A note on Stokes’ hypothesis

Buresti, Guido

doi:10.1007/s00707-015-1380-9

Cited by 37 publications

(18 citation statements)

References 13 publications

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“…This equation is solved in the upper half-space y ≥ 0 with the condition u(0, t) = 1 in a dimension of space; in two dimensions of space (x, y) with periodic conditions in x the solution is the same. According to the values of the attenuation coefficient of transverse waves α t the solution presents different behaviors, the spatial evolutions of the solutions are presented on the figure (2) for three values of the coefficient α t . It should be noted that, given the recent origin of the physical model, the values of this coefficient are not known; it is a physical parameter which must be measured experimentally or determined from the measures of attenuation in the various media.…”

Section: Transverse Diffusion Paradoxmentioning

confidence: 99%

An alternative to the concept of continuous medium

Caltagirone

2021

Acta Mech

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Discrete mechanics proposes an alternative formulation of the equations of mechanics where the Navier-Stokes and Navier-Lamé equations become approximations of the equation of discrete motion. It unifies the fields of fluid and solid mechanics by extending the fields of application of these equations to all space and time scales. This article presents the essential differences induced by the abandonment of the notion of continuous medium and global frame of reference. The results of the mechanics of continuous medium validated by fluid and solid observations are not questioned. The concept of continuous medium is not invalidated, the discrete formulation proposed simply widens the spectrum of the applications of the classical equations.The discrete equation of motion introduces several important modifications, in particular the fundamental law of the dynamics on an element of volume becomes a law of conservation of the accelerations on an edge. The acceleration considered as an absolute quantity is written as a sum of two components, one soledoidal the other irrotational according to a local orthogonal Helmholtz-Hodge decomposition. The mass is abandoned and replaced by the compression and rotation energies represented by the scalar and vectorial potentials of the acceleration. The equation of motion and all the physical parameters are expressed only with two fundamental units, those of length and time. The essential differences between the two approaches are listed and some of them are discussed in depth. This is particularly the case with the known paradoxes of the Navier-Stokes equation or the importance of inertia for the Navier-Lamé equation.

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Section: Transverse Diffusion Paradoxmentioning

confidence: 99%

An alternative to the concept of continuous medium

Caltagirone

2021

Acta Mech

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“…However, Stokes himself had serious doubts about this assumption. [44][45][46] In fact, all the experiments that are available contradict the Stokes assumption. Note that Reynolds, in literature, 4 used Stokes' assumption, see equation (9).…”

Section: Orcid Idmentioning

confidence: 99%

On lower-dimensional models in lubrication, Part A: Common misinterpretations and incorrect usage of the Reynolds equation

Almqvist

Burtseva

Rajagopal

et al. 2020

Proceedings of the Institution of Mechanical Engineers, Part J:

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Most of the problems in lubrication are studied within the context of Reynolds’ equation, which can be derived by writing the incompressible Navier-Stokes equation in a dimensionless form and neglecting terms which are small under the assumption that the lubricant film is very thin. Unfortunately, the Reynolds equation is often used even though the basic assumptions under which it is derived are not satisfied. One example is in the mathematical modelling of elastohydrodynamic lubrication (EHL). In the EHL regime, the pressure is so high that the viscosity changes by several orders of magnitude. This is taken into account by just replacing the constant viscosity in either the incompressible Navier-Stokes equation or the Reynolds equation by a viscosity-pressure relation. However, there are no available rigorous arguments which justify such an assumption. The main purpose of this two-part work is to investigate if such arguments exist or not. In Part A, we formulate a generalised form of the Navier-Stokes equation for piezo-viscous incompressible fluids. By dimensional analysis of this equation we, thereafter, show that it is not possible to obtain the Reynolds equation, where the constant viscosity is replaced with a viscosity-pressure relation, by just neglecting terms which are small under the assumption that the lubricant film is very thin. The reason is that the lone assumption that the fluid film is very thin is not enough to neglect the terms, in the generalised Navier-Stokes equation, which are related to the body forces and the inertia. However, we analysed the coefficients in front of these (remaining) terms and provided arguments for when they may be neglected. In Part B, we present an alternative method to derive a lower-dimensional model, which is based on asymptotic analysis of the generalised Navier-Stokes equation as the film thickness goes to zero.

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“…This closure is calculated using velocity derivatives. 25,26 This makes the shear stress tensor dependent on the viscosity in linear (i.e. Newtonian) or implicitly nonlinear (i.e.…”

Section: Introductionmentioning

confidence: 99%

Computational fluid dynamics simulations of cerebral aneurysm using Newtonian, power-law and quasi-mechanistic blood viscosity models

Saqr

2020

Proc Inst Mech Eng H

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Cerebral aneurysm is a fatal neurovascular disorder. Computational fluid dynamics simulation of aneurysm haemodynamics is one of the most important research tools which provide increasing potential for clinical applications. However, computational fluid dynamics modelling of such delicate neurovascular disorder involves physical complexities that cannot be easily simplified. Recently, it was shown that the Newtonian simplification used to close the shear stress tensor of the Navier–Stokes equation is not sufficient to explore aneurysm haemodynamics. This article explores the differences between the latter simplification, non-Newtonian power-law model and a newly proposed quasi-mechanistic model. The modified Krieger model, which treats blood as a suspension of plasma and particles, was implemented in computational fluid dynamics context here for the first time and is made available to the readers in a C# code in the supplementary material of this article. Two middle-cerebral artery and two anterior-communicating artery aneurysms, all ruptured, were utilized here as case studies. It was shown that the modified Krieger model had higher sensitivity for wall shear stress calculations in comparison with the other two models. The modified Krieger model yielded lower wall shear stress values consistently in comparison with the other two models. Moreover, the modified Krieger model has generally predicted higher pressure in the aneurysm models. Based on published aneurysm rupture studies, it is believed that ruptured aneurysms are usually correlated with lower wall shear stress values than unruptured ones. Therefore, this work concludes that the modified Krieger model is a potential candidate for providing better clinical relevance to aneurysm computational fluid dynamics simulations.

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A note on Stokes’ hypothesis

Cited by 37 publications

References 13 publications

An alternative to the concept of continuous medium

An alternative to the concept of continuous medium

On lower-dimensional models in lubrication, Part A: Common misinterpretations and incorrect usage of the Reynolds equation

Computational fluid dynamics simulations of cerebral aneurysm using Newtonian, power-law and quasi-mechanistic blood viscosity models

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