Instability Of Couette Flow Research Articles

Low-Dimensional Analysis and Numerical Simulation of the Incompressible Flow Between Two Concentric Rotating Cylinders

There has been a very large number of experimental and theoretical studies of flow between concentric rotating cylinders in the century, since these pioneering works, the instability of Couette flow and Taylor vortex flow, now known as the “Couette–Taylor problem” is a paradigm of nonlinear problem. In order to explore the transition way of Couette–Taylor flow from laminar to turbulence and the characteristics of chaotic attractors in the turbulent regime of Couette–Taylor flow, dynamical behaviors and numerical simulation of Couette–Taylor flow have been studied by using low-dimensional model analysis method in this paper. Dynamical behaviors of a three-model Lorenz-like system of Couette–Taylor flow have been discussed, such as the stability of equilibrium, presence of limit cycles, occurrence of bifurcation and chaos, as well as the analysis of global stability etc. Moreover, using these results we explain successive transitions of Couette–Taylor flow from Laminar flow to turbulence in the experiment. By means of numerical simulation results of bifurcation diagram, Lyapunov exponent spectrum and Poincare map we analyze complex dynamic behaviors of the system from the bifurcation transition to chaos.

Linear stability of spiral and annular Poiseuille flow for small radius ratio

For the radius ratio $\eta\,{\equiv}\,R_i/R_o\,{=}\,0.1$ and several rotation rate ratios $\mu\,{\equiv}\,\Omega_o/\Omega_i$, we consider the linear stability of spiral Poiseuille flow (SPF) up to ${\hbox {\it Re}}\,{=}\,10^5$, where $R_i$ and $R_o$ are the radii of the inner and outer cylinders, respectively, ${\hbox {\it Re}}\,{\equiv}\,\overline V_Z(R_o\,{-}R_i)/\nu$ is the Reynolds number, $\Omega_i$ and $\Omega_o$ are the (signed) angular speeds of the inner and outer cylinders, respectively, $\nu$ is the kinematic viscosity, and $\overline V_Z$ is the mean axial velocity. The Re range extends more than three orders of magnitude beyond that considered in the previous $\mu\,{=}\,0$ work of Recktenwald et al. (Phys. Rev. E, vol. 48, 1993, p. 444). We show that in the non-rotating limit of annular Poiseuille flow, linear instability does not occur below a critical radius ratio $\hat\eta\,{\approx}\,0.115$. We also establish the connection of the linear stability of annular Poiseuille flow for $0\,{<}\,\eta\,{\leq}\,\hat\eta$ at all Re to the linear stability of circular Poiseuille flow ($\eta\,{=}\,0$) at all Re. For the rotating case, with $\mu\,{=}\,{-}1$, ${-}\,0.5$, ${-}\,0.25$, 0 and 0.2, the stability boundaries, presented in terms of critical Taylor number ${\hbox {\it Ta}}\,{\equiv}\,\Omega_i(R_o\,{-}R_i)^2/\nu$ versus Re, show that the results are qualitatively different from those at larger $\eta$. For each $\mu$, the centrifugal instability at small Re does not connect to a high-Re Tollmien–Schlichting-like instability of annular Poiseuille flow, since the latter instability does not exist for $\eta\,{<}\,\hat\eta$. We find a range of Re for which disconnected neutral curves exist in the $k$–Ta plane, which for each non-zero $\mu$ considered, lead to a multi-valued stability boundary, corresponding to two disjoint ranges of stable Ta. For each counter-rotating ($\mu\,{<}\,0$) case, there is a finite range of Re for which there exist three critical values of Ta, with the upper branch emanating from the ${\hbox {\it Re}}\,{=}\,0$ instability of Couette flow. For the co-rotating ($\mu\,{=}\,0.2$) case, there are two critical values of Ta for each Re in an apparently semi-infinite range of Re, with neither branch of the stability boundary intersecting the Re = 0 axis, consistent with the classical result of Synge that Couette flow is stable with respect to all small disturbances if $\mu\,{>}\,\eta^2$, and our earlier results for $\mu\,{>}\,\eta^2$ at larger $\eta$.

Centrifugal instability of Couette flow over a wavy wall

Linear stability of Couette flow over a wavy wall is considered. It is shown that centrifugal effects may create an instability that leads to the formation of streamwise vortices. The conditions leading to the onset of the instability depend on the amplitude and the wavelength of the waviness and can be expressed in terms of the critical Reynolds number. The global critical conditions describing the minimum critical Reynolds number and the associated wave number of wall waviness required to create the instability for the specified amplitude of the waviness are also given.

Mechanism of elastic instability in Couette flow of polymer solutions: Experiment

Experiments on flow stability and pattern formation in Couette flow between two cylinders with highly elastic polymer solutions are reported. It is found that the flow instabilities are determined by the elastic Deborah number, De, and the polymer concentration only, while the Reynolds number becomes completely irrelevant. A mechanism of such “purely elastic” instability was suggested a few years ago by Larson, Shaqfeh, and Muller [J. Fluid Mech. 218, 573 (1990)], referred to as LMS. It is based on the Oldroyd-B rheological model and implies a certain functional relation between De at the instability threshold and the polymer contribution to the solution viscosity, ηp/η, that depends on the polymer concentration. The elastic force driving the instability arises when perturbative elongational flow in radial direction is coupled to the strong primary azimuthal shear. This force is provided by the “hoop stress” that develops due to stretching of the polymer molecules along the curved streamlines. It is found experimentally that the elastic instability leads to a strongly nonlinear flow transition. Therefore, the linear consideration by LMS is expanded to include finite amplitude velocity perturbations. It is shown that the nature of the elastic force implies major asymmetry between inflow and outflow in finite amplitude secondary flows. This special feature is indeed exhibited by the experimentally observed flow patterns. For one of the flow patterns it is also shown that the suggested elastic force should be quite efficient in driving it, which is important evidence for the validity of the mechanism proposed by LMS. Further, the predicted relation between De and ηp/η is tested. At fixed ηp/η the elastic instability is found to occur at constant Deborah number in a broad range of the solution relaxation times in full agreement with the theoretical prediction. The experimentally found dependence of the Deborah number on ηp/η also agrees with the theoretical prediction rather well if a proper correction for the shear thinning is made. This provides further support to the proposed instability mechanism.

Stability of Couette flow of an ideal fluid with free boundaries

The linear and nonlinear stage of development of instability of Couette flow with two free boundaries is studied. It is established that instability occurs only for long waves, and the critical wave number is computed. In the presence of surface-tension forces, instability is preserved only at Weber numbersWe ≤ 1/3.

The effect of a radial temperature gradient on the stability of Couette flow: Non-axisymmetric modes analysis

The effect of a radial temperature gradient on the character of the critical disturbances responsible for the onset of instability of Couette flow is examined. The stability equations with respect to non-axisymmetric disturbances are derived and solved numerically for the narrow gap case. For counter-rotating cylinders, we found that the critical non-axisymmetric disturbances of the isothermal flow become axisymmetric when the outer cylinder temperature θ 2 is higher than the inner cylinder temperature θ 1 by a certain critical value. For the stationary outer cylinder, the critical axisymmetric disturbances become non-axisymmetric when θ 2 is less than θ 1 by a certain critical value. These critical values are calculated for fluids with different Prandtl numbers.

Kelvin–Helmholtz instability of Couette flow between vertical walls with a free surface

A novel type of Kelvin–Helmholtz instability model is developed from hydrodynamic theory. The classical Kelvin–Helmholtz instability involves a horizontal interface between two fluids with different parallel, uniform, horizontal velocities. If the upper fluid is a gas with a much smaller density than the lower fluid which is a liquid, then the phase velocity of the critical disturbance equals the liquid’s velocity, so that the liquid sees a standing interfacial wave. The inertial force driving the interfacial instability involves only the gas, no matter how small its density is. In a much more realistic flow model, the liquid velocity at the free surface is not uniform, but varies across the free surface. The disturbance phase velocity can only equal the liquid velocity at one point, while liquid on either side of this point moves faster or slower than the wave. The inertial forces in the liquid then dominate and the gas plays a negligible role. The concept is developed from a Couette flow hydrodynamic model where the fluid flows between two parallel vertical walls with a free surface. The importance of a nonuniform liquid velocity is demonstrated. This modified theory will be applied in future work to study the ejection instability at the interface of the liquid metal and inert cover gas in sliding electrical contacts.

Orientational instabilities in Couette flow of non-flow-aligning nematic liquid crystals.

A numerical solution of the continuum equations for nonaligning nematic liquid crystals in cylindrical Couette flow is obtained. The initial orientation of the director is radial and therefore contained in the plane of the shear. The stability of the solution is analyzed for a series of sets of material constants in order to investigate the dependence of the stability threshold with temperature. The results are compared with experiments.

A linear analysis of instabilities in Couette flow of nematic liquid crystals

This paper presents a new method for determining the critical threshold values at which both homogeneous and roll-type instabilities may appear when thin layers of nematic liquid crystal are subjected to Couette flow. The anisotropic axis of the material is initially uniformly aligned parallel to the axial direction and the effect of a stabilising, axially oriented, uniform magnetic field is allowed for in calculations. Carrying out a linear stability analysis of the continuum equations, we employ a Fourier series method to derive an expression for determining the critical shear rate at which instability occurs. Thresholds for both homogeneous and roll instabilities are obtained and their variation with rotation rate and applied magnetic field strength is examined for the material MBBA.

A theoretical investigation of Benard-Couette instabilities in nematic liquid crystals

Employing continuum theory, this paper investigates the onset of homogeneous and roll instabilities in Couette flow of a nematic liquid crystal when a radial temperature gradient is present. The arrangement considered is that in which the anisotropic axis of the liquid crystal is initially aligned parallel to the axis of the circular cylinders. Using an orthonormalisation method, the authors present accurate numerical results for the shear rate thresholds as a function of the applied thermal gradient. It is shown that a positive thermal gradient reduces the roll threshold but that a thermal gradient of either sign has no effect on the homogeneous threshold. They therefore predict a value phi T for the radial temperature gradient at which there is a transition from a homogeneous to a roll-type instability. In addition they demonstrate the important effect rotation has through the centrifugal force.

A theoretical investigation of the Pieranski—Guyon instability in Couette flow of nematic liquid crystals

This paper examines the onset of homogeneous and roll type instabilities in Couette flow of a nematic liquid crystal with the anisotropic axis of these transversely isotropic liquids uniformly aligned in the axial direction. The stabilising influence of an axially oriented magnetic field is also considered Within the context of a linear stability analysis, an orthonormalisation method is employed to compute accurate numerical results from the continuum equations for the critical shear rates at which instability becomes possible. The dependance of the thresholds upon magnetic field strength is investigated leading to a prediction of the value at which a change in the nature of the instability occurs. The influence of superposed rotation upon this transition value is also examined.

Instability of Couette flow in a rotating fluid and origin of Langmuir circulations

A simple calculation shows that plane Couette flow in a rotating inviscid neutrally buoyant fluid is unstable to roll perturbations. The linear shear flow may exist in nature as a boundary layer flow or a thermal wind. The results of the calculations are compared with observations of wind streaks on the ocean and cloud streets in the atmosphere.

Thermal instability of Couette flow

SummaryThe present paper investigates the thermal instability of a non-homogeneous fluid rotating between two co-axial cylinders when the inner cylinder is being heated uniformly. The conditions are established under which the oscillatory and non-oscillatory modes exist and further it has been shown that the oscillatory modes are amplified due to the adverse temperature gradient. In the case of non-oscillatory modes, sufficient conditions for stability and instability are obtained.

On the spiral form of instability of couette flow of a viscous fluid

The instability of Couette flow caused by vortices with spiral symmetry is studied; the equations of motion in a suitable “spiral” metric turn out to be analogous with those describing vortices with axial symmetry. The approximate solution is obtained by discretization of the differential equations. Special numerical techniques reduce drastically the dimensions of the matrices involved in the calculations. The Taylor number for transition is computed; comparisons with known results are established. The results are shown in a number of graphs.

Stability of Spatially Periodic Supercritical Flows in Hydrodynamics

Recently, Eckhaus developed a theory for a class of nonlinear stability problems which can be formulated in terms of a scalar partial differential equation with quadratic nonlinearities. It is demonstrated that Eckhaus' work on the development and stability of periodic solutions can be extended to a class of nonlinear matrix partial differential equations. The equations governing axisymmetric viscous flow between concentric rotating cylinders belong to the class of equations considered. When the Taylor number T is slightly above the minimum critical value Tc there exists an interval of possible equilibrium flows (Taylor-vortex flows) growing out of the instability of Couette flow. It is shown that within the interval of possible Taylor-vortex flows, there exists a subinterval of stable vortex flows.

Experiments on the stability of viscous flow between rotating cylinders III. Enhancement of stability by modulation

The onset of instability in Couette flow can be inhibited by modulating the rate of rotation of the inner cylinder. The origin of the inhibition has been shown experimentally to be due to the viscous wave propagated across the annulus by the modulated cylinder.

Asymmetric Inviscid Modes of Instability in Couette Flow

An approximate solution of the equation which governs the inviscid modes of instability of steady flow between concentric rotating cylinders is derived. The problem is treated in the narrow-gap approximation, and the solution is applicable over the entire range of opposite rotation rates, but is limited to the treatment of disturbances which are nearly axially symmetric. Numerical results are given for the limiting cases where the ratio of angular velocities of the cylinders approaches zero and minus infinity. It is found that the growth rates of unstable modes which are very nearly symmetric are always less than the growth rate of the corresponding axisymmetric mode.

On the reversibility of the transition past instability in Couette flow

We have used a small rotating-cylinder viscometer in which the inner cylinder rotates to measure the torque transmitted by fluid flow in the laminar and supercritical range of speeds. We find that when the radii of the cylinders are nearly equal, the torque is a unique function of the Reynolds number of the flow regardless of the manner in which the flow is established, provided only that sufficient time is allowed for equilibrium to be reached. In particular we find that the critical speed for transition to cellular flow is identical with the speed at which cellular flow reverts to laminar flow. When the gap between the cylinders is large a small hysteresis in the torque near the critical speed is observed. Some observations are reported which suggest that the transitions in hydromagnetic and spiral flow between rotating cylinders are also reversible.

Instability in Couette flow of solutions of macromolecules

In Couette flow between coaxial cylinders, ring-shaped vortices, illustrated schematically in figure 1, develop at a Reynolds number which depends strongly on the geometry of the system, as treated analytically for Newtonian fluids by Chandrasekhar (1958). The problem was first considered by Taylor (1923) who, examining simple (micromolecular) liquids, arrived at an equation for the critical Reynolds numberReiabove which instability (vortex formation) would occur when the inner cylinder rotated and the outer is stationary, and when the gap is small compared to the radius.

Inviscid modes of instability in Couette flow

An exact solution is presented for the inviscid modes of instability in Couette flow. When the cylinders rotate in the same direction, only one mode of instability is present. It corresponds to the convective modes found in the existing viscous treatments of the problem. When the cylinders rotate in opposite directions, however, two types of instability are present, the second being of an oscillatory type. These results suggest that although the “principle of exchange of stabilities” may be valid when the cylinders rotate in the same direction, it may not be valid when they rotate in opposite directions.