Thermofluiddynamics: Do We Solve the Right Kind of Equations?

Durst, F.; Gomes, J. F.; Sambasivam, R.

doi:10.1615/ichmt.2006.turbulheatmasstransf.10

Cited by 21 publications

(38 citation statements)

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“…In order to compare the transformed stress tensor with the Korteweg stress tensor, it is convenient to write the re-casted momentum balance equation (12) in the following form:…”

Section: Re-casted Momentum Balance Equationmentioning

confidence: 99%

“…In the last decade, the classical Navier-Stokes equations have received a number of modifications/ extensions based on the diffusion transport of mass [1,[8][9][10][11][12][13][14]. In particular, Brenner [8,9] first proposed a set of modifications to the equations based on the physical arguments of thermophoretic motion in gases.…”

Section: Introductionmentioning

confidence: 99%

“…A more general form of friction matrix that includes the basic fact that particles operate diffusive motions, leads to a revised set of transport equations that contain two velocities, which are very similar in nature to the volume and mass velocities. Durst et al [12] based their arguments on that the absence of mass diffusion terms in the continuity equation contradicts constitutive relations for momentum and heat diffusion: when the fluid properties changes spatially in the presence of momentum and heat diffusions, there should also be present the mass diffusion. They later derived the Extended Navier-Stokes equations [12,13,17] based on the mass-diffusion controlled formalism.…”

mentioning

confidence: 99%

“…Durst et al [12] based their arguments on that the absence of mass diffusion terms in the continuity equation contradicts constitutive relations for momentum and heat diffusion: when the fluid properties changes spatially in the presence of momentum and heat diffusions, there should also be present the mass diffusion. They later derived the Extended Navier-Stokes equations [12,13,17] based on the mass-diffusion controlled formalism. A late suggestion to substitute the Navier-Stokes is given by Svärd [18].Generally, there are a number of experimental data that standard Navier-Stokes fail to predict.…”

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confidence: 99%

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Recasting Navier–Stokes equations

et al. 2019

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Classical Navier-Stokes equations fail to describe some flows in both the compressible and incompressible configurations. In this article, we propose a new methodology based on transforming the fluid mass velocity vector field to obtain a new class of continuum models. We uncover a class of continuum models which we call the re-casted Navier-Stokes. They naturally exhibit the physics of previously proposed models by different authors to substitute the original Navier-Stokes equations. The new models unlike the conventional Navier-Stokes appear as more complete forms of mass diffusion type continuum flow equations. They also form systematically a class of thermomechanically consistent hydrodynamic equations via the original equations. The plane wave analysis is performed to check their linear stability under small perturbations, which confirms that all re-casted models are spatially and temporally stable like their classical counterpart. We then use the Rayleigh-Brillouin scattering experiments to demonstrate that the re-casted equations may be better suited for explaining some of the experimental data where original Navier-Stokes equations fail.Öttinger [11] also proposed earlier a substitute for the classical Navier-Stokes in his phenomenological GENERIC formalism. In the GENERIC formulation, Öttinger [11] demonstrated that by assuming the row and the column, which are associated with the mass density in the friction matrix to be identically zero, leads to the conventional Navier-Stokes equations. A more general form of friction matrix that includes the basic fact that particles operate diffusive motions, leads to a revised set of transport equations that contain two velocities, which are very similar in nature to the volume and mass velocities. Durst et al [12] based their arguments on that the absence of mass diffusion terms in the continuity equation contradicts constitutive relations for momentum and heat diffusion: when the fluid properties changes spatially in the presence of momentum and heat diffusions, there should also be present the mass diffusion. They later derived the Extended Navier-Stokes equations [12,13,17] based on the mass-diffusion controlled formalism. A late suggestion to substitute the Navier-Stokes is given by Svärd [18].Generally, there are a number of experimental data that standard Navier-Stokes fail to predict. Shock wave structure predictions are a ubiquitous example of Navier-Stokes failure [4,6,19,20]. Experimental data of water flows in carbon nanotubes or confined pores is another research topic showing large deviations from the classical Hagen-Poiseuille equation [21,22]. A convincing theoretical interpretation of this data is still lacking. Leaving aside non-linear configurations, conventional Navier-Stokes equations also fail to describe some of the linear flow problems accurately. For example, it is unsuccessful in describing the actual spectrum shapes in the Rayleigh-Brillouin scattering problem [23][24][25][26][27][28].In the current work, we provide new insights into the ...

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“…In order to compare the transformed stress tensor with the Korteweg stress tensor, it is convenient to write the re-casted momentum balance equation (12) in the following form:…”

Section: Re-casted Momentum Balance Equationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Recasting Navier–Stokes equations

et al. 2019

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“…A new approach has been proposed in a series of publications by Durst and coworkers (see, e.g., Refs. ). The underlying assumption in this theory is that the pressure gradient in the direction of the compressible flow provides an additional diffusive mode of mass transport, which is accounted for analytically by adding an extra term to the CNSE.…”

Section: Introductionmentioning

confidence: 97%

An analytical solution to the extended Navier–Stokes equations using the Lambert W function

Jaishankar

McKinley

2014

AIChE Journal

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Micro channel gas flows are of importance in a range Micro Electro Mechanical Systems (MEMS). Due to the confined geometry of the channel, the mean free path of the gas can be comparable to the characteristic length of the micro channel, leading to slip-like flow behavior and strong diffusion-enhanced transport of mass and momentum. The Extended Navier Stokes Equations (ENSE) have successfully been used to model the slip behavior in rarefied gas flows as well as quantitatively predict the mass flow rate through the channel, without introducing any ad hoc parameters. The model involves accounting for self-diffusion due to local pressure gradients, and decomposing the total transport as the additive sum of individual convective and diffusive terms. In this paper, we show that it is possible to obtain exact analytical solutions to the ENSE equation set for the pressure and velocity field using the Lambert W function. We find that diffusive contributions to the total transport are only dominant for low average pressures and low pressure drops across the micro channel. For large inlet pressures, we show that the expressions involving the Lambert function predict steep gradients in the pressure and velocity localized near the channel exit. Using a limit analysis, we extract a characteristic length for this boundary layer. Our analytical results are validated by numerical calculations as well as experimental results available in the literature.

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Pressure-driven diffusive gas flows in micro-channels: from the Knudsen to the continuum regimes

Dongari

Sharma

Durst

2008

Microfluid Nanofluid

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108

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Despite the enormous scientific and technological importance of micro-channel gas flows, the understanding of these flows, by classical fluid mechanics, remains incomplete including the prediction of flow rates. In this paper, we revisit the problem of micro-channel compressible gas flows and show that the axial diffusion of mass engendered by the density (pressure) gradient becomes increasingly significant with increased Knudsen number compared to the pressure driven convection. The present theoretical treatment is based on a recently proposed modification (Durst et al. in Proceeding of the international symposium on turbulence, heat and mass transfer, Dubrovnik, 3-18 September, pp [25][26][27][28][29] 2006) of the Navier-Stokes equations that include the diffusion of mass caused by the density and temperature gradients. The theoretical predictions using the modified NavierStokes equations are found to be in good agreement with the available experimental data spanning the continuum, transition and free-molecular (Knudsen) flow regimes, without invoking the concept of Maxwellian wall-slip boundary condition. The simple theory also results in excellent agreement with the results of linearized Boltzmann equations and Direct Simulation Monte Carlo (DSMC) method. Finally, the theory explains the Knudsen minimum and suggests the design of future micro-channel flow experiments and their employment to complete the present days understanding of micro-channel flows.

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Thermofluiddynamics: Do We Solve the Right Kind of Equations?

Cited by 21 publications

References 0 publications

Recasting Navier–Stokes equations

Recasting Navier–Stokes equations

An analytical solution to the extended Navier–Stokes equations using the Lambert W function

Pressure-driven diffusive gas flows in micro-channels: from the Knudsen to the continuum regimes

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