Filling the gap between transient and steady shear rheology of aqueous graphene oxide dispersions

Even though the rheological behavior of aqueous graphene oxide (G-O) dispersions has been shown to be strongly time-dependent, only few transient measurements have been reported in the literature. In this work, we attempt to fill the gap between transient and steady shear rheological characterizations of aqueous G-O dispersions in the concentration range of 0.004 < ? <?3.5 wt%, by conducting comprehensive rheological measurements, including oscillatory shear flow, transient shear flow, and steady shear flow. Steady shear measurements have been performed after the evaluation of transient properties of the G-O dispersions, to assure steady-state conditions. We identify the critical concentration ? _c =?0.08 wt% (where G-O sheets start to interact) from oscillatory shear experiments. We find that the rheology of G-O dispersions strongly depends on the G-O concentration ?. Transient measurements of shear viscosity and first normal stress difference suggest that G-O dispersions behave like nematic polymeric liquid crystals at ?/? _c =?25, in agreement with other work reported in the literature. G-O dispersions also display a transition from negative to positive values of the first normal stress difference with increasing shear rates. Experimental findings of aqueous graphene oxide dispersions are compared and discussed with models and experiments reported for nematic polymeric liquid crystals, laponite, and organoclay dispersions.     

Extracting extensional properties through excess pressure drop estimation in axisymmetric contraction and expansion flows for constant shear viscosity,extension strain-hardening fluids

In this study, hyperbolic contraction–expansion flow (HCF) devices have been investigated with the specific aim of devising new experimental measuring systems for extensional rheological properties. To this end, a hyperbolic contraction–expansion configuration has been designed to minimize the influence of shear in the flow. Experiments have been conducted using well-characterized model fluids, alongside simulations using a viscoelastic White–Metzner/FENE-CR model and finite element/finite volume analysis. Here, the application of appropriate rheological models to reproduce quantitative pressure drop predictions for constant shear viscosity fluids has been investigated, in order to extract the relevant extensional properties for the various test fluids in question. Accordingly, experimental evaluation of the hyperbolic contraction–expansion configuration has shown rising corrected pressure drops with increasing elastic behaviour (De=0～16), evidence which has been corroborated through numerical prediction. Moreover, theoretical to predicted solution correspondence has been established between extensional viscosity and first normal stress difference. This leads to a practical means to measure extensional viscosity for elastic fluids, obtained through the derived pressure drop data in these HCF devices.     

Characterization of Viscoelasticity Due to Shear Wave Propagation: a Comparison of Existing Methods Based on Computational Simulation and Experimental Data

Shear wave propagation causes microvibrations within a medium; measuring the wave attenuation coefficient, α, and phase velocity, c _s , the medium shear modulus, μ, and shear viscosity, η, are determined based on a viscoelastic model that includes both c _s and α. The present work compares the performances of nine processing methods, based on cross-correlation and quadrature demodulation, used to extract the motion waveform from a sequence of radio-frequency (RF) echo signals from the medium. Kalman filtering determined the amplitude and the phase of the extracted motion waveform. The comparisons were done with regard to computational simulation and experiments with a gel phantom. Estimates obtained for μ and η of the medium considered different conditions for the vibration amplitude and the signal-to-noise ratio (SNR) of the RF echo signals and the waveform extracted by means of single frequency and shear wave dispersion ultrasound vibration (SDUV) methods. According to the simulated results, the cross-correlation-based processing techniques are more precise and accurate in comparison to quadrature demodulation techniques. The results for c _s , α, μ and η of the phantom and those obtained under the same setup conditions for experimental and computational tests agree with each other. Comparing the estimates based on single frequency and SDUV techniques, they presented similar performances at high SNR of the RF echo signal. On the other hand, the former technique prevailed for low SNR.     

Experimental study on rheological and thermophysical properties of seawater with surfactant additive—part I: rheological properties

The rheological properties of seawater with the addition of surfactant additive (cetyltrimethyl ammonium chloride (CTAC)/sodium salicylate (NaSal)) are measured at different temperatures, including shear viscosity and first normal stress difference (N₁). The effects of the temperature, the salts, and CTAC/NaSal concentration on the rheological properties of test solutions are investigated, and the corresponding influence mechanisms are analyzed. It shows that the addition of salt can decrease the shear viscosities of the solutions, and also decrease N₁ and even eliminate the sharp augment of N₁ above a certain shear rate. The growing elasticity can be characterized by the increase of the initial shear rate for shear-thickening inception. High temperature can also remove the sharp increase of N₁ with salt. Nevertheless, the increase of CTAC/NaSal concentration can withstand the elimination of the sharp augment of N₁.     

Self-similar solutions in the problem of generalized vortex diffusion

The analysis of the group properties and the search for self-similar solutions in problems of mathematical physics and continuum mechanics have always been of interest, both theoretical and applied [1–3]. Self-similar solutions of parabolic problems that depend only on a variable of the type η = x/√t are classical fundamental solutions of the one-dimensional linear and nonlinear heat conduction equations describing numerous physical phenomena with initial discontinuities on the boundary [4]. In this study, the term “generalized vortex diffusion” is introduced in order to unify the different processes in mechanics modeled by these problems. Here, vortex layer diffusion and vortex filament diffusion in a Newtonian fluid [5] can serve as classical hydrodynamic examples. The cases of self-similarity with respect to the variable η are classified for fairly general kinematics of the processes, physical nonlinearities of the medium, and types of boundary conditions at the discontinuity points. The general initial and boundary value problem thus formulated is analyzed in detail for Newtonian and non-Newtonian power-law fluids and a medium similar in behavior to a rigid-ideally plastic body. New self-similar solutions for the shear stress are derived.     

Turbulent Drag Reduction by a Near Wall Surface Tension Active Interface

In this work we study the turbulence modulation in a viscosity-stratified two-phase flow using Direct Numerical Simulation (DNS) of turbulence and the Phase Field Method (PFM) to simulate the interfacial phenomena. Specifically we consider the case of two immiscible fluid layers driven in a closed rectangular channel by an imposed mean pressure gradient. The present problem, which may mimic the behaviour of an oil flowing under a thin layer of different oil, thickness ratio h₂/h₁ =?9, is described by three main flow parameters: the shear Reynolds number Re _τ (which quantifies the importance of inertia compared to viscous effects), the Weber number We (which quantifies surface tension effects) and the viscosity ratio λ = ν₁/ν₂ between the two fluids. For this first study, the density ratio of the two fluid layers is the same (ρ₂ = ρ₁), we keep Re _τ and We constant, but we consider three different values for the viscosity ratio: λ =?1, λ =?0.875 and λ =?0.75. Compared to a single phase flow at the same shear Reynolds number (Re _τ =?100), in the two phase flow case we observe a decrease of the wall-shear stress and a strong turbulence modulation in particular in the proximity of the interface. Interestingly, we observe that the modulation of turbulence by the liquid-liquid interface extends up to the top wall (i.e. the closest to the interface) and produces local shear stress inversions and flow recirculation regions. The observed results depend primarily on the interface deformability and on the viscosity ratio between the two fluids (λ).     

Effects of the Reynolds number on a scale-similarity model of Lagrangian velocity correlations in isotropic turbulent flows

A scale-similarity model of a two-point two-time Lagrangian velocity correlation(LVC) was originally developed for the relative dispersion of tracer particles in isotropic turbulent flows(HE, G. W., JIN, G. D., and ZHAO, X. Scale-similarity model for Lagrangian velocity correlations in isotropic and stationary turbulence. Physical Review E, 80, 066313(2009)). The model can be expressed as a two-point Eulerian space correlation and the dispersion velocity V. The dispersion velocity denotes the rate at which one moving particle departs from another fixed particle. This paper numerically validates the robustness of the scale-similarity model at high Taylor micro-scale Reynolds numbers up to 373, which are much higher than the original values(R_λ = 66, 102). The effect of the Reynolds number on the dispersion velocity in the scale-similarity model is carefully investigated. The results show that the scale-similarity model is more accurate at higher Reynolds numbers because the two-point Lagrangian velocity correlations with different initial spatial separations collapse into a universal form compared with a combination of the initial separation and the temporal separation via the dispersion velocity.Moreover, the dispersion velocity V normalized by the Kolmogorov velocity V_η≡η/τ_η in which η and τ_η are the Kolmogorov space and time scales, respectively, scales with the Reynolds number R_λ as V/V_η∝ R_λ~(1.39) obtained from the numerical data.     

A generalized equation for rheology of emulsions and suspensions of deformable particles subjected to simple shear at low Reynolds number

We present analyses to provide a generalized rheological equation for suspensions and emulsions of non-Brownian particles. These multiparticle systems are subjected to a steady straining flow at low Reynolds number. We first consider the effect of a single deformable fluid particle on the ambient velocity and stress fields to constrain the rheological behavior of dilute mixtures. In the homogenization process, we introduce a first volume correction by considering a finite domain for the incompressible matrix. We then extend the solution for the rheology of concentrated system using an incremental differential method operating in a fixed and finite volume, where we account for the effective volume of particles through a crowding factor. This approach provides a self-consistent method to approximate hydrodynamic interactions between bubbles, droplets, or solid particles in concentrated systems. The resultant non-linear model predicts the relative viscosity over particle volume fractions ranging from dilute to the the random close packing in the limit of small deformation (capillary or Weissenberg numbers) for any viscosity ratio between the dispersed and continuous phases. The predictions from our model are tested against published datasets and other constitutive equations over different ranges of viscosity ratio, volume fraction, and shear rate. These comparisons show that our model, is in excellent agreement with published datasets. Moreover, comparisons with experimental data show that the model performs very well when extrapolated to high capillary numbers (C a?1). We also predict the existence of two dimensionless numbers; a critical viscosity ratio and critical capillary numbers that characterize transitions in the macroscopic rheological behavior of emulsions. Finally, we present a regime diagram in terms of the viscosity ratio and capillary number that constrains conditions where emulsions behave like Newtonian or Non-Newtonian fluids.     

Large amplitude oscillatory shear rheology of three different shear-thickening particle dispersions

We present a large amplitude oscillatory shear rheology (LAOS) investigation of three different shear-thickening particle dispersions - fumed silica in polyethylene oxide (FLOC), fumed silica in polypropylene glycol (HydroC), and cornstarch in water (JAM). These systems shear-thicken by three different mechanisms - shear-induced formation of particle clusters flocculated by polymer bridging, hydrocluster formation, and jamming. The viscoelastic non-linearities of the three fluids were studied as a function of strain and strain-rate space through the use of Lissajous-Bowditch curves and local nonlinear viscoelastic moduli of an oscillatory shear cycle. The nonlinear behaviors of the three fluids were compared and contrasted to understand the nonlinear shear-thickening mechanism of each. Both HydroC and JAM dispersions were found to exhibit strong strain stiffening of the elastic moduli and strain thickening of the loss moduli behavior associated with possible hydrocluster formation and particle jamming. However, the FLOC dispersion, in contrast, showed strong strain softening and strain thinning behavior at large strain amplitudes associated with yielding of the microstructure. The expected thickening of the loss modulus of FLOC in LAOS with increasing strain was not observed even though viscosity of FLOC was found to shear-thicken in steady-shear measurements. This disagreement is likely due to very large strain amplitudes required for shear-thickening to occur by shear-induced polymer bridging mechanism. The hypothesis was confirmed through stress growth experiments. Conversely, the HydroC and JAM dispersions required relatively small applied strains for shear-thickening to occur by hydrocluster and jamming mechanism. The comparison of local intra-cycle nonlinearity through Lissajous-Bowditch plots and nonlinear viscoelastic parameters indicated that the elastic nonlinearities of all three systems are primarily driven by a strong dependence on the magnitude of the applied strain-rates within an oscillatory cycle rather than the amplitude of the applied strain. A close inspection of the LAOS data reveals strong differences in the viscoelastic nonlinearities of these three different shear-thickening dispersions which can be used to create a nonlinear rheological fingerprint for each and offers valuable new insights into the nonlinear dynamics associated with each of the shear-thickening mechanisms.     

Elastomers in large-amplitude oscillatory uniaxial extension

In this work, we subject elastomers to a fixed pre-stretch in uniaxial extension, ε _p , upon which a large-amplitude, ε ₀, oscillatory uniaxial extensional (LAOE) deformation is superposed. We find that if both ε _p and ε ₀ are large enough, the stress responds with a rich set of higher harmonics, both even and odd. We further find the Lissajous-Bowditch plots of our measured stress responses versus uniaxial strain to be without twofold symmetry and, specifically, to be shaped like convex bananas. Our new continuum model for this behavior combines a new nonlinear spring, in parallel with a Newtonian dashpot, and we call this the Voigt model with strain-hardening. We consider this three-parameter (Young’s modulus, viscosity, and strain-hardening coefficient) model to be the simplest relevant one for the observed convex bananas. We fit the parameters to both our LAOE measurements and then to our uniaxial elongation measurements at constant extension rate. We develop analytical expressions for the Fourier components of the stress response, parts both in-phase and out-of-phase with the extensional strain, for the zeroth, first, second, and third harmonics. We find that the part of the second harmonic that is out-of-phase with the strain must be negative for proper banana convexity.     

Operator method for solving the plane elasticity problem for a strip with periodic cuts

In the present paper, we use the conformal mapping z/c = ζ?2a sin ζ (a, c?const, ζ = u + iv) of the strip {|v| ≤ v ₀, |u| < ∞} onto the domain D, which is a strip with symmetric periodic cuts. For the domain D, in the orthogonal system of isometric coordinates u, v, we solve the plane elasticity problem. We seek the biharmonic function in the form F = C ψ ₀ + S ψ*₀ + x(C ψ ₁ ? S ψ ₂) + y(C ψ ₂ + S ψ ₁), where C(v) and S(v) are the operator functions described in [1] and ψ ₀(u), …, ψ ₂(u) are the desired functions. The boundary conditions for the function F posed for v = ±v ₀ are equivalent to two operator equations for ψ ₁(u) and ψ ₂(u) and to two ordinary differential equations of first order for ψ ₀(u) and ψ*₀(u) [2]. By finding the functions ψ _j (u) in the form of trigonometric series with indeterminate coefficients and by solving the operator equations, we obtain infinite systems of linear equations for the unknown coefficients. We present an efficient method for solving these systems, which is based on studying stable recursive relations. In the present paper, we give an example of analysis of a specific strip (a = 1/4, v ₀ = 1) loaded on the boundary v = v ₀ by a normal load of intensity p. We find the particular solutions corresponding to the extension of the strip by the longitudinal force X ^∞ and to the transverse and pure bending of the strip due to the transverse force Y ^∞ and the constant moment M ^∞, respectively. We also present the graphs of normal and tangential stresses in the transverse cross-section x = 0 and study the stress concentration effect near the cut bottom.     

Brinkman Flow of a Viscous Fluid Through a Spherical Porous Medium Embedded in Another Porous Medium

An analytical investigation for a two-dimensional steady, viscous, and incompressible flow past a permeable sphere embedded in another porous medium is presented using the Brinkman model, assuming a uniform shear flow far away from the sphere. Semi-analytical solutions of the problem are derived and relevant quantities such as velocities and shearing stresses on the surface of the sphere are obtained. The streamlines inside and outside the sphere and the radial velocity are shown in several graphs for different values of the porous parameters \({\sigma _1 =(\mu /\tilde {\mu }) (a/\sqrt{K_1 })}\) and \({\sigma _2 =(\mu /\tilde {\mu }) (a/\sqrt{K_2 })}\) , where a is the radius of the sphere, μ is the dynamic viscosity of the fluid, \({\tilde {\mu }}\) is an effective or Brinkman viscosity, while K ₁ and K ₂ are the permeabilities of the two porous media. It is shown that the dimensionless shearing stress on the sphere is periodic in nature and its absolute value increases with an increase of both porous parameters σ ₁ and σ ₂.     

Rheology and structural changes of plasticized zeins in the molten state

Zeins, storage proteins from maize, are suitable for making biobased thermoplastic materials. The rheological behavior of a commercial zein plasticized with 20 w% glycerol was studied in the molten state by steady-state flow experiments in extrusion conditions and oscillatory rheometry. For low residence times, a shear-thinning viscoelastic behavior was observed, with G″ exceeding G′. After 300 s at 130 °C, the complex viscosity |η ^?|?=?7?×?10³ ω ^?0.46 was found to be similar to that of thermoplastic polymer melts used in fused deposition modeling. However, the ratio between the exponents of the power laws describing G′(ω) and G″(ω) did not meet the typical value of 2 for entangled polymer melts. Moreover, for longer residence times, the viscosity increased and a gelation phenomenon was observed with a crossing over of G′(ω) and G″(ω). Gel times ranged from 6000 s at 120 °C to 1700 s at 150 °C. The evolution of the macromolecular structure assessed by sodium dodecyl sulfate-polyacrylamide gel electrophoresis and high-performance size exclusion chromatography suggested that this gelation phenomenon involves various types of covalent and non-covalent cross-links. Disulfide bonds played a significant role in gelation kinetics despite a very low cysteine residue content in the protein primary structure (about 1 mol%). These results suggested that plasticized zeins initially behave like a low-viscosity non-entangled polymer melt, before cross-linking progressively led to a continuous network.     

Power-law creep and residual stresses in a carbopol gel

We report on the interplay between creep and residual stresses in a carbopol microgel. When a constant shear stress σ is applied below the yield stress σ _y, the strain is shown to increase as a power law of time, γ(t) = γ ₀ + (t/τ) ^α , with an exponent α = 0.39 ± 0.04 that is strongly reminiscent of Andrade creep in hard solids. For applied shear stresses lower than some typical value σ _c ? 0.2σ _y, the microgel experiences a more complex, anomalous creep behavior, characterized by an initial decrease of the strain, that we attribute to the existence of residual stresses of the order of σ _c that persist after a rest time under a zero shear rate following preshear. The influence of gel concentration on creep and residual stresses are investigated as well as possible aging effects. We discuss our results in light of previous works on colloidal glasses and other soft glassy systems.     

Direct Numerical Simulation and Theory of a Wall-Bounded Flow with Zero Skin Friction

We study turbulent plane Couette-Poiseuille (CP) flows in which the conditions (relative wall velocity ΔU _w ≡ 2U _w , pressure gradient dP/dx and viscosity ν) are adjusted to produce zero mean skin friction on one of the walls, denoted by APG for adverse pressure gradient. The other wall, FPG for favorable pressure gradient, provides the friction velocity u _τ , and h is the half-height of the channel. This leads to a one-parameter family of one-dimensional flows of varying Reynolds number Re ≡ U _w h/ν. We apply three codes, and cover three Reynolds numbers stepping by a factor of two each time. The agreement between codes is very good, and the Reynolds-number range is sizable. The theoretical questions revolve around Reynolds-number independence in both the core region (free of local viscous effects) and the two wall regions. The core region follows Townsend’s hypothesis of universal behavior for the velocity and shear stress, when they are normalized with u _τ and h; on the other hand universality is not observed for all the Reynolds stresses, any more than it is in Poiseuille flow or boundary layers. The FPG wall region obeys the classical law of the wall, again for velocity and shear stress. For the APG wall region, Stratford conjectured universal behavior when normalized with the pressure gradient, leading to a square-root law for the velocity. The literature, also covering other flows with zero skin friction, is ambiguous. Our results are very consistent with both of Stratford’s conjectures, suggesting that at least in this idealized flow turbulence theory is successful like it was for the classical logarithmic law of the wall. We appear to know the constants of the law within a 10% bracket. On the other hand, that again does not extend to Reynolds stresses other than the shear stress, but these stresses are passive in the momentum equation.     

Decay of Almost Periodic Solutions¶of Conservation Laws

We consider the asymptotic behavior of solutions of systems of inviscid or viscous conservation laws in one or several space variables, which are almost periodic in the space variables in a generalized sense introduced by Stepanoff and Wiener, which extends the original one of H. Bohr. We prove that if u(x,t) is such a solution whose inclusion intervals at time t, with respect to ?>0, satisfy l _epsiv;(t)/t→0 as t→∞, and such that the scaling sequence u ^T (x,t)=u(T x,T t) is pre-compact as t→∞ in L _loc ¹(? ^{d +1} ₊, then u(x,t) decays to its mean value \(\), which is independent of t, as t→∞. The decay considered here is in L ¹ _loc of the variable ξ≡x/t, which implies, as we show, that \(\) as t→∞, where M _x denotes taking the mean value with respect to x. In many cases we show that, if the initial data are almost periodic in the generalized sense, then so also are the solutions. We also show, in these cases, how to reduce the condition on the growth of the inclusion intervals l _?(t) with t, as t→∞, for fixed ? > 0, to a condition on the growth of l _?(0) with ?, as ?→ 0, which amounts to imposing restrictions only on the initial data. We show with a simple example the existence of almost periodic (non-periodic) functions whose inclusion intervals satisfy any prescribed growth condition as ?→ 0. The applications given here include inviscid and viscous scalar conservation laws in several space variables, some inviscid systems in chromatography and isentropic gas dynamics, as well as many viscous 2 × 2 systems such as those of nonlinear elasticity and Eulerian isentropic gas dynamics, with artificial viscosity, among others. In the case of the inviscid scalar equations and chromatography systems, the class of initial data for which decay results are proved includes, in particular, the L ^∞ generalized limit periodic functions. Our procedures can be easily adapted to provide similar results for semilinear and kinetic relaxations of systems of conservation laws.     

Buckling and its effect on the confined flow of a model capsule suspension

The rheology of confined flowing suspensions, such as blood, depends upon the dynamics of the components, which can be particularly rich when they are elastic capsules. Using boundary integral methods, we simulate a two-dimensional model channel through which flows a dense suspension of fluid-filled capsules. A parameter of principal interest is the equilibrium membrane perimeter, parameterized by ξ_o, which ranges from round capsules with ξ_o=1.0 to ξ_o=3.0 capsules with a dog-bone-like equilibrium shape. It is shown that the minimum effective viscosity occurs for ξ_o≈1.6, which forms a biconcave equilibrium shape, similar to a red blood cell. The rheological behavior changes significantly over this range; transitions are linked to specific changes in the capsule dynamics. Most noteworthy is an abrupt change in behavior for ξ_o≈2.0, which correlates with the onset of capsule buckling. The buckled capsules have a more varied orientation and make significant rotational (rotlet) contributions to the capsule–capsule interactions.     

Shear flow of a Newtonian fluid over a quiescent generalized Newtonian fluid

The flow of an upper shear-driven Newtonian fluid above an otherwise still non-Newtonian fluid is considered. The lower fluid is modelled as a generalized Newtonian fluid and set into motion by interfacial shear. By means of similarity transformations, the governing partial differential equations for the two-fluid problem transform exactly into two sets of ordinary differential equations coupled only at the interface. The successful transformation of the two-fluid problem is applied to the particular case when the lower fluid obeys power-law rheology. The resulting three-parameter problem is solved numerically for some different parameter combinations by means of a direct integration approach with the density ratio fixed to unity. We observed that the interfacial velocities decreased with increasing values of the power-law index n in the range from 0.6 to 1.4 whereas the shear-induced motion of the lower fluid penetrates far deeper into a shear-thinning (n < 1) than into a shear-thickening (n > 1) fluid. This phenomenon is ascribed to a corresponding increase of the non-linear viscosity function with lower n-values.     

Application of the γ-Reθ Transition Model to Simulations of the Flow Past a Circular Cylinder

Based on the finite volume method, the flow past a two-dimensional circular cylinder at a critical Reynolds number (Re = 8.5 × 10⁵) was simulated using the Navier-Stokes equations and the γ-Re_θ transition model coupled with the SST k ? ω turbulence model (hereinafter abbreviated as γ-Re_θ model). Considering the effect of free-stream turbulence intensity decay, the SST k ? ω turbulence model was modified according to the ambient source term method proposed by Spalart and Rumsey, and then the modified SST k ? ω turbulence model is coupled with the γ-Re_θ transition model (hereinafter abbreviated as γ-Re_θ-SR model). The flow past a circular cylinder at different inlet turbulence intensities were simulated by the γ-Re_θ-SR model. At last, the flow past a circular cylinder at subcritical, critical and supercritical Reynolds numbers were each simulated by the γ-Re_θ-SR model, and the three flow states were analyzed. It was found that compared with the SST k ? ω turbulence model, the γ-Re_θ model could simulate the transition of laminar to turbulent, resulting in better consistency with experimental result. Compared with the γ-Re_θ model, for relatively high inlet turbulence intensities, the γ-Re_θ-SR model could better simulate the flow past a circular cylinder; however the improvement almost diminished for relatively low inlet turbulence intensities The γ-Re_θ-SR model could well simulate the flow past a circular cylinder at subcritical, critical and supercritical Reynolds numbers.     

New Periodic Solutions for Newtonian n-Body Problems with Dihedral Group Symmetry and Topological Constraints

In this paper, we prove the existence of a family of new non-collision periodic solutions for the classical Newtonian n-body problems. In our assumption, the \({n=2l \geqq 4}\) particles are invariant under the dihedral rotation group D_l in \({\mathbb{R}^3}\) such that, at each instant, the n particles form two twisted l-regular polygons. Our approach is the variational minimizing method and we show that the minimizers are collision-free by level estimates and local deformations.