperatures, at a constant gravitational potential water cools as it compresses and heats as it expands. If one ignores the gravitational energy, transport of enthalpy alone leads to water heating by 2 \(^\circ \) C per kilometer as it is brought up from depth. The corrected energy balance transports methalpy, which is enthalpy plus gravitational potential energy. Although the simpler form leads to small changes in the temperature profile for typical simulations, there are several instances where this effect may prove to be important. The most important impact of the erroneous form is probably in the field of geothermal energy production, where the creation of a few degrees of heat in a simulation could lead to miscalculation of power plant efficiencies.     

On the Darcy–Brinkman–Boussinesq Flow in a Thin Channel with Irregularities

In this paper, we investigate the effects of a small boundary perturbation on the non-isothermal fluid flow through a thin channel filled with porous medium. Starting from the Darcy–Brinkman–Boussinesq system and employing asymptotic analysis, we derive a higher-order effective model given by the explicit formulae. To observe the effects of the boundary irregularities, we numerically visualize the asymptotic approximation for the temperature, whereas the justification and the order of accuracy of the model is provided by the theoretical error analysis.

     

On the Darcy–Brinkman Flow Through a Channel with Slightly Perturbed Boundary

The goal of this paper is to study the effects of a slightly perturbed boundary on the Darcy–Brinkman flow through a porous channel. We start from a rectangular domain and then perturb the upper part of its boundary by the product of the small parameter \(\epsilon \) and arbitrary smooth function h. Using asymptotic analysis with respect to \(\epsilon \), the effective model has been formally derived. Being in the form of the explicit formulae for the velocity and pressure, the asymptotic approximation clearly shows the nonlocal effects of the small boundary perturbation. The error analysis is also conducted providing the order of accuracy of the asymptotic solution.     

Vapor Flow with Evaporation–Condensation on Solid Particles

Strongly nonequilibrium vapor (gas) flows in a region filled by solid particles are considered with allowance for particlesize variation due to evaporation–condensation on the particle surface. The study is performed by directly solving the kinetic Boltzmann equation with allowance for the transformation of the distribution function of gas molecules due to their interaction with dust particles.     

Analysis of the Cahn–Hilliard Equation with a Relaxation Boundary Condition Modeling the Contact Angle Dynamics

We analyze the Cahn–Hilliard equation with a relaxation boundary condition modeling the evolution of an interface in contact with the solid boundary. An L ^∞ estimate is established which enables us to prove the global existence of the solution. We also study the sharp interface limit of the system. The dynamic of the contact point and the contact angle are derived and the results are compared with the numerical simulations.     

Vortex Dynamics for the Nonlinear Maxwell–Klein–Gordon Equation

We derive the vortex dynamics for the nonlinear Maxwell–Klein–Gordon equation with the Ginzburg–Landau type potential. In particular, we consider the case when the external electric fields are of order \({O( | \log \epsilon |^{\frac{1}{2}})}\). We study the convergence of the space–time Jacobian \({\partial_t \psi \cdot i \nabla \psi}\) as an interaction term between the vortices and electric fields. An explicit form of the limiting vector measure is shown.     

On the Beavers–Joseph Interface Condition for Non-parallel Coupled Channel Flow over a Porous Structure at High Reynolds Numbers

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Sloping Saturated–Unsaturated Flow with Outflow at Seepage Face

Analytic series solutions are constructed for the phreatic free surface problem of two-dimensional steady downslope saturated–unsaturated flow, with water exiting at a seepage face. The region in free parameter space is delineated for which the water table intersects the upper surface, and the steady state with uniform constant irrigation rate, ceases to exist. The flow solution is extended to a case of the domain being more general than a parallelogram, with the upper and lower boundaries being piecewise linear. This geometry resembles that of large-scale rainfall simulators that are designed to test slope stability of wetted soil beds.     

Subsonic Flow for the Multidimensional Euler–Poisson System

We establish the existence and stability of subsonic potential flow for the steady Euler–Poisson system in a multidimensional nozzle of a finite length when prescribing the electric potential difference on a non-insulated boundary from a fixed point at the exit, and prescribing the pressure at the exit of the nozzle. The Euler–Poisson system for subsonic potential flow can be reduced to a nonlinear elliptic system of second order. In this paper, we develop a technique to achieve a priori \({C^{1,\alpha}}\) estimates of solutions to a quasi-linear second order elliptic system with mixed boundary conditions in a multidimensional domain enclosed by a Lipschitz continuous boundary. In particular, we discovered a special structure of the Euler–Poisson system which enables us to obtain \({C^{1,\alpha}}\) estimates of the velocity potential and the electric potential functions, and this leads us to establish structural stability of subsonic flows for the Euler–Poisson system under perturbations of various data.     

Stability of the Front under a Vlasov–Fokker–Planck Dynamics

We consider a kinetic model for a system of two species of particles interacting through a long range repulsive potential and a reservoir at given temperature. The model is described by a set of two coupled Vlasov–Fokker–Plank equations. The important front solution, which represents the phase boundary, is a stationary solution on the real line with given asymptotic values at infinity. We prove the asymptotic stability of the front for small symmetric perturbations.     

Dynamics of a mass–spring–pendulum system with vastly different frequencies

We investigate the dynamics of a simple pendulum coupled to a horizontal mass?Cspring system. The spring is assumed to have a very large stiffness value such that the natural frequency of the mass?Cspring oscillator, when uncoupled from the pendulum, is an order of magnitude larger than that of the oscillations of the pendulum. The leading order dynamics of the autonomous coupled system is studied using the method of Direct Partition of Motion (DPM), in conjunction with a rescaling of fast time in a manner that is inspired by the WKB method. We particularly study the motions in which the amplitude of the motion of the harmonic oscillator is an order of magnitude smaller than that of the pendulum. In this regime, a pitchfork bifurcation of periodic orbits is found to occur for energy values larger that a critical value. The bifurcation gives rise to nonlocal periodic and quasi-periodic orbits in which the pendulum oscillates about an angle between zero and ??/2 from the down right position. The bifurcating periodic orbits are nonlinear normal modes of the coupled system and correspond to fixed points of a Poincare map. An approximate expression for the value of the new fixed points of the map is obtained. These formal analytic results are confirmed by comparison with numerical integration.     

Ginzburg–Landau Vortex Dynamics with Pinning and Strong Applied Currents

We study a mixed heat and Schrödinger Ginzburg–Landau evolution equation on a bounded two-dimensional domain with an electric current applied on the boundary and a pinning potential term. This is meant to model a superconductor subjected to an applied electric current and electromagnetic field and containing impurities. Such a current is expected to set the vortices in motion, while the pinning term drives them toward minima of the pinning potential and “pins” them there. We derive the limiting dynamics of a finite number of vortices in the limit of a large Ginzburg–Landau parameter, or \({\varepsilon \to 0}\) , when the intensity of the electric current and applied magnetic field on the boundary scale like \({|{\rm log} \, \varepsilon|}\) . We show that the limiting velocity of the vortices is the sum of a Lorentz force, due to the current, and a pinning force. We state an analogous result for a model Ginzburg–Landau equation without magnetic field but with forcing terms. Our proof provides a unified approach to various proofs of dynamics of Ginzburg–Landau vortices.     

Dynamics in two nonsmooth predator–prey models with threshold harvesting

Considering a good pest control program should reduce the pest to levels acceptable to the public, we investigate the threshold harvesting policy on pests in two predator–prey models. Both models are nonsmooth and the aim of this paper is to provide how threshold harvesting affects the dynamics of the two systems. When the harvesting threshold is larger than some positive level, the harvesting does not affect the ecosystem; when the harvesting threshold is less than the level, the model has complex dynamics with multiple coexistence equilibria, limit cycle, bistability, homoclinic orbit, saddle-node bifurcation, transcritical bifurcation, subcritical and supercritical Hopf bifurcation, Bogdanov–Takens bifurcation, and discontinuous Hopf bifurcation. Firstly, we provide the complete stability analysis and bifurcation analysis for the two models. Furthermore, some numerical simulations are given to illustrate our results. Finally, it is found that harvesting lowers the level of both species for natural enemy–pest system while raises the densities of both species for the pest–crop system. It is seen that the threshold harvesting policy of the enemy system is more effective than the crop system.     

Flow Through Super-elliptic Ducts Filled with a Darcy–Brinkman Medium

The fully developed flow and constant flux heat transfer in super-elliptic ducts filled with a porous (Darcy–Brinkman) medium are studied. Super-elliptic ducts resemble rectangular ducts with rounded corners. An efficient Ritz method is used to determine the velocity and temperature fields. Extensive tables for friction factor–Reynolds number product and Nusselt number are given.     

Dynamics of worm-like micelles: the Cox–Merz rule

The viscoelastic behaviour of worm-like micelles in small-amplitude oscillatory, steady simple shear and uniaxial extensional flows are analyzed with a model that couples the Oldroyd-B constitutive equation with a kinetic equation that accounts for the structural changes induced by the flow. In some cases, the constitutive equation predicts a viscoelastic behaviour that is consistent with the Cox–Merz rule. Departures from this rule are also predicted. Experimental data obtained for two worm-like micellar systems indicate that in these solutions, the Cox–Merz rule is not usually followed, in agreement with the predictions of our model. In uniaxial extensional flow, the model predicts a strain hardening in the extensional viscosity at low extensional rates and a strain-thinning at high extensional rates.     

Slip–Brinkman Flow Through Corrugated Microannulus with Stationary Random Roughness

This paper presents a boundary perturbation method of the Brinkman-extended Darcy model to investigate the flow in corrugated microannuli cylindrical tubes with slip surfaces. The stationary random model is used to mimic the surface roughness of the cylindrical walls. The tube is filled with a porous medium. We shall consider the two cases where corrugations are either perpendicular or parallel to the flow, and particular attention is given to the effect of the phase shift. The effects of the corrugations on the flow rate and pressure gradient are investigated as functions of wavelength, the permeability of the medium, the radius ratio and the slip parameter. Particular surface roughnesses are examined as special cases of stationary random surface. It is found that the effect of the partial slip is significant on the corrugation functions. The limiting cases of Stokes and Darcy’s flows and no-slip case are discussed.     

Flow around λ wings with a bend in the leading edge

In recent years a considerable number of studies have been published on flow around wings at high supersonic velocities. The researches have been conducted in two directions: there are studies of hypersonic flow around wings of traditional shape and a search is carried out for new types of lay-out which possess optimal aerodynamic characteristics. The second direction relates to the numerous studies of flow around wings with shaped transverse cross sections [1–7]. The calculation of the aerodynamic quality of a shaped delta wing composed of plane surfaces on the basis of the relationships on an oblique shock [1, 2], from the results of experiments on the pressure distribution and from weight tests [3, 4], showed that the shaped wing has a higher quality than the plane delta wing.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 171–175, January–February, 1985.     

Dynamics of a delayed diffusive predator–prey model with hyperbolic mortality

This paper is devoted to consider a time-delayed diffusive prey–predator model with hyperbolic mortality. We focus on the impact of time delay on the stability of positive constant solution of delayed differential equations and positive constant equilibrium of delayed diffusive differential equations, respectively, and we investigate the similarities and differences between them. Our conclusions show that when time delay continues to increase and crosses through some critical values, a family of homogenous and inhomogeneous periodic solutions emerge. Particularly, we find the minimum value of time delay, which is often hard to be found. We also consider the nonexistence and existence of steady state solutions to the reaction–diffusion model without time delay.