Classical Navier-Stokes Theory Research Articles

Transition from laminar to turbulent pipe flow as a process of growing material instabilities

Experiments show that a laminar pipe flow starts to change to turbulence at Reynolds numbers around 2000. However, the classical Navier-Stokes theory fails to explain such a transition; according to it the pipe flow should always be stable and laminar. In the present work, it is hypothesized that the transition to turbulence occurs due to material rather than kinematic instabilities of the flow, and the classical Navier-Stokes theory is enhanced with a description of viscous strength. Numerical simulations of the pipe flow with and without the viscous strength are performed. The results show that material instabilities due to the viscous strength gradually develop into a turbulent flow, in agreement with the experimental observations. In the absence of viscous strength, the pipe flow remains laminar in disagreement with experiments. It is concluded that the material instabilities can generally compete with the kinematic ones in the process of transition from laminar to turbulent flow.

Flow past a porous plate of non-Newtonian fluids with implicit shear stress shear rate relationships

One of the simplest problems in the classical Navier–Stokes theory that illustrates the development of boundary layers is the asymptotic suction problem (Schlichting (1960)). This problem has been studied within the context of many different non-Newtonian fluid models and in this note we consider it within the context of a new class of constitutive relations that has been put into place. Depending on the values of the parameters, this class of constitutive relations includes models for both shear thinning and shear thickening fluids, and truly implicit shear stress shear rate relationships. For any set of the material parameters we are able to find exact solutions to the governing equations, study their stability, show that suction has a stabilizing effect, and delineate the development of boundary layers.

A discontinuous variational principle implying a non-equilibrium dispersion relation for damped acoustic waves

The discontinuous Lagrangian approach, allowing for a variational description of irreversible phenomena in continuum theory such as viscosity and thermal conductivity, is utilised for the analysis of damped acoustic waves. Starting from a Lagrangian for general viscous flow theory, by linearisation of the resulting Euler–Lagrange equations and performing an ensemble average, a single wave equation for the density perturbations is obtained, being the one resulting from classical Navier–Stokes theory with an additional term due to thermodynamic non-equilibrium. By considering harmonic waves, the respective non-classical dispersion relation and its implications are analysed.

Analytical solution of Newtonian nanofluid flow in a tapered artery based on a consistent couple stress theory

Reduction of the stenosis in arteries is one of the most important applications of nanoparticles in medicine. Thus, investigating nanoparticles in stenosed arteries is a subject attracting so much attention from the researchers nowadays. However, considering the small size of arteries, it seems that non-classical theories demonstrate better results when compared to the classical Navier-Stokes theory. The present paper aims at investigating the Newtonian nanofluid flow through divergent and convergent, non-tapered and tapered arteries with mild stenosis according to the consistent couple stress theory. Concentration, temperature, and velocity profiles, as well as the impact of various parameters such as the height of the stenosis, the shape of the stenosis curve, Brownian distribution parameter, thermophoresis distribution parameter, Darcy number, and the length scale were calculated for all the three geometries. Results demonstrate that an increase in the height of the stenosis and an increase in the Darcy coefficient, respectively, lead to a decrease and an increase in the axial velocity in all three artery geometries. Moreover, an increase in the impact of the length scale leads to a decrease in the axial velocity. The impact of the length scale on the velocity profile shows the significance of the length parameter on the flow within the geometries with small sizes which is not present in the classical Navier-Stokes theory.

Rapidity Correlation Structures from Causal Hydrodynamics

Viscous diffusion can broaden the rapidity dependence of two-particle transverse momentum fluctuations. Surprisingly, measurements at RHIC by the STAR collaboration demonstrate that this broadening is accompanied by the appearance of unanticipated structure in the rapidity distribution of these fluctuations in the most central collisions. Although a first order classical Navier-Stokes theory can roughly explain the rapidity broadening, it cannot explain the additional structure. We propose that the rapidity structure can be explained using the second order causal Israel-Stewart hydrodynamics with stochastic noise.

A micropolar fluid model of blood flow through a tapered artery with a stenosis

A nonlinear two-dimensional micropolar fluid model for blood flow in a tapered artery with a single stenosis is considered. This model takes into account blood rheology in which blood consists of microelements suspended in plasma. The classical Navier–Stokes theory is inadequate to describe the microrotations or particles' spin of such suspension in a viscous medium. The governing equations involving unsteady nonlinear partial differential equations are solved using a finite difference scheme. A quantitative analysis performed through numerical computation shows that the axial velocity profile and the flow rate decrease and the wall shear stress increases once the artery is narrower in the presence of the polar effect. Furthermore, the taper angle certainly bears the potential to influence the velocity and the flow characteristics to considerable extent

Unsteady Mixed Convection Boundary Layer from a Circular Cylinder in a Micropolar Fluid

Most industrial fluids such as polymers, liquid crystals, and colloids contain suspensions of rigid particles that undergo rotation. However, the classical Navier-Stokes theory normally associated with Newtonian fluids is inadequate to describe such fluids as it does not take into account the effects of these microstructures. In this paper, the unsteady mixed convection boundary layer flow of a micropolar fluid past an isothermal horizontal circular cylinder is numerically studied, where the unsteadiness is due to an impulsive motion of the free stream. Both the assisting (heated cylinder) and opposing cases (cooled cylinder) are considered. Thus, both small and large time solutions as well as the occurrence of flow separation, followed by the flow reversal are studied. The flow along the entire surface of a cylinder is solved numerically using the Keller-box scheme. The obtained results are compared with the ones from the open literature, and it is shown that the agreement is very good.

Viscous properties of isotropic fluids composed of linear molecules: Departure from the classical Navier-Stokes theory in nano-confined geometries

In this paper we present equilibrium molecular-dynamics results for the shear, rotational, and spin viscosities for fluids composed of linear molecules. The density dependence of the shear viscosity follows a stretched exponential function, whereas the rotational viscosity and the spin viscosities show approximately power-law dependencies. The frequency-dependent shear and spin viscosities are also studied. It is found that viscoelastic behavior is first manifested in the shear viscosity and that the real part of the spin viscosities features a maximum for nonzero frequency. The calculated transport coefficients are used together with the extended Navier-Stokes equations to investigate the effect of the coupling between the intrinsic angular momentum and linear momentum for highly confined fluids. Both steady and oscillatory flows are studied. It is shown, for example, that the fluid flow rate for Poiseuille flow is reduced by up to 10% in a 2 nm channel for a buta-triene fluid at density 236 kg m(-3) and temperature 306 K. The coupling effect may, therefore, become very important for nanofluidic applications.

A numerical method for a second-gradient theory of incompressible fluid flow

This work concerns the development of a finite-element method for discretizing a recent second-gradient theory for the flow of incompressible fluids. The new theory gives rise to a flow equation involving higher-order gradients of the velocity field and introduces an accompanying length scale and boundary conditions. Finite-element methods based on similar equations involving fourth-order differential operators typically rely on C 1-continuous basis functions or a mixed approach, both of which entail certain implementational difficulties. Here, we examine the adaptation of a relatively inexpensive, non-conforming method based on C 0-continuous basis functions. We first develop the variational form of the method and establish consistency. The method weakly enforces continuity of the vorticity, traction, and hypertraction across interelement boundaries. Stabilization is achieved via Nitsche’s method. Further, pressure stabilization scales with the higher-order moduli, so that the classical formulation is recovered as a particular limit. The numerical method is verified for the problem of steady, plane Poiseuille flow. We then provide several numerical examples illustrating the robustness of the method and contrasting the predictions to those provided by classical Navier–Stokes theory.

Physical properties of sulfur near the polymerization transition.

Acoustical measurements, electron spin resonance, and Raman spectroscopy have been employed to probe sulfur over the temperature range 80-180 degrees C, which includes the polymerization transition and the supercooled liquid state. Acoustical properties (sound velocity, absorption, and impedance) have been studied with both longitudinal and transverse waves at frequencies between 500 kHz and 22 MHz. The results confirm that polymeric sulfur is a solution of long chain molecules in monomeric solvent, and that the polymerization transition is not a second-order phase transition, as was proposed theoretically. Sulfur is a viscous liquid, but not viscoelastic, both below and above the polymerization transition temperature. It is shown that the classical Navier-Stokes theory is not applicable to the sound absorption in liquid sulfur in the highly viscous state.

Poiseuille flow of a micropolar fluid

We use non-equilibrium molecular dynamics simulations to study the flow of a micropolar fluid and to test an extended Navier-Stokes theory (ENS) for such fluids. The angular streaming velocity (which is of course missing in the classical Navier-Stokes theory) and the translational streaming velocity are found to be in good agreement with the predictions of ENS theory. Besides, owing to molecular rotation, the translational streaming velocity profile is shown to deviate from the classical parabolic profile. Finally, temperature profiles calculated using three different expressions (a kinetic translational, a kinetic rotational and a recently derived configurational expression) are found to be in excellent agreement, demonstrating that the equipartition principle still holds in this non-equilibrium system. No deviation from the classical quartic temperature profile is observed.

Laminar fluid behavior in microchannels using micropolar fluid theory

In this paper, we describe microchannel fluid behavior using a numerical model based on micropolar fluid theory and experimentally verify the model using micromachined channels. The micropolar fluid theory augments the laws of classical continuum mechanics by incorporating the effects of fluid molecules on the continuum. The behavior of fluids was studied using surface micromachined rectangular metallic pipette arrays. Each array consisted of 5 or 7 pipettes with widths varying from 50 to 600 μm and heights ranging from 20 to 30 μm. A downstream port for static pressure measurement was used to eliminate entrance effects. A controllable syringe pump was used to provide flow while a differential pressure transducer was used to record pressure drop. The experimental data obtained for water showed an increase in the Darcy friction factor when compared to traditional macroscale theory, especially at the lower Reynolds number flows. The numerical model of the micropolar fluid theory predicted experimental data better than the classical Navier–Stokes theory and the model compares favorably with the currently available experimental data.

Departure from Navier-Stokes hydrodynamics in confined liquids

In this work we use nonequilibrium molecular dynamics (NEMD) to simulate an atomic liquid undergoing gravity-fed flow down a narrow channel. We compare the simulation results against the predictions of classical Navier-Stokes theory for two different channel widths. For a channel width of 5.1 molecular diameters, we find that the velocity profile deviates significantly from the hydrodynamic prediction. The shape of this velocity profile is found to be independent of the applied field (pressure gradient). We find that the heat flux profile does not agree with the cubic profile predicted by Navier-Stokes hydrodynamics, but shows significant oscillations located about one molecular diameter from the walls. This result differs from the earlier work of Todd and Evans [B. D. Todd and D. J. Evans, J. Chem. Phys. 103, 9804 (1995)], in which an assumption of a purely quadratic velocity profile resulted in very weak oscillations in the heat flux. We find that in narrow channels the viscosity cannot be described by a linear, local constitutive relation. However, classical Navier-Stokes behavior is approached for a channel width of g\ensuremath{\sim}10 molecular diameters.

Start-up flows of second grade fluids in domains with one finite dimension

A number of unidirectional transient flows of a second grade fluid in a domain with one finite dimension are studied. The method of integral transforms (Fourier, Hankel or Laplace) is applied to obtain exact solutions. A general theorem on start-up flows for second grade fluids is presented that allows us to determine unidirectional flows of second grade fluids once the corresponding solution is known within the context of the Navier-Stokes theory. In the process of obtaining solutions for the fluid of second grade, we find several new exact solutions within the context of the classical Navier-Stokes theory.

Viscous Multipolar Fluids-Physical Background and Mathematical Theory-

We introduce the theory of multipolar fluids in which constitutive laws depend linearly not only on the first spatial gradients of velocity as in classical Navier-Stokes theory of newtonian fluids but also on its higher order spatial gradients up to the order 2k − 1, k = 2, 3,… Such fluids are called k-polar fluids. A thermodynamic theory of the constitutive equations satisfying the second law of thermodynamics and the principle of material frame indifference is developed. Special thermodynamic processes as isothermal, barotropic, adiabatic and general heat-conductive motion for compressible multipolar fluids are studied. It is well known that there does not exist adequate existence theory for compressible newtonian fluids. We given a consistent theory for compressible multipolar fluids in two or three dimensions, i. e. we prove the global in time existence of weak solutions for the initial boundary value problems in bounded domains for the systems of partial differential equations describing isothermal, barotropic, adiabatic and general compressible motion. Under some assumptions on the regularity of the initial data and external forces, we prove existence of strong solutions, uniqueness and regularity. Some other properties as e. g. cavitation of density are discussed. We put stress on the lowest possible polarity of the fluid. In the isothermal case we consider the polarity k ≧ 2 and in barotropic and heat-conductive gas the polarity k ≧ 3.

Plane interior boundary-value problems in microcontinuum fluid mechanics

This paper examines the fundamental question of existence and uniqueness of solutions of two plane interior boundary-value problems encountered in the field of microcontinuum fluid mechanics. These problems are analyzed using potential theoretic methods and with the aid of singular integral equations. The results obtained have their counterparts in the classical Navier-Stokes theory.

Navier-Stokes Calculations of Argon Shock Wave Structure

The classical Navier-Stokes theory was used to calculate profiles of plane shock waves in argon. The viscosity-temperature relationship employed in the calculation was based on a Lennard-Jones 6–12 intermolecular potential function for which the parameters were derived from experimental viscosity measurements up to 1100°K. With this viscosity function, the maximum-slope shock thicknesses obtained were smaller than previously calculated assuming viscosity proportional to T0.816. Also it was found that the maximum-slope thickness based on the density profile was less than that based on the velocity profile. The density profile thicknesses calculated with the Lennard-Jones viscosity function were the same order of magnitude as, but not in quantitative agreement with experimental density thicknesses obtained with the optical reflectivity technique. It was found that Navier-Stokes profiles become asymmetric as the shock strength is increased. However, the calculated asymmetry would have a negligible effect on thickness measurements by the optical reflectivity method.