Effects of unrelaxed stress tension on the weakly nonlinear instability of viscoelastic sheets

2016 ◽  
Vol 28 (10) ◽  
pp. 104104 ◽  
Author(s):  
Luo Xie ◽  
Li-Jun Yang ◽  
Qing-Fei Fu ◽  
Li-Zi Qin
Keyword(s):  
Nonlinear Instability ◽  
Weakly Nonlinear

Weakly nonlinear instability of planar viscoelastic sheets

2015 ◽  
Vol 27 (1) ◽  
pp. 013103 ◽  
Author(s):  
Chen Wang ◽  
Li-jun Yang ◽  
Luo Xie ◽  
Pi-min Chen
Keyword(s):  
Nonlinear Instability ◽  
Weakly Nonlinear

Nonlinear instability of a contact line driven by gravity

2000 ◽  
Vol 413 ◽  
pp. 355-378 ◽  
Author(s):  
SERAFIM KALLIADASIS
Keyword(s):  
Thin Film ◽  
Differential Equation ◽  
Partial Differential Equation ◽  
Dynamical Systems ◽  
Contact Line ◽  
Nonlinear Instability ◽  
Inclined Plane ◽  
Translational Invariance ◽  
Weakly Nonlinear ◽  
Fingering Instability

A thin liquid mass of fixed volume spreading under the action of gravity on an inclined plane develops a fingering instability at the front. In this study we consider the motion of a viscous sheet down a pre-wetted plane with a large inclination angle. We demonstrate that the instability is a phase instability associated with the translational invariance of the system in the direction of flow and we analyse the weakly nonlinear regime of the instability by utilizing methods from dynamical systems theory. It is shown that the evolution of the fingers is governed by a Kuramoto–Sivashinsky-type partial differential equation with solution a saw-tooth pattern when the inclined plane is pre-wetted with a thin film, while the presence of a thick film suppresses fingering.

scholarly journals Linear and Weakly Nonlinear Instability of Shallow Mixing Layers with Variable Friction

2018 ◽  
Vol 2018 ◽  
pp. 1-10
Author(s):  
Irina Eglite ◽  
Andrei Kolyshkin
Keyword(s):  
Unstable Mode ◽  
Landau Equation ◽  
Main Channel ◽  
Mixing Layers ◽  
Resistance Force ◽  
Nonlinear Instability ◽  
Ginzburg Landau ◽  
Weakly Nonlinear ◽  
Variable Friction ◽  
Linear Stability Problem

Linear and weakly nonlinear instability of shallow mixing layers is analysed in the present paper. It is assumed that the resistance force varies in the transverse direction. Linear stability problem is solved numerically using collocation method. It is shown that the increase in the ratio of the friction coefficients in the main channel to that in the floodplain has a stabilizing influence on the flow. The amplitude evolution equation for the most unstable mode (the complex Ginzburg–Landau equation) is derived from the shallow water equations under the rigid-lid assumption. Results of numerical calculations are presented.

scholarly journals Nonlinear development of subsonic modes on compressible mixing layers: a unified strongly nonlinear critical-layer theory

2008 ◽  
Vol 614 ◽  
pp. 105-144 ◽  
Author(s):  
CLIFFORD A. SPARKS ◽  
XUESONG WU
Keyword(s):  
Reynolds Numbers ◽  
Mixing Layers ◽  
Critical Layer ◽  
Instability Wave ◽  
Nonlinear Instability ◽  
Weakly Nonlinear ◽  
Strongly Nonlinear ◽  
High Reynolds Numbers ◽  
Pole Singularity ◽  
High Reynolds

This paper is concerned with the nonlinear instability of compressible mixing layers in the regime of small to moderate values of Mach numberM, in which subsonic modes play a dominant role. At high Reynolds numbers of practical interest, previous studies have shown that the dominant nonlinear effect controlling the evolution of an instability wave comes from the so-called critical layer. In the incompressible limit (M= 0), the critical-layer dynamics are strongly nonlinear, with the nonlinearity being associated with the logarithmic singularity of the velocity fluctuation (Goldstein & Leib,J. Fluid Mech.vol. 191, 1988, p. 481). In contrast, in the fully compressible regime (M=O(1)), nonlinearity is associated with a simple-pole singularity in the temperature fluctuation and enters in a weakly nonlinear fashion (Goldstein & Leib,J. Fluid Mech.vol. 207, 1989, p. 73). In this paper, we first consider a weakly compressible regime, corresponding to the distinguished scalingM=O(ε1/4), for which the strongly nonlinear structure persists but is affected by compressibility at leading order (where ε ≪ 1 measures the magnitude of the instability mode). A strongly nonlinear system governing the development of the vorticity and temperature perturbation is derived. It is further noted that the strength of the pole singularity is controlled byT′c, the mean temperature gradient at the critical level, and for typical base-flow profilesT′cis small even whenM=O(1). By treatingT′cas an independent parameter ofO(ε1/2), we construct a composite strongly nonlinear theory, from which the weakly nonlinear result forM=O(1) can be derived as an appropriate limiting case. Thus the strongly nonlinear formulation is uniformly valid forO(1) Mach numbers. Numerical solutions show that this theory captures the vortex roll-up process, which remains the most prominent feature of compressible mixing-layer transition. The theory offers an effective tool for investigating the nonlinear instability of mixing layers at high Reynolds numbers.

Weakly Nonlinear Instability of a Liquid Jet in a Viscous Gas

1992 ◽  
Vol 59 (2S) ◽  
pp. S291-S296 ◽  
Author(s):  
E. A. Ibrahim ◽  
S. P. Lin
Keyword(s):  
Growth Rate ◽  
Normal Mode ◽  
Interfacial Stress ◽  
Linear Stability Theory ◽  
Liquid Jet ◽  
Nonlinear Instability ◽  
Weakly Nonlinear ◽  
Viscous Gas ◽  
Jet Breakup ◽  
Stress Fluctuation

The weakly nonlinear instability of a viscous liquid jet emanated into a viscous gas contained in a coaxial vertical circular pipe is investigated as an initial-value problem. The linear stability theory predicted that the jet may become unstable either due to capillary pinching or due to interfacial stress fluctuation. The results of nonlinear stability analysis shows no tendency of supercritical stability for both of the linearly unstable modes. In fact, the nonlinear growth rate of the disturbance is faster than the exponential growth rate of the linear normal mode disturbance for the same flow parameters. Moreover, the most amplified linear normal mode disturbance evolves nonlinearly into a nonsinusoidal wave of shorter wavelength. No nonlinear instability is found for the linearly stable disturbances. Thus, while the linear theory is adequate for the prediction of the onset of jet breakup, nonlinear theory is required to describe the outcome of the jet breakup.

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