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 Weakly Nonlinear Magnetic Convection in a Nonuniformly Rotating Electrically Conductive Medium Under the Action of Modulation of External Fields

2020 ◽  
Keyword(s):  
Magnetic Field ◽  
Angular Velocity ◽  
Landau Equation ◽  
Temperature Modulation ◽  
External Fields ◽  
Ginzburg Landau ◽  
Electrically Conductive ◽  
Weakly Nonlinear ◽  
Conductive Fluid ◽  
Ginzburg Landau Equations

In this paper we studied the weakly nonlinear stage of stationary convective instability in a nonuniformly rotating layer of an electrically conductive fluid in an axial uniform magnetic field under the influence of: a) temperature modulation of the layer boundaries; b) gravitational modulation; c) modulation of the magnetic field; d) modulation of the angular velocity of rotation. As a result of applying the method of perturbation theory for the small parameter of supercriticality of the stationary Rayleigh number nonlinear non-autonomous Ginzburg-Landau equations for the above types of modulation were obtaned. By utilizing the solution of the Ginzburg-Landau equation, we determined the dynamics of unsteady heat transfer for various types of modulation of external fields and for different profiles of the angular velocity of the rotation of electrically conductive fluid.

Hydromagnetic Stability of a Thin Viscoelastic Magnetic Fluid on Coating Flow Using Landau Equation

2016 ◽  
Vol 32 (5) ◽  
pp. 643-651 ◽  
Author(s):  
C.-K. Chen ◽  
M.-C. Lin
Keyword(s):  
Viscoelastic Fluid ◽  
Nonlinear Stability ◽  
Film Flow ◽  
Landau Equation ◽  
Ginzburg Landau ◽  
Viscoelastic Parameter ◽  
Weakly Nonlinear ◽  
Linear Mode ◽  
Coating Flow ◽  
The One

AbstractThis paper investigates the weakly nonlinear stability of a thin axisymmetric viscoelastic fluid with hydromagnetic effects on coating flow. The governing equation is resolved using long-wave perturbation method as part of an initial value problem for spatial periodic surface waves with the Walter's liquid B type fluid. The most unstable linear mode of a film flow is determined by Ginzburg-Landau equation (GLE). The coefficients of the GLE are calculated numerically from the solution of the corresponding stability problem on coating flow. The effect of a viscoelastic fluid under an applied magnetic field on the nonlinear stability mechanism is studied in terms of the rotation number, Ro, viscoelastic parameter, k, and the Hartmann constant, m. Modeling results indicate that the Ro, k and m parameters strongly affect the film flow. Enhancing the magnetic effects is found to stabilize the film flow when the viscoelastic parameter destabilizes the one in a thin viscoelastic fluid.

Weakly Nonlinear Oscillatory Convection in a Rotating Fluid Layer Under Temperature Modulation

2016 ◽  
Vol 138 (5) ◽  
Author(s):  
Palle Kiran ◽  
B. S. Bhadauria
Keyword(s):  
Heat Transport ◽  
Fluid Layer ◽  
Landau Equation ◽  
Complex Form ◽  
Oscillatory Mode ◽  
Temperature Modulation ◽  
Rotating Fluid ◽  
Nonlinear Stability Analysis ◽  
Ginzburg Landau ◽  
Weakly Nonlinear

A study of thermal instability driven by buoyancy force is carried out in an initially quiescent infinitely extended horizontal rotating fluid layer. The temperature at the boundaries has been taken to be time-periodic, governed by the sinusoidal function. A weakly nonlinear stability analysis has been performed for the oscillatory mode of convection, and heat transport in terms of the Nusselt number, which is governed by the complex form of Ginzburg–Landau equation (CGLE), is calculated. The influence of external controlling parameters such as amplitude and frequency of modulation on heat transfer has been investigated. The dual effect of rotation on the system for the oscillatory mode of convection is found either to stabilize or destabilize the system. The study establishes that heat transport can be controlled effectively by a mechanism that is external to the system. Further, the bifurcation analysis also presented and established that CGLE possesses the supercritical bifurcation.

Throughflow and G-Jitter Effects on Oscillatory Convection in a Rotating Porous Medium

2020 ◽  
Vol 12 (6) ◽  
pp. 781-791
Author(s):  
S. H. Manjula ◽  
Palle Kiran ◽  
B. S. Bhadauria
Keyword(s):  
Heat Transfer ◽  
Porous Medium ◽  
Heat Transport ◽  
Landau Equation ◽  
Nonlinear Stability Analysis ◽  
Periodic Mode ◽  
Ginzburg Landau ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Analysis ◽  
The Impact

The impact of vertical throughflow and g-jitter effect on rotating porous medium is investigated. A feeble nonlinear stability analysis associate to complex Ginzburg-Landau equation (CGLE) has been studied. This weakly nonlinear analysis performed for a periodic mode of convection and quantified heat transport in terms of the Nusselt number, which is governed by the non-autonomous advanced CGLE. Each idea, rotation and throughflow is used as an external mechanism to the system either to extend or decrease the heat transfer. The results of amplitude and frequency of modulation on heat transport are analyzed and portrayed graphically. Throughflow has dual impact on heat transfer either to increase or decrease heat transfer in the system. Particularly the outflow enhances and inflow diminishes the heat transfer. High centrifugal rates promote heat transfer and low centrifugal rates diminish heat transfer. The streamlines and isotherms area portrayed graphically, the results of rotation and throughflow on isotherms shows convective development.

A quasi-steady approach to the instability of time-dependent flows in pipes

2002 ◽  
Vol 465 ◽  
pp. 301-330 ◽  
Author(s):  
M. S. GHIDAOUI ◽  
A. A. KOLYSHKIN
Keyword(s):  
Experimental Data ◽  
Laminar Flow ◽  
Asymptotic Expansions ◽  
Unstable Mode ◽  
Landau Equation ◽  
Base Flow ◽  
Neutral Stability ◽  
Weakly Nonlinear ◽  
One Dimensional ◽  
Weakly Nonlinear Analysis

Asymptotic solutions for unsteady one-dimensional axisymmetric laminar flow in a pipe subject to rapid deceleration and/or acceleration are derived and their stability investigated using linear and weakly nonlinear analysis. In particular, base flow solutions for unsteady one-dimensional axisymmetric laminar flow in a pipe are derived by the method of matched asymptotic expansions. The solutions are valid for short times and can be successfully applied to the case of an arbitrary (but unidirectional) axisymmetric initial velocity distribution. Excellent agreement between asymptotic and analytical solutions for the case of an instantaneous pipe blockage is found for small time intervals. Linear stability of the base flow solutions obtained from the asymptotic expansions to a three-dimensional perturbation is investigated and the results are used to re-interpret the experimental results of Das & Arakeri (1998). Comparison of the neutral stability curves computed with and without the planar channel assumption shows that this assumption is accurate when the ratio of the unsteady boundary layer thickness to radius (i.e. δ1/R) is small but becomes unacceptable when this ratio exceeds 0.3. Both the current analysis and the experiments show that the flow instability is non-axisymmetric for δ1/R = 0.55 and 0.85. In addition, when δ1/R = 0.18 and 0.39, the neutral stability curves for n = 0 and n = 1 are found to be close to one another at all times but the most unstable mode in these two cases is the axisymmetric mode. The accuracy of the quasi-steady assumption, employed both in this research and in that of Das & Arakeri (1998), is supported by the fact that the results obtained under this assumption show satisfactory agreement with the experimental features such as type of instability and spacing between vortices. In addition, the computations show that the ratio of the rate of growth of perturbations to the rate of change of the base flow is much larger than 1 for all cases considered, providing further support for the quasi-steady assumption. The neutral stability curves obtained from linear stability analysis suggest that a weakly nonlinear approach can be used in order to study further development of instability. Weakly nonlinear analysis shows that the amplitude of the most unstable mode is governed by the complex Ginzburg–Landau equation which reduces to the Landau equation if the amplitude is a function of time only. The coefficients of the Landau equation are calculated for two cases of the experimental data given by Das & Arakeri (1998). It is shown that the real part of the Landau constant is positive in both cases. Therefore, finite-amplitude equilibrium is possible. These results are in qualitative agreement with experimental data of Das & Arakeri (1998).

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.

scholarly journals The Complex Ginzburg Landau Model for an Oscillatory Convection in a Rotating Fluid Layer

2020 ◽  
Vol 25 (1) ◽  
pp. 75-91
Author(s):  
S.H. Manjula ◽  
P. Kiran ◽  
P. Raj Reddy ◽  
B.S. Bhadauria
Keyword(s):  
Heat Transfer ◽  
Thermal Instability ◽  
Fluid Layer ◽  
Landau Equation ◽  
Finite Amplitude ◽  
Oscillatory Mode ◽  
Stationary Mode ◽  
Ginzburg Landau ◽  
Weakly Nonlinear ◽  
Speed Modulation

AbstractA weakly nonlinear thermal instability is investigated under rotation speed modulation. Using the perturbation analysis, a nonlinear physical model is simplified to determine the convective amplitude for oscillatory mode. A non-autonomous complex Ginzburg-Landau equation for the finite amplitude of convection is derived based on a small perturbed parameter. The effect of rotation is found either to stabilize or destabilize the system. The Nusselt number is obtained numerically to present the results of heat transfer. It is found that modulation has a significant effect on heat transport for lower values of ωf while no effect for higher values. It is also found that modulation can be used alternately to control the heat transfer in the system. Further, oscillatory mode enhances heat transfer rather than stationary mode.

scholarly journals Study of Weakly nonlinear Mass transport in Newtonian Fluid with Applied Magnetic Field under Concentration/Gravity modulation

2019 ◽  
Vol 8 (1) ◽  
pp. 513-522 ◽  
Author(s):  
Om Prakash Keshri ◽  
Vinod K. Gupta ◽  
Anand Kumar
Keyword(s):  
Mass Transport ◽  
Fluid Layer ◽  
Landau Equation ◽  
Periodic Part ◽  
Gravity Modulation ◽  
Nonlinear Stability Analysis ◽  
Ginzburg Landau ◽  
Weakly Nonlinear ◽  
Mass Transfer Effect ◽  
Time Periodic

Abstract In the present paper, a weakly nonlinear stability analysis is used to analyze the effect of time-periodic concentration/gravity modulation on mass transport. We have considered an infinite horizontal fluid layer with constant appliedmagnetic flux salted from above, subjected to an imposed time-periodic boundary concentration (ITBC) or gravity modulation (ITGM). In the case of ITBC, the concentration gradient between the plates of the fluid layer consists of a steady part and a time-dependent oscillatory part. The concentration of both walls is modulated. In the case of ITGM, the gravity fleld consists of two parts: a constant part and an externally imposed time periodic part, which can be realized by oscillating the fluid layer. We have expanded the infinitesimal disturbances in terms of power series of an amplitude of modulation, which is assumed to be small. Ginzburg-Landau equation is derived for dinding the rate of mass transfer. Effect of various parameters on the mass transport is also discussed. It is found that the mass transport can be controlled by suitably adjusting the frequency and amplitude of modulation.

Dynamic bifurcations and pattern formation in melting-boundary convection

2011 ◽  
Vol 686 ◽  
pp. 77-108 ◽  
Author(s):  
G. M. Vasil ◽  
M. R. E. Proctor
Keyword(s):  
Pattern Formation ◽  
General Feature ◽  
Fluid Layer ◽  
Landau Equation ◽  
Ginzburg Landau ◽  
Nonlinear Convection ◽  
Weakly Nonlinear ◽  
Dynamic Bifurcation ◽  
Dynamic Bifurcations ◽  
New System

AbstractWe consider weakly nonlinear convection in a fluid layer with a melting top boundary. This leads us to derive a new set of non-autonomous envelope equations as a dynamic generalization to the well-known Ginzburg–Landau equation. However, this new system possesses a number of interesting properties not found in systems close to a traditional dynamic bifurcation, because it involves the interaction of two destabilizing mechanisms. We investigate the system both analytically and numerically; specifically, we find the robust ‘locking in’ of spatially complex patterns, and show this is a general feature of systems of this nature.

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