Stability of plane Couette flow of a power-law fluid past a neo-Hookean solid at arbitrary Reynolds number

2017 ◽  
Vol 29 (7) ◽  
pp. 074106 ◽  
Author(s):  
D. Giribabu ◽  
V. Shankar
Keyword(s):  
Reynolds Number ◽  
Couette Flow ◽  
Power Law ◽  
Plane Couette Flow ◽  
Power Law Fluid

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

scholarly journals Non-modal instability in plane Couette flow of a power-law fluid

2011 ◽  
Vol 676 ◽  
pp. 145-171 ◽  
Author(s):  
R. LIU ◽  
Q. S. LIU
Keyword(s):  
Couette Flow ◽  
Power Law ◽  
Energy Method ◽  
Stability Theory ◽  
Initial Conditions ◽  
Shear Thinning ◽  
Plane Couette Flow ◽  
Power Law Fluid ◽  
The Stability ◽  
Thinning Effect

In this paper, we study the linear stability of a plane Couette flow of a power-law fluid. The influence of shear-thinning effect on the stability is investigated using the classical eigenvalue analysis, the energy method and the non-modal stability theory. For the plane Couette flow, there is no stratification of viscosity. Thus, for the stability problem the stress tensor is anisotropic aligned with the strain rate perturbation. The results of the eigenvalue analysis and the energy method show that the shear-thinning effect is destabilizing. We focus on the effect of non-Newtonian viscosity on the transition from laminar flow towards turbulence in the framework of non-modal stability theory. Response to external excitations and initial conditions has been studied by examining the ε-pseudospectrum and the transient energy growth. For both Newtonian and non-Newtonian fluids, it is found that there can be a rather large transient growth even though the linear operator of the Couette flow has no unstable eigenvalue. The results show that shear-thinning significantly increases the amplitude of response to external excitations and initial conditions.

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).

Onset of Instability of Power-Law Fluid in Couette Flow

1971 ◽  
Vol 97 (1) ◽  
pp. 1-12
Author(s):  
Yun-Sheng Yu ◽  
Dan B. McVickar
Keyword(s):  
Couette Flow ◽  
Power Law ◽  
Power Law Fluid

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 A doubly localized equilibrium solution of plane Couette flow

2014 ◽  
Vol 750 ◽  
Author(s):  
E. Brand ◽  
J. F. Gibson
Keyword(s):  
Reynolds Number ◽  
Couette Flow ◽  
Equilibrium Solution ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Edge States ◽  
Plane Couette Flow ◽  
Turbulent Spot ◽  
Navier Stokes Equations ◽  
Exponential Decay Rate

AbstractWe present an equilibrium solution of plane Couette flow that is exponentially localized in both the spanwise and streamwise directions. The solution is similar in size and structure to previously computed turbulent spots and localized, chaotically wandering edge states of plane Couette flow. A linear analysis of dominant terms in the Navier–Stokes equations shows how the exponential decay rate and the wall-normal overhang profile of the streamwise tails are governed by the Reynolds number and the dominant spanwise wavenumber. Perturbations of the solution along its leading eigenfunctions cause rapid disruption of the interior roll-streak structure and formation of a turbulent spot, whose growth or decay depends on the Reynolds number and the choice of perturbation.

A resonance mechanism in plane Couette flow

1980 ◽  
Vol 98 (1) ◽  
pp. 149-159 ◽  
Author(s):  
L. HÅKan Gustavsson ◽  
Lennart S. Hultgren
Keyword(s):  
Reynolds Number ◽  
Couette Flow ◽  
Velocity Component ◽  
Three Dimensional ◽  
Phase Speed ◽  
Maximum Amplitude ◽  
Streamwise Velocity ◽  
Plane Couette Flow ◽  
Mean Velocity ◽  
Streamwise Velocity Component

The temporal evolution of small three-dimensional disturbances on viscous flows between parallel walls is studied. The initial-value problem is formally solved by using Fourier–Laplace transform techniques. The streamwise velocity component is obtained as the solution of a forced problem. As a consequence of the three-dimensionality, a resonant response is possible, leading to algebraic growth for small times. It occurs when the eigenvalues of the Orr–Sommerfeld equation coincide with the eigenvalues of the homogeneous operator for the streamwise velocity component. The resonance has been investigated numerically for plane Couette flow. The phase speed of the resonant waves equals the average mean velocity. The wavenumber combination that leads to the largest amplitude corresponds to structures highly elongated in the streamwise direction. The maximum amplitude, and the time to reach this maximum, scale with the Reynolds number. The aspect ratio of the most rapidly growing wave increases with the Reynolds number, with its spanwise wavelength approaching a constant value of about 3 channel heights.

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