Subcritical bifurcation of shear-thinning plane Poiseuille flows

2011 ◽  
Vol 686 ◽  
pp. 272-298 ◽  
Author(s):  
A. Chekila ◽  
C. Nouar ◽  
E. Plaut ◽  
A. Nemdili
Keyword(s):  
Travelling Waves ◽  
Nonlinear Effects ◽  
Expansion Method ◽  
Shear Thinning ◽  
Finite Amplitude ◽  
Rheological Parameters ◽  
Weakly Nonlinear ◽  
Shear Thinning Fluids ◽  
Wide Range ◽  
Threshold Amplitude

AbstractIn a recent article (Nouar, Bottaro & Brancher, J. Fluid. Mech., vol. 592, 2007, pp. 177–194), a linear stability analysis of plane Poiseuille flow of shear-thinning fluids has been performed. The authors concluded that the viscosity stratification delays the transition and that is important to account for the viscosity perturbation. The current paper focuses on the first-principles understanding of the influence of the viscosity stratification and the nonlinear variation of the effective viscosity $\ensuremath{\mu} $ with the shear rate $\dot {\gamma } $ on the flow stability with respect to a finite-amplitude perturbation. A weakly nonlinear analysis, using the amplitude expansion method, is adopted as a first approach to study nonlinear effects. The bifurcation to two-dimensional travelling waves is studied. For the numerical computations, the shear-thinning behaviour is described by the Carreau model. The rheological parameters are varied in a wide range. The results indicate that (i) the nonlinearity of the viscous terms tends to reduce the viscous dissipation and to accelerate the flow, (ii) the harmonic generated by the nonlinearity $\ensuremath{\mu} (\dot {\gamma } )$ is smaller and in opposite phase to that generated by the quadratic nonlinear inertial terms and (iii) with increasing shear-thinning effects, the bifurcation becomes highly subcritical. Consequently, the magnitude of the threshold amplitude of the perturbation, beyond which the flow is nonlinearly unstable, decreases. This result is confirmed by computing higher order-Landau constants.

Finite-amplitude convection in the presence of an unsteady shear flow

1995 ◽  
Vol 287 ◽  
pp. 225-249 ◽  
Author(s):  
Philip Hall
Keyword(s):  
Shear Flow ◽  
Critical Size ◽  
Low Frequency ◽  
Nonlinear Effects ◽  
Finite Amplitude ◽  
Present Calculation ◽  
Wide Range ◽  
Prandtl Numbers ◽  
Boussinesq Fluid

The effect of an unsteady shear flow on the planform of convection in a Boussinesq fluid heated from below is investigated. In the absence of the shear flow it is well-known, if non-Boussinesq effects can be neglected, that convection begins in the form of a supercritical bifurcation to rolls. Subcritical convection in the form of say hexagons can be induced by non-Boussinesq behaviour which destroys the symmetry of the basic state. Here it is found that the symmetry breaking effects associated with an unsteady shear flow are not sufficient to cause subcritical convection so the problem reduces to the determination of how the orientations of roll cells are modified by an unsteady shear flow. Recently Kelly & Hu (1993) showed that such a flow has a significant stabilizing effect on the linear stability problem and that, for a wide range of Prandtl numbers, the effect is most pronounced in the low-frequency limit. In the present calculation it is shown that the stabilizing effects found by Kelly & Hu (1993) do survive for most frequencies when nonlinear effects and imperfections are taken into account. However a critical size of the frequency is identified below which the Kelly & Hu (1993) conclusions no longer carry through into the nonlinear regime. For frequencies of size comparable with this critical size it is shown that the convection pattern changes in time. The cell pattern is found to be extremely complicated and straight rolls exist only for part of a period.

Disturbing vortices

2001 ◽  
Vol 426 ◽  
pp. 95-133 ◽  
Author(s):  
N. J. BALMFORTH ◽  
STEFAN G. LLEWELLYN SMITH ◽  
W. R. YOUNG
Keyword(s):  
Critical Radius ◽  
Nonlinear Effects ◽  
Rotation Frequency ◽  
Finite Amplitude ◽  
Matched Asymptotic Expansion ◽  
Critical Amplitude ◽  
Subsequent Evolution ◽  
Weakly Nonlinear ◽  
Vorticity Distribution ◽  
The Core

Inviscid spatially compact vortices (such as the Rankine vortex) have discrete Kelvin modes. For these modes, the critical radius, at which the rotation frequency of the wave matches the angular velocity of the fluid, lies outside the vortex core. When such a vortex is not perfectly compact, but has a weak vorticity distribution beyond the core, these Kelvin disturbances are singular at the critical radius and become ‘quasi-modes’. These are not true eigenmodes but have streamfunction perturbations that decay exponentially with time while the associated vorticity wraps up into a tight spiral without decay. We use a matched asymptotic expansion to derive a simplified description of weakly nonlinear, externally forced quasi-modes.We consider the excitation and subsequent evolution of finite-amplitude quasi- modes excited with azimuthal wavenumber 2. Provided the forcing amplitude is below a certain critical amplitude, the quasi-mode decays and the disturbed vortex returns to axisymmetry. If the amplitude of the forcing is above critical, then nonlinear effects arrest the decay and cat's eye patterns form. Thus the vortex is permanently deformed into a tripolar structure.

Effects of Shear Thinning on Taylor-Couette Flow: A Model for Synovial Fluid Flow

2006 ◽  
Author(s):  
Nariman Ashrafi
Keyword(s):  
Couette Flow ◽  
Shear Thinning ◽  
Taylor Number ◽  
Taylor Vortex ◽  
Narrow Gap ◽  
Shear Thinning Fluids ◽  
Wide Range ◽  
Taylor Couette ◽  
The Stability ◽  
Taylor Couette Flow

The effect of shear thinning on the stability of the Taylor-Couette flow (TCF) is explored for a Carreau-Bird fluid in the narrow-gap limit to simulate journal bearings in general. Also considered is the changing eccentricity to cover a wide range of applied situations such as bearings and even articulation of human joints. Here, a low-order dynamical system is obtained from the conservation of mass and momentum equations. In comparison with the Newtonian system, the present equations include additional nonlinear coupling in the velocity components through the viscosity. It is found that the critical Taylor number, corresponding to the loss of stability of the base (Couette) flow becomes lower s the shear-thinning effect increases. Similar to Newtonian fluids, there is an exchange of stability between the Couette and Taylor vortex flows. However, unlike the Newtonian model, the Taylor vortex cellular structure loses its stability in turn as the Taylor number reaches a critical value. At this point, A Hopf bifurcation emerges, which exists only for shear-thinning fluids. Variation of stresses in the narrow gap has been evaluated with significant applications in the non-Newtonian lubricant.

scholarly journals How dimensional analysis allows to go beyond Metzner–Otto concept for non-Newtonian fluids

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Guillaume Delaplace ◽  
Romain Jeantet ◽  
Richard Grenville ◽  
Gérard Cuvelier ◽  
Karine Loubiere
Keyword(s):  
Shear Rate ◽  
Dimensional Analysis ◽  
Batch Reactor ◽  
Mixing Time ◽  
Theoretical Background ◽  
Newtonian Fluids ◽  
Rheological Parameters ◽  
Laminar Regime ◽  
Shear Thinning Fluids ◽  
Wide Range

AbstractThe concept of Metzner and Otto was initially developed for correlating power measurements in stirred vessels for shear-thinning fluids in the laminar regime with regard to those obtained for Newtonian liquids. To get this overlap, Metzner and Otto postulated and determined an “effective shear rate” which was proportional to the rotational speed of the impeller Although it was not based on a strong theoretical background, it was rapidly admitted as a practical engineering approach and was extended for seeking out a “Newtonian correspondence” with non-Newtonian results (i.e. different classes of fluids). This was applied in a variety of tank processes even for predicting heat transfer or mixing time, which stretches far away from the frame initially envisaged by Metzner and Otto themselves. This paper aimed to show how dimensional analysis offers a theoretically founded framework to address this issue without the experimental determination of effective quantities. This work also aimed to enlarge the underlying questions to any process in which a variable material property exists and impacts the process. For that purpose, the pending questions of Metzner and Otto concept were first reminded (i.e. dependence of the Metzner–Otto constant to rheological parameters, physical meaning of the effective shear rate, etc.). Then, the theoretical background underlying the dimensional analysis was described and applied to the case of variable material properties (including non-Newtonian fluids), by introducing in particular the concept of material similarity. Finally, two examples were proposed to demonstrate how the rigorous framework associated with the dimensional analysis is a powerful method to exceed the concept of Metzner and Otto and can be adapted beyond the Ostwald–de Waele power law model to a wide range of non-Newtonian fluids in various processes, without being restricted to batch reactor and laminar regime.

scholarly journals Weakly nonlinear analysis of Rayleigh–Bénard convection in shear-thinning fluids: nature of the bifurcation and pattern selection

2015 ◽  
Vol 767 ◽  
pp. 696-734 ◽  
Author(s):  
M. Bouteraa ◽  
C. Nouar ◽  
E. Plaut ◽  
C. Metivier ◽  
A. Kalck
Keyword(s):  
Heat Transfer ◽  
Heat Transfer Coefficient ◽  
Transfer Coefficient ◽  
Nonlinear Analysis ◽  
Critical Rayleigh Number ◽  
Shear Thinning ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Analysis ◽  
Shear Thinning Fluids ◽  
The Heat Transfer Coefficient

AbstractA linear and weakly nonlinear analysis of convection in a layer of shear-thinning fluids between two horizontal plates heated from below is performed. The objective is to examine the effects of the nonlinear variation of the viscosity with the shear rate on the nature of the bifurcation, the planform selection problem between rolls, squares and hexagons, and the consequences on the heat transfer coefficient. Navier’s slip boundary conditions are used at the top and bottom walls. The shear-thinning behaviour of the fluid is described by the Carreau model. By considering an infinitesimal perturbation, the critical conditions, corresponding to the onset of convection, are determined. At this stage, non-Newtonian effects do not come into play. The critical Rayleigh number decreases and the critical wavenumber increases when the slip increases. For a finite-amplitude perturbation, nonlinear effects enter in the dynamic. Analysis of the saturation coefficients at cubic order in the amplitude equations shows that the nature of the bifurcation depends on the rheological properties, i.e. the fluid characteristic time and shear-thinning index. For weakly shear-thinning fluids, the bifurcation is supercritical and the heat transfer coefficient increases, as compared with the Newtonian case. When the shear-thinning character is large enough, the bifurcation is subcritical, pointing out the destabilizing effect of the nonlinearities arising from the rheological law. Departing from the onset, the weakly nonlinear analysis is carried out up to fifth order in the amplitude expansion. The flow structure, the modification of the viscosity field and the Nusselt number are characterized. The competition between rolls, squares and hexagons is investigated. Unlike Albaalbaki & Khayat (J. Fluid. Mech., vol. 668, 2011, pp. 500–550), it is shown that in the supercritical regime, only rolls are stable near onset.

scholarly journals Linearly polarized waves of finite amplitude in pre-strained elastic materials

2019 ◽  
Vol 475 (2226) ◽  
pp. 20180891 ◽  
Author(s):  
Edvige Pucci ◽  
Giuseppe Saccomandi ◽  
Luigi Vergori
Keyword(s):  
Principal Direction ◽  
Elastic Solid ◽  
Nonlinear Effects ◽  
Finite Amplitude ◽  
Theory Of Elasticity ◽  
Deformation Tensor ◽  
Transverse Waves ◽  
Weakly Nonlinear ◽  
Linearly Polarized ◽  
Propagation Of Waves

We study the propagation of linearly polarized transverse waves in a pre-strained incompressible isotropic elastic solid. Both finite and small-but-finite amplitude waves are examined. Irrespective of the magnitude of the wave amplitude, these waves may propagate only if the (unit) normal to the plane spanned by the directions of propagation and polarization is a principal direction of the left Cauchy–Green deformation tensor associated with the pre-strained state. A rigorous asymptotic analysis of the equations governing the propagation of waves of small but finite amplitude reveals that the time scale over which the nonlinear effects become significant depends on the direction along which the wave travels. Moreover, we design theoretically an experimental procedure to determine the Landau constants of the fourth-order weakly nonlinear theory of elasticity.

scholarly journals Mixing Enhancement of Non-Newtonian Shear-Thinning Fluid for a Kenics Micromixer

Micromachines ◽  
2021 ◽  
Vol 12 (12) ◽  
pp. 1494
Author(s):  
Abdelkader Mahammedi ◽  
Naas Toufik Tayeb ◽  
Kwang-Yong Kim ◽  
Shakhawat Hossain
Keyword(s):  
Reynolds Number ◽  
Stokes Equations ◽  
Fluid Dynamic ◽  
Reynolds Numbers ◽  
Shear Thinning ◽  
Navier Stokes ◽  
Mixing Enhancement ◽  
Pressure Losses ◽  
Shear Thinning Fluids ◽  
Wide Range

In this work, a numerical investigation was analyzed to exhibit the mixing behaviors of non-Newtonian shear-thinning fluids in Kenics micromixers. The numerical analysis was performed using the computational fluid dynamic (CFD) tool to solve 3D Navier-Stokes equations with the species transport equations. The efficiency of mixing is estimated by the calculation of the mixing index for different cases of Reynolds number. The geometry of micro Kenics collected with a series of six helical elements twisted 180° and arranged alternately to achieve the higher level of chaotic mixing, inside a pipe with a Y-inlet. Under a wide range of Reynolds numbers between 0.1 to 500 and the carboxymethyl cellulose (CMC) solutions with power-law indices among 1 to 0.49, the micro-Kenics proves high mixing Performances at low and high Reynolds number. Moreover the pressure losses of the shear-thinning fluids for different Reynolds numbers was validated and represented.

EHL Circular Contact Film Thickness Correction Factor for Shear-Thinning Fluids

2008 ◽  
Vol 130 (4) ◽  
Author(s):  
Punit Kumar ◽  
M. M. Khonsari
Keyword(s):  
Film Thickness ◽  
Correction Factor ◽  
Point Contact ◽  
Elastohydrodynamic Lubrication ◽  
Shear Thinning ◽  
Correction Factors ◽  
Shear Thinning Fluids ◽  
Pure Rolling ◽  
Wide Range ◽  
Circular Contact

An extensive set of full elastohydrodynamic lubrication point contact simulations has been used to develop correction factors to account for the effect of shear-thinning lubricant behavior on the central and minimum film thickness in circular contacts under pure rolling condition. The film thickness for a shear-thinning lubricant can be easily obtained by dividing the corresponding Newtonian film thickness by the appropriate correction factor. Comparisons of the film thickness values obtained using the correction factors have been matched with the published experimental results pertaining to shear-thinning lubricants with a variety of realistic flow and piezoviscous properties under a wide range of operating speed. The good agreement between them establishes the validity and versatility of the correction factors developed in this paper.

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