Multi-Fluid Theory and Cosmology: A Convective Variational Approach to Interacting Dark-Sector Bob Osano 1,2 and Timothy Oreta 1 1 Cosmology and Gravity Group, Department of Mathematics and Applied Mathematics, University of Cape Town (UCT), Rondebosch 7701, Cape Town, South Africa 2 Centre for Higher Education Development, University of Cape Town (UCT), Rondebosch 7701,Cape Town, South Africa (Dated: August 2, 2018) Abstract This article examines the foundation of the recently developed relativistic variational formalism[1]. Our work is heavily based on [2, 27] which extends this approach to the multi-fluid theory and examines its utility in astrophysics and cosmology. Unlike the extension to the formalism mentioned above, that looks at the general interaction between different types of matter, we use the formalism to examine the interaction involving, ordinary matter, dark matter (DM) and dark energy (DE). We focus on an entrainment phenomena involving the dark-sector constituents. 1. INTRODUCTION The theory of relativistic fluids has received consider- able attention for many years following the seminal work of Landau and Lif tshitz[3]. Interest in relativistic fluids is largely driven by its utility in astrophysics and cos- mology and the need to resolve some of the outstand- ing questions in these areas of study. The mathematical modelling process that is carried out in these studies, and which take into account physically plausible scenar- ios, presents several challenges. These fall into two broad categories; conceptual and theoretical. An awareness of the differences between these two categories is important as it provides a guide to a modeller when making deci- sions on assumptions that go into the modelling process, and when analysing the model. It is therefore, worth reviewing the differences between the two categories. The conceptual related issues have to do with the diffi- culty in identifying specific and measurable variables that give rise to a framework for characterising relativistic flu- ids. Issues classified under theoretical category have to do with the foundational theories, which in this study are theories of fluids, general relativity (GR), and ther- modynamics as applied to environments where there are more than one species of fluids. Connected to this is what might be termed as the unifying framework of how the species are treated; either single-fluid or multi-fluid. For example, the nature of interactions between the different species may affect how the mixture flows, effects that may only be captured in the multi-fluid treatment and not in the single-fluid theory. Examples of these are dissipa- tion and entrainment. Dissipative effects are known to be common and are found in flows involving heat flow in the presence of thermal resistance, in fluid flows with viscos- ity, diffusion, chemical reactions, and electric current flow in resistive media. These examples and others manifest in the lab environment, in astrophysics and predictably in cosmology. Dissipation has successfully been incor- porated and examined in the modelling of N ewtonian or non-relativistic fluids. But the same cannot be said of relativistic fluids (as pointed out by [2, 4, 5] for example). The obvious question is, what motivates the need to in- corporate dissipation in relativistic fluids and how would one go about doing this? The authors of [6] are prompted by the need to develop a formalism that could be used to study gravitational radiations emanating from compact objects; neutron stars in particular. Radiative processes in some of these astrophysical objects are known to be influenced by dissipation. Dissipation is often largely ne- glected in cosmology [7–9], but there are processes that occur during structure formation, and during reheating epochs in the early universe that suggest that dissipa- tion may play a role and hence should ideally be taken into account. The same can be said of heat flow in gen- eral [10–12] and DM dynamics [13, 14]. In order to ac- count for these, one needs to develop a formalism that incorporates them. Entrainment is understood to be the quantification of the ease with which neighbouring fluids species are able to move relative to each other. Unlike dissipation, entrainment is much less known or studied particularly for the subclass of relativistic fluids. We are motivated by the need to examine multi-fluid and entrainment effects involving the DM and DE. As pointed out in [15], the most interesting development in classical relativistic fluids dynamics is the consider- ation of multi-fluid systems that is composed of elements whose collective dynamics involve a superfluid, heat flow or the treatment of electromagnetic charge as a dynami- cal variable [2, 16, 17]. These developments are allowing such systems to be used to study a wider range of rel- evant phenomena. These developments have, however, been patchy and a general theory remains incomplete in at least two different respects. On one hand, they re- quire the inclusion of dissipation and on the other, the coupling of dissipation to electromagnetism. These de- velopments hold the key to the greater applicability of the fluid theory in both astrophysics and cosmology and need examination. Single-fluid approximation has suc- cessfully played a crucial role in our cosmological mod- elling of the universe. It is our contention that multi-fluid approximation is the more appropriate for cosmological modelling. The single-fluid approximation is ideally the limit of multi-fluid approximation. In this regard, the 1 arXiv:1807.00768v2 [gr-qc] 31 Jul 2018