Stability of Couette flow in nematic liquid crystals

1978 ◽  
Vol 89 (2) ◽  
pp. 273-303 ◽  
Author(s):  
E. Dubois-Violette ◽  
P. Manneville
Keyword(s):  
Liquid Crystals ◽  
Couette Flow ◽  
Time Constant ◽  
Rotation Axis ◽  
Taylor Instability ◽  
Plane Shear ◽  
Velocity Fluctuations ◽  
Inertial Coupling ◽  
Cylindrical Couette Flow ◽  
The Stability

We study the stability of the cylindrical Couette flow in nematics when the director is parallel to the rotation axis. The contribution of the inertial coupling of velocity fluctuations (responsible for the Taylor instability in isotropic liquids) is shown to be destabilizing when the inner cylinder rotates faster than the outer one. However, the instability remains driven by the mechanisms first discovered by Pieranski & Guyon for the plane shear case and quite specific to nematics. This mechanism couples the different orientation fluctuations via viscous torques and the corresponding threshold is given by \[ s\tau_0\sim 1, \] where τ0 is the time constant for the diffusion of orientation fluctuations. The contribution of inertia terms is measured by 2ωm τv, where τv is the time constant for the diffusion of velocity fluctuations. In usual nematics one has τv/τ0 ∼ 10−5 so that corrections due to rotation are small in general. At different stages of the discussion differences between the case of nematics and that of isotropic liquids are pointed out. We also study the possibility of an oscillatory instability when α3 is positive and large, where no stationary instability can occur.

The stability of the Couette flow of helium II

1988 ◽  
Vol 197 ◽  
pp. 551-569 ◽  
Author(s):  
C. F. Barenghi ◽  
C. A. Jones
Keyword(s):  
Couette Flow ◽  
Fluid Model ◽  
Taylor Instability ◽  
Mutual Friction ◽  
Helium Ii ◽  
Two Fluid Model ◽  
The Stability ◽  
Superfluid Vortices ◽  
Two Fluid

The stability of Couette flow in HeII is considered by an analysis of the HVBK equations. These equations are based on the Landau two-fluid model of HeII and include mutual friction between the normal and superfluid components, and the vortex tension due to the presence of superfluid vortices. We find that the vortex tension strongly affects the nature of the Taylor instability at temperatures below ≈ 2.05 K. The effect of the vortex tension is to make non-axisymmetric modes the most unstable, and to make the critical axial wavelength very long.We compare our results with experiments.

Stability of modulated finite-gap cylindrical Couette flow: linear theory

1981 ◽  
Vol 108 ◽  
pp. 19-42 ◽  
Author(s):  
S. Carmi ◽  
J. I. Tustaniwskyj
Keyword(s):  
Unsteady Flow ◽  
Couette Flow ◽  
Floquet Theory ◽  
Analytic Solution ◽  
Outer Cylinder ◽  
Three Dimensional ◽  
Linear Perturbation ◽  
Cylindrical Couette Flow ◽  
The Stability ◽  
The Galerkin Method

The linear stability of an extensively modulated cylindrical Couette flow is investigated in the finite-gap range. A closed form analytic solution is obtained for the basic unsteady flow after modulation is introduced through the boundary conditions. The general linear perturbation equations for three-dimensional disturbances are then derived and subsequently solved using the Galerkin method with the stability analysed by the Floquet theory. Modulation is found to destabilize the flow in most cases and results compare very favourably with the ones obtained experimentally. Stabilization is possible only for some cases of outer cylinder modulation.

Effect of axial flow on viscoelastic Taylor–Couette instability

1998 ◽  
Vol 360 ◽  
pp. 341-374 ◽  
Author(s):  
M. D. GRAHAM
Keyword(s):  
Couette Flow ◽  
Poiseuille Flow ◽  
Axial Flow ◽  
Weissenberg Number ◽  
Scaling Analysis ◽  
Plane Shear ◽  
Weakly Nonlinear ◽  
Elastic Stresses ◽  
Circular Couette Flow ◽  
The Stability

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.

A Liquid Metal Flow Between Two Coaxial Cylinders System

2019 ◽  
Vol 11 (1) ◽  
pp. 79-86
Author(s):  
Abdelkrim Merah ◽  
Ridha Kelaiaia ◽  
Faiza Mokhtari
Keyword(s):  
Couette Flow ◽  
Metal Flow ◽  
Three Dimensional ◽  
Liquid Sodium ◽  
Taylor Vortex ◽  
Turbulent Regime ◽  
Coaxial Cylinders ◽  
Concentric Cylinders ◽  
Ideal Tool ◽  
The Stability

Abstract The Taylor-Couette flow between two rotating coaxial cylinders remains an ideal tool for understanding the mechanism of the transition from laminar to turbulent regime in rotating flow for the scientific community. We present for different Taylor numbers a set of three-dimensional numerical investigations of the stability and transition from Couette flow to Taylor vortex regime of a viscous incompressible fluid (liquid sodium) between two concentric cylinders with the inner one rotating and the outer one at rest. We seek the onset of the first instability and we compare the obtained results for different velocity rates. We calculate the corresponding Taylor number in order to show its effect on flow patterns and pressure field.

Tertiary solutions and their stability in rotating plane Couette flow

1998 ◽  
Vol 358 ◽  
pp. 357-378 ◽  
Author(s):  
M. NAGATA
Keyword(s):  
Reynolds Number ◽  
Couette Flow ◽  
Stability Boundary ◽  
Streamwise Vortex ◽  
Reynolds Numbers ◽  
Time Dependent ◽  
Plane Couette Flow ◽  
Streamwise Direction ◽  
The Stability ◽  
Oscillatory Instabilities

The stability of nonlinear tertiary solutions in rotating plane Couette flow is examined numerically. It is found that the tertiary flows, which bifurcate from two-dimensional streamwise vortex flows, are stable within a certain range of the rotation rate when the Reynolds number is relatively small. The stability boundary is determined by perturbations which are subharmonic in the streamwise direction. As the Reynolds number is increased, the rotation range for the stable tertiary motions is destroyed gradually by oscillatory instabilities. We expect that the tertiary flow is overtaken by time-dependent motions for large Reynolds numbers. The results are compared with the recent experimental observation by Tillmark & Alfredsson (1996).

scholarly journals A frictional Cosserat model for the slow shearing of granular materials

2002 ◽  
Vol 457 ◽  
pp. 377-409 ◽  
Author(s):  
L. SRINIVASA MOHAN ◽  
K. KESAVA RAO ◽  
PRABHU R. NOTT
Keyword(s):  
Angular Velocity ◽  
Granular Material ◽  
Shear Layer ◽  
Granular Materials ◽  
Layer Thickness ◽  
Couette Flow ◽  
Velocity Fields ◽  
Cosserat Continuum ◽  
Cosserat Model ◽  
Cylindrical Couette Flow

A rigid-plastic Cosserat model for slow frictional flow of granular materials, proposed by us in an earlier paper, has been used to analyse plane and cylindrical Couette flow. In this model, the hydrodynamic fields of a classical continuum are supplemented by the couple stress and the intrinsic angular velocity fields. The balance of angular momentum, which is satisfied implicitly in a classical continuum, must be enforced in a Cosserat continuum. As a result, the stress tensor could be asymmetric, and the angular velocity of a material point may differ from half the local vorticity. An important consequence of treating the granular medium as a Cosserat continuum is that it incorporates a material length scale in the model, which is absent in frictional models based on a classical continuum. Further, the Cosserat model allows determination of the velocity fields uniquely in viscometric flows, in contrast to classical frictional models. Experiments on viscometric flows of dense, slowly deforming granular materials indicate that shear is confined to a narrow region, usually a few grain diameters thick, while the remaining material is largely undeformed. This feature is captured by the present model, and the velocity profile predicted for cylindrical Couette flow is in good agreement with reported data. When the walls of the Couette cell are smoother than the granular material, the model predicts that the shear layer thickness is independent of the Couette gap H when the latter is large compared to the grain diameter dp. When the walls are of the same roughness as the granular material, the model predicts that the shear layer thickness varies as (H/dp)1/3 (in the limit H/dp [Gt ] 1) for plane shear under gravity and cylindrical Couette flow.

Stability of Shear-Thickening Liquids Between Rotating Cylinders

2010 ◽  
Author(s):  
Nariman Ashrafi ◽  
Habib Karimi Haghighi
Keyword(s):  
Couette Flow ◽  
Dynamical Behavior ◽  
Shear Thickening ◽  
Shear Thinning ◽  
Taylor Vortex ◽  
Shear Dependent Viscosity ◽  
Shear Thinning Fluids ◽  
Shear Thickening Fluids ◽  
The Stability ◽  
Dependent Viscosity

The effects of nonlinearities on the stability are explored for shear thickening fluids in the narrow-gap limit of the Taylor-Couette flow. It is assumed that shear-thickening fluids behave exactly as opposite of shear thinning ones. A dynamical system is obtained from the conservation of mass and momentum equations which include nonlinear terms in velocity components due to the shear-dependent viscosity. It is found that the critical Taylor number, corresponding to the loss of stability of Couette flow becomes higher as the shear-thickening effects increases. Similar to the shear thinning case, the Taylor vortex structure emerges in the shear thickening flow, however they quickly disappear thus bringing the flow back to the purely azimuthal flow. Naturally, one expects shear thickening fluids to result in inverse dynamical behavior of shear thinning fluids. This study proves that this is not the case for every point on the bifurcation diagram.

scholarly journals On the linear stability of compressible plane Couette flow

1994 ◽  
Vol 258 ◽  
pp. 131-165 ◽  
Author(s):  
Peter W. Duck ◽  
Gordon Erlebacher ◽  
M. Yousuff Hussaini
Keyword(s):  
Linear Stability ◽  
Couette Flow ◽  
Small Amplitude ◽  
Reynolds Numbers ◽  
Inviscid Limit ◽  
Plane Couette Flow ◽  
Moving Wall ◽  
Type Boundary ◽  
The Stability ◽  
Radiation Type

The linear stability of compressible plane Couette flow is investigated. The appropriate basic velocity and temperature distributions are perturbed by a small-amplitude normal-mode disturbance. The full small-amplitude disturbance equations are solved numerically at finite Reynolds numbers, and the inviscid limit of these equations is then investigated in some detail. It is found that instabilities can occur, although the corresponding growth rates are often quite small; the stability characteristics of the flow are quite different from unbounded flows. The effects of viscosity are also calculated, asymptotically, and shown to have a stabilizing role in all the cases investigated. Exceptional regimes to the problem occur when the wave speed of the disturbances approaches the velocity of either of the walls, and these regimes are also analysed in some detail. Finally, the effect of imposing radiation-type boundary conditions on the upper (moving) wall (in place of impermeability) is investigated, and shown to yield results common to both bounded and unbounded flows.

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