Linear stability analysis of plane Couette flow with viscous heating

1996 ◽  
Vol 8 (7) ◽  
pp. 1802-1813 ◽  
Author(s):  
Chao‐Sheng Yueh ◽  
Cheng‐I Weng
Keyword(s):  
Stability Analysis ◽  
Linear Stability ◽  
Couette Flow ◽  
Linear Stability Analysis ◽  
Viscous Heating ◽  
Plane Couette Flow

scholarly journals Bioconvection under uniform shear: linear stability analysis

2013 ◽  
Vol 738 ◽  
pp. 522-562 ◽  
Author(s):  
Yongyun Hwang ◽  
T. J. Pedley
Keyword(s):  
Stability Analysis ◽  
Shear Rate ◽  
Linear Stability ◽  
Couette Flow ◽  
Linear Stability Analysis ◽  
Plane Couette Flow ◽  
Shear Rates ◽  
Uniform Shear ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

AbstractThe role of uniform shear in bioconvective instability in a shallow suspension of swimming gyrotactic cells is studied using linear stability analysis. The shear is introduced by applying a plane Couette flow, and it significantly disturbs gravitaxis of the cell. The unstably stratified basic state of the cell concentration is gradually relieved as the shear rate is increased, and it even becomes stably stratified at very large shear rates. Stability of the basic state is significantly changed. The instability at high wavenumbers is drastically damped out with the shear rate, while that at low wavenumbers is destabilized. However, at very large shear rates, the latter is also suppressed. The most unstable mode is found as a pair of streamwise uniform rolls aligned with the shear, analogous to Rayleigh–Bénard convection in plane Couette flow. To understand these findings, the physical mechanism of the bioconvective instability is reexamined with several sets of numerical experiments. It is shown that the bioconvective instability in a shallow suspension originates from three different physical processes: gravitational overturning, gyrotaxis of the cell and negative cross-diffusion flux. The first mechanism is found to rule the behaviour of low-wavenumber instability whereas the last two mechanisms are mainly associated with high-wavenumber instability. With the increase of the shear rate, the former is enhanced, thereby leading to destabilization at low wavenumbers, whereas the latter two mechanisms are significantly suppressed. For streamwise varying perturbations, shear with sufficiently large rates is also found to play a stabilizing role as in Rayleigh–Bénard convection. However, at small shear rates, it destabilizes these perturbations through the mechanism of overstability discussed by Hill, Pedley and Kessler (J. Fluid Mech., vol. 208, 1989, pp. 509–543). Finally, the present findings are compared with a recent experiment by Croze, Ashraf and Bees (Phys. Biol., vol. 7, 2010, 046001) and they are in qualitative agreement.

Stability and bifurcations in stratified Taylor–Couette flow

2000 ◽  
Vol 419 ◽  
pp. 93-124 ◽  
Author(s):  
F. CATON ◽  
B. JANIAUD ◽  
E. J. HOPFINGER
Keyword(s):  
Stability Analysis ◽  
Bifurcation Diagram ◽  
Linear Stability ◽  
Couette Flow ◽  
Linear Stability Analysis ◽  
Rotation Rate ◽  
Axisymmetric Flow ◽  
Global Bifurcation ◽  
Internal Gravity Waves ◽  
Fluid Exchange

In this article we present new experimental and theoretical results which were obtained for the flow between two concentric cylinders, with the inner one rotating and in the presence of an axial, stable density stratification. This system is characterized by two control parameters: one destabilizing, the rotation rate of the inner cylinder; and the other stabilizing, the stratification.Two oscillatory linear stability analyses assuming axisymmetric flow conditions are presented. First an eigenmode linear stability analysis is performed, using the small-gap approximation. The solutions obtained give insight into the instability mechanisms and indicate the existence of a confined internal gravity wave mode at the onset of instability. In the second stability analysis, only diffusion is neglected, predicting accurately the instability threshold as well as the critical pulsation for all the stratifications used in the experiments.Experiments show that the basic, purely azimuthal flow (circular Couette flow) is destabilized through a supercritical Hopf bifurcation to an oscillatory flow of confined internal gravity waves, in excellent agreement with the linear stability analysis. The secondary bifurcation, which takes the system to a pattern of drifting non-axisymmetric vortices, is a saddle-node bifurcation. The proposed bifurcation diagram shows a global bifurcation, and explains the discrepancies between previous experimental and numerical results. For slightly larger values of the rotation rate, weakly turbulent spectra are obtained, indicating an early appearance of weak turbulence: stationary structures and defects coexist. Moreover, in this regime, there is a large distribution of structure sizes. Visualizations of the next regime exhibit constant-wavelength structures and fluid exchange between neighbouring cells, similar to wavy vortices. Their existence is explained by a simple energy argument.The generalization of the bifurcation diagram to hydrodynamic systems with one destabilizing and one stabilizing control parameter is discussed. A qualitative argument is derived to discriminate between oscillatory and stationary onset of instability in the general case.

scholarly journals Competition between the centrifugal and strato-rotational instabilities in the stratified Taylor–Couette flow

2018 ◽  
Vol 840 ◽  
pp. 5-24 ◽  
Author(s):  
Junho Park ◽  
Paul Billant ◽  
Jong-Jin Baik ◽  
Jaemyeong Mango Seo
Keyword(s):  
Stability Analysis ◽  
Linear Stability ◽  
Couette Flow ◽  
Linear Stability Analysis ◽  
Reynolds Numbers ◽  
Axisymmetric Mode ◽  
Stability Threshold ◽  
Taylor Couette ◽  
The Stability ◽  
Taylor Couette Flow

The stably stratified Taylor–Couette flow is investigated experimentally and numerically through linear stability analysis. In the experiments, the stability threshold and flow regimes have been mapped over the ranges of outer and inner Reynolds numbers: $-2000<Re_{o}<2000$ and $0<Re_{i}<3000$, for the radius ratio $r_{i}/r_{o}=0.9$ and the Brunt–Väisälä frequency $N\approx 3.2~\text{rad}~\text{s}^{-1}$. The corresponding Froude numbers $F_{o}$ and $F_{i}$ are always much smaller than unity. Depending on $Re_{o}$ (or equivalently on the angular velocity ratio $\unicode[STIX]{x1D707}=\unicode[STIX]{x1D6FA}_{o}/\unicode[STIX]{x1D6FA}_{i}$), three different regimes have been identified above instability onset: a weakly non-axisymmetric mode with low azimuthal wavenumber $m=O(1)$ is observed for $Re_{o}<0$ ($\unicode[STIX]{x1D707}<0$), a highly non-axisymmetric mode with $m\sim 12$ occurs for $Re_{o}>840$ ($\unicode[STIX]{x1D707}>0.57$) while both modes are present simultaneously in the lower and upper parts of the flow for $0\leqslant Re_{o}\leqslant 840$ ($0\leqslant \unicode[STIX]{x1D707}\leqslant 0.57$). The destabilization of these primary modes and the transition to turbulence as $Re_{i}$ increases have been also studied. The linear stability analysis proves that the weakly non-axisymmetric mode is due to the centrifugal instability while the highly non-axisymmetric mode comes from the strato-rotational instability. These two instabilities can be clearly distinguished because of their distinct dominant azimuthal wavenumber and frequency, in agreement with the recent results of Park et al. (J. Fluid Mech., vol. 822, 2017, pp. 80–108). The stability threshold and the characteristics of the primary modes observed in the experiments are in very good agreement with the numerical predictions. Moreover, we show that the centrifugal and strato-rotational instabilities are observed simultaneously for $0\leqslant Re_{o}\leqslant 840$ in the lower and upper parts of the flow, respectively, because of the variations of the local Reynolds numbers along the vertical due to the salinity gradient.

The Instability of Flow Through a Slowly Diverging Pipe With Viscous Heating

2011 ◽  
Vol 133 (7) ◽  
Author(s):  
Kirti Chandra Sahu
Keyword(s):  
Stability Analysis ◽  
Prandtl Number ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Energy Equation ◽  
Pipe Wall ◽  
Negligible Effect ◽  
Viscous Heating ◽  
Fluid Viscosity ◽  
Flow Through

The nonparallel linear stability analysis of flow through a slowly diverging pipe undergoing viscous heating is considered. The pipe wall is maintained at constant temperatures and Nahme’s law is applied to model the temperature dependence of the fluid viscosity. A one-parameter family of velocity profiles for the basic state is obtained for small angles of divergence. The nonparallel stability equations for the disturbance velocity coupled to a linearized energy equation are derived and solved using a spectral collocation method. Our results indicate that increasing viscous heating, characterized by increasing Nahme number, is destabilizing. The Prandtl number has a negligible effect on the linear stability characteristics. The Grashof number stablizes the flow for Gr>106, below which it has a negligible effect.

The effect of viscous heating on the stability of Taylor–Couette flow

2002 ◽  
Vol 462 ◽  
pp. 111-132 ◽  
Author(s):  
U. A. AL-MUBAIYEDH ◽  
R. SURESHKUMAR ◽  
B. KHOMAMI
Keyword(s):  
Stability Analysis ◽  
Secondary Flow ◽  
Linear Stability ◽  
Couette Flow ◽  
Temperature Difference ◽  
Time Dependent ◽  
Viscous Heating ◽  
Taylor Couette ◽  
The Stability ◽  
Taylor Couette Flow

The influence of viscous heating on the stability of Taylor–Couette flow is investigated theoretically. Based on a linear stability analysis it is shown that viscous heating leads to significant destabilization of the Taylor–Couette flow. Specifically, it is shown that in the presence of viscous dissipation the most dangerous disturbances are axisymmetric and that the temporal characteristic of the secondary flow is very sensitive to the thermal boundary conditions. If the temperature difference between the two cylinders is small, the secondary flow is stationary as in the case of isothermal Taylor–Couette flow. However, when the temperature difference between the two cylinders is large, time-dependent secondary states are predicted. These linear stability predictions are in agreement with the experimental observations of White & Muller (2000) in terms of onset conditions as well as the spatiotemporal characteristics of the secondary flow. Nonlinear stability analysis has revealed that over a broad range of operating conditions, the bifurcation to the time-dependent secondary state is subcritical, while stationary states result as a consequence of supercritical bifurcation. Moreover, the supercritically bifurcated stationary state undergoes a secondary bifurcation to a time-dependent flow. Overall, the structure of the time-dependent state predicted by the analysis compares very well with the experimental observations of White & Muller (2000) that correspond to slowly moving vortices parallel to the cylinder axis. The significant destabilization observed in the presence of viscous heating arises as the result of the coupling of the perturbation velocity and the base-state temperature gradient that gives rise to fluctuations in the radial temperature distribution. Due to the thermal sensitivity of the fluid these fluctuations greatly modify the fluid viscosity and reduce the dissipation of disturbances provided by the viscous stress terms in the momentum equation.

Generalized hydrodynamics and linear stability analysis of cylindrical Couette flow of a dilute Lennard–Jones fluid

1993 ◽  
Vol 71 (11-12) ◽  
pp. 518-536 ◽  
Author(s):  
Roger E. Khayat ◽  
Byung Chan Eu
Keyword(s):  
Stability Analysis ◽  
Linear Stability ◽  
Couette Flow ◽  
Linear Stability Analysis ◽  
Hydrodynamic Theory ◽  
Navier Stokes ◽  
Marginal Stability ◽  
Lennard Jones ◽  
Stability Point ◽  
Cylindrical Couette Flow

Linear stability analysis is carried out for cylindrical Couette flow of a Lennard–Jones fluid in the density range from the dense liquid to the dilute gas regime. Generalized hydrodynamic equations are used to calculate marginal stability curves and compare them with those obtained by using the Navier–Stokes–Fourier equations for compressible fluids and also for incompressible fluids. In the low Reynolds or Mach number regime, if the Knudsen number is sufficiently low, the marginal stability curves calculated by the generalized hydrodynamic theory coincide, within numerical errors, with those based on the Navier–Stokes theory. But there are considerable deviations between them in the regimes beyond those mentioned earlier, since nonlinear effects manifest themselves in the laminar mean flow through the nonlinear dissipation term and normal stresses. There are three marginal stability curves obtained in contrast to the Navier–Stokes theory, which yields only two. The previously observed phase-transition-like behavior in fluid variables and the slip phenomenon are found to occur beyond the hydrodynamic stability point. The disturbance entropy production associated with the Taylor–Couette vortices is calculated to first order in disturbances in flow variables and is found to decrease as the number of vortices increases and thereby the dynamic structure is progressively more organized.

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