Numerical predictions of backward-facing step flows in microchannels using extended Navier–Stokes equations

Sambasivam, R.; Chakraborty, Suman; Durst, F.

doi:10.1007/s10404-013-1254-1

Cited by 8 publications

(3 citation statements)

References 15 publications

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“…7 Replot of data in Fig. 6, but FMP-Data plotted for W = 0.0004 was successfully demonstrated by Sambasivam et al [17], who obtained numerically the results shown in Table 2 and in Fig. 8 below: Comparisons of the results Table 2 and in Fig.…”

Section: Analytical Treatmentsmentioning

confidence: 62%

Treatments of Micro-channel Flows Revisited: Continuum Versus Rarified Gas Considerations

Durst

Filimonov

Sambasivam

2020

J. Inst. Eng. India Ser. C

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There are numerous treatments of micro-channel flows available that point out the breakdown of the Navier–Stokes equations under molecular flow conditions when continuum conditions should still apply. The wrong conclusion regarding the validity of the Navier–Stokes equations comes about from the missing mass diffusion terms in the continuity equation. This term also affects the Navier–Stokes equations and yields improved theoretical results for flows with strong pressure and temperature gradients. Rarified gas treatments are often applied, although the fluid mechanics conditions are still such, that continuum equations should work. In the paper, the missing mass diffusion term is added, and it is shown that those extended fluid mechanics equation (EFME) allows micro-channel flows to be treated in the so-called slip regime. Hence, the continuum approach for flow treatment holds in micro-channel flows, in the flow regimes where modeling of wall interactions is applied these days. The paper also describes the treatments of micro-channel flows in the slip-flow regime by the rarified gas flow treatment method of Shapiro and Seleznev, and the results are compared with the corresponding results of the EFME. Good agreement was obtained, but differences exist regarding the wall interactions, which are explained, and suggested to use both methods to obtain a deeper insight into molecule–wall interactions in micro-channel flows. The EFME claim that large pressure and temperature gradients are the reasons for the differences between experimental and theoretical results of micro-channel flows. Such differences also exist in other fluid flows with strong property gradients, e.g., in shock waves. Flows of this kind are also treated in this paper in a way to show that the EFME have a wide range of applications.

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Section: Analytical Treatmentsmentioning

confidence: 62%

Treatments of Micro-channel Flows Revisited: Continuum Versus Rarified Gas Considerations

Durst

Filimonov

Sambasivam

2020

J. Inst. Eng. India Ser. C

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“…At steady state and in the absence of any temperature gradients, the system of equations for a single species referred to for compactness as the ENSE are given by

\frac{\partial (ρ u_{i}^{normalT})}{\partial x_{i}} = 0

\frac{\partial}{\partial x_{i}} true(ρ u_{i}^{normalT} u_{i}^{normalT} true) = - \frac{\partial P}{\partial x_{j}} - \frac{\partial}{\partial x_{i}} true(τ_{i j}^{C} - \frac{2}{3} δ_{i j} {\overset{m}{true}}_{k}^{normalD} u_{k}^{normalC} true)

τ_{i j}^{normalC} = - μ true(\frac{\partial u_{i}^{normalC}}{\partial x_{j}} + \frac{\partial u_{j}^{normalC}}{\partial x_{i}} true) + \frac{2}{3} μ δ_{i j} \frac{\partial u_{k}^{normalC}}{\partial x_{k}}

along with the ideal gas equation of state

ρ = \frac{P}{R T}

…”

Section: The Extended Navier–stokes Equationsmentioning

confidence: 99%

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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“…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.…”

mentioning

confidence: 99%

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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Numerical predictions of backward-facing step flows in microchannels using extended Navier–Stokes equations

Cited by 8 publications

References 15 publications

Treatments of Micro-channel Flows Revisited: Continuum Versus Rarified Gas Considerations

Treatments of Micro-channel Flows Revisited: Continuum Versus Rarified Gas Considerations

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

Recasting Navier–Stokes equations

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