Periodic Shear Flow Research Articles

Normal deformation of a soft lipid-based microtube in response to the flow generated by a tangential motion of a nearby microbead

In this study, we experimentally examine the deformation of a micrometer-scale soft membrane in response to a periodic shear flow. We fabricate long and straight cylindrical-shaped soft microtubes from lipid bilayers as part of this study. A microtube with a diameter of about 1–5 μm is moved parallel to its longitudinal axes. At the same time, a polystyrene microbead, trapped by optical tweezers, is forced to stay near the external surface of the microtube. We study the induced shape deformation in the microtube for different shearing parameters. The effects related to the size of the bead, its distance to the surface of the microtube, and their relative speed are analyzed. The overall results qualitatively demonstrate the impact of shearing on the shape of the soft membrane.

Effective Lewis number and burning speed for flames propagating in small-scale spatio-temporal periodic flows

Propagation of premixed flames having thick reaction zones in rapidly-varying, small-scale, zero-mean, spatio-temporal periodic flows is considered. Techniques of large activation energy asymptotics and homogenization theory are used to determine the effective Lewis number Leeff and the effective burning speed ratio ST/SL, which are influenced by the flow through flow-enhanced diffusion. The resultant effective diffusivity matrix is, in general, neither a scalar nor a diagonal matrix and therefore induces anisotropic effects on the propagation of multi-dimensional flames. As the flow Peclet number Pe becomes large, the flow-enhanced fuel diffusion coefficient and the thermal diffusivity behave respectively like (PeLe)σ and Peσ, where Le is the Lewis number and σ≤2 is a constant which depends on the flow and the direction of flame propagation. The maximal value σ=2 is achieved for steady, unidirectional, spatially periodic shear flows, while for steady two-dimensional square vortices, we have σ=1/2. In general, the constant σ is determined by solving a linear partial differential equation. The scaling laws for the diffusion coefficients lead to corresponding scaling laws for the effective Lewis number and the effective burning speed ratio of the form Leeff≃Le1−σ and ST/SL∼(Pe/Le)σ/2. Effects of thermal expansion and volumetric heat loss on the flame are also briefly discussed. In particular, it is shown that the quenching limit is enlarged by a factor 1/Leσ for Le<1 and diminished by the same factor for Le>1, due to the flow-enhanced diffusion. The potential implications of the results to better understand turbulent combustion are discussed. A special emphasis is placed on the dependence of the flame on Le in the presence of high-intensity, small-scale flows. In particular, it is shown that this dependence is intimately linked to the flow through Taylor-dispersion like enhanced diffusion, rather than through the traditional molecular diffusion coupled with curvature effects. The flow-dependent effective Lewis number identified may also provide an explanation to the peculiar experimental observation that turbulence appears to facilitate ignition in Le>1 mixtures and to inhibit it in Le<1 mixtures. Novelty and significance statementAn original study, combining asymptotic analysis and homogenization theory, is applied to describe flame propagation in small-scale, spatio-temporal periodic flow fields. Scaling laws are derived for the effective burning speed and the effective Lewis number for high-intensity small-scale flows, which are useful to better understand the behaviour of turbulent premixed flames in the distributed reaction zone regime. The formula for the flow-dependent effective Lewis number identified herein may explain the peculiar experimental observation that turbulence appears to facilitate ignition in Le>1 mixtures and to inhibit it in Le<1 mixtures. The high-intensity small-scale flows are shown to increase the quenching limit due to volumetric heat losses in Le<1 mixtures and decrease it in Le>1 mixtures

Exponential mixing for random dynamical systems and an example of Pierrehumbert

We consider the question of exponential mixing for random dynamical systems on arbitrary compact manifolds without boundary. We put forward a robust, dynamics-based framework that allows us to construct space-time smooth, uniformly bounded in time, universal exponential mixers. The framework is then applied to the problem of proving exponential mixing in a classical example proposed by Pierrehumbert in 1994, consisting of alternating periodic shear flows with randomized phases. This settles a longstanding open problem on proving the existence of a space-time smooth (universal) exponentially mixing incompressible velocity field on a two-dimensional periodic domain while also providing a toolbox for constructing such smooth universal mixers in all dimensions.

Non-ideal instabilities in sinusoidal shear flows with a streamwise magnetic field

We investigate the linear stability of a sinusoidal shear flow with an initially uniform streamwise magnetic field in the framework of incompressible magnetohydrodynamics (MHD) with finite resistivity and viscosity. This flow is known to be unstable to the Kelvin–Helmholtz instability in the hydrodynamic case. The same is true in ideal MHD, where dissipation is neglected, provided the magnetic field strength does not exceed a critical threshold beyond which magnetic tension stabilizes the flow. Here, we demonstrate that including viscosity and resistivity introduces two new modes of instability. One of these modes, which we refer to as an Alfvénic Dubrulle–Frisch instability, exists for any non-zero magnetic field strength as long as the magnetic Prandtl number ${{{Pm}}} &lt; 1$ . We present a reduced model for this instability that reveals its excitation mechanism to be the negative eddy viscosity of periodic shear flows described by Dubrulle &amp; Frisch (Phys. Rev. A, vol. 43, 1991, pp. 5355–5364). Finally, we demonstrate numerically that this mode saturates in a quasi-stationary state dominated by counter-propagating solitons.

Enhanced diffusivity and skewness of a diffusing tracer in the presence of an oscillating wall

We develop a theory of enhanced diffusivity and skewness of the longitudinal distribution of a diffusing tracer advected by a periodic time-varying shear flow in a straight channel. Although applicable to general fluid flow, we restrict the examples of our theory to the tracer advected by flows that are induced by a periodically oscillating wall in a Newtonian fluid between two infinite parallel plates as well as flow in an infinitely long duct. We first derived the formula of the flow produced by the wall motions. Second, we calculate the second Aris moment for all time and its long-time limiting effective diffusivity as a function of the geometrical parameters, frequency, viscosity, and diffusivity. Using a new formalism based upon the Helmholtz operator we establish a new single series formula for the variance. We show that the viscous-dominated limit results in a linear shear layer for which the effective diffusivity is bounded. For finite viscosities, the enhanced diffusion diverges in the high-frequency limit. We present a study of the effective diffusivity surface as a function of the non-dimensional parameters which shows how a maximum can exist for various parameter sweeps. Physical experiments are performed in water using particle tracking velocimetry to quantitatively measure the fluid flow. Using fluorescein dye as the passive tracer, we document that the theory is quantitatively accurate. Specifically, image analysis suggests that the distribution variance be measured using the full width at half maximum is robust to noise. Further, we show that the scalar skewness is zero for linear shear flows at all times, whereas for the nonlinear Stokes layer, the skewness sign can be controlled through the oscillating phase. Last, for single-frequency wall motion, the long-time skewness decays at the faster rate as compared with the case with steady flow.

Numerical study on droplet deformation in periodic pulsatile shear flow and effects of inertia

Abstract Single droplet deformation in pulsatile shear flow is numerically studied by using meso-scopic dissipative particle dynamics method. Numerical model and simulation method are fully validated with three typical cases in which transient flow behaviors are well captured. Cases in quasi-Stokes regime are first investigated with flow Reynolds number to be less than O ( 1 0 − 2 ) . In early stage, droplet deforms gradually in the manner of periodic oscillation but irreversible. Later, droplet deformation gets into a stable state of oscillation and becomes reversible. The mean deformation parameter in stable state is frequency independent and its relationship to mean Capillary number is close to the trend in steady shear. Cases with finite inertia are studied with flow Reynolds number to be approximately to 1. In contrast, the inertial effects make the mean deformation parameters have a strong frequency dependence. The shear and relax tests are conducted to explain above dynamic behavior of droplet deformation. It is found that the droplet deformations in shear and relax are antisymmetric in quasi-Stokes regime while in finite inertia regime droplet deforms slower in shear process than it is in relax process. The frequency dependent behavior of droplet in pulsatile shear might provide a reliable strategy in the control of droplet deformation.

Influence of gas expansion on the propagation of a premixed flame in a spatially periodic shear flow

It has been previously demonstrated that thermal gas expansion might have a role in boundary layer flashback of premixed turbulent flames [Gruber et al., J Fluid Mech 2012], inducing local flow-reversal in the boundary layer's low-velocity streaks on the reactants’ side of the flame and facilitating its upstream propagation. We perform a two-dimensional numerical investigation of the interaction between a periodic shear flow and a laminar premixed flame. The periodic shear is a simplified model for the oncoming prolonged streamwise velocity streaks with alternating regions of high and low velocities found in turbulent boundary layers in the vicinity of the walls. The parametric study focuses on the amplitude and wavelength of the periodic shear flow and on the gas expansion ratio (unburnt-to-burnt density ratio). With the increase of the amplitudes of the periodic shear flow and of the gas expansion, the curved flame velocity increases monotonically. The flame velocity dependence on the periodic shear wavelength is non-monotonic, which is consistent with previous theoretical studies of curved premixed flame velocity. The flame shape that is initially formed by the oncoming periodic shear appears to be metastable. At a later stage of the flame propagation, the flame shape transforms into the stationary one dominated by the Darrieus-Landau instability.

On the existence and monotonicity of curved fronts in a periodic shear flow

On the existence and monotonicity of curved fronts in a periodic shear flow

Enhanced multi-relaxation-time lattice Boltzmann model by entropic stabilizers.

The difficulty of choice of relaxation rates in multi-relaxation-time lattice Boltzmann model (MRT-LBM) is surmounted by solution of least-square problem of entropic stabilizer equations. Relaxation rates in the enhanced MRT-LBM are evolving with time rather than remain constants. To derive entropic stabilizer equations, nonequilibrium population is split into different modes in terms of column vectors in the inverse transform matrix. The entropic stabilizer equations are achieved by minimization of H-function. Different moment representations in MRT-LBM, such as Gram-Schmidt orthogonal moment, natural moment, and central moment, are tested for double periodic shear flow, shock tube problem, and lid-driven cavity flow, which demonstrates the potential of enhanced MRT-LBM.

Flame propagation through a spatially periodic shear flow: Extinction vs. transition to detonation

The problem of flame propagation through a spatially periodic shear flow is revisited. When the flow intensity exceeds a certain critical level the corrugated flame either quenches or transitions to detonation. The quenching takes place for relatively small-scale flows where the impact of the flame stretch is strong. In large-scale flows the stretch weakens, while the impact of precompression increases, and the flame transitions to detonation, provided the flow intensity is high enough. In the parametric space a change of the dynamical picture occurs in an abrupt nearly jumpwise manner.

Poisson structure of the three-dimensional Euler equations in Fourier space

We derive a simple Poisson structure in the space of Fourier modes for the vorticity formulation of the Euler equations on a three-dimensional periodic domain. This allows us to analyse the structure of the Euler equations using a Hamiltonian framework. The Poisson structure is valid on the divergence free subspace only, and we show that using a projection operator it can be extended to be valid in the full space. We then restrict the simple Poisson structure to the divergence-free subspace on which the dynamics of the Euler equations take place, reducing the size of the system of ordinary differential equations by a third. The projected and the restricted Poisson structures are shown to have the helicity as a Casimir invariant. We conclude by showing that periodic shear flows in three dimensions are equilibria that correspond to singular points of the projected Poisson structure, and hence that the usual approach to study their nonlinear stability through the energy-Casimir method fails.

The stability of curved fronts in a periodic shear flow

Recently, El Smaily et al. (2011) proved the existence of curved fronts in a periodic shear flow, which satisfy some “conical” conditions at infinity in the whole plane R2. In this paper, we study the stability of these curved fronts. In fact, we prove that all the curved fronts are exponentially stable provided that the initial perturbations decay exponentially with an appropriate rate in the lower cone.

A spectral condition for Liouville-type result of monostable KPP equation in periodic shear flows

This paper is devoted to providing a simple condition, in term of spectral theory, that characterizes existence/nonexistence and uniqueness of positive bounded solution to $$\begin{aligned} \nabla \cdot [n(y)\nabla u(x,y)] +\alpha (y)\partial _x u+\beta (y)\cdot \nabla _y u+f(x,y,u)=0 \quad (x,y)\in \mathbb {R}\times \mathbb {R}^{N-1},\qquad \end{aligned}$$ (0.1) where f is of monostable KPP type nonlinearity and periodic in y. Our contribution answers a conjecture raised by Prof. H. Berestycki: which suitable assumption can impose at infinity that characterizes existence/nonexistence and uniqueness of (0.1) instead of the followings \(\liminf _{|z|\rightarrow \infty }\partial _u f(z,0)>0\) as in Berestycki et al. (Ann Mat Pura Appl 186(4):469–507, 2007) and \(\limsup _{|z|\rightarrow \infty }\partial _u f(z,0)<0\) as in Berestycki et al. (Bull Math Biol 71:399, 2008) and Berestycki and Rossi (Discret Contin Dyn Syst Ser B 21:41–67, 2008) but allow \(\partial _u f(z,0)\) to change sign all the way as \(|z|\rightarrow \infty \)? Our result is simply based on maximum principle and complements to those in Berestycki et al. (Ann Mat Pura Appl 186:469–507, 2007; Bull Math Biol 71:399, 2008), Berestycki and Rossi (Discret Contin Dyn Syst Ser B 21:41–67, 2008) and Vo (J Differ Equ 259:4947–4988, 2015).

The stability of stratified spatially periodic shear flows at low Péclet number

This work addresses the question of the stability of stratified, spatially periodic shear flows at low Péclet number but high Reynolds number. This little-studied limit is motivated by astrophysical systems, where the Prandtl number is often very small. Furthermore, it can be studied using a reduced set of “low-Péclet-number equations” proposed by Lignières [“The small-Péclet-number approximation in stellar radiative zones,” Astron. Astrophys. 348, 933–939 (1999)]. Through a linear stability analysis, we first determine the conditions for instability to infinitesimal perturbations. We formally extend Squire’s theorem to the low-Péclet-number equations, which shows that the first unstable mode is always two-dimensional. We then perform an energy stability analysis of the low-Péclet-number equations and prove that for a given value of the Reynolds number, above a critical strength of the stratification, any smooth periodic shear flow is stable to perturbations of arbitrary amplitude. In that parameter regime, the flow can only be laminar and turbulent mixing does not take place. Finding that the conditions for linear and energy stability are different, we thus identify a region in parameter space where finite-amplitude instabilities could exist. Using direct numerical simulations, we indeed find that the system is subject to such finite-amplitude instabilities. We determine numerically how far into the linearly stable region of parameter space turbulence can be sustained.

Particle dispersion in sheared suspensions: Crucial role of solid-solid contacts

We performed high resolution measurements of the dynamics of non-Brownian and neutrally buoyant particles subjected to a periodic shear flow under low Reynolds number conditions. By changing the particle roughness and showing that it significantly affects the particle motion, we provide direct evidence that particle solid-solid contacts occur in viscous suspensions and strongly influence the particle dynamics. An accurate prediction of the particle trajectories is obtained with a minimal model that solely includes normal lubrication interactions and a frictionless contact force.

High-frequency viscosity of a dilute suspension of elongated particles in a linear shear flow between two walls

AbstractA general expression for the effective viscosity of a dilute suspension of arbitrary-shaped particles in linear shear flow between two parallel walls is derived in terms of the induced stresslets on particles. This formula is applied to $N$-bead rods and to prolate spheroids with the same length, aspect ratio and volume. The effective viscosity of non-Brownian particles in a periodic shear flow is considered here. The oscillating frequency is high enough for the particle orientation and centre-of-mass distribution to be practically frozen, yet small enough for the flow to be quasi-steady. It is known that for spheres, the intrinsic viscosity $[{\it\mu}]$ increases monotonically when the distance $H$ between the walls is decreased. The dependence is more complex for both types of elongated particles. Three regimes are theoretically predicted here: (i) a ‘weakly confined’ regime (for $H&gt;l$, where $l$ is the particle length), where $[{\it\mu}]$ is slightly larger for smaller $H$; (ii) a ‘semi-confined’ regime, when $H$ becomes smaller than $l$, where $[{\it\mu}]$ rapidly decreases since the geometric constraints eliminate particle orientations corresponding to the largest stresslets; (iii) a ‘strongly confined’ regime when $H$ becomes smaller than 2–3 particle widths $d$, where $[{\it\mu}]$ rapidly increases owing to the strong hydrodynamic coupling with the walls. In addition, for sufficiently slender particles (with aspect ratio larger than 5–6) there is a domain of narrow gaps for which the intrinsic viscosity is smaller than that in unbounded fluid.

Estimating the effective viscosity of bubble suspensions in oscillatory shear flows by means of ultrasonic spinning rheometry

We have proposed a novel methodology using ultrasonic velocity profiling to estimate the effective viscosity of bubble suspensions that are accompanied by non-equilibrium bubble deformations in periodic shear flows. The methodology was termed “ultrasonic spinning rheometry” and validated on measurement of the effective viscosity of particle suspensions that has a semi-empirical formula giving good estimation of the actual viscosity. The results indicated that the proposed technique is valid for particle volume fractions below 3.0 %. Applying this to bubble suspensions suggested that the effective value of temporal variations in the capillary number, $$\hbox{Ca}_{\rm rms}$$ , is an important indicator to distinguish regimes in estimating the effective viscosity: Unsteady flows having larger $$\hbox{Ca}_{\rm rms}$$ number than the critical capillary number for the deformation of bubbles are categorized into Regime 2 that includes both highly unsteady conditions and large steady deformation of bubbles.

Homogenization Driven by a Fractional Brownian Motion: The Shear Layer Case

We consider a passive scalar in a periodic shear flow perturbed by an additive fractional noise with the Hurst exponent $H\in(0,1)$. We establish a diffusive homogenization limit for the tracer when the Hurst exponent $H\in(0,1/2)$. We also identify an intermediate range of times when the tracer behaves diffusively even when $H\in(1/2,1)$. The proof is based on an auxiliary limit theorem for an additive functional of a fractional Brownian motion.

Transformation of internal solitary waves at the &amp;quot;deep&amp;quot; and &amp;quot;shallow&amp;quot; shelf: satellite observations and laboratory experiment

Abstract. An interaction of internal solitary waves with the shelf edge in the time periods related to the presence of a pronounced seasonal pycnocline in the Red Sea and in the Alboran Sea is analysed via satellite photos and SAR images. Laboratory data on transformation of a solitary wave of depression while passing along the transverse bottom step were obtained in a tank with a two-layer stratified fluid. The certain difference between two characteristic types of hydrophysical phenomena was revealed both in the field observations and in experiments. The hydrological conditions for these two processes were named the "deep" and the "shallow" shelf respectively. The first one provides the generation of the secondary periodic short internal waves – "runaway" edge waves – due to change in the polarity of a part of a soliton approaching the shelf normally. Another one causes a periodic shear flow in the upper quasi-homogeneous water layer with the period of incident solitary wave. The strength of the revealed mechanisms depends on the thickness of the water layer between the pycnocline and the shelf bottom as well as on the amplitude of the incident solitary wave.

Irreversibility and chaos: Role of lubrication interactions in sheared suspensions

We investigate non-Brownian particles suspended in a periodic shear-flow using simulations. Following Metzger and Butler [Phys. Rev. E 82, 051406 (2010)], we show that the chaotic dynamics arising from lubrication interactions are too weak to generate an observable particle dispersion. The irreversibility observed in periodic flow is dominated by contact interactions. Nonetheless, we show that lubrication interactions must be included in the calculation to obtain results that agree with experiments.