Extended Navier Stokes Equations Research Articles

Turbulence calculation based on the extended Navier–Stokes equations

In this study, phenomenological observations and the Kreuer interpretation of the origin of viscosity were used to develop a computational method for solving the turbulence problem of incompressible viscous Newtonian fluids based on extended Navier–Stokes (N–S) equations. The shear process in fluid flow was hypothesized to be accompanied by eddy formation, and the effects of eddies on the convection and diffusion were considered. The classical N–S equations were improved to obtain extended N–S equations. The extended equations are closed, and the sources of the velocity fluctuations are explicitly considered to be additional convection and diffusion. The extended equations are compatible with the classical N–S equations; thus, they can describe laminar and turbulent flows in a unified manner. In fluid flow simulations, the equations describing the mean flow quantities could be directly obtained from the extended N–S equations without any additional turbulence models. A numerical investigation was carried out to verify the extended equations by exploring the flow over a cube placed in a channel. The simulation results were compared with both the large eddy simulation and experimental results.

Extensions to the Navier–Stokes equations

Historically, the mass conservation and the classical Navier–Stokes equations were derived in the co-moving reference frame. It is shown that the mass conservation and Navier–Stokes equations are Galilean invariant—they are valid in any arbitrary inertial reference frame. From the mass conservation and Navier–Stokes equations, we can derive a wave equation, which contains the speed of pressure wave as its parameter. This parameter is independent of the speed of the source—the fluid element velocity. The speed of pressure wave is determined from the thermodynamic equation of state of the fluid, which is reference frame independent. It is well known that Lorentz transformation ensures wave speed invariant in all inertial frames, and the Lorentz invariance holds for different inertial observers. Based on these arguments, general Navier–Stokes equations (conservation law for the energy–momentum) can be written in any arbitrary inertial reference frame, they are transformed from one reference frame into another with the help of the Lorentz transformation. The key issue is that the Lorentz factor is parametrized by the local Mach number. In the instantaneous co-moving reference frame, these equations will degrade to the classical Navier–Stokes equations—the limit of the non-relativistic ones. These extended equations contain a square of the Lorentz factor. When the local Mach number is equal to one (the Lorentz factor approaches infinity), the extended Navier–Stokes equations will embody an intrinsic singularity, meaning that the transitions from the subsonic flow to the supersonic flow will happen. For the subsonic flow, the square of the Lorentz factor is positive, while for the supersonic flow, the square of the Lorentz factor becomes a negative number, which represents that the speed of sound cannot travel upstream faster than the flow velocity.

Finite ion size and ion permittivity effects on gel electrophoresis of a soft particle

Electrophoresis of a soft particle embedded in a hydrogel medium is studied by considering the hydrated ions as charged dielectric spheres of finite radius. The Nernst-Planck equation for the transport of ions is modified to take into account the ion steric repulsion and the ion-solvent dielectric interactions (MNP-model). The effective polarizability of the hydrated ions lowers the dielectric permittivity of the ionic solution and consequently, the permittivity of the medium varies with local ionic concentration. This spatial variation of the permittivity creates a Born energy difference in ions leading to a Born force experienced by the ions. In addition, ions experience a dielectrophoretic force due to the polarization under the imposed external electric field. The ion steric repulsion are modeled by the Carnahan-Starling equation of state. The convection of ions are governed by the Brinkman extended Navier-Stokes equations. The modified Nernst-Planck equation coupled with the Brinkman Navier-Stokes equations and the equation for electric field are solved numerically through a control volume approach. The ion steric interactions and ion-solvent interactions create a saturation of mobile ions in the polyelectrolyte layer (PEL) of the soft particle, which leads to an attenuation of the counterion condensation of the PEL. This short range steric repulsion and ion-solvent interactions have significance for higher bulk ionic concentration as well as higher range of charge density of the soft particle. These interactions are elucidated by comparing with the mobility obtained by the standard model (PNP-model). The Dukhin number and the effective charge density of the PEL are determined to illustrate the mobile ion saturation. The discrepancy between the present modified model from the standard model magnifies for the case of salts of multivalent counterions.

Extended Navier–Stokes Equations in the Framework of Higher-Order Generalized Hydrodynamics

Extended Navier–Stokes Equations in the Framework of Higher-Order Generalized Hydrodynamics

Alternative constitutive relation for momentum transport of extended Navier–Stokes equations* *Project supported by the National Natural Science Foundation of China–Outstanding Youth Foundation (Grant No. 51522903), the National Natural Science Foundation of China (Grant Nos. 11602276 and 51479094), and the Fund from the Key Laboratory for Mechanics in Fluid Solid Coupling Systems of the Chinese Academy of Sciences.

The classical Navier–Stokes equation (NSE) is the fundamental partial differential equation that describes the flow of fluids, but in certain cases, like high local density and temperature gradient, it is inconsistent with the experimental results. Some extended Navier–Stokes equations with diffusion terms taken into consideration have been proposed. However, a consensus conclusion on the specific expression of the additional diffusion term has not been reached in the academic circle. The models adopt the form of the generalized Newtonian constitutive relation by substituting the convection velocity with a new term, or by using some analogy. In this study, a new constitutive relation for momentum transport and a momentum balance equation are obtained based on the molecular kinetic theory. The new constitutive relation preserves the symmetry of the deviation stress, and the momentum balance equation satisfies Galilean invariance. The results show that for Poiseuille flow in a circular micro-tube, self-diffusion in micro-flow needs considering even if the local density gradient is very low.

Influence of slip velocity at the core of a diffuse soft particle and ion partition effects on mobility.

Nonlinear effects on the electrophoresis of a soft particle, consisting of a rigid hydrophobic core coated with a diffuse polymer layer (PEL) suspended in an electrolyte medium, are studied. The impact of the ion partitioning effect arising due to the Born energy difference between the PEL and the electrolyte is approximated based on the equilibrium Boltzmann equation, with which the ion distribution and hence, the charge density is modified. The equations describing the electrokinetic transport comprising the Darcy-Brinkman extended Navier-Stokes equations which includes the ion partitioning effect coupled with the modified Nernst-Planck equations and Poisson equations for electric field are solved numerically. The present numerical model for the soft particle compares well with the existing theoretical solutions and experimental results in the limiting cases. A deviation from existing simplified models based on the Boltzmann distribution of ions occurs when the Debye layer polarization, relaxation and the electroosmosis induced by the PEL immobile charge become significant. The hydrophobicity of the inner core strongly influences the nonlinear electrokinetic effects by modifying the Debye layer, electroosmotic flow in the PEL and surface conduction. The results indicate that the ion partitioning can significantly increase the electrophoretic mobility of the soft particle by attenuating the shielding effect. When the Debye layer is in the order of the particle size the hydrophobicity of the core surface and the ion partitioning effect manifest the surface conduction, which implies that the Boltzmann distribution of ions is no longer valid. The core hydrophobicity and ion partitioning effect have influence on the condensation of the PEL immobile charge, which creates a significant impact on the mobility.

Modified Boltzmann equation and extended Navier–Stokes equations

We refine the derivation of the Boltzmann equation by considering that the molecules passing through the interfaces of a volume element of physical space and velocity space exhibit different velocity distribution functions and number densities. The resulting equation has a time parameter close to the relaxation time and degenerates into the conventional Boltzmann equation when this parameter takes a value of zero. By considering the macroscopic averaging of mass, momentum, and energy, the corresponding continuity, momentum, and energy equations are obtained. Compared with the extended Navier–Stokes equations, the momentum and energy equations contain additional terms to represent the external forces.

Development of An Integrated Numerical Model for Simulating Wave Interaction with Permeable Submerged Breakwaters Using Extended Navier–Stokes Equations

An integrated two-dimensional vertical (2DV) model was developed to investigate wave interactions with permeable submerged breakwaters. The integrated model is capable of predicting the flow field in both surface water and porous media on the basis of the extended volume-averaged Reynolds-averaged Navier–Stokes equations (VARANS). The impact of porous medium was considered by the inclusion of the additional terms of drag and inertia forces into conventional Navier–Stokes equations. Finite volume method (FVM) in an arbitrary Lagrangian–Eulerian (ALE) formulation was adopted for discretization of the governing equations. Projection method was utilized to solve the unsteady incompressible extended Navier–Stokes equations. The time-dependent volume and surface porosities were calculated at each time step using the fraction of a grid open to water and the total porosity of porous medium. The numerical model was first verified against analytical solutions of small amplitude progressive Stokes wave and solitary wave propagation in the absence of a bottom-mounted barrier. Comparisons showed pleasing agreements between the numerical predictions and analytical solutions. The model was then further validated by comparing the numerical model results with the experimental measurements of wave propagation over a permeable submerged breakwater reported in the literature. Good agreements were obtained for the free surface elevations at various spatial and temporal scales, velocity fields around and inside the obstacle, as well as the velocity profiles.

Heat Transfer of Gas Flow through Microchannels using Extended Navier–Stokes Equations

In this paper, the temperature profile of hydrodynamically fully developed, laminar, nonisothermal, and incompressible gas flow through rectangular microchannels is modelled using Extended Navier–Stokes Equation (ENSE). A semianalytic solution is obtained for the proposed model in its total velocity form, using integral transform technique. Further, the relation between Nusselt number, Knudsen number, and Prandtl number under various aspect ratios was analysed. The results obtained show excellent agreement with existing measurements.

Geostrophic Flow and Wind-Driven Ocean Currents Depending on the Spatial Dimensionality of the Medium

An approach based on noninteger fractional dimension is used to derive extended forms of the Navier–Stokes equations in order to describe geostrophic flow and wind-driven ocean current motion. The equations give rise to several features not obtained in the conventional formalism, which are discussed in some detail. This study demonstrates that the spatial dimensionality of the medium plays a crucial role in rotating fluids and ocean physics, and besides the extended Navier–Stokes equations could be used to describe turbulent ocean physical models.

A fractal model for real gas transport in porous shale

A model for real gas flow in shale gas matrices is proposed and consists of two main steps: (a) developing a microscopic (single pore) model for a real gas flow by generalizing our previously reported Extended Navier‐Stokes Equations (ENSE) method and (b) by using fractal theory concepts, up‐scaling the single pore model to the macroscopic scale. A prominent feature of the up‐scaled model is a predictor for the apparent permeability (AP). Both models are successfully validated with experimental data. The impact of the deviation of the gas behavior from ideality (real gas effect) on the gas transport mechanisms is investigated. The effect of the structural parameters (porosity Ф, the maximum pore diameter Dmax, and the minimum pore diameter Dmin) of the shale matrix on the apparent permeability is studied and a sensitivity analysis is performed to evaluate the significance of the parameters for gas transport. We find that (1) the real gas transport models for a single pore and porous shale matrix are both reliable and reasonable; (2) the real gas effect affects the thermodynamic parameters of the free gas and the adsorption and transport capacity of the adsorbed gas; (3) the real gas effect decreases the effective permeability for convective flow and surface diffusion; i.e., the derivation degree of the effective permeability for bulk diffusion and Knudsen diffusion increases with increasing pressure but presents a bathtub shape when the pore diameter is smaller than 10 nm; and (4) the apparent permeability increases with Ф, Dmax, and Dmin. It is more sensitive to Dmax, followed by the porosity. Dmin has a minor impact. © 2016 American Institute of Chemical Engineers AIChE J, 63: 1430–1440, 2017

A fractal permeability model for shale gas flow through heterogeneous matrix systems

Natural gas flow in shale matrices consisting of both organic and inorganic components was modeled using fractal concepts. The expression of the apparent permeability (AP) of such shale systems was derived following three main steps: (a) modeling real gas flow in a single pore by generalizing our previously reported Extended Navier-Stokes Equations method, then (b) using fractal theory concepts to obtain the apparent permeability for both the organic and inorganic-matter cells, and finally (c) upscaling the AP model to the sample scale, accounting for the heterogeneous distribution of the organic matter. The up-scaled AP model is more realistic, because it considers not only the varying cross-section shapes and tortuosity of the pores, but also the characteristic flow mechanisms and multi-scale pore size distribution in heterogeneous shale matrix systems. The model was successfully validated with experimental data from real samples. The effects of the organic matter and shape of the pores on the AP are investigated. The sensitivity of the AP to the total organic carbon (TOC), porosity, pore shape, and structural parameters of both organic and inorganic systems is studied. The results indicated that the AP would be overestimated by up to 24.1% if the characteristics of the organic matrix are ignored, and the AP proves more sensitive to the effect of the pore shape in the inorganic matrix. In addition, the top three key parameters affecting the AP are pore shape, maximum pore diameter in the inorganic matrix, and porosity. The AP is more sensitive to the pore size within the inorganic matrix than within the organic matter. The proposed model provides some theoretical and technical support for shale gas simulations.

A diffusion–viscous flow model for simulating shale gas transport in nano-pores

A model for gas flow in nano-pores was developed based on the extended Navier–Stokes equations with the assumption of neglecting adsorption and desorption. The model describes multiple flow regimes, including continuum regime, slip flow regime, transition regime, as well as molecular regime. The total mass flux includes a convective motion term and a diffusion mass transport term. The latter was obtained as a weighted superposition of bulk and Knudsen diffusion. The mass flux, contributed by different transport mechanisms, was analyzed by varying the Knudsen number (ratio between mean free path of the molecules and the pore diameter). The effect of the ratio coefficient (the power-law exponent in the relation between bulk and Knudsen diffusion), the pore size and the pressure on the gas transport was investigated using the proposed model. The predictions of the newly developed model are in good agreement with the Direct Simulation Monte Carlo (DSMC) method and with the results of the experiments. Our results show that: (1) As the Knudsen number increases, the contribution of viscous flow to gas transport decreases monotonically, bulk diffusion increases to a peak, and then decreases, whereas the Knudsen diffusion increases monotonically. (2) For a larger ratio coefficient, the bulk diffusion changes more rapidly and the peak is higher. The changing trend of the bulk diffusion is opposite to the Knudsen diffusion. (3) The pressure range in which the viscous flow dominates becomes larger and the peak of bulk diffusion is smaller in larger pores. When the pore pressure is higher, the viscous flow and the bulk diffusion tend to dominate.

A particle–grid method for Euler–Lagrange approach

In Euler–Lagrange approach, the physical values of fluid and solid phases are computed in a common grid, namely fluid grid. The refinement of the fluid grid resolution beyond the particle scale leads to inconsistency with the basis of using extended Navier–Stokes equations for gas–solid flow (Anderson et al., 1967). In this study, an additional particle–grid is applied, in which the physical values of solid phase are computed. To investigate the influence of the particle–grid application on the simulation accuracy, the numerical results obtained by Euler–Lagrange approach combined with a deterministic collision model (known also as Discrete Element Method (DEM)) are validated with measurements obtained from a lab-scale spouted fluidized bed. The results confirm that the particle–grid method allows the variation of the fluid grid resolution independent of the particle size and consequently improves the calculation accuracy. In the second part of this work, the simulation results obtained from the extended Euler–Lagrange/DEM model are compared with the simulation results obtained from the Euler–Lagrange approach combined with a stochastic collision model. Two different fluidization mass flow rates are considered to analyse the ability of the used simulation approaches to predict the hydrodynamic behaviour of the gas spouted fluidized bed. The results show that both techniques can reproduce the right fluidization regimes including the bubble size and the bed expansion. Deviations from the experimental data in the jet zone and during the final stage of the bubble formation are, however, observed. The reasons of these discrepancies in predicting the dynamic behaviour of the bed, the advantages and limitations of the two approaches are demonstrated.

Extended Navier–Stokes equations, ultrasonic absorption and shock structure

The Navier–Stokes (N–S) equations for a steady or unsteady, compressible, continuum flow are modified. The extension is based on a Stokesian fluid with a single nonlinear term in an isotropic stress, rate-of-deformation relation. This is the simplest possible nonlinear extension that also satisfies the second law of thermodynamics. The transport coefficient of this term is referred to as the third viscosity coefficient. In the extended version, the momentum and energy equations each contain a nonlinear term that is proportional to this new coefficient. These terms are significant only when the velocity gradient is extremely large. They are inconsequential, e.g., in a laminar boundary layer. Nevertheless, there are flows where the extended version of the N–S equations is relevant. The first of these is an ultrasonic, unsteady, one-dimensional flow, which is used for evaluating the bulk viscosity. In this case, the linearized N–S equations become singular as the ultrasonic frequency increases toward infinity. When the frequency is sufficiently large, nonlinear terms in the extended N–S equations need to be retained. The terms that are proportional to the third viscosity coefficient increase in importance, relative to linear terms, as the fourth power of the frequency. A second example is shock wave structure. A model is established and numerically solved for the normalized density derivative. Results are compared with corresponding measurements for argon when the upstream Mach number is 1.058 and 1.23. Good agreement between the extended N–S predictions and measurements is obtained for both Mach numbers with a single, but extremely small, value for the third viscosity coefficient. An important difference between conventional and extended N–S shock structure solutions is that the extended-model solution depends on the upstream pressure, whereas the conventional solution does not.

Molecular Dynamics Study of Nanoconfined Water Flow Driven by Rotating Electric Fields under Realistic Experimental Conditions

In our recent work, J. Chem. Phys. 2013, 138, 154712, we demonstrated the feasibility of unidirectional pumping of water, exploiting translational-rotational momentum coupling using nonequilibrium molecular dynamics simulations. Flow can be sustained when the fluid is driven out of equilibrium by an external spatially uniform rotating electric field and confined between two planar surfaces exposing different degrees of hydrophobicity. The permanent dipole moment of water follows the rotating field, thus inducing the molecules to spin, and the torque exerted by the field is continuously injected into the fluid, enabling a steady conversion of spin angular momentum into linear momentum. The translational-rotational coupling is a sensitive function of the rotating electric field parameters. In this work, we have found that there exists a small energy dissipation region attainable when the frequency of the rotating electric field matches the inverse of the dielectric relaxation time of water and when its amplitude lies in a range just before dielectric saturation effects take place. In this region, that is, when the frequency lies in a small window of the microwave region around ∼20 GHz and amplitude ∼0.03 V Å(-1), the translational-rotational coupling is most effective, yielding fluid velocities of magnitudes of ∼2 ms(-1) with only moderate fluid heating. In this work, we also confine water to a realistic nanochannel made of graphene giving a hydrophobic surface on one side and β-cristobalite giving a hydrophilic surface on the other, reproducing slip-and-stick velocity boundary conditions, respectively. This enables us to demonstrate that in a realistic environment, the coupling can be effectively exploited to achieve noncontact pumping of water at the nanoscale. A quantitative comparison between nonequilibrium molecular dynamics and analytical solutions of the extended Navier-Stokes equations, including an external rotating electric field has been performed, showing excellent agreement when the electric field parameters match the aforementioned small energy dissipation region.

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

Microchannel gas flows are of importance in a wide range of microelectro mechanical devices. In these flows, the mean free path of the gas can be comparable to the characteristic length of the microchannel, leading to strong diffusion‐enhanced transport of momentum. Numerical solutions to the extended Navier–Stokes equations (ENSE) have successfully modeled such microchannel flows. Analytical solutions to the ENSE for the pressure and velocity fields using the Lambert W function are derived. We find that diffusive contributions to the total transport are only dominant for low average pressures and low pressure drops across the microchannel. For large inlet pressures, we show that the expressions involving the Lambert W function predict steep gradients in the pressure and velocity localized near the channel exit. We extract a characteristic length for this boundary layer. Our analytical results are validated by numerical and experimental results available in the literature. © 2014 American Institute of Chemical Engineers AIChE J, 60: 1413–1423, 2014

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

Flows in microchannels were successfully predicted, in the past, both analytically and numerically, employing the extended Navier–Stokes equations (ENSE). In ENSE, the self-diffusion transport of mass, together with the resulting momentum and heat transport, is taken into account properly and the same is omitted in the classical Navier–Stokes equations. The ENSE have been employed here to numerically predict backward-facing step flows in microchannels, and the predictions are summarized in this paper. The results obtained by employing ENSE are compared with the available literature data computed by both direct simulation Monte Carlo and slip-velocity-based simulations. The good agreement of the present results with those given in the literature evidently points out that the ENSE can be applied to gas flows through complex microchannel geometries.

Electrokinetics of a charged permeable porous aggregate in an aqueous medium

The combined effects of hydrodynamical flow and electric field on the transport of permeable aggregate with fixed charged density are analyzed. Electrophoretic separation is often combined with a hydrodynamically driven flow to speed-up the transport of colloidal particles. The electrophoresis of the aggregate is governed by the interaction of electroosmotic flow in and around the particle, its hydrodynamical friction and the electric force experienced by the particle. The imposed convection of the aggregate further modifies the double layer polarization (DLP) and counterion penetration compared to the electrophoresis case. We have analyzed the effects of convection on DLP and diffusion dominated counterions penetration for a wide range of key parameters such as, permeability, fixed charge density and Debye layer thickness. Our numerical model is based on the non-linear Nernst–Planck equations for ion transport coupled with the Darcy–Brinckman extended Navier–Stokes equations and the Poisson equation for electric field. The electroosmotic flow within and around the aggregate produces a retardation effect which results in the drag experienced by the aggregate being higher than the corresponding Stokes drag of an uncharged aggregate. The induced electric field, which is opposite to the direction of an applied electric field, grows with the rise of the translation speed of the aggregate. At large convection speed, the strong momentum of the aggregate nullifies the retarding effect due to electroosmosis. The electroosmosis of the electrolyte has a significant impact on the dynamics of the aggregate when the Debye layer thickness is high. The impact of electroosmosis enhances for a higher charge density as well as a higher permeability of the particle when the Debye layer is thick.

Effects of electroosmosis and counterion penetration on electrophoresis of a positively charged spherical permeable particle

We have studied the electrophoresis of a positively charged spherical porous particle (polyelectrolyte) through an electrolyte solution. The migration of the positively charged particle leads to a deformation of the anionic cloud around it. The hydrodynamics is governed by the interaction of electroosmotic flow in and around the particle, its hydrodynamic friction and the electric force experienced by the particle. Our numerical model is based on the non-linear Nernst–Planck equations for ion transport coupled with the Darcy–Brinkman extended Navier–Stokes equations and the Poisson equation for an electric field. The effect of convection and an imposed electric field on the ion transport is included in the present analysis. A control volume approach over a staggered grid arrangement is adopted for computations. The permeability of the particle causes electroosmosis within the particle, which in addition to the electroosmosis in the double layer and the induced pressure gradient due to the migration of the particle creates a double layer polarization (DLP) effect. The DLP effect induces an electric field, which retards the applied field strength. The penetration of counterions inside the particle causes a neutralization of the particle charges, which reduces the electric force exerted on the particle. The distribution of counterions, net charge density and the effective volume charge density of the particle is evaluated to analyze the dependence of the electric force and hence the mobility on electrokinetic parameters, such as the electrolyte concentration, permeability and fixed charge density of the particle. Our results show that the counterion penetration for a high electrolyte concentration or low permeability causes a significant reduction in the effective charge density of the particle, which results in low mobility. However, the double layer and surface potential assumes symmetry i.e., the DLP effect is negligible for a thin EDL or low permeable cases. The fixed charge density of the particle increases the electric body force linearly until the counterion condensation sets-in at a large value of the fixed charge density.