scholarly journals Perturbations of the coupled Jeffery–Stokes equations

2011 ◽  
Vol 681 ◽  
pp. 622-638
Author(s):  
STEPHEN MONTGOMERY-SMITH
Keyword(s):  
Spectral Analysis ◽  
Stokes Equation ◽  
Stokes Equations ◽  
Volume Ratio ◽  
Hydrodynamic Interactions ◽  
Perturbation Equation ◽  
Poor Predictor ◽  
Theoretical Predictions ◽  
Dilute Suspensions ◽  
Jeffery’S Equation

This paper seeks to provide clues as to why experimental evidence for the alignment of slender fibres in semi-dilute suspensions under shear flows does not match theoretical predictions. This paper posits that the hydrodynamic interactions between the different fibres that might be responsible for the deviation from theory, can at least partially be modelled by the coupling between Jeffery's equation and Stokes' equation. It is proposed that if the initial data are slightly non-uniform, in that the probability distribution of the orientation has small spatial variations, then there is feedback via Stokes' equation that causes these non-uniformities to grow significantly in short amounts of time, so that the standard uncoupled Jeffery's equation becomes a poor predictor when the volume ratio of fibres to fluid is not extremely low. This paper provides numerical evidence, involving spectral analysis of the linearization of the perturbation equation, to support this theory.

scholarly journals Generalized Navier–Stokes Equations and Dynamics of Plane Molecular Media

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 288
Author(s):  
Alexei Kushner ◽  
Valentin Lychagin
Keyword(s):  
Carbon Dioxide ◽  
Internal Structure ◽  
Stokes Equation ◽  
Hydrogen Cyanide ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations

The first analysis of media with internal structure were done by the Cosserat brothers. Birkhoff noted that the classical Navier–Stokes equation does not fully describe the motion of water. In this article, we propose an approach to the dynamics of media formed by chiral, planar and rigid molecules and propose some kind of Navier–Stokes equations for their description. Examples of such media are water, ozone, carbon dioxide and hydrogen cyanide.

A note on the solution of the Navier-Stokes equations for a spherically symmetric expansion into a very low pressure

1973 ◽  
Vol 59 (2) ◽  
pp. 391-396 ◽  
Author(s):  
N. C. Freeman ◽  
S. Kumar
Keyword(s):  
Shock Wave ◽  
Reynolds Number ◽  
Stokes Equation ◽  
Stokes Equations ◽  
Low Pressure ◽  
Navier Stokes ◽  
Spherically Symmetric ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
Type Flow

It is shown that, for a spherically symmetric expansion of a gas into a low pressure, the shock wave with area change region discussed earlier (Freeman & Kumar 1972) can be further divided into two parts. For the Navier–Stokes equation, these are a region in which the asymptotic zero-pressure behaviour predicted by Ladyzhenskii is achieved followed further downstream by a transition to subsonic-type flow. The distance of this final region downstream is of order (pressure)−2/3 × (Reynolds number)−1/3.

Fast forced liquid film spreading on a substrate: flow, heat transfer and phase transition

2010 ◽  
Vol 656 ◽  
pp. 189-204 ◽  
Author(s):  
ILIA V. ROISMAN
Keyword(s):  
Heat Transfer ◽  
Phase Transition ◽  
Stokes Equations ◽  
Substrate Surface ◽  
Reynolds Numbers ◽  
Drop Impact ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Theoretical Predictions ◽  
Viscous Drop

This theoretical study is devoted to description of fluid flow and heat transfer in a spreading viscous drop with phase transition. A similarity solution for the combined full Navier–Stokes equations and energy equation for the expanding lamella generated by drop impact is obtained for a general case of oblique drop impact with high Weber and Reynolds numbers. The theory is applicable to the analysis of the phenomena of drop solidification, target melting and film boiling. The theoretical predictions for the contact temperature at the substrate surface agree well with the existing experimental data.

scholarly journals Remarks on the Systems of Semilinear Fractional Rayleigh-Stokes Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Le Dinh Long
Keyword(s):  
Cauchy Problem ◽  
Fourier Analysis ◽  
Stokes Equation ◽  
Stokes Equations ◽  
Newtonian Fluids ◽  
System Of Equations ◽  
Main Technique ◽  
The Cauchy Problem ◽  
The Stability

In this paper, we study the Cauchy problem for a system of Rayleigh-Stokes equations. In this system of equations, we use derivatives in the classical Riemann-Liouville sense. This system has many applications in some non-Newtonian fluids. We obtained results for the existence, uniqueness, and frequency of the solution. We discuss the stability of the solutions and find the solution spaces. Our main technique is to use the Banach mapping theorem combined with some techniques in Fourier analysis.

scholarly journals Gyrotactic swimmers in turbulence: shape effects and role of the large-scale flow

2018 ◽  
Vol 856 ◽  
Author(s):  
M. Borgnino ◽  
G. Boffetta ◽  
F. De Lillo ◽  
M. Cencini
Keyword(s):  
Large Scale ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Small Scale ◽  
Navier Stokes Equations ◽  
Preferential Sampling ◽  
Wide Range ◽  
Shape Effects ◽  
Dilute Suspensions ◽  
Large Scale Flow

We study the dynamics and the statistics of dilute suspensions of gyrotactic swimmers, a model for many aquatic motile microorganisms. By means of extensive numerical simulations of the Navier–Stokes equations at different Reynolds numbers, we investigate preferential sampling and small-scale clustering as a function of the swimming (stability and speed) and shape parameters, considering in particular the limits of spherical and rod-like particles. While spherical swimmers preferentially sample local downwelling flow, for elongated swimmers we observe a transition from downwelling to upwelling regions at sufficiently high swimming speed. The spatial distribution of both spherical and elongated swimmers is found to be fractal at small scales in a wide range of swimming parameters. The direct comparison between the different shapes shows that spherical swimmers are more clusterized at small stability and speed numbers, while for large values of the parameters elongated cells concentrate more. The relevance of our results for phytoplankton swimming in the ocean is briefly discussed.

A Comparison Between Full and Simplified Navier-Stokes Equation Solutions for Rotating Blade Cascade Flow on S1 Stream Surface of Revolution

1985 ◽  
Author(s):  
Chen Naixing ◽  
Zhang Fengxian
Keyword(s):  
Stokes Equation ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
Stream Surface ◽  
Surface Of Revolution ◽  
Rotating Blade ◽  
Blade Cascade ◽  
Cascade Flow

A method for solving the Navier-Stokes equations of the rotating blade cascade flow on S1 stream surface of revolution is developed in the present paper. In this paper a complete set of full and simplified Navier-Stokes equations is given which includes stream-function equation, energy equation and entropy equation, equation of state for a perfect gas, formula for estimating density and formulas for calculating viscous forces, work done by viscous force, dissipation function and heat-transfer term. A comparison between the full and the simplified Navier-Stokes equations is made. The viscous terms of both full and simplified Navier-Stokes equation solutions are also compared in the present paper. The comparison shows that the simplified Navier-Stokes equations are applicable.

Transonic nozzle flow of dense gases

1993 ◽  
Vol 247 ◽  
pp. 661-688 ◽  
Author(s):  
A. Kluwick
Keyword(s):  
Stokes Equations ◽  
Equations Of State ◽  
Weak Shock ◽  
Inviscid Limit ◽  
Navier Stokes ◽  
Perturbation Equation ◽  
Navier Stokes Equations ◽  
Sonic Point ◽  
Specific Heats ◽  
Dense Gases

The paper deals with the flow properties of dense gases in the throat area of slender nozzles. Starting from the Navier–Stokes equations supplemented with realistic equations of state for gases which have relatively large specific heats a novel form of the viscous transonic small-perturbation equation is derived. Evaluation of the inviscid limit of this equation shows that three sonic points rather than a single sonic point may occur during isentropic expansion of such media, in contrast to the case of perfect gases. As a consequence, a shock-free transition from subsonic to supersonic speeds cannot, in general, be achieved by means of a conventional converging–diverging nozzle. Nozzles leading to shock-free flow fields must have an unusual shape consisting of two throats and an intervening antithroat. Additional new results include the computation of the internal thermoviscous structure of weak shock waves and a phenomenon referred to as impending shock splitting. Finally, the relevance of these results to the description of external transonic flows is discussed briefly.

A framework for optimising aspects of rotor blades

2011 ◽  
Vol 115 (1165) ◽  
pp. 147-161 ◽  
Author(s):  
C. S. Johnson ◽  
G. N. Barakos
Keyword(s):  
Objective Function ◽  
Stokes Equations ◽  
Rotor Blades ◽  
Navier Stokes ◽  
Computational Framework ◽  
Navier Stokes Equations ◽  
Theoretical Predictions ◽  
Artificial Neural ◽  
Optimisation Methods ◽  
Selection Of

AbstractThis work presents a computational framework for the optimisation of various aspects of rotor blades. The proposed method employs CFD combined with artificial neural networks, employed as metamodels, and optimisation methods based on genetic algorithms. To demonstrate this approach, two examples have been used, one is the optimal selection of 4- and 5-digit NACA aerofoils for rotor sections and the other is the optimisation of linear blade twist for rotors in hover. For each case, an objective function was created and the meta-model was subsequently used to evaluate this objective function during the optimisation process. The obtained results agree with real world design examples and theoretical predictions. For the selected cases, the artificial neural network was found to perform adequately though the results required a substantial amount of data for training. The genetic algorithm was found to be very effective in identifying a set of near-optimal parameters. The main CPU cost was associated with the population of the database necessary for the meta-models and this task required CFD computations based on the Reynolds-averaged Navier-Stokes equations. The framework is general enough to allow for several design or optimisation tasks to be carried out and it is based on open-source code made available by the authors.

scholarly journals About local fractional three-dimensional compressible Navier-Stokes equations in Cantor-type cylindrical co-ordinate system

2016 ◽  
Vol 20 (suppl. 3) ◽  
pp. 847-851 ◽  
Author(s):  
Guo-Ping Gao ◽  
Carlo Cattani ◽  
Xiao-Jun Yang
Keyword(s):  
Fractional Derivative ◽  
Stokes Equation ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
Local Fractional Derivative

In this article, we investigate the local fractional 3-D compressible Navier-Stokes equation via local fractional derivative. We use the Cantor-type cylindrical co-ordinate method to transfer 3-D compressible Navier-Stokes equation from the Cantorian co-ordinate system to the Cantor-type cylindrical co-ordinate system.

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