Deon Solomons

Bounce conditions in f(R) 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 bounce solutions. Furthermore, unlike the case of general relativity, a bounce appears to be possible in open and flat cosmologies.

Bounce behaviour in Kantowski–Sachs and Bianchi cosmologies

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.

A Solution of the Convective-Diffusion Equation for Solute Mass Transfer inside a Capillary Membrane Bioreactor

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.

Classical and quantum wormholes with perfect fluids and scalar fields

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.

Integrability conditions of quasi-Newtonian cosmologies in modified gravity

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 f(R) 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–Lemaitre–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 πab takes in f(R) gravity, the general integrability conditions in this case are satisfied.

A Finite-Difference Solution of Solute Transport through a Membrane Bioreactor

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.

Euclidean quantum wormholes with scalar fields

We consider the quantum analogs of Euclidean wormholes obtained by Carlini and Mijic (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.

Structure formation in modified gravity

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.