Planar Shear Flow Research Articles

A new algorithm for unrestricted duration nonequilibrium molecular dynamics simulations of planar elongational flow

We apply the spatially and temporally periodic boundary conditions devised by Kraynik and Reinelt (Int. J. Multiphase Flow 18 (1992) 1045) to an atomic fluid undergoing planar elongational flow and describe several algorithms which efficiently apply the required nonstandard periodic boundary conditions. These periodic boundary conditions guarantee unrestricted simulation times, and are simply implemented if a rotational transformation is first applied to the coordinates of all atoms such that one of the cell boundaries aligns with the direction of elongation. While in the transformed frame one can apply either Lagrangian rhomboid (LR) or “deforming brick” (DB) periodic boundary conditions to all particle coordinates and relative distances. The latter (DB) scheme turns out to be very similar in form to standard Lees-Edwards periodic boundary conditions for planar shear flow, and both the LR and DB schemes are shown to be equivalent and numerically highly efficient.

Wall slip in the molecular dynamics simulation of thin films of hexadecane

A molecular dynamics simulation of a thin liquid film as it is sheared between two planar walls is reported. The model liquid is composed of linear chain molecules of hexadecane (C16H34) with intramolecular architecture such as bond stretching, angle bending and dihedral potentials included in the model. Designing a model that can mimic the planar shear flow enables us to study important questions on the effects of the wall properties on the slip between the liquid film and the wall. Different properties of the wall such as wall density, wall stiffness and wall–fluid interaction strength have been studied to determine the slip between the wall and fluid. The slip has been investigated for strong and weak adsorbing surfaces at various shear rates. The results emphasize the importance of adsorption on the degree of slip. The dependence of slip on the film thickness is also demonstrated.

Three-Dimensional Coherent States in Plane Shear Flows

Three-dimensional steady states in plane Couette flow and traveling-wave solutions in plane Poiseuille flow are calculated for stress boundary conditions. The procedure is tied to a self-sustaining mechanism associated with the coherent structures that have been observed in turbulent shear flows. The exact states in both types of flow are remarkably similar to each other and to the coherent structures. They survive down to Reynolds numbers below the critical value for turbulence onset. These results suggest that the underlying process is generic and fundamental to both transitional and developed turbulence.

Effect of axial flow on viscoelastic Taylor–Couette instability

Viscoelastic flow instabilities can arise from gradients in elastic stresses in flows with curved streamlines. Circular Couette flow displays the prototypical instability of this type, when the azimuthal Weissenberg number Weθ is O(ε−1/2), where ε measures the streamline curvature. We consider here the effect of superimposed steady axial Couette or Poiseuille flow on this instability. For inertialess flow of an upper-convected Maxwell or Oldroyd-B fluid in the narrow gap limit (ε[Lt ]1), the analysis predicts that the addition of a relatively weak steady axial Couette flow (axial Weissenberg number Wez=O(1)) can delay the onset of instability until Weθ is significantly higher than without axial flow. Weakly nonlinear analysis shows that these bifurcations are subcritical. The numerical results are consistent with a scaling analysis for Wez[Gt ]1, which shows that the critical azimuthal Weissenberg number for instability increases linearly with Wez. Non-axisymmetric disturbances are very strongly suppressed, becoming unstable only when ε1/2Weθ= O(We2z). A similar, but smaller, stabilizing effect occurs if steady axial Poiseuille flow is added. In this case, however, the bifurcations are converted from subcritical to supercritical as Wez increases. The observed stabilization is due to the axial stresses introduced by the axial flow, which overshadow the destabilizing hoop stress. If only a weak (Wez[les ]1) steady axial flow is added, the flow is actually slightly destabilized. The analysis also elucidates new aspects of the stability problems for plane shear flows, including the exact structure of the modes in the continuous spectrum, and illustrates the connection between these problems and the viscoelastic circular Couette flow.

An analysis of rotating shear flow using linear theory and DNS and LES results

The development of turbulence is investigated in the presence of a mean plane shear flow (rate S) rotating with angular velocity vector (rate Ω) perpendicular to its plane. An important motivation was generalizing the work by Lee, Kim & Moin (1990) to rotating shear flow, in particular detailed comparisons of homogeneous rapid distortion theory (RDT) results and the databases of homogeneous and channel flow direct numerical simulations (DNS). Linear analysis and related RDT are used starting from the linearized equations governing the fluctuating velocity field. The parameterization based on the value of the Bradshaw–Richardson number B=R(1+R) (with R=−2Ω/S) is checked against complete linear solutions. Owing to the pressure fluctuation, the dynamics is not governed entirely by the parameter B, and the subsequent breaking of symmetry (between the R and −1 −R cases) is investigated. New analytical solutions for the ‘two-dimensional energy components’ [Escr ](l)ij =Eij(kl=0, t) (i.e. the limits at kl=0 of the one-dimensional energy spectra) are calculated by inviscid and viscous RDT, for various ratios Ω/S and both streamwise l=1 and spanwise l=3 directions. Structure effects (streak-like tendencies, dimensionality) in rotating shear flow are discussed through these quantities and more conventional second-order statistics. In order to compare in a quantitative way RDT solutions for single-point statistics with available large-eddy simulation (LES) data (Bardina, Ferziger & Reynolds 1983), an ‘effective viscosity’ model (following Townsend) is used, yielding an impressive agreement.

Lyapunov Instability of the Boundary-Driven Chernov–Lebowitz Model for Stationary Shear Flow

We report on the computation of full Lyapunov spectra of the boundary-driven Chernov–Lebowitz model for stationary planar shear flow. The Lyapunov exponents are calculated with a recently developed formalism for systems with elastic hard collisions. Although the Chernov–Lebowitz model is strictly energy conserving, any phase-space volume is subjected to a contraction due to the reflection rules of the hard disks colliding with the walls. Consequently, the sum of Lyapunov exponents is negative. As expected for an inhomogeneously driven system, the Lyapunov spectra do not obey the conjugate pairing rule. The external driving makes the system less chaotic, which is reflected in a decrease of the Kolmogorov–Sinai entropy if the driving is increased.

On the Existence of Infinitely Many Limit Points on the Solution Branch of Planar Shear Flow of Nematic Liquid Crystals

We use the Ericksen–Leslie equations to investigate the solution branch of simple planar shear flow between moving parallel plates for non-flow-orienting nematic liquid crystals. We show the existence of infinitely many limit points on the solution branch of simple planar shear flow. Hence the tumbling phenomena observed in experiments can be explained as the exchange of stability of the planar configuration by another planar configuration with a more distorted director field.

The role of tank-treading motions in the transverse migration of a spheroidal vesicle in a shear flow

The behavior of a spheroidal vesicle, in a plane shear flow bounded from one side by a wall, is analysed when the distance from the wall is much larger than the spheroid radius. It is found that tank treading motions produce a transverse drift away from the wall, proportional to the spheroid eccentricity and the inverse square of the distance from the wall. This drift is independent of inertia, and is completely determined by the characteristics of the vesicle membrane. The relative strength of the contribution to drift from tank-treading motions and from the presence of inertial corrections, is discussed.

Chevron patterns in liquid crystal flows

We study plane shear and Poiseuille flows of uniaxially nematic liquid crystals for large values of the Ericksen number, ε. This is a non-dimensional quantity that renders rigorous notions of fast flow regimes. (The former corresponds to large values of ε.) The model employed is that due to Ericksen for liquid crystals with variable degree of orientation (Ericksen, 1991). The constitutive equations derived by Kuzuu and Doi (1984) and Marrucci (1981), play an essential role in the analysis. Variables of the problem include the velocity field, v, the pressure, p, the scalar order parameter, s, and the director, n. Recent experimental work involving, both, low molecular weight liquid crystals as well as polymers, has pointed out that the positivity of the ratio α 2 α 3 of the Leslie coefficients does not necessarily guarantee uniform molecular alignment in shear flow regimes, as previously believed. In particular, for fast regimes, the flow breaks up into stripped configurations, chevrons parallel to the velocity field. Taking such observations into account, here, we consider regimes that fulfill the alignment condition, and show that for large values of ε, solutions exhibiting disclinations in the flow region can be found. Moreover we point out that across singular lines, the director experiences jump discontinuities of approximately ±45° to the direction of flow. We construct solutions that present domain structures in the form of stripes parallel to the flow direction, on the plane of shear. The interface between two such stripes corresponds to a nearly isotropic layer.

The effects of inertia and finite amplitude on oscillatory plane shear flow of K-BKZ fluids such as LDPE melts

An asymptotic analysis of oscillatory plane shear flow is justified in the context of a separable K-BKZ model for LDPE melts (see Luo and Tanner, Int. J. Numer. Methods Eng., 25 (1988) 9–22). Thus for oscillations of moderate amplitude an asymptotic approximation to the constitutive law may be substituted to within an error bound determined by material constants. A non-dimensional form of the equations of motion is then obtained. These equations involve a frequency dependent parameter R ∗ with upper and lower limits in the nature of a Mach number and Reynolds number respectively, which is a measure of the influence of inertia, and another parameter D, which is a measure of the influence of non-linearity in the constitutive law. Solutions to these equations are then sought in the form of a perturbation series in the parameter D and shear profiles are plotted using data supplied by Luo and Tanner. A consistency test for the ‘gap loading’ assumption made in experimental analysis is suggested. Although the degree of non-linearity and inertia present may be varied independently, it is shown that they are interactive in their effects on shear wave profiles.

Non-equilibrium modifications of the nuclear equation of state

Non-equilibrium modifications of the nuclear equation of state are assessed in the framework of Extended Irreversible Thermodynamics. The expected size of the non-equilibrium corrections to quantities like the nuclear-matter incompressibility K and the liquid-vapor critical temperature T c are compared to the uncertainties associated at present with the usual equilibrium values. The quantitative estimates are obtained in situations characterized by a stationary planar shear flow of realistic magnitude, or by a stationary temperature gradient.

Determination of Internal Deformation Field in Asphalt Cores Using X-Ray Computer Tomography

Although existing nondestructive evaluation methods and transducers provide useful quantitative information on composite materials, they measure only macroscopic deformations and, often, only after the strain has exceeded a certain predetermined threshold. It is well established, however, that maintenance intervention is more effective if applied early in the deformation process. A new method for computing the microscopic internal displacement fields associated with permanent deformations of three-dimensional asphalt-aggregate cores with complex internal structure and satisfying the small gradient approximation of continuum mechanics is presented. The displacement fields are computed from a sequence of three-dimensional X-ray computed tomography images, obtained using a new imaging protocol developed specifically for mass-fraction and mix-density estimates of composite cores. By assuming that the image intensity of the tomographic images represents a certain conserved property that is incompressible, a constrained nonlinear regression model for motion estimation is developed. Successive linear approximation is then employed and each linear subsidiary problem is solved using variational calculus. The resulting Euler-Lagrange equations are approximated and solved using finite differencing methods and a conjugate gradient algorithm in a multiresolution framework. The method is validated using pairs of synthetic images of plane shear flow. The three-dimensional displacement field in the interior of a cylindrical asphalt/aggregate core loaded to a state of permanent deformation is calculated using this method.

Determination of Internal Deformation Field in Asphalt Cores Using X-Ray Computer Tomography

Although existing nondestructive evaluation methods and transducers provide useful quantitative information on composite materials, they measure only macroscopic deformations and, often, only after the strain has exceeded a certain predetermined threshold. It is well established, however, that maintenance intervention is more effective if applied early in the deformation process. A new method for computing the microscopic internal displacement fields associated with permanent deformations of three-dimensional asphalt-aggregate cores with complex internal structure and satisfying the small gradient approximation of continuum mechanics is presented. The displacement fields are computed from a sequence of three-dimensional X-ray computed tomography images, obtained using a new imaging protocol developed specifically for mass-fraction and mix-density estimates of composite cores. By assuming that the image intensity of the tomographic images represents a certain conserved property that is incompressible, a constrained nonlinear regression model for motion estimation is developed. Successive linear approximation is then employed and each linear subsidiary problem is solved using variational calculus. The resulting Euler-Lagrange equations are approximated and solved using finite differencing methods and a conjugate gradient algorithm in a multiresolution framework. The method is validated using pairs of synthetic images of plane shear flow. The three-dimensional displacement field in the interior of a cylindrical asphalt/aggregate core loaded to a state of permanent deformation is calculated using this method.

Unified turbulence closure model for axisymmetric and planar free shear flows

A new two-equation turbulence model based on the exact turbulent kinetic energy and the variance of vorticity or enstrophy equations is developed and used to calculate planar and axisymmetric free shear flows. It is shown that one set of model constants reproduces available growth rates and similarity profiles of velocity and shear stress. In general, agreement is well within the scatter of experimental data.

Calculation of 3D internal displacement fields from 3D X-ray computer tomographic images

We present a new method for computing the internal displacement fields associated with permanent deformations of 3D composite objects with complex internal structure for fields satisfying the small displacement gradient approximation of continuum mechanics. We compute the displacement fields from a sequence of 3D X-ray computed tomography (CT) images. By assuming that the intensity of the tomographic images represents a conserved property which is incompressible, we develop a constrained nonlinear regression model for estimation of the displacement field. Successive linear approximation is then employed and each linear subsidiary problem is solved using variational calculus. We approximate the resulting Euler-Lagrange equations using a finite set of linear equations using finite differencing methods. We solve these equations using a conjugate gradient algorithm in a multiresolution framework. We validate our method using pairs of synthetic images of plane shear flow. Finally, we determine the 3D displacement field in the interior of a cylindrical asphalt/aggregate core loaded to a state of permanent deformation.

Preferential concentration of particles by turbulence

Preferential concentration describes the accumulation of dense particles within specific regions of the instantaneous turbulence field. This phenomenon occurs in dilute particle-laden flows with particle time constants of the same order as an appropriately chosen turbulence time scale. The mechanisms which drive preferential concentration are centrifuging of particles away from vortex cores and accumulation of particles in convergence zones. Experimental and numerical studies are reviewed which demonstrate that preferential concentration occurs in a wide range of flows including plane and axisymmetric free shear flows, wall-bounded flows, homogeneous turbulent flows, and complex shear flows. The same basic mechanisms are active in all these flows but the specific effects and the definition of an appropriate time scale change from problem to problem. Preferential concentration has been shown to have a significant effect on droplet combustion, aerosol particle settling and turbulence modification by particles and is expected to be significant in numerous other applications. Current models describing preferential concentration and its global effects on the flow are inadequate.

Orientational instability of nematics under oscillatory flow

Generalizing a recently introduced approximation scheme valid when the director relaxation time is large compared to the inverse frequency of an oscillatory flow, we derive equations for the time-averaged torques on the director for a prescribed plane shear flow. Whereas for a linear flow field (simple Couette flow) the average torques vanish, one has for Poiseuille flow (and more general flow fields) torques that tend to orient the director essentially perpendicular to the flow plane. For flow-aligning materials the orientation parallel to the flow is also (weakly) stable. Including the effect of homeotropic surface alignment we estimate the threshold of the oscillation amplitude for the out-of-plane transition. The results are essentially recovered (and improved) by direct numerical simulations

The deceleration of a cylinder in an infinite viscous fluid

The retardation of the rotation of a cylinder in an infinite viscous fluid is investigated in two cases: (1) the retardation by the action of the viscous shear on the cylinder surface and the fluid is initially at rest and (2) the retardation by the action of the viscous shear and by a simple plane shear flow at infinity whose velocity components are U x = KY and U y = 0 . Calculations show exponential decay in the form Case 1 ω = ω 0 exp − (8πμt/m) Case 2 ω = −K/2 + (ω 0 + K/2) exp −(8πμt/m) where ω = initial angular velocity of the cylinder μ = coefficient of viscosity of the fluid m= mass of the cylinder per unit legth t= time

The deceleration of a sphere in an infinite viscous fluid

In this paper, the retardation of the rotation of a homogeneous sphere about its diameter submerged in an infinite viscous incompressible fluid is investigated in two cases. 1. (1) The retardation by the action of the viscous shear on the sphere surface, and the fluid is initially at rest. 2. (2) The retardation by the action of the viscous shear and by a simple plane shear flow at infinity whose velocity components are U x = KY and U y = 0 . Calculations show exponential decay in the form Case 1 ω = ω 0 exp − Πμa 40 + 15Π 12M t ω = − 40 K (40 + 15 M) + ω 0+ 40 K (40 + 15 Π) exp S ω 0 = initial angular velocity of the cylinder μ = coefficient of viscosity of the fluid M= mass of the sphere a = radius of the sphere t = time S = −Πμa 40 + 15Π 12M t .

Self-diffusion of particles in an alternatively sheared macroscopic suspension

In this communication, we present the preliminary results of a study of the migration motions of macroscopic spherical particles (diameter 45 μm) in a suspension submitted to an alternating plane shear flow. The shear flow is applied in two directions which are both perpendicular and in phase quadrature (circularly polarized shear). By following the particles in time and analysing their trajectory, we can compute the mean-square displacement σ 2 = ) 2 > which shows a linear behaviour with time. We interpret this diffusive behaviour as a result of the hydrodynamic interactions experienced by each particle from the other particles of the suspension. These interactions come from inertial corrections to the applied flow, of non-zero mean, over long times in comparison with the period of the alternating shear. We propose a simple model, based on macroscopic Brownian-like diffusion to give an estimate of the long-time self-diffusion coefficient