Weak selection and the separation of eco-evo time scales using perturbation analysis

We show that under the assumption of weak frequency-dependent selection a wide class of population dynamical models can be analysed using perturbation theory. The inner solution corresponds to the ecological dynamics, where to zeroth order, the genotype frequencies remain constant. The outer solution provides the evolutionary dynamics and corresponds, to zeroth order, to a generalisation of the replicator equation. We apply this method to a model of public goods dynamics and show that the error between the composite solution, which describes the dynamics for all times, and the solution to the full model scales linearly with the strength of selection.

Solution to Propagation of Shock Wave Based on Continuum Hypothesis

Based on the continuum hypothesis, problem of compressing wave was studied analytically. By exploring the temperature distribution, the propagation velocity, and the thickness of the transition region, the developing process of compressing wave turning into shock wave was revealed, and the following conclusions were reached: (1) the governing equations of compressing wave can be turned into two autonomous equations, and the phase diagram can be used as an effective tool for the analysis of the compressing wave; (2) the solution of compressing wave was composed of two parts: the inner solution and the outer solution; when Pr=3/4, the analytical inner solution can be obtained; (3) as the disturbance velocity increases, the thickness of compressing wave will decrease, and the propagation velocity of the compressing wave will increase, and the compressing wave will become shock wave when the disturbance velocity is large enough; (4) velocity and temperature across compressing wave change monotonically and continuously, but the entropy generation changing tendency is closely related to Pr; therefore, the inner solution reveals the mechanism of irreversibility happening in compressing waves.

Whitham modulation theory for the Ostrovsky equation

This paper derives the Whitham modulation equations for the Ostrovsky equation. The equations are then used to analyse localized cnoidal wavepacket solutions of the Ostrovsky equation in the weak rotation limit. The analysis is split into two main parameter regimes: the Ostrovsky equation with normal dispersion relevant to typical oceanic parameters and the Ostrovsky equation with anomalous dispersion relevant to strongly sheared oceanic flows and other physical systems. For anomalous dispersion a new steady, symmetric cnoidal wavepacket solution is presented. The new wavepacket can be represented as a solution of the modulation equations and an analytical solution for the outer solution of the wavepacket is given. For normal dispersion the modulation equations are used to describe the unsteady finite-amplitude wavepacket solutions produced from the rotation-induced decay of a Korteweg–de Vries solitary wave. Again, an analytical solution for the outer solution can be given. The centre of the wavepacket closely approximates a train of solitary waves with the results suggesting that the unsteady wavepacket is a localized, modulated cnoidal wavetrain. The formation of wavepackets from solitary wave initial conditions is considered, contrasting the rapid formation of the packets in anomalous dispersion with the slower formation of unsteady packets under normal dispersion.

Impact of Acetic Acid on the Survival ofL. plantarumupon Microencapsulation by Coaxial Electrospraying

In this work, coaxial electrospraying was used for the first time to microencapsulate probiotic bacteria, specificallyLactobacillus plantarum, within edible protein particles with the aim of improving their resistance to in vitro digestion. The developed structures, based on an inner core of whey protein concentrate and an outer layer of gelatin, were obtained in the presence of acetic acid in the outer solution as a requirement for the electrospraying of gelatin. Despite the limited contact of the inner suspension and outer solution during electrospraying, the combination of the high voltage used during electrospraying with the presence of acetic acid was found to have a severe impact on the lactobacilli, not only decreasing initial viability but also negatively affecting the survival of the bacteria during storage and their resistance to different stress conditions, including simulated in vitro digestion.

Adapting a Linear Potential Theory Solver for the Outer Hull to Account for Fluid Dynamics in Tanks

The linear boundary value problem for the wave dynamics inside a tank is very similar to the solution for the outer hull. Because of this, the boundary value solver for the outer hull can be re-used for the tank. The oscillating hydrostatic pressure in the tank may also be calculated in the same way as for the outer hull. Thereby, the hydrostatic coefficients from the tank can also be obtained from the outer solution. This makes it, in principle, easy to adapt outer solution computer code to also account for the inner solutions for all the tanks. The procedure is discussed by Newman (2005). We have used it in a different way, isolating the tank solution into more flexible independent sub-runs. This approach provides part-results for the tanks, like added mass and restoring from the tanks. It also has numerical benefits, with the possibility to reuse the calculations for tanks of equal geometrical shape. We have also extended the procedure to account for full tanks without waves and restoring effects. The linear tank fluid dynamics is programmed into a quite general hydrodynamic frequency domain solver, with the possibility of automatic transferring of local loads to structural (FEM) analysis. Results for local loads are presented. A simpler method of quasi-static loading in tanks is discussed, with comparison to the present method. Effects on global motions and local pressure coming from the tank dynamics contributions are pointed out, such as the shifted resonance of the vessel and the added mass which differs from rigid masses of the tanks.

Matched asymptotic solutions for turbulent plumes

Recent analytical investigations have shown that the vertical evolution of turbulent plumes variables can be derived straightforwardly from the knowledge of a single function $\Gamma (z)$ (called the plume function) which is the solution of a nonlinear differential equation. This article presents matched asymptotic solutions of this equation in the cases corresponding to highly lazy or highly forced plumes. First, it is shown that, far from the source, the asymptotic expression of the plume function can be derived by means of a perturbation method based on a Padé-like approximation. The resulting outer solution is invariant under translation (with respect to the vertical coordinate) so that we are led to the classical problem concerning the location of the plume (asymptotic) virtual origin. In order to determine this virtual origin location as a function of the conditions at the source, the far-field asymptotic solution is matched to an inner expansion of the solution which is valid near the source. Comparisons between these asymptotic solutions and numerical results are finally made in order to test their validity.

Re-entrant corner flows of Oldroyd-B fluids

The method of matched asymptotic expansions is used to construct solutions for the planar steady flow of Oldroyd-B fluids around re-entrant corners of angles π / α (1/2≤ α <1). Two types of similarity solutions are described for the core flow away from the walls. These correspond to the two main dominant balances of the constitutive equation, where the upper convected derivative of stress either dominates or is balanced by the upper convected derivative of the rate of strain. The former balance gives the incompressible Euler or inviscid flow equations and the latter balance the incompressible Navier–Stokes equations. The inviscid flow similarity solution for the core is that first derived by Hinch (Hinch 1993 J. Non-Newtonian Fluid Mech. 50 , 161–171) with a core stress singularity that depends upon the corner angle and radial distance as O ( r −2(1− α ) ) and a velocity behaviour that vanishes as O ( r α (3− α )−1 ). Extending the analysis of Renardy (Renardy 1995 J. Non-Newtonian Fluid Mech. 58 , 83–39), this outer solution is matched to viscometric wall behaviour for both upstream and downstream boundary layers. This structure is shown to hold for the majority of the retardation parameter range. In contrast, the similarity solution associated with the Navier–Stokes equations has a velocity behaviour O ( r λ ) where λ ∈(0,1) satisfies a nonlinear eigenvalue problem, dependent upon the corner angle and an associated Reynolds number defined in terms of the ratio of the retardation and relaxation times. This similarity solution is shown to hold as an outer solution and is matched into stress boundary layers at the walls which recover viscometric behaviour. However, the matching is restricted to values of the retardation parameter close to the relaxation parameter. In this case the leading order core stress is Newtonian with behaviour O ( r −(1− λ ) ).