An exact solution of the 3-D Navier–Stokes equation

Muriel, A.

doi:10.1016/j.rinp.2011.04.002

Cited by 17 publications

(10 citation statements)

References 1 publication

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“…Muriel [9,10] used the time evolution of a one-particle distribution function initialized with spatially uniform data along with a choice of a Gaussian pair potential between particles to arrive at a divergence-free velocity field, but he encountered problems in the integration of the NSE to obtain a scalar pressure field. He concludes with a suggestion that the pressure field is a higher-order tensor field [2].…”

Section: Literature Review On Extant Solutions and Modelsmentioning

confidence: 99%

“…However, when that is done, different results are obtained from integrating in either of the two spatial directions. Muriel [9] encountered this problem, and he suggests that the pressure is a higher-order tensor which converges to a scalar value in a long-time limit. However, the current literature on pressure appears to suggest otherwise [2,4,24].…”

Section: Conservation Of Momentummentioning

confidence: 99%

“…CMI has been at the forefront of the coordination of the worldwide search for the general solution(s) (or proof of no solution(s)) to these relations. A check on the status of the problem on its website (accessed on November 28, 2021) still reveals it to be unsolved despite attempts by various researchers [2,8,[8][9][10][11][12][13][14][15][16][17][18][19][20][21]. However, with limiting but suitable assumptions and approximations, exact and approximate analytical solutions of these relations have been obtained for some special flows.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A Closed-Form Analytical Solution to the Complete Three-Dimensional Unsteady Compressible Navier-Stokes Equation

Amoloye

2021

Preprint

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The three main approaches to exploring fluid dynamics are actual experiments, numerical simulations, and theoretical solutions. Numerical simulations and theoretical solutions are based on the continuity equation and Navier-Stokes equations (NSE) that govern experimental observations of fluid dynamics. Theoretical solutions can offer huge advantages over numerical solutions and experiments in the understanding of fluid flows and design. These advantages are in terms of cost and time consumption. However, theoretical solutions have been limited by the prized NSE problem that seeks a physically consistent solution than what classical potential theory (CPT) offers. Therefore, the current author embarked on a doctoral research on the refinement of CPT. He introduced the Refined Potential Theory (RPT) that provides the Kwasu function as a physically consistent solution to the NSE problem. The Kwasu function is a viscous scalar potential function that captures known and observable unsteady features of experimentally observed wall bounded flows including flow separation, wake formation, vortex shedding, compressibility effects, turbulence and Reynolds-number-dependence. It is appropriately defined to combine the properties of a three-dimensional potential function to satisfy the inertia terms of the NSE and the features of a stream function to satisfy the continuity equation, the viscous vorticity equation and the viscous terms of the NSE. RPT has been verified and validated against experimental and numerical results of incompressible unsteady sub-critical Reynolds number flows on stationary finite circular cylinder, sphere and spheroid. It is concluded that the Kwasu function is a physically consistent and closed-form analytical solution to the NSE problem.

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Section: Literature Review On Extant Solutions and Modelsmentioning

confidence: 99%

Section: Conservation Of Momentummentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A Closed-Form Analytical Solution to the Complete Three-Dimensional Unsteady Compressible Navier-Stokes Equation

Amoloye

2021

Preprint

View full text Add to dashboard Cite

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Section: Literature Review On Extant Solutions and Modelsmentioning

confidence: 99%

A Closed-Form Analytical Solution to the Complete Three-Dimensional Unsteady Compressible Navier-Stokes Equations

Amoloye

2021

Preprint

View full text Add to dashboard Cite

The three main approaches in fluid dynamics are actual experiments, numerical simulations, and theoretical solutions. Numerical simulations and theoretical solutions are based on the continuity equation and Navier-Stokes equations (NSE) that govern experimental observations of fluid dynamics.Theoretical solutions can offer huge advantages over numerical solutions and experiments in the understanding of fluid flows and design. These advantages are in terms of cost and time consumption. However, theoretical solutions have been limited by the prized NSE problem that seeks a physically consistent solution than what classical potential theory (CPT) offers. Therefore, the current author refined CPT. He introduced refined potential theory (RPT) that provides a viscous potential/stream function as a physically consistent solution to the NSE problem. This function captures observable unsteady flow features including separation, wake, vortex shedding, compressibility, turbulence, and Reynolds-number-dependence. It appropriately combines the properties of a three-dimensional potential function that satisfy the inertia terms of NSE and the features of a stream function that satisfy the continuity equation, the viscous vorticity equation, and the viscous terms of NSE. RPT has been verified and validated against experimental and numerical results of incompressible unsteady sub-critical Reynolds number flows on stationary finite circular cylinder, sphere, and spheroid.

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“…This last approach has been used in a great number of physical models and consequent engineering applications such as, for example, Fick's law, Fourier's law, Navier's law, the Smoluchowsky correction to Fick's law, the Einstein-Smoluchovsky relation, the Ornstein-Uhlenbeck process, the Kossakowski-Lindblad equation, the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy, etc. [104][105][106][107][108][109][110][111][112][113][114][115][116][117][118][119][120].…”

Section: The Generalized Gouy-stodola Approach: the Entropy Generatiomentioning

confidence: 99%

The Gouy-Stodola Theorem in Bioenergetic Analysis of Living Systems (Irreversibility in Bioenergetics of Living Systems)

Lucia

2014

Energies

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Thermodynamics studies the transformations of energy occurring in open systems. Living systems, with particular reference to cells, are complex systems in which energy transformations occur. Thermo-electro-chemical processes and transports occur across their border, the cells membranes. These processes take place with important differences between healthy and diseased states. In particular, different thermal and biochemical behaviours can be highlighted between these two states and they can be related to the energy transformations inside the living systems, in particular the metabolic behaviour. Moreover, living systems waste heat. This heat is the consequence of the internal irreversibility. Irreversibility is effectively studied by using the Gouy-Stodola theorem. Consequently, this approach can be introduced in the analysis of the states of living systems, in order to obtain a unifying approach to study them. Indeed, this approach allows us to consider living systems as black boxes and analyze only the inflows and outflows and their changes in relation to the modification of the environment, so information on the systems can be obtained by analyzing their behaviour in relation to the modification of external perturbations. This paper presents a review of the recent results obtained in the thermodynamics analysis of cell systems.

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An exact solution of the 3-D Navier–Stokes equation

Cited by 17 publications

References 1 publication

A Closed-Form Analytical Solution to the Complete Three-Dimensional Unsteady Compressible Navier-Stokes Equation

A Closed-Form Analytical Solution to the Complete Three-Dimensional Unsteady Compressible Navier-Stokes Equation

A Closed-Form Analytical Solution to the Complete Three-Dimensional Unsteady Compressible Navier-Stokes Equations

The Gouy-Stodola Theorem in Bioenergetic Analysis of Living Systems (Irreversibility in Bioenergetics of Living Systems)

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