A new method to obtain the Carnahan–Starling equation and its generalization

Nikolaev, Pavel

doi:10.3103/s0027134916050131

Cited by 10 publications

(3 citation statements)

References 48 publications

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“…The problem of determining the pressure of the unsheared fluid thus reduces to the computation of the equation of state of a hard-sphere fluid, a widely studied subject, that still received a fair amount of attention in the last few years (see [71][72][73][74][75][76][77][78] and references therein). Indeed, if the Carnahan-Starling expression is often considered as the best compromise between simplicity and precision, it is not the most accurate in our regime of interest (ϕ 0.4).…”

Section: An Equation Of State For Inelastic Hard Spheresmentioning

confidence: 99%

Integration through transients approach to the $μ(\mathcal{I})$ rheology

Coquand,

Sperl,

Kranz

2020

Preprint

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This work generalises the granular integration through transients formalism introduced by Kranz et al. [Phys. Rev. Lett. 121, 148002 (2018)] to the determination of the pressure. We focus on the Bagnold regime, and provide theoretical support to the empirical µ(I) rheology laws, that have been successfully applied in many granular flow problems. In particular, we confirm that the interparticle friction is irrelevant in the regime where the µ(I) laws apply.

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Section: An Equation Of State For Inelastic Hard Spheresmentioning

confidence: 99%

Integration through transients approach to the $μ(\mathcal{I})$ rheology

Coquand,

Sperl,

Kranz

2020

Preprint

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“…We found to our surprise (and to my horror) that the fundament transport law called the Starling equation in the physiology literature for more than a century (since 1896 (226)) did not have a mathematically satisfying derivation despite the best efforts of many (227)(228)(229)(230)(231)(232)(233).…”

Section: Fluid Flow Starling Equationmentioning

confidence: 93%

Structural Analysis of Fluid Flow in Complex Biological Systems

Eisenberg¹

2022

Preprint

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Biology is about structure. Structures within structures. Organs within animals, tissues within organs, cells within tissues, and molecules, often proteins within cells. The structures are so complex that they can only be described by numbers. No numbers are of more importance than those that describe proteins. The numbers that describe coordinates of its atoms, often determined by Patterson functions (which are inverse Fourier Transforms of intensities) of crystal diffraction. Without these numbers, structural biology would hardly exist. Without numbers, engineering would not exist. Numbers are surely needed by the engineers who produce the x-rays diffracting from atoms of protein crystals. Devices of engineering have function. They are built to implement equations. Engineering devices use structures to implement equations, when power is supplied at the right places, that produces appropriate flows. Flows are the essence of life. Equilibrium means death in most living systems. Flows in biological structures are hard to analyze because we do not know input output equations in advance. Sometimes we do not know their function. Flows, forces, and structures of life (like those of engineering) are related by field equations of conservation laws, partial differential equations, constrained by location and properties of structures. Constraints are boundary conditions located on the complicated domain of biological structure. Dealing with this complexity is simplified if one systematically analyzes structure using the most general field theory known, electricity described by the Maxwell equations, without significant known error. Currents are involved because flows of biology usually involve migration of charges, convection of water and solutes, diffusion of ions that form the plasma of life, and their interactions. Interactions can dominate function. Here I show how a few complex structures can be understood in engineering detail. This approach may be useful in dealing with biological and medical issues in many other cases. In one limited case—the clearance of a toxic waste (potassium ions) from the optic nerve—this approach seems to have succeeded.

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“…We found to our surprise (and to my horror) that the fundament transport law called the Starling equation in the physiology literature for more than a century (since 1896 226 ) did not have a mathematically satisfying derivation despite the best efforts of many. [227][228][229][230][231][232][233] The central issue, from our point of view, was the classical treatment of ionic solutions as simple liquids (nearly ideal gases, so well presented in previous work [30][31][32][33][34] ). It seemed clear to us that ionic solutions had too many too complex forms of energy (and flow and dissipation) to fit comfortably in a tradition that so thoroughly depended on the theory of ideal (i.e., perfect) gases (made of particles that did not interact at all, and had no internal structure) at equilibrium (with zero flows of all components) from which it grew.…”

Section: Fluid Flow Starling Equationmentioning

confidence: 99%

Structural analysis of fluid flow in complex biological systems

Eisenberg

2023

MAIO

View full text Add to dashboard Cite

Biology is about structure. Structures within structures. Organs within animals, tissues within organs, cells within tissues, and molecules, often proteins within cells. The structures are so complex that they can only be described by numbers. No numbers are of more importance than those that describe proteins. The numbers that describe coordinates of its atoms, often determined by Patterson functions (which are inverse Fourier Transforms of intensities) of crystal diffraction. Without these numbers, structural biology would hardly exist. Without numbers, engineering would not exist. Numbers are surely needed by the engineers who produce the x-rays diffracting from atoms of protein crystals. Devices of engineering have function. They are built to implement equations. Engineering devices use structures to implement equations, when power is supplied at the right places, that produces appropriate flows. Flows are the essence of life. Equilibrium means death in most living systems. Flows in biological structures are hard to analyze because we do not know input output equations in advance. Sometimes we do not know the function of the structures. Flows, forces, and structures of life (like those of engineering) are related by field equations of conservation laws, partial differential equations, constrained by location and properties of structures. Constraints are boundary conditions located on the complicated domain of biological structure. The hierarchy of structures allows a handful of atoms (in proteins and nucleic acids) to control macroscopic function. Dealing with this complexity is simplified if one systematically analyzes structure using the most general field theory known, electricity described by the Maxwell equations, without significant known error. Currents are involved because flows of biology usually involve migration of charges, convection of water and solutes, diffusion of ions that form the plasma of life, and their interactions. Interactions can dominate function. Here I show how a few complex structures can be understood in engineering detail. This approach may be useful in dealing with biological and medical issues in many other cases. In one limited case—the clearance of a toxic waste (potassium ions) from the optic nerve—this approach seems to have succeeded.

show abstract

A new method to obtain the Carnahan–Starling equation and its generalization

Cited by 10 publications

References 48 publications

Integration through transients approach to the $μ(\mathcal{I})$ rheology

Integration through transients approach to the $μ(\mathcal{I})$ rheology

Structural Analysis of Fluid Flow in Complex Biological Systems

Structural analysis of fluid flow in complex biological systems

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