Planar Shear Flow Research Articles

Assessment of phenomenological models for viscosity of liquids based on nonequilibrium atomistic simulations of copper

The shear viscosity of liquid copper is studied using nonequilibrium molecular-dynamics simulations under planar shear flow conditions. We examined variation of viscosity as function of shear rate at a range of pressures (ca. 0 - 40 GPa). We analyzed these results using eight different phenomenological models and find that the observed non-Newtonian behavior is best described by the Powell-Eyring (PE) model: eta(gamma) = (eta(0)-eta(infinity))sinh(-1)(taugamma)(taugamma) + eta(infinity), where gamma is the shear rate. Here eta(0) (the zero-shear-rate viscosity) extracted from the PE fit is in excellent agreement with available experimental data. The relaxation time tau from the PE fit describes the shear response to an applied stress. This provides the framework for interpreting the shear flow phenomena in complex systems, such as liquid metal and amorphous metal alloys.

Nested separatrices in simple shear flows: The effect of localized disturbances on stagnation lines

The effects of localized two-dimensional disturbances on the structure of shear flows featuring a stagnation line are investigated. A simple superposition of a planar Couette flow and Moffatt’s [J. Fluid Mech. 18, 1–18 (1964)] streamfunction for the decay of a disturbance between infinite stationary parallel plates shows that in general the stagnation line is replaced by a chain of alternating elliptic and hyperbolic stagnation points with a separation equal to 2.78 times the half-gap between the plates. The flow structure associated with each saddle point consists of a homoclinic separatrix and two other separatrices which locally diverge but become parallel far from the disturbance. This basic structure repeats to give a sequence of nested separatrices permitting the streamfunction to approach that of simple shear flow far from the disturbance. Using the finite-element method, the specific disturbance caused by a stationary cylinder placed on the stagnation line is considered, and results confirm the existence of the stagnation point chain, with computed separations and velocity damping ratios in very good agreement with those obtained from the Couette-Moffatt superposition. Numerical solutions also illustrate that while Reynolds number greatly affects the stagnation point separation and velocity damping ratio, these two quantities are the same for any pair of adjacent stagnation points in a given chain. Insight gained from the analysis of planar shear flows is applied to the flow in a half-filled horizontal annulus between rotating coaxial cylinders, and is used to explain why only certain flow patterns from the range of mathematically possible structures arise in previous numerical solutions. By way of contrast, the concentric annulus solution is then perturbed to allow for a small eccentricity. The nonuniformity of the intercylinder gap is shown to destroy the chain of stagnation points, but also to unfold additional flow structures not realizable when the gap is uniform.

Rheophysics of dense granular materials: Discrete simulation of plane shear flows

We study the plane shear flow of a dense assembly of dissipative disks using discrete simulation and prescribing the pressure and the shear rate. Those shear states are steady and uniform, and become intermittent in the quasistatic regime. In the limit of rigid grains, the shear state is determined by a single dimensionless number, called the inertial number I , which describes the ratio of inertial to pressure forces. Small values of I correspond to the quasistatic critical state of soil mechanics, while large values of I correspond to the fully collisional regime of kinetic theory. When I increases in the intermediate dense flow regime, we measure an approximately linear decrease of the solid fraction from the maximum packing value, and an approximately linear increase of the effective friction coefficient from the static internal friction value. From those dilatancy and friction laws, we deduce the constitutive law for dense granular flows, with a plastic Coulomb term and a viscous Bagnold term. The mechanical characteristics of the grains (restitution, friction, and elasticity) have a small influence in the dense flow regime. Finally, we show that the evolution of the relative velocity fluctuations and of the contact force anisotropy as a function of I provides a simple explanation of the friction law.

On the theory of the shear-induced isotropic-to-nematic phase transition of side chain liquid-crystalline polymers

The shear-induced isotropic-to-nematic phase transition of side chain liquid-crystalline polymers is studied theoretically. A modification of the previous models of main-chain liquid crystals to the case of side chain liquid-crystalline polymers is proposed. Orientational and rheological properties of the model are studied in plane-shear flow. Predictions of the present model agree qualitatively with experimental results (Pujolle-Robic, Noirez in Nature 409:167, 2001).

Shear flow in nematic liquid crystals: Fréedericksz transition as a bifurcation

We numerically investigate the Fréedericksz transition for steady state plane shear flow of nematic liquid crystals between two parallel plates in the presence of external magnetic fields. Three typical configurations with the external field in the plane of the flow and perpendicular to it, in the plane and along the flow, and where it is perpendicular to the plane of the flow are considered. In each case, the Fréedericksz transition is studied as a bifurcation problem. Beginning with a steady state shear flow, solutions corresponding to slowly increasing magnetic fields and those corresponding to fields which are suddenly turned on at a given intensity are studied. For a typical idealized nematic, we show that the symmetric pitch-fork bifurcation in the absence of shear becomes a transcritical bifurcation from the trivial solution in one configuration while in another it resembles a disconnected pitch-fork where the turning point of the disconnected branch is a generic singularity in the absence of symmetry or a trivial solution.

Velocity Profiles in Repulsive Athermal Systems under Shear

We conduct molecular dynamics simulations of athermal systems undergoing boundary-driven planar shear flow in two and three spatial dimensions. We find that these systems possess nonlinear mean velocity profiles when the velocity u of the shearing wall exceeds a critical value u(c). Above u(c), we also show that the packing fraction and mean-square velocity profiles become spatially dependent with dilation and enhanced velocity fluctuations near the moving boundary. In systems with overdamped dynamics, u(c) is only weakly dependent on packing fraction phi. However, in systems with underdamped dynamics, u(c) is set by the speed of shear waves in the material and tends to zero as phi approaches phi(c), which is near random close packing at small damping. For underdamped systems with phi<phi(c), u(c) is zero; thus they possess nonlinear velocity profiles at any nonzero u.

Attractor dimension estimate for plane shear flow of micropolar fluid with free boundary

This research is motivated by a problem from lubrication theory. We consider a free boundary problem of a two-dimensional boundary-driven micropolar fluid flow. The existence of a unique global-in-time solution of the problem and the global attractor for the associated semigroup are known. In this paper we estimate the dimension of the global attractor in terms of the given data and the geometry of the domain of the flow by establishing a new version of the Lieb–Thirring inequality with constants depending explicitly on the geometry of the domain. We also obtain some new estimates for the Navier–Stokes shear flows. Copyright © 2005 John Wiley & Sons, Ltd.

Rheology of dense polydisperse granular fluids under shear

The solution of the Enskog equation for the one-body velocity distribution of a moderately dense arbitrary mixture of inelastic hard spheres undergoing planar shear flow is described. A generalization of the Grad moment method, implemented by means of a novel generating function technique, is used so as to avoid any assumptions concerning the size of the shear rate. The result is illustrated by using it to calculate the pressure, normal stresses, and shear viscosity of a model polydisperse granular fluid in which grain size, mass, and coefficient of restitution vary among the grains. The results are compared to a numerical solution of the Enskog equation as well as molecular-dynamics simulations. Most bulk properties are well described by the Enskog theory and it is shown that the generalized moment method is more accurate than the simple (Grad) moment method. However, the description of the distribution of temperatures in the mixture predicted by Enskog theory does not compare well to simulation, even at relatively modest densities.

Structure and rheology of model-ferrofluids under shear flow

Nonequilibrium simulations of ferromagnetic colloids in the presence of a magnetic field and plane shear flow are performed. Results for the nonequilibrium magnetization and the nonequilibrium structure factor are presented. We observe that the nonequilibrium magnetization and the magnetoviscosity are enhanced due to dipolar interactions. Structure formation due to magnetic field and shear flow are observed in qualitative agreement with experimental results.

Publisher’s Note: “Polymer drag reduction in exact coherent structures of plane shear flow” [Phys. Fluids 16, 3470 (2004)

First Page

Transient growth and minimal defects: Two possible initial paths of transition to turbulence in plane shear flows

Two possible initial paths of transition to turbulence in simple wall-bounded shear flows are examined by looking at the development in space of infinitesimal disturbances. The first is the—by-now-classical—transient growth scenario which may have an important role in the bypass transition of flows for which traditional eigenmode analysis predicts asymptotic stability. It is studied by means of a simplified parabolic model justified by the underlying physics of the problem; results for optimal disturbances and maximum transient growth are found in excellent agreement with computations based on the full Orr–Sommerfeld/Squire equations. The second path starts with the exponential amplification, in nominally subcritical conditions, of modal disturbances superposed to base flows mildly distorted compared to their idealized counterparts. Such mean flow distortions might arise from the presence of unwanted external forcing related, for example, to the experimental environment. A technique is described that is capable of providing the worst case distortion of fixed norm for any ideal base flow, i.e., that base flow modification capable of maximizing the amplification rate of a given instability mode. Both initial paths considered here provide feasible initial conditions for the transition process, and it is likely that in most practical situations algebraic and exponential growth mechanisms are concurrently at play in destabilizing plane shear flows.

Polymer drag reduction in exact coherent structures of plane shear flow

Recently discovered traveling-wave solutions to the Navier–Stokes equations in plane shear geometries provide model flows for the study of turbulent drag reduction by polymer additives. These solutions, or “exact coherent states” (ECS), qualitatively capture the dominant structure of the near-wall buffer region of shear turbulence, i.e., counter-rotating pairs of streamwise-aligned vortices flanking a low-speed streak in the streamwise velocity. The optimum length scales for the ECS match well the length scales of the turbulent coherent structures and evidence suggests that the ECS underlie the dynamics of these structures. We study here the effect of viscoelasticity on these states. The changes to the velocity field for the viscoelastic ECS, where the FENE-P model calculates the polymer stress, mirror the modifications seen in experiments of fully turbulent flows of polymer solutions at low to moderate levels of drag reduction: drag is reduced, streamwise velocity fluctuations increase while wall-normal fluctuations decrease, and smaller wavelength structures are suppressed. These modifications to the ECS are due to the suppression of the streamwise vortices. The polymer molecules become highly stretched in the wavy, streamwise streaks, where the flow is predominately elongational, then relax as they move from the streaks into and around the streamwise vortices, where the flow is predominately rotational. This relaxation of the polymer molecules produces a force that directly opposes the fluid motion in the vortices, weakening them. Since the pressure fluctuations have their greatest magnitude (i.e., they are most negative) in the cores of the vortices, a reduction in vortex strength leads to a decrease in the magnitude of the pressure fluctuations. The pressure fluctuations redistribute energy from the streamwise velocity fluctuations to the Reynolds shear stress, so a decrease in their magnitude leads to a reduction in turbulent drag. For the viscoelastic ECS, we also find that after the onset of drag reduction (at Weissenberg number, We≈7) there is a dramatic increase in the critical wall-normal length scale at which the ECS can exist. This sharp increase in length scale mirrors experimental observations and is also consistent with the observed shift to higher Reynolds numbers of the transition to turbulence in polymer solutions.

A study of mechanical shear bands in liquids at high pressure

Shear localization has been observed in liquid lubricants flowing at high pressure. Two kinds have been identified: for stress-controlled plane Couette shear at high values of Nahme No., thermal runaway can result in the appearance of a band at mid-plane where the temperature and shear rate become localized. Alternatively, at high pressure and high shear stress, intermittent slip can be visualized along inclined bands for both planar Couette shear and squeeze film flows. These mechanical shear bands are responsible for the rate independent behavior that characterizes the traction transferred across the lubricant film in concentrated contact. We consider three models for shear band formation: Mohr–Coulomb slip, loss of ellipticity of Navier–Stokes equations, and non-monotonic stress–strain rate response. The observations that slip occurs for a ratio of shear to normal stress of about 0.1 and that bands form in two characteristic directions support both the Mohr–Coulomb failure criterion description and a change of character of the governing differential equations. Some evidence of constitutive instability is also presented; however, band orientation does not support this model.

An experimental study on vortex–flame interaction of a spark-ignited flame with coherent structure in a plane premixed shear flow

In order to elucidate the enhancement mechanisms of turbulent flame propagation, it is indispensable to examine and clarify the process of vortex–flame interaction in detail. This is because it constitutes one of the key processes in the promotion of flame propagation velocity. In this investigation, a series of large-scale organized eddies formed in the plane propane–air premixed shear flow are selected as an objective eddy field. Two devised techniques are introduced and combined with each other; one is the acoustic excitation of the shear flow, the other is the simultaneous two-directional high-speed schlieren photography. The former makes the organized eddy formation extremely regular and enables synchronized ignition with a high-speed video camera. The latter provides the opportunity to analyze two different directional views of a single propagating flame. In this study a spark-ignited flame, which is initiated at the center of a premixed organized eddy and is propagating along the axis, is optically observed and analyzed by using the proposed technique. It is shown that the propagation velocity along the axis of the eddy is promoted by the vortex bursting and is about 80% greater than that in the radial direction. It is also shown that after completing combustion in the radial direction within an organized eddy, the flame travels to the boundary regions between two adjacent eddies on the upstream and downstream sides, mainly promoted by the rolling-up motion in the coherent structure. These results suggest that there exist two kinds of vortex–flame interactions of a spark-ignited flame with coherent structure in the plane shear flow.

Structural Changes in Block Copolymer Solutions under Shear Flow as Determined by Non‐Equilibrium Molecular Dynamics

AbstractSummary: A non‐equilibrium molecular dynamics computer simulation on microsegregated solutions of symmetrical diblock copolymers is reported. As the polymer concentration increases, the system undergoes phase transitions in the following order: body centered cubic (BCC) micelles, hexagonal (HEX) cylinders, gyroid (GYR) bicontinuous networks and lamellae (L), which are the same morphology reported for block copolymer melts. Structural classification is based on the patterns of the anisotropic static structure factor and characteristic 3‐dimensional images. The systems in the BCC micellar (ρσ3 = 0.3) and HEX cylindrical (ρσ3 = 0.4) phases were then subjected to a steady planar shear flow. In weak shear flow, the segregated domains in both systems tend to rearrange into sliding parallel close‐packed layers with their normal in the direction of the shear gradient. At higher shear rates, both systems adopt a perpendicular lamellar structure with the normal along the neutral direction. A further increase in the shear rate results in a decrease in lamellar spacing without any further structural transitions. Two critical shear rate values that correspond to the demarcation of different structural behaviors were found.Shear‐induced BCC‐LAM phase transition.imageShear‐induced BCC‐LAM phase transition.

Energy dissipation in body-forced plane shear flow

We study the problem of body-force driven shear flows in a plane channel of width l with free-slip boundaries. A mini-max variational problem for upper bounds on the bulk time averaged energy dissipation rate epsilon is derived from the incompressible Navier-Stokes equations with no secondary assumptions. This produces rigorous limits on the power consumption that are valid for laminar or turbulent solutions. The mini-max problem is solved exactly at high Reynolds numbers Re = U*l/nu, where U is the rms velocity and nu is the kinematic viscosity, yielding an explicit bound on the dimensionless asymptotic dissipation factor beta=epsilon*l/U^3 that depends only on the ``shape'' of the shearing body force. For a simple half-cosine force profile, for example, the high Reynolds number bound is beta <= pi^2/sqrt{216} = .6715... . We also report extensive direct numerical simulations for this particular force shape up to Re approximately 400; the observed dissipation rates are about a factor of three below the rigorous high-Re bound. Interestingly, the high-Re optimal solution of the variational problem bears some qualitative resemblence to the observed mean flow profiles in the simulations. These results extend and refine the recent analysis for body-forced turbulence in J. Fluid Mech. 467, 289-306 (2002).

Staggered circular cylinders immersed in a uniform planar shear flow

Two circular cylinders of equal diameter, arranged in staggered configurations with dimensionless centre-to-centre pitch ratios of P/ D=1.125 and 1.25, were immersed in a uniform planar shear flow with a dimensionless shear parameter of K=0.05 and a Reynolds number of Re=5.0×10 4. The mean aerodynamic forces and the vortex shedding frequencies were measured for the upstream and downstream cylinders at each pitch ratio. Under uniform flow conditions, where there is no shear, the cylinders behave as a single bluff body with a single vortex shedding process. The aerodynamic force coefficients on both cylinders, and the Strouhal number, undergo complex changes as the incidence angle is varied from α=0° to 90°. At several critical incidence angles there are discontinuous changes in, or extreme values of, the aerodynamic force coefficients. These are related to the shear layer dynamics between the upstream and downstream cylinders at lower incidence angles, and the changes to the gap flow between the cylinders and the effect of base bleed at higher incidence angles. The effects of the uniform planar shear flow were mostly seen in the aerodynamic forces, particularly for the mean drag coefficient, where there was a reduction in drag for the upstream cylinder at all incidence angles, a reduction in drag for the downstream cylinder at higher incidence angles, small changes in the lift coefficient on the upstream cylinder, and shifts in the critical incidence angles. In contrast, the behaviour of the Strouhal number was generally unchanged, although a small reduction in Strouhal number was observed. Planar shear flow more strongly influenced the staggered cylinders at higher incidence angles, where it had an influence on the direction and strength of the gap flow, seen in the movement of two critical incidence angles, which was accompanied by a weakening of the vortex shedding process.

A planar simple shear test and flow behavior in a superplastic Al-Mg alloy

Superplasticity is generally studied by performing tensile and gas-pressure-bulge tests. In formed parts, however, a variety of strain states, including in-plane shear, are encountered. The understanding of the mechanical response in shear is helpful in the study of superplastic metal forming. In this study, a device for a planar simple shear test was designed and used to perform tests on a superplastic Al-Mg alloy sheet at the elevated temperatures of 500 °C (773K) and 550 °C (823K). In such a test, the incremental rotation of the principal strain axes and specimen-end effects during deformation can complicate the determination of true mechanical response. The possible approximations regarding the strain state in the specimen gage have been investigated. The σe-ee curves developed based on a simple-shear assumption show a lower flow stress than that under uniaxial tension, and strain hardening is related to dynamic grain growth. The rate of strain hardening at a fixed \(\dot \varepsilon \)e level is essentially the same for both uniaxial tension and shear, but the difference in the effective stress between uniaxial tension and shear depends upon strain rate and temperature. This study marks the first known attempt to characterize large strain response for superplastic metals under conditions of simple shear.

On uniform planar shear flow around a circular cylinder at subcritical Reynolds number

An experimental investigation was conducted of a circular cylinder immersed in a uniform planar shear flow, where the approach velocity varies across the diameter of the cylinder. The study was motivated by some apparent discrepancies between numerical and experimental studies of the flow, and the general lack of experimental data, particularly in the subcritical Reynolds number regime. Of interest was the effect of shear on the vortex-shedding frequency and the mean aerodynamic forces, including the direction and origin of the steady mean lift force experienced by the cylinder, which has been the subject of contradictory results in the literature and for which measurements have rarely been reported. The circular cylinder was tested at Reynolds numbers from Re=4.0×10 4 to 9.0×10 4, and the dimensionless shear parameter ranged from K=0.02 to 0.07, which corresponds to a flow with low to moderate shear. The results from both the present and previous studies show that low to moderate shear causes a small increase in the Strouhal number, an increase in the mean base pressure coefficient, a reduction in the mean drag force coefficient, and a small mean lift force directed towards the low-velocity side. These effects are consistent with a narrowing, or reduction in width, of the cylinder near wake, which is supported by an asymmetric mean static pressure distribution on the surface of the cylinder.

Homotopy of exact coherent structures in plane shear flows

Three-dimensional steady states and traveling wave solutions of the Navier–Stokes equations are computed in plane Couette and Poiseuille flows with both free-slip and no-slip boundary conditions. They are calculated using Newton’s method by continuation of solutions that bifurcate from a two-dimensional streaky flow then by smooth transformation (homotopy) from Couette to Poiseuille flow and from free-slip to no-slip boundary conditions. The structural and statistical connections between these solutions and turbulent flows are illustrated. Parametric studies are performed and the parameters leading to the lowest onset Reynolds numbers are determined. In all cases, the lowest onset Reynolds number corresponds to spanwise periods of about 100 wall units. In particular, the rigid-free plane Poiseuille flow traveling wave arises at Reτ=44.2 for Lx+=273.7 and Lz+=105.5, in excellent agreement with observations of the streak spacing. A simple one-dimensional map is proposed to illustrate the possible nature of the “hard” transition to shear turbulence and connections with the unstable exact coherent structures.