The stability of plane Couette flow with viscous heating

1973 ◽  
Vol 57 (4) ◽  
pp. 651-670 ◽  
Author(s):  
Peter C. Sukanek ◽  
Charles A. Goldstein ◽  
Robert L. Laurence
Keyword(s):  
Couette Flow ◽  
Maximum Shear Stress ◽  
Negative Real Part ◽  
Navier Stokes ◽  
Exponential Dependence ◽  
Viscous Heating ◽  
Thermal Mode ◽  
Neutral Stability ◽  
Plane Couette Flow ◽  
The Stability

An investigation of the stability of plane Couette flow with viscous heating of a Navier–Stokes–Pourier fluid with an exponential dependence of viscosity upon temperature is presented. Using classical small perturbation theory, the stability of the flow can be described by a sixth-order set of coupled ordinary differential equations. Using Galerkin's method, these equations are reduced to an algebraic eigenvalue problem. An eigenvalue with a negative real part means that the flow is unstable.Neutral stability curves are determined at Brinkman numbers of 15, 19, 25, 30,40,80 and 600 for Prandtl numbers of 1, s and 50. A Brinkman number of 19 corresponds approximately to the maximum shear stress which can be applied to the system.The results indicate that four different modes of instability occur: one termed an inviscid mode, arising from an inflexion point in the primary flow; a viscous mode, due to the stratification of viscosity in the flow field and an associated diffusive mechanism; a coupling mode, resulting from the convective and viscous dissipation terms in the energy equation; and finally a purely thermal mode.

The stability of plane Couette flow of a power-law fluid with viscous heating

2007 ◽  
Vol 19 (9) ◽  
pp. 094107 ◽  
Author(s):  
Nabil T. M. Eldabe ◽  
M. F. El-Sabbagh ◽  
M. A.-S. El-Sayed(Hajjaj)
Keyword(s):  
Couette Flow ◽  
Power Law ◽  
Viscous Heating ◽  
Plane Couette Flow ◽  
Power Law Fluid ◽  
The Stability

On the behaviour of small disturbances in plane Couette flow with a temperature gradient

1965 ◽  
Vol 286 (1404) ◽  
pp. 117-128 ◽  
Keyword(s):  
Rayleigh Number ◽  
Couette Flow ◽  
Reynolds Numbers ◽  
Neutral Stability ◽  
Two Dimensional ◽  
Plane Couette Flow ◽  
Infinitesimal Disturbance ◽  
Prandtl Numbers ◽  
The Stability ◽  
Small Disturbances

The stability of plane Couette flow with a heated lower plate is considered with respect to a two-dimensional infinitesimal disturbance. The eigenvalues are found with the aid of a digital computer as the latent roots of a matrix. Neutral stability curves for various Prandtl numbers at Reynolds numbers up to 150 are obtained by a second method. It is found that the principle of the exchange of stabilities does not hold for this problem. With the aid of Squire’s transformation the conclusion is drawn that all fluids will become unstable at the same value of the Rayleigh number irrespective of whether shear is present or not.

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 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.

scholarly journals The linear instability of the stratified plane Couette flow

2018 ◽  
Vol 853 ◽  
pp. 205-234 ◽  
Author(s):  
Giulio Facchini ◽  
Benjamin Favier ◽  
Patrice Le Gal ◽  
Meng Wang ◽  
Michael Le Bars
Keyword(s):  
Reynolds Number ◽  
Stability Analysis ◽  
Numerical Simulations ◽  
Couette Flow ◽  
Vertical Direction ◽  
Stability Diagram ◽  
Linear Instability ◽  
Plane Couette Flow ◽  
Horizontal Shear ◽  
The Stability

We present the stability analysis of a plane Couette flow which is stably stratified in the vertical direction orthogonal to the horizontal shear. Interest in such a flow comes from geophysical and astrophysical applications where background shear and vertical stable stratification commonly coexist. We perform the linear stability analysis of the flow in a domain which is periodic in the streamwise and vertical directions and confined in the cross-stream direction. The stability diagram is constructed as a function of the Reynolds number $Re$ and the Froude number $Fr$, which compares the importance of shear and stratification. We find that the flow becomes unstable when shear and stratification are of the same order (i.e. $Fr\sim 1$) and above a moderate value of the Reynolds number $Re\gtrsim 700$. The instability results from a wave resonance mechanism already known in the context of channel flows – for instance, unstratified plane Couette flow in the shallow-water approximation. The result is confirmed by fully nonlinear direct numerical simulations and, to the best of our knowledge, constitutes the first evidence of linear instability in a vertically stratified plane Couette flow. We also report the study of a laboratory flow generated by a transparent belt entrained by two vertical cylinders and immersed in a tank filled with salty water, linearly stratified in density. We observe the emergence of a robust spatio-temporal pattern close to the threshold values of $Fr$ and $Re$ indicated by linear analysis, and explore the accessible part of the stability diagram. With the support of numerical simulations we conclude that the observed pattern is a signature of the same instability predicted by the linear theory, although slightly modified due to streamwise confinement.

scholarly journals Visualizing the geometry of state space in plane Couette flow

2008 ◽  
Vol 611 ◽  
pp. 107-130 ◽  
Author(s):  
J. F. GIBSON ◽  
J. HALCROW ◽  
P. CVITANOVIĆ
Keyword(s):  
State Space ◽  
Couette Flow ◽  
Invariant Manifolds ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Space Representation ◽  
Plane Couette Flow ◽  
Periodic Cell ◽  
State Space Representation ◽  
Dimensional State Space

Motivated by recent experimental and numerical studies of coherent structures in wall-bounded shear flows, we initiate a systematic exploration of the hierarchy of unstable invariant solutions of the Navier–Stokes equations. We construct a dynamical 105-dimensional state-space representation of plane Couette flow at Reynolds number Re = 400 in a small periodic cell and offer a new method of visualizing invariant manifolds embedded in such high dimensions. We compute a new equilibrium solution of plane Couette flow and the leading eigenvalues and eigenfunctions of known equilibria at this Re and cell size. What emerges from global continuations of their unstable manifolds is a surprisingly elegant dynamical-systems visualization of moderate-Re turbulence. The invariant manifolds partially tessellate the region of state space explored by transiently turbulent dynamics with a rigid web of symmetry-induced heteroclinic connections.

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