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

2014 ◽  
Vol 764 ◽  
pp. 133-147 ◽  
Author(s):  
François Feuillebois ◽  
Maria L. Ekiel-Jeżewska ◽  
Eligiusz Wajnryb ◽  
Antoine Sellier ◽  
Jerzy Bławzdziewicz
Keyword(s):  
Aspect Ratio ◽  
Shear Flow ◽  
Intrinsic Viscosity ◽  
Effective Viscosity ◽  
Geometric Constraints ◽  
Particle Orientation ◽  
Hydrodynamic Coupling ◽  
Linear Shear ◽  
Dilute Suspension ◽  
Particle Length

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 $H>l$, 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.

High-frequency effective viscosity of a dilute suspension of particles in Poiseuille flow between parallel walls

2016 ◽  
Vol 800 ◽  
pp. 111-139
Author(s):  
François Feuillebois ◽  
Maria L. Ekiel-Jeżewska ◽  
Eligiusz Wajnryb ◽  
Antoine Sellier ◽  
Jerzy Bławzdziewicz
Keyword(s):  
Aspect Ratio ◽  
Intrinsic Viscosity ◽  
Poiseuille Flow ◽  
High Frequency ◽  
Effective Viscosity ◽  
Volume Fraction ◽  
Positive Contribution ◽  
Important Conclusion ◽  
Dilute Suspension ◽  
Quadrupole Term

It is shown that the formal expression for the effective viscosity of a dilute suspension of arbitrary-shaped particles in Poiseuille flow contains a novel quadrupole term, besides the expected stresslet. This term becomes important for a very confined geometry. For a high-frequency flow field (in the sense used in Feuillebois et al. (J. Fluid Mech., vol. 764, 2015, pp. 133–147), the suspension rheology is Newtonian at first order in volume fraction. The effective viscosity is calculated for suspensions of $N$-bead rods and of prolate spheroids with the same length, volume and aspect ratio (up to 6), entrained by the Poiseuille flow between two infinite parallel flat hard walls. The numerical computations, based on solving the Stokes equations, indicate that the quadrupole term gives a significant positive contribution to the intrinsic viscosity $[{\it\mu}]$ if the distance between the walls is less than ten times the particle width, or less. It is found that the intrinsic viscosity in bounded Poiseuille flow is generally smaller than the corresponding value in unbounded flow, except for extremely narrow gaps when it becomes larger because of lubrication effects. The intrinsic viscosity is at a minimum for a gap between walls of the order of 1.5–2 particle width. For spheroids, the intrinsic viscosity is generally smaller than for chains of beads with the same aspect ratio, but when normalized by its value in the bulk, the results are qualitatively the same. Therefore, a rigid chain of beads can serve as a simple model of an orthotropic particle with a more complicated shape. The important conclusion is that the intrinsic viscosity in shear flow is larger than in the Poiseuille flow between two walls, and the difference is significant even for relatively wide channels, e.g. three times wider than the particle length. For such confined geometries, the hydrodynamic interactions with the walls are significant and should be taken into account.

The effect of shear flow on the rotational diffusion of a single axisymmetric particle

2015 ◽  
Vol 772 ◽  
pp. 42-79 ◽  
Author(s):  
Brian D. Leahy ◽  
Donald L. Koch ◽  
Itai Cohen
Keyword(s):  
Aspect Ratio ◽  
Shear Flow ◽  
Intrinsic Viscosity ◽  
Strain Amplitude ◽  
Rotational Diffusion ◽  
Oscillatory Shear ◽  
Time Dependent ◽  
Continuous Shear ◽  
Orientation Distributions ◽  
Particle Aspect Ratio

Understanding the orientation dynamics of anisotropic colloidal particles is important for suspension rheology and particle self-assembly. However, even for the simplest case of dilute suspensions in shear flow, the orientation dynamics of non-spherical Brownian particles are poorly understood. Here we analytically calculate the time-dependent orientation distributions for non-spherical axisymmetric particles confined to rotate in the flow–gradient plane, in the limit of small but non-zero Brownian diffusivity. For continuous shear, despite the complicated dynamics arising from the particle rotations, we find a coordinate change that maps the orientation dynamics to a diffusion equation with a remarkably simple ratio of the enhanced rotary diffusivity to the zero shear diffusion: $D_{eff}^{r}/D_{0}^{r}=(3/8)(p-1/p)^{2}+1$, where $p$ is the particle aspect ratio. For oscillatory shear, the enhanced diffusion becomes orientation dependent and drastically alters the long-time orientation distributions. We describe a general method for solving the time-dependent oscillatory shear distributions and finding the effective diffusion constant. As an illustration, we use this method to solve for the diffusion and distributions in the case of triangle-wave oscillatory shear and find that they depend strongly on the strain amplitude and particle aspect ratio. These results provide new insight into the time-dependent rheology of suspensions of anisotropic particles. For continuous shear, we find two distinct diffusive time scales in the rheology that scale separately with aspect ratio $p$, as $1/D_{0}^{r}p^{4}$ and as $1/D_{0}^{r}p^{2}$ for $p\gg 1$. For oscillatory shear flows, the intrinsic viscosity oscillates with the strain amplitude. Finally, we show the relevance of our results to real suspensions in which particles can rotate freely. Collectively, the interplay between shear-induced rotations and diffusion has rich structure and strong effects: for a particle with aspect ratio 10, the oscillatory shear intrinsic viscosity varies by a factor of ${\approx}2$ and the rotational diffusion by a factor of ${\approx}40$.

scholarly journals Behavior of active filaments near solid-boundary under linear shear flow

Soft Matter ◽  
2019 ◽  
Vol 15 (19) ◽  
pp. 4008-4018 ◽  
Author(s):  
Shalabh K. Anand ◽  
Sunil P. Singh
Keyword(s):  
Steady State ◽  
Shear Flow ◽  
Couette Flow ◽  
Solid Boundary ◽  
State Behavior ◽  
Linear Shear ◽  
Dilute Suspension ◽  
Steady State Behavior

The steady-state behavior of a dilute suspension of self-propelled filaments confined between planar walls subjected to Couette-flow is reported herein.

scholarly journals Fluid-Structure Energy Transfer of a Tensioned Beam Subject to Vortex-Induced Vibrations in Shear Flow

2011 ◽  
Author(s):  
Remi Bourguet ◽  
Michael S. Triantafyllou ◽  
Michael Tognarelli ◽  
Pierre Beynet
Keyword(s):  
Energy Transfer ◽  
Shear Flow ◽  
Cross Flow ◽  
Reynolds Numbers ◽  
Structural Vibrations ◽  
Fluid Structure ◽  
Vortex Induced Vibrations ◽  
Linear Shear ◽  
Structural Responses ◽  
Lock In

The fluid-structure energy transfer of a tensioned beam of length to diameter ratio 200, subject to vortex-induced vibrations in linear shear flow, is investigated by means of direct numerical simulation at three Reynolds numbers, from 110 to 1,100. In both the in-line and cross-flow directions, the high-wavenumber structural responses are characterized by mixed standing-traveling wave patterns. The spanwise zones where the flow provides energy to excite the structural vibrations are located mainly within the region of high current where the lock-in condition is established, i.e. where vortex shedding and cross-flow vibration frequencies coincide. However, the energy input is not uniform across the entire lock-in region. This can be related to observed changes from counterclockwise to clockwise structural orbits. The energy transfer is also impacted by the possible occurrence of multi-frequency vibrations.

Stable equilibrium configurations of an oblate capsule in simple shear flow

2016 ◽  
Vol 791 ◽  
pp. 738-757 ◽  
Author(s):  
C. Dupont ◽  
F. Delahaye ◽  
D. Barthès-Biesel ◽  
A.-V. Salsac
Keyword(s):  
Aspect Ratio ◽  
Shear Flow ◽  
Simple Shear ◽  
Capillary Number ◽  
Stable Equilibrium ◽  
Viscosity Ratio ◽  
Shear Plane ◽  
Initial Orientation ◽  
Simple Shear Flow ◽  
Mechanical Equilibrium

The objective of the paper is to determine the stable mechanical equilibrium states of an oblate capsule subjected to a simple shear flow, by positioning its revolution axis initially off the shear plane. We consider an oblate capsule with a strain-hardening membrane and investigate the influence of the initial orientation, capsule aspect ratio$a/b$, viscosity ratio${\it\lambda}$between the internal and external fluids and the capillary number$Ca$which compares the viscous to the elastic forces. A numerical model coupling the finite element and boundary integral methods is used to solve the three-dimensional fluid–structure interaction problem. For any initial orientation, the capsule converges towards the same mechanical equilibrium state, which is only a function of the capillary number and viscosity ratio. For$a/b=0.5$, only four regimes are stable when${\it\lambda}=1$: tumbling and swinging in the low and medium$Ca$range ($Ca\lesssim 1$), regimes for which the capsule revolution axis is contained within the shear plane; then wobbling during which the capsule experiences precession around the vorticity axis; and finally rolling along the vorticity axis at high capillary numbers. When${\it\lambda}$is increased, the tumbling-to-swinging transition occurs for higher$Ca$; the wobbling regime takes place at lower$Ca$values and within a narrower$Ca$range. For${\it\lambda}\gtrsim 3$, the swinging regime completely disappears, which indicates that the stable equilibrium states are mainly the tumbling and rolling regimes at higher viscosity ratios. We finally show that the$Ca$–${\it\lambda}$phase diagram is qualitatively similar for higher aspect ratio. Only the$Ca$-range over which wobbling is stable increases with$a/b$, restricting the stability ranges of in- and out-of-plane motions, although this phenomenon is mainly visible for viscosity ratios larger than 1.

scholarly journals A Variational Approach and Finite Element Implementation for Swelling of Polymeric Hydrogels Under Geometric Constraints

2010 ◽  
Vol 77 (6) ◽  
Author(s):  
Min Kyoo Kang ◽  
Rui Huang
Keyword(s):  
Finite Element ◽  
Aspect Ratio ◽  
Numerical Simulations ◽  
Variational Approach ◽  
Chemical Potential ◽  
Polymer Network ◽  
Material Model ◽  
Geometric Constraints ◽  
Free Energy Function ◽  
Finite Element Implementation

A hydrogel consists of a cross-linked polymer network and solvent molecules. Depending on its chemical and mechanical environment, the polymer network may undergo enormous volume change. The present work develops a general formulation based on a variational approach, which leads to a set of governing equations coupling mechanical and chemical equilibrium conditions along with proper boundary conditions. A specific material model is employed in a finite element implementation, for which the nonlinear constitutive behavior is derived from a free energy function, with explicit formula for the true stress and tangent modulus at the current state of deformation and chemical potential. Such implementation enables numerical simulations of hydrogels swelling under various constraints. Several examples are presented, with both homogeneous and inhomogeneous swelling deformation. In particular, the effect of geometric constraint is emphasized for the inhomogeneous swelling of surface-attached hydrogel lines of rectangular cross sections, which depends on the width-to-height aspect ratio of the line. The present numerical simulations show that, beyond a critical aspect ratio, creaselike surface instability occurs upon swelling.

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