Abstract:
The locomotion of microorganisms in fluids is ubiquitous and plays an important role in numerous biological processes. In this chapter we present an overview of theoretical modeling for low-Reynolds-number locomotion.

Abstract:
The 3+1 (canonical) decomposition of all geometries admitting two-dimensional space-like surfaces is exhibited. A proposal consisting of a specific re-normalization {\bf Assumption} and an accompanying {\bf Requirement} is put forward, which enables the canonical quantization of these geometries. The resulting Wheeler-deWitt equation is based on a re-normalized manifold parameterized by three smooth scalar functionals. The entire space of solutions to this equation is analytically given, exploiting the freedom left by the imposition of the {\bf Requirement} and contained in the third functional.

Abstract:
The objective of the present paper is to develop a theoretical model describing the evolution of a turbulent wake behind a towed sphere in a stably stratified fluid at large Froude and Reynolds numbers. The wake flow is considered as a quasi two-dimensional (2-D) turbulent jet flow whose dynamics is governed by the momentum transfer from the mean flow to a quasi-2-D sinuous mode growing due to hydrodynamic instability. The model employs a quasi-linear approximation to describe this momentum transfer. The model scaling coefficients are defined with the use of available experimental data, and the performance of the model is verified by comparison with the results of a direct numerical simulation of a 2-D turbulent jet flow. The model prediction for the temporal development of the wake axis mean velocity is found to be in good agreement with the experimental data obtained by Spedding (1997).

Abstract:
The canonical analysis and subsequent quantization of the (2+1)-dimensional action of pure gravity plus a cosmological constant term is considered, under the assumption of the existence of one spacelike Killing vector field. The proper imposition of the quantum analogues of the two linear (momentum) constraints reduces an initial collection of state vectors, consisting of all smooth functionals of the components (and/or their derivatives) of the spatial metric, to particular scalar smooth functionals. The demand that the midi-superspace metric (inferred from the kinetic part of the quadratic (Hamiltonian) constraint) must define on the space of these states an induced metric whose components are given in terms of the same states, which is made possible through an appropriate re-normalization assumption, severely reduces the possible state vectors to three unique (up to general coordinate transformations) smooth scalar functionals. The quantum analogue of the Hamiltonian constraint produces a Wheeler-DeWitt equation based on this reduced manifold of states, which is completely integrated.

Abstract:
We discuss conceptual aspects of neutrino oscillations with the main emphasis on the field-theoretical approach. This approach includes the neutrino source and detector processes and allows to obtain the neutrino transition or survival probabilities as cross sections derived from the Feynman diagram of the combined source - detection process. In this context, the neutrinos which are supposed to oscillate appear as propagators of the neutrino mass eigenfields, connecting the source and detection processes. We consider also the question why the canonical neutrino oscillation formula is so robust against corrections and discuss the nature of the oscillating neutrino state emerging in the field-theoretical approach.

Abstract:
We define restricted ensembles, called pico-canonical ensembles, for a statistical-mechanical description of the metastable and glassy phases. In this approach, time-evolution is Markovian, with temperature dependent rates. Below a particular glass-temperature, the system is strongly non-ergodic, and the phase space breaks up into a large number of mutually disconnected sectors. Averages are calculated over states within one such sector, and then averaged over sectors. As a soluble example, we calculate these explicitly for a one dimensional lattice gas with nearest neighbor couplings.

Abstract:
The nonlinear evolution of the magnetorotational instability (MRI) in weakly ionized accretion disks, including the effect of the Hall term and ohmic dissipation, is investigated using local three-dimensional MHD simulations and various initial magnetic field geometries. When the magnetic Reynolds number, Re_M \equiv v_A^2 / \eta \Omega (where v_A is the Alfven speed, \eta the magnetic diffusivity, and \Omega the angular frequency), is initially larger than a critical value Re_{M, crit}, the MRI evolves into MHD turbulence in which angular momentum is transported efficiently by the Maxwell stress. If Re_M < Re_{M, crit}, however, ohmic dissipation suppresses the MRI, and the stress is reduced by several orders of magnitude. The critical value is in the range of 1 - 30 depending on the initial field configuration. The Hall effect does not modify the critical magnetic Reynolds number by much, but enhances the saturation level of the Maxwell stress by a factor of a few. We show that the saturation level of the MRI is characterized by v_{Az}^2 / \eta \Omega, where v_{Az} is the Alfven speed in the nonlinear regime along the vertical component of the field. The condition for turbulence and significant transport is given by v_{Az}^2 / \eta \Omega \gtrsim 1, and this critical value is independent of the strength and geometry of the magnetic field or the size of the Hall term. If the magnetic field strength in an accretion disk can be estimated observationally, and the magnetic Reynolds number v_A^2 / \eta \Omega is larger than about 30, this would imply the MRI is operating in the disk.

Abstract:
Several authors have now suggested that some interstellar clouds above the plane of the Galaxy are interacting with the Reynolds' layer, the warm ionized gas extending well above (H~910 pc) the Galactic plane (Reynolds 1993). Characterizing the interaction between these clouds and their surroundings should be useful in understanding one source of interstellar turbulence: vertical shear flows. This paper discusses how studies of the morphology and drag coefficient of falling clouds might be used to constrain the Reynolds number for the flow, and hence the effective viscosity of the warm ionized medium. If arguments based on morphology are correct, the effective viscosity of the warm ionized medium is significantly higher than the classical values. Possible resolutions to this problem are suggested.

Abstract:
(abridged) Context. Turbulent fluxes of angular momentum and heat due to rotationally affected convection play a key role in determining differential rotation of stars. Here we perform a systematic comparison between Cartesian and spherical geometries as a function of the rotation rate. Aims. We extend the earlier studies by using spherical wedges to obtain turbulent angular momentum and heat transport as functions of the rotation rate from stratified convection. We compare results from spherical and Cartesian models in the same parameter regime. In particular, we want to clarify whether the sharp equatorial profile of the horizontal Reynolds stress found in earlier Cartesian models is reproduced in spherical models. Methods. We employ direct numerical simulations of turbulent convection. In order to reach as high spatial resolution as possible in the spherical runs, we model only parts of the latitude and longitude. The rotational influence, measured by the Coriolis number, is varied from zero to roughly seven, which is the regime that is likely to be realised in the solar convection zone. Cartesian simulations are performed in overlapping parameter regimes. Results. For slow rotation we find that the radial and latitudinal turbulent angular momentum fluxes are directed inward and equatorward, respectively. In the rapid rotation regime the radial flux changes sign in contradiction with theory. The latitudinal flux remains mostly equatorward and develops a maximum close to the equator. In Cartesian simulations this peak can be explained by the strong `banana cells'. The latitudinal heat flux is mostly equatorward for slow rotation but changes sign for rapid rotation. The rotation profiles vary from anti-solar (slow equator) for slow and intermediate rotation to solar-like (fast equator) for rapid rotation. The solar-like profiles are dominated by the Taylor--Proudman balance.

Abstract:
Equations for correlation functions, here referred to as Reynolds- Kraichnan-Lewis equations (RKLE), are considered and their wide application is indicated. Perturbation and non-perturbation solutions are given. To elucidate a closure problem - various forms of equations are presented. Exact, closed equations for the projected correlation functions are derived.