We investigate the integrability conditions of a class of shear-free perfect-fluid cosmological models within the framework of anisotropic fluid sources, applying our results to [Formula: see text] dark energy models. Generalizing earlier general relativistic results for timelike geodesics, we extend the potential and acceleration terms of the quasi-Newtonian formulation of integrable dust cosmological models about a linearized Friedmann–Lemaître–Robertson–Walker background and derive the equations that describe their dynamical evolutions. We show that in general, models with an anisotropic fluid source are not consistent, but because of the particular form the anisotropic stress [Formula: see text] takes in [Formula: see text] gravity, the general integrability conditions in this case are satisfied.

The current paper presents a theoretical analysis of the transport of solutes through a fixed-film membrane bioreactor (MBR), immobilised with an active biocatalyst. The dimensionless convection-diffusion equation with variable coefficients was solved analytically and numerically for concentration profiles of the solutes through the MBR. The analytical solution makes use of regular perturbation and accounts for radial convective flow as well as axial diffusion of the substrate species. The Michaelis-Menten (or Monod) rate equation was assumed for the sink term, and the perturbation was extended up to second-order. In the analytical solution only the first-order limit of the Michaelis-Menten equation was considered; hence the linearized equation was solved. In the numerical solution, however, this restriction was lifted. The solution of the nonlinear, elliptic, partial differential equation was based on an implicit finite-difference method (FDM). An upwind scheme was employed for numerical stability. The resulting algebraic equations were solved simultaneously using the multivariate Newton-Raphson iteration method. The solution allows for the evaluation of the effect on the concentration profiles of (i) the radial and axial convective velocity, (ii) the convective mass transfer rates, (iii) the reaction rates, (iv) the fraction retentate, and (v) the aspect ratio.

This paper presents an analytical model of substrate mass transfer through the lumen of a membrane bioreactor. The model is a solution of the convective-diffusion equation in two dimensions using a regular perturbation technique. The analysis accounts for radial-convective flow as well as axial diffusion of the substrate specie. The model is applicable to the different modes of operation of membrane bioreactor (MBR) systems (e.g., dead-end, open-shell, or closed-shell mode), as well as the vertical or horizontal orientation. The first-order limit of the Michaelis-Menten equation for substrate consumption was used to test the developed model against available analytical results. The results obtained from the application of this model, along with a biofilm growth kinetic model, will be useful in the derivation of an efficiency expression for enzyme production in an MBR.

We pursue a (1+3)-covariant analysis of cosmological peculiar velocity of pressure-free matter induced by the matter density perturbations in modified f(R) gravity theories. Instead of working in a quasi-Newtonian Eulerian frame, we select a curvature comoving Lagrangian frame, which means the shear and heat energy flux are non-zero. This is part of ongoing work on the end-structure of a generic over-density in f(R) gravity.

We investigate the conditions for a bounce to occur in Friedmann Robertson -Walker cosmologies for the class of fourth order gravity theories. The general bounce criterion is determined and constraints on the parameters of three specific models are given in order to obtain bounces solutions. Furthermore, unlike the case of General Relativity a bounce appears to be possible in open and flat cosmologies.

Innovation in biotechnology research has resulted in a number of fungi being identified for diverse industrial applications. Much research has been done in developing optimised membrane bioreactor (MBR) systems for the cultivation of these fungi as a consequence of their potent industrial applications. This research has been hampered by the lack of a thorough understanding of the fluid mechanics through these devices. In this article, analytical and numerical solutions to the Navier–Stokes equations were developed to describe the hydrostatic pressure and velocity profiles in a single fiber membrane gradostat reactor (SFMGR). A generic equation for low wall Reynolds number (Rew=ρvwrH/μ) flows was developed and solved for the case of negligible angular variations of the flow profiles. The mathematical expressions were proposed as solutions to transient state, laminar, incompressible, viscous and isothermal flow through a membrane with a variable hydraulic permeability. These profiles were developed for the lumen and shell sides, taking into account the osmotic pressure and gel formation that occurs when solute particles are rejected on the membrane. The models developed are applicable to different orientations and configurations. A numerical scheme, with a complete stability analysis, was developed to complement the analytical models. The models were tested on a vertically orientated MGR, operated in the dead-end mode. The model solutions gave profiles that are in agreement with the experimental results.

Many cosmological scenarios envisage either a bounce of the universe at early times, or collapse of matter locally to form a black hole which re-expands into a new expanding universe region. Energy conditions preclude this happening for ordinary matter in general relativistic universes, but scalar or dilatonic fields can violate some of these conditions, and so could possibly provide bounce behaviour. In this paper we show that such bounces cannot occur in Kantowski–Sachs models without violating the reality condition . This also holds true for other isotropic spatially homogeneous Bianchi models, with the exception of closed Friedmann–Robertson–Walker and Bianchi IX models; bounce behaviour violates the weak energy condition ρ ⩾ 0 and ρ + p ⩾ 0. We turn to the Randall–Sundrum type braneworld scenario for a possible resolution of this problem.

We investigate the conditions for a bounce to occur in Friedmann–Robertson–Walker cosmologies for the class of fourth-order gravity theories. The general bounce criterion is determined and constraints on the parameters of three specific models are given in order to obtain bounce solutions. Furthermore, unlike the case of general relativity, a bounce appears to be possible in open and flat cosmologies.

We investigate the conditions for a bounce to occur in Friedmann-Robertson-Walker cosmologies for the class of fourth order gravity theories. The general bounce criterion is determined and constraints on the parameters of three specific models are given in order to obtain bounces solutions. It is found that unlike the case of General Relativity, a bounce appears to be possible in open and flat cosmologies.

In Randall-Sundrum type braneworld cosmologies, the dynamical equations on the three-brane differ from the general relativity equations by terms that carry the effects of embedding and of the free gravitational field in the five-dimensional bulk. In a FRW ansatze for the metric, we present two methods for deriving inflationary solutions to the covariant non-linear dynamical equations for the gravitational and matter fields on the brane. In the first approach we examine the constraints on the dynamical relationship between the cosmological scale factor and the scalar field driving self-interaction potential, imposed by the weak energy condition. We then investigate inflationary solutions obtained from a scalar field superpotential. Both these techniques for solving the braneworld field equations are illustrated by flat curvature models.

We examine the string cosmology equations with a dilaton potential in the context of the pre-big-bang scenario with the desired scale factor duality, and give a generic algorithm for obtaining solutions with appropriate evolutionary properties. This enables us to find pre-big-bang type solutions with suitable dilaton behavior that are regular at $t=0,$ thereby solving the graceful exit problem. However, to avoid fine-tuning of initial data, an ``exotic'' equation of state is needed that relates the fluid properties to the dilaton field. We discuss why such an equation of state should be required for reliable dilaton behavior at late times.

We examine the string cosmology equations with a dilaton potential, contrasting the requirements that the equations and their solutions have the desired a → 1/a pre-big bang symmetry, and give a generic algorithm for obtaining solutions with desired evolutionary properties. This enables us to find pre-big bang type solutions that are regular at t = 0, and with suitable dilaton behaviour. However to avoid fine tuning of initial data, an ‘exotic’ equation of state is needed that relates the fluid properties to the dilaton field. We discuss why such an equation of state should be required for reliable dilaton behaviour at late times.

We consider the quantum analogs of Euclidean wormholes obtained by Carlini and Mijić (CM), who analytically continued recollapsing closed universe models. Using a perfect fluid matter source, we obtain asymptotically Euclidean (AE) wormholes when the strong energy condition is satisfied. Such wormholes are found to be consistent with the Hawking–Page (HP) conjecture for quantum wormholes as solutions of the Wheeler–DeWitt (WDW) equation. By simulating the equation of state of a perfect fluid with a real scalar field, quantum wormholes are also found when the strong energy condition is violated, although generally not AE. The non-AE solutions are interpreted as excited states of the quantum wormhole spectrum. Our results give support to the claim of HP that quantum wormhole solutions are a fairly general property of the WDW equation for various matter sources. Matter sources giving quantum wormholes could now include those expected in low energy effective string theory: a dilaton scalar field with exponential potential. This is unlike the classical case where such matter sources do not allow wormhole solutions. We finally contrast quantum wormhole solutions with other boundary conditions of quantum cosmology describing an inflationary earlier behavior and a resulting large Lorentzian universe phase.

Euclidean wormholes are obtained by the analytic continuation of closed recollapsing Friedmann-Robertson-Walker universes. We demonstrate this for a perfect fluid satisfying the strong energy condition. The quantum versions of such wormholes are consistent with the Hawking-Page (HP) conjecture for quantum wormholes as solutions of the Wheeler-DeWitt equation. We contrast this with a classical change of signature approach which, although might be consistent with the existence of classical wormholes for a given definition of the energy-momentum tensor of the fluid, upon quantization gives everywhere oscillatory wave functions which do not satisfy the HP conjecture.

We consider the quantum analogues of wormholes obtained by Carlini and Miji\'c (CM), who analytically continued closed universe models. To obtain wormholes when the strong energy condition ($\gamma>2/3$) is satisfied, we are able to simplify the Wheeler-DeWitt (WDW) equation by using an equivalent scalar potential which is a function of the scale factor. Such wormholes are found to be consistent with the Hawking-Page (HP) conjecture for quantum wormholes as solutions of the WDW equation. In addition to the CM type wormholes, for a scalar field realization of the potential in the WDW equation we also obtain quantum wormholes when the strong energy condition is violated. This violation can be up to an arbitrary large distance from the wormhole throat, before the violation eventually has to be relaxed in order to have a flat Euclidean space time. These results give support to the claim of HP that wormhole solutions are a fairly general property of the WDW equation. However, by allowing such solutions one might be precluding other more important properties such as a Lorentzian behaviour and a possible inflationary earlier stage of our universe.

We find the quantum analogues of wormholes obtained by Carlini and Miji´c (CM), who analytically continued closed universe models. The CM requirement that the strong energy condition (γ > 2/3) be satisfied is shown to be consistent with the Hawking-Page conjecture for quantum wormholes as solutions of the Wheeler-DeWitt equation. The presence of a cosmological constant Λ violates such a condition and so prevents wormholes occurring. It is therefore inconsistent to invoke these wormholes to make Λ a dynamical variable: used in arguments which suggest Λ → 0. We analyse instead a simple model with just Λ present. Differing results are obtained depending on the boundary conditions applied. In the Euclidean regime only the Hartle-Hawking boundary condition gives the factor exp(1/Λ) but is badly behaved for negative Λ. Tunneling boundary conditions suggest an initially large value for Λ. Whereas in the Lorentzian region all boundary conditions suggest an initially large value of Λ for spatial curvature k = 1. This differs from the previously obtained result of Strominger[28] for such models.

We find the quantum analogues of Carlini-Mijic wormholes and consider their application to the cosmological constant problem. In a simple model with $Λ$ only we differ with the results of Strominger.

We find the quantum analogues of wormholes obtained by Carlini and Miji´c (CM), who analytically continued closed universe models. The CM requirement that the strong energy condition ( γ > 2 / 3) be satisfied is shown to be consistent with the Hawking-Page conjecture for quantum wormholes as solutions of the Wheeler-DeWitt equation. The presence of a cosmological constant Λ violates such a condition and so prevents wormholes occurring. It is therefore inconsistent to invoke these wormholes to make Λ a dynamical variable: used in ar-guments which suggest Λ → 0. We analyse instead a simple model with just Λ present. Differing results are obtained depending on the boundary conditions applied. In the Euclidean regime only the Hartle-Hawking boundary condition gives the factor exp (1 / Λ) but is badly behaved for negative Λ . Tunneling boundary conditions suggest an initially large value for Λ. Whereas in the Lorentzian region all boundary conditions suggest an initially large value of Λ for spatial curvature k = 1. This differs from the previously obtained result of Strominger[28] for such models.