There are many fluid flow problems involving geometries for which a nonorthogonal curvilinear coordinate system may be the most suitable. To the authors’ knowledge, the Navier–Stokes equations for an incompressible fluid formulated in terms of an arbitrary nonorthogonal curvilinear coordinate system have not been given explicitly in the literature in the simplified form obtained herein. The specific novelty in the equations derived here is the use of the general Laplacian in arbitrary nonorthogonal curvilinear coordinates and the simplification arising from a Ricci identity for Christoffel symbols of the second kind for flat space. Evidently, however, the derived equations must be consistent with the various general forms given previously by others. The general equations derived here admit the well-known formulae for cylindrical and spherical polars, and for the purposes of illustration, the procedure is presented for spherical polar coordinates. Further, the procedure is illustrated for a nonorthogonal helical coordinate system. For a slow flow for which the inertial terms may be neglected, we give the harmonic equation for the pressure function, and the corresponding equation if the inertial effects are included. We also note the general stress boundary conditions for a free surface with surface tension. For completeness, the equations for a compressible flow are derived in an appendix.

Macroscale “continuum” level predictions are made by a new way to construct computationally efficient “wrappers” around fine-scale, microscopic, detailed descriptions of dynamical systems, such as molecular dynamics. It is often significantly easier to code a microscale simulator with periodicity: so the challenge addressed here is to develop a scheme that uses only a given periodic microscale simulator; specifically, one for atomistic dynamics. Numerical simulations show that applying a suitable proportional controller within “action regions” of a patch of atomistic simulation effectively predicts the macroscale transport of heat. Theoretical analysis establishes that such an approach will generally be effective and efficient, and also determines good values for the strength of the proportional controller. This work has the potential to empower systematic analysis and understanding at a macroscopic system level when only a given microscale simulator is available.

The human retina is supplied by two vascular systems: the highly vascular choroidal, situated behind the retina; and the retinal, which is dependent on the restriction that the light path must be minimally disrupted. Between these two circulations, the avascular retinal layers depend on diffusion of metabolites through the tissue. Oxygen supply to these layers may be threatened by diseases affecting microvasculature, for example diabetes and hypertension, which may ultimately cause loss of sight.

An accurate model of retinal blood flow will therefore facilitate the study of retinal oxygen supply and, hence, the complications caused by systemic vascular disease. Here, two simple models of the blood flow and exchange of hydrogen with the retina are presented and compared qualitatively with data obtained from experimental measurements. The models capture some interesting features of the exchange and highlight effects that will need to be considered in a more sophisticated model and in the interpretation of experimental results.