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Finite volume methods (FVMs) constitute a popular class of methods for the numerical simulation of fluid flows. Among the various components of these methods, the discretisation of the gradient operator has received less attention despite its fundamental importance with regards to the accuracy of the FVM. The most popular gradient schemes are the divergence theorem (DT) (or Green-Gauss) scheme, and the least-squares (LS) scheme. Both are widely believed to be secondorder accurate, but the present study shows that in fact the common variant of the DT gradient is second-order accurate only on structured meshes whereas it is zeroth-order accurate on general unstructured meshes, and the LS gradient is second-order and first-order accurate, respectively. This is explained through a theoretical analysis and is confirmed by numerical tests. The schemes are then used within a FVM to solve a simple diffusion equation on unstructured grids generated by several methods; the results reveal that the zeroth-order accuracy of the DT gradient is inherited by the FVM as a whole, and the discretisation error does not decrease with grid refinement. On the other hand, use of the LS gradient leads to second-order accurate results, as does the use of alternative, consistent, DT gradient schemes, including a new iterative scheme that makes the common DT gradient consistent at almost no extra cost. The numerical tests are performed using both an in-house code and the popular public domain PDE solver OpenFOAM. This is the accepted version of the article published in:

The aim of the current study was the introduction and standardization of two experimental conditions for dynamic pupillometry. Pupillometry is a method that can provide valuable data concerning the functioning of the autonomous nervous system. The system for recording the pupil reaction was developed in the Laboratory of Clinical Neurophysiology of the 1st Department of Neurology of Aristotle University of Thessaloniki, in co-operation with the Laboratory of Fluid Mechanics of the Aristotle University of Thessaloniki. This system is fully automated. It includes an infra-red video camera, which has the capacity to record in complete darkness, and an SLE (clinical photic stimulator) lamp. A software application automatically performed all the procedures. During the first experiment, one flash was administered. During the second experiment, a series of 25 flashes (1 Hz frequency) was administered. Fifty physically and mentally healthy subjects aged 23-48 years took part in the study. Means, standard deviations and ranges for all variables characterizing normal subjects during both experimental conditions are reported. Test/re-test results and comparisons of the two eyes are also reported. The combined use of these two experimental conditions in dynamic pupillometry may be a very useful tool in medical research. There are already reports on the usefulness of pupillometry in the research of various diseases, including depression and Alzheimer's disease. It is expected that it will also be a valuable research tool in the study of diabetes, alcoholism, myasthenia gravis, cancer, multiple sclerosis, etc.

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