Nonlinear Shear Viscosity Research Articles

Global existence of entropy-weak solutions to the compressible Navier–Stokes equations with non-linear density dependent viscosities

In this paper, we considerably extend the results on global existence of entropy-weak solutions to the compressible Navier–Stokes system with density dependent viscosities obtained, independently (using different strategies) by Vasseur–Yu [Invent. Math. 206 (2016) and arXiv:1501.06803 (2015)] and by Li–Xin [arXiv:1504.06826 (2015)]. More precisely, we are able to consider a physical symmetric viscous stress tensor \sigma = 2 \mu(\rho) \,{\mathbb D}(u) +(\lambda(\rho) \operatorname{div} u - P(\rho) \operatorname {Id} where {\mathbb D}(u) = [\nabla u + \nabla^T u]/2 with shear and bulk viscosities (respectively \mu(\rho) and \lambda(\rho) ) satisfying the BD relation \lambda(\rho)=2(\mu'(\rho)\rho - \mu(\rho)) and a pressure law P(\rho)=a\rho^\gamma (with a>0 a given constant) for any adiabatic constant \gamma>1 . The non-linear shear viscosity \mu(\rho) satisfies some lower and upper bounds for low and high densities (our result includes the case \mu(\rho)= \mu\rho^\alpha with 2/3 < \alpha < 4 and \mu>0 constant). This provides an answer to a longstanding question on compressible Navier–Stokes equations with density dependent viscosities, mentioned for instance by F. Rousset [Bourbaki 69ème année, 2016–2017, exp. 1135].

Effects of contraction ratio on non-linear dynamics of semi-dilute, highly polydisperse PAAm solutions in microfluidics

Highly non-linear flows of semi-dilute ( 3.3 ⩽ c / c ∗ ⩽ 16.6 ) polyacrylamide (PAAm) aqueous solutions through three micro-fabricated 4:1:4, 8:1:8 and 16:1:16 contraction–expansion geometries have been investigated by micro-particle imaging velocimetry (μ-PIV) and pressure drop measurements. A set of quantitative data for PAAm solutions are obtained from: Gel Permeation Chromatography (GPC) analysis; rheometric characterization using Vilastic-3 and piezoelastic axial vibrator (PAV) with frequency up to 6650 Hz for linear viscoelasticity measurements, and using a microfluidic device for the measurement of the nonlinear shear viscosity. With three different flow geometries, and a wide range of Elasticity numbers ( 14.7 ⩽ El ⩽ 476.8 ) and Weissenberg numbers ( 1.4 ⩽ Wi ⩽ 131.7 ) , different nonlinear flow regimes are identified and correlated to vortex growth mechanisms. Results of similar El flows were mapped in a contraction ratio – Wi diagram. We found that under a given flow geometry tuning El can either increase or decrease the stability of the nonlinear flow structures. In addition to Wi and Re, the viscoelastic flow phenomena are dependent on the contraction and aspect ratios of flow geometry, and also sensitive to the molecular weight distribution of polymer. These results are of significance to the further development of microfluidic technologies.

Impurity in a granular gas under nonlinear Couette flow

We study in this work the transport properties of an impurity immersed in a granular gasunder stationary nonlinear Couette flow. The starting point is a kinetic model forlow-density granular mixtures recently proposed by the authors (Vega Reyes et al 2007 Phys. Rev. E 75 061306). Two routes have been considered. First, a hydrodynamic ornormal solution is found by exploiting a formal mapping between the kinetic equations forthe gas particles and for the impurity. We show that the transport properties of theimpurity are characterized by the ratio between the temperatures of the impurity and gasparticles and by five generalized transport coefficients: three related to the momentum flux(a nonlinear shear viscosity and two normal stress differences) and two related to the heatflux (a nonlinear thermal conductivity and a cross-coefficient measuring a component of theheat flux orthogonal to the thermal gradient). Second, by means of a Monte Carlosimulation method we numerically solve the kinetic equations and show that ourhydrodynamic solution is valid in the bulk of the fluid when realistic boundary conditionsare used. Furthermore, the hydrodynamic solution applies to arbitrarily (inside thecontinuum regime) large values of the shear rate, of the inelasticity, and of therest of the parameters of the system. Preliminary simulation results of the trueBoltzmann description show the reliability of the nonlinear hydrodynamic solution ofthe kinetic model. This shows again the validity of a hydrodynamic descriptionfor granular flows, even under extreme conditions, beyond the Navier–Stokesdomain.

Structure of a semidilute polymer solution under steady shear

We use Brownian dynamics computer simulations to investigate the structure of a semidilute polymer solution undergoing a steady, uniform shear flow. We find that the contributions to structure factor from intra- and interchain correlations, which cancel each other almost completely for an equilibrium semidilute solution, are modified in different ways by the shear flow. Incomplete cancellation of these contributions leads to anisotropic patterns that resemble those observed in light scattering experiments on sheared semidilute solutions [Wu et al., Phys. Rev. Lett. 66, 2408 (1991)]. For small wave vectors the structure factor change is dominated by the interchain contribution. We also monitor the distortion of the pair correlation function and show that for small distances it is dominated by the intrachain contribution. Finally, we investigate nonlinear shear viscosity and find that, like the short-distance part of the distortion of the pair correlation function, it is predominantly of intrachain origin.

Analytical solution for the Hagen–Poiseuille flow of a hard sphere gas with nonlinear shearviscosity

We study the Hagen–Poiseuille flow in cylindrical geometry of a Zwanzig fluid, for whichthe shear viscosity is a highly nonlinear function of the shear rate. It is shownthat the problem has a real solution only if the radial distance from the flowcentreline is smaller than a critical value, showing that such a constitutive model isnot always compatible with the flow under a constant pressure gradient. Thisleads to the introduction of scenarios such as shear banding, shear-induced phasetransitions, and wall slip, which are generally believed to occur only in complexfluids.

Molecular drag–strain coupling in branched polymer melts

The “pom-pom” model of McLeish and Larson [J. Rheol. 42, 81–110 (1998)] provides a simple molecular theory for the nonlinear rheology of long chain branched polymer melts. A feature of this model is a maximum stretch for the branched molecules. Sharp transitions were predicted in the extensional viscosity at this maximum stretch. We introduce a simple treatment of the coupling between relaxed and unrelaxed polymer segments at branch points. The branch point is allowed to move in a quadratic localizing potential of unknown strength. Taking account of this effect smoothes the sharp transitions of the model and accounts for the extensional viscosity of “pom-pom” model polymers at their maximum stretch. The result is an improved multimode pom-pom fit for low-density polyethylene rheology. By fitting the nonlinear extensional viscosity, quantitative predictions are made for the nonlinear steady shear viscosity and transient first normal stress difference in shear.

On the validity of a variational principle for multicomponent systems

The validity of a variational principle for nonequilibrium steady states proposed by Evans and Baranyai [Phys. Rev. Lett. 67, 2597 (1991)] is investigated in the case of a dilute binary mixture described by the well-known Groos–Krook kinetic model. We construct a perturbation solution around the unconstrained shear flow state and evaluate the phase-space compression factor, the temperature ratios, and the nonlinear shear viscosity up to the first-order approximation. All these quantities are nonlinear functions of the shear rate and the parameters of the mixture (particle masses, concentrations, and force constants). It is shown that this principle does not hold exactly, although deviations from it are small in some situations for not very large shear rates. The calculations presented here extend previous results derived for a single dilute gas.

Does the Gaussian thermostat maximize the phase-space compression factor?

A recent hypothesis of D. J. Evans and A. Baranyai according to which the Gaussian thermostat maximizes the average phase-space compression factor λ in nonequilibrium steady states is analyzed for a dilute gas under uniform shear flow. Three routes have been followed: (i) an exact solution of the Bhatnagar-Gross-Krook kinetic equation for arbitrary shear rate, (ii) an exact solution of the Boltzmann equation through super-Burnett order, and (iii) a numerical solution of the Boltzmann equation for finite shear rates. The results show that the above hypothesis does not exactly hold for arbitrary shear rates, although the thermostat that maximizes λ is close to the Gaussian one. In addition, the influence of the thermostat considered on the nonlinear shear viscosity is also analyzed.

Combined heat and momentum transport in a dilute gas

The infinite hierarchy of moment equations derived from the Boltzmann equation for Maxwell molecules is analyzed in the case of steady planar Couette flow. It is proved that a solution exists that is consistent with the following hydrodynamic profiles: p=nkBT=const, T∂ux/∂y=const, (T∂/∂y)2T=const. In general, the velocity moments of order k are polynomials of degree k−2 in a scaled space variable s∝∫T−1dy. The momentum and energy transport are described by a nonlinear shear viscosity η(a)=η(0)Fη(a) and a nonlinear thermal conductivity κ(a)=κ(0)Fκ(a), respectively, where a≡∂ux/∂s is the (constant) reduced shear rate. By performing a perturbation expansion in powers of a, it is found that Fη(a)=1−1.472a2+𝒪(a4) and Fκ(a)=1−3.226a2+𝒪(a4). These numerical values are compared with those obtained from the BGK and the Liu kinetic models.

A ?zero-parameter? constitutive relation for simple shear viscoelasticity

Based on the Cox-Merz rule and Eyring's expression for the nonlinear shear viscosity, a Wagner-type constitutive relation with no nontrivial adjustable parameters is proposed for simple shear viscoelasticity. The predictions for a number of non-steady shear flows are worked out analytically. It is shown that most features of shear viscoelasticity are reproduced by the model.

Nonlinear shear viscosity in two dimensions.

Application de la methode de la fonction de correlation en transitoire au calcul de la viscosite de cisaillement non lineaire du fluide a 2 dimensions de disques mous

Asymptotic nonlinear stress tensor in small periodic systems undergoing Couette flow.

In this paper we describe a method for calculating the nonlinear shear viscosity at arbitrarily small strain rates. Although these calculations are consistent with previously observed enhanced long-time tail behavior at higher strain rates, they also show that in small periodic systems classical Burnett behavior is the ultimate small-field nonlinear response.

Viscoelasticity of homogeneous polymer melts near a critical point

The anomalous contribution to the shear stress caused by critical fluctuations in homogeneous polymer melts is investigated. Mean field expressions for the critical contributions to the stress tensor, dynamic shear modulus, and nonlinear steady shear viscosity are derived. These expressions are applied to the case of a homogeneous diblock copolymer near its microphase separation transition. The low frequency scaling behavior of the storage modulus is found to be G′(ω)∼ω2a−5/2, while the loss modulus is predicted to scale like G″(ω)∼ωa−3/2. The dimensionless parameter a is defined by a=2(χN)s−2χN, where χN is the product of the Flory interaction parameter and the number of Kuhn segments per chain, and (χN)s is the value of χN on the spinodal computed by Leibler. The linear steady shear viscosity is also found to be singular as a→0. A comparison is made between the present theoretical predictions and the recent experiments of Bates on 1,4/1,2 polybutadiene block copolymers.

Theory of dynamic shear viscosity and normal stress coefficients of dense fluids

On the basis of the nonlinear evolution equation for the stress tensor derived from the kinetic equation for dense simple fluids, we examine the shear rate and frequency dependence of the shear viscosity and normal stress coefficients. An analytic expression for nonlinear shear viscosity has been obtained, which is in agreement with nonequilibrium molecular dynamic simulation results. Based on the material functions obtained in the analysis, a theory of rheological corresponding states is developed by showing that there exists a set of reduced material functions which depend on the reduced shear rate, reduced frequency, reduced static normal coefficients and limiting values of the reduced imaginary parts of dynamic normal stress coefficients. It is also shown that the dynamic normal stress coefficients can change its sign as the critical value of the frequency is crossed.

Mode-coupling contributions to the nonlinear shear viscosity.

Three different results for the nonanalytic dependence of the shear viscosity on shear rate have been given. This discrepancy is resolved by an independent calculation based on the nonlinear Navier-Stokes-Langevin equations. The relationship to previous calculations and reasons for the differences are described.

Dynamics of concentration fluctuations for polydimethylsiloxane-diethyl carbonate

The scaled decay rate ${\ensuremath{\Gamma}}^{*}[=\frac{{\ensuremath{\Gamma}}^{c}}{\overline{R}{k}_{B}}\frac{T{k}^{3}}{\ensuremath{\eta}}(T)]$ of the system polydimethylsiloxane-diethyl carbonate, where $\ensuremath{\eta}(T)$ is the shear viscosity, $k$ the transfer wave vector, and $\overline{R}$ a universal constant, is in agreement with the mode-coupling theory in the range of $0.2\ensuremath{\lesssim}k\ensuremath{\xi}\ensuremath{\lesssim}3$, and in disagreement with the renormalization-group calculation of critical dynamics with $\overline{R}=1.2{(6\ensuremath{\pi})}^{\ensuremath{-}1}$. The nonlinear shear viscosity due to shear gradients is in agreement with the theoretical prediction by Oxtoby.

Nonlinear shear viscosity and long time tails

A theoretical connection between nonlinear shear viscosity and the long time tail of the equilibrium stress-stress correlation function is pointed out. The connection is a consequence of the Goddard-Miller rheological equation of state which takes into account the angular rotation of a fluid in steady uniform shear.

Nonlinear shear viscosity of a gas

This article contains a derivation of the shear rate dependence of the viscosity of a gas in steady Couette flow. The viscosity is found from a solution of the Bhatnagar-Gross-Krook model kinetic equation, taking into account the viscous heating of the gas. The results are in reasonably good agreement with computer molecular dynamic experiments by Naitoh and Ono.

Non-newtonian effect and long-range correlation in shear flow in two dimensions

A steady state of a two-dimensional incompressible fluid with shear flow is considered. The nonlinear (non-newtonian) shear viscosity is obtained and the correlation of velocity field fluctuations is found to become long-ranged.

Kinetic theory of nonlinear viscous flow in two and three dimensions

On the basis of a nonlinear kinetic equation for a moderately dense system of hard spheres and disks it is shown that shear and normal stresses in a steady-state, uniform shear flow contain singular contributions of the form ¦X¦3/2 for hard spheres, or ¦X¦ log ¦X¦ for hard disks. HereX is proportional to the velocity gradient in the shear flow. The origin of these terms is closely related to the hydrodynamic tails t−d/2 in the current-current correlation functions. These results also imply that a nonlinear shear viscosity exists in two-dimensional systems. An extensive discussion is given on the range ofX values where the present theory can be applied, and numerical estimates of the effects are given for typical circumstances in laboratory and computer experiments.