AMPLITUDE EQUATION IN POLYMERIC FLUID CONVECTION

1994 ◽  
Vol 04 (05) ◽  
pp. 1347-1351 ◽  
Author(s):  
J. MARTÍNEZ-MARDONES ◽  
R. TIEMANN ◽  
W. ZELLER ◽  
C. PÉREZ-GARCÍA
Keyword(s):  
Nonlinear Analysis ◽  
Standing Waves ◽  
Amplitude Equation ◽  
Oscillatory Convection ◽  
Amplitude Equations ◽  
Numerical Techniques ◽  
Polymeric Fluids ◽  
Polymeric Fluid ◽  
Convective Instabilities ◽  
Fluid Convection

The convective instabilities in viscoelastic polymeric Oldroyd-B models are studied. First, the nonlinear analysis of the stationary and oscillatory convection is carried out. Then, in the scope of weak nonlinear analysis, the coefficients of the amplitude equations are evaluated, in order to be in condition to estimate the possible behavior of stationary patterns and also travelling and standing waves. The values of these coefficients are determined by means of analytical and numerical techniques for convection in polymeric fluids.

Interactions between steady and oscillatory convection in mushy layers

2010 ◽  
Vol 645 ◽  
pp. 411-434 ◽  
Author(s):  
PETER GUBA ◽  
M. GRAE WORSTER
Keyword(s):  
Standing Waves ◽  
Mushy Layer ◽  
Two Dimensional ◽  
Oscillatory Convection ◽  
Mushy Layers ◽  
Theoretical Predictions ◽  
Convection Patterns ◽  
The Stability ◽  
Nonlinear Solutions ◽  
Steady Convection

We study nonlinear, two-dimensional convection in a mushy layer during solidification of a binary mixture. We consider a particular limit in which the onset of oscillatory convection just precedes the onset of steady overturning convection, at a prescribed aspect ratio of convection patterns. This asymptotic limit allows us to determine nonlinear solutions analytically. The results provide a complete description of the stability of and transitions between steady and oscillatory convection as functions of the Rayleigh number and the compositional ratio. Of particular focus are the effects of the basic-state asymmetries and non-uniformity in the permeability of the mushy layer, which give rise to abrupt (hysteretic) transitions in the system. We find that the transition between travelling and standing waves, as well as that between standing waves and steady convection, can be hysteretic. The relevance of our theoretical predictions to recent experiments on directionally solidifying mushy layers is also discussed.

Amplitude Equations from Spatiotemporal Binary-Fluid Convection Data

1999 ◽  
Vol 83 (17) ◽  
pp. 3422-3425 ◽  
Author(s):  
Henning U. Voss ◽  
Paul Kolodner ◽  
Markus Abel ◽  
Jürgen Kurths
Keyword(s):  
Amplitude Equations ◽  
Binary Fluid ◽  
Fluid Convection

Flow Characteristics of Microglass Fiber Suspension in Polymeric Fluids in Spherical Gaps

2010 ◽  
Vol 132 (5) ◽  
Author(s):  
Hiroshi Yamaguchi ◽  
Xin-Rong Zhang ◽  
Xiao-Dong Niu ◽  
Yuta Ito
Keyword(s):  
Reynolds Number ◽  
Couette Flow ◽  
Flow Characteristics ◽  
Flow Region ◽  
Fiber Suspension ◽  
Polymeric Fluids ◽  
Polymeric Fluid ◽  
Aspect Ratios ◽  
Gap Ratio ◽  
Microglass Fibers

An experimental study is carried out to investigate the effects of microglass fiber suspensions in the non-Newtonian fluids in a gap between an inner rotating sphere and an outer whole stationary sphere. In the experiments, the microglass fibers with different aspect ratios are mixed with a macromolecule polymeric fluid to obtain different suspension fluids. For comparison, a Newtonian fluid and the non-Newtonian polymeric fluid are also studied. The stationary torques of the inner sphere that the test fluids acted on are measured under conditions of various concentric spherical gaps and rotational Reynolds numbers. It is found that the polymeric fluid could be governed by the Couette flow at a gap ratio of less than 0.42 and the Reynolds number of less than 100, while the fiber-suspended polymeric fluids could expand the Couette flow region more than the Reynolds number of 100 at the same gap ratios.

A New Mechanical Algorithm for Calculating the Amplitude Equation of the Reaction-Diffusion Systems

2012 ◽  
Vol 3 (4) ◽  
pp. 21-28
Author(s):  
Houye Liu ◽  
Weiming Wang
Keyword(s):  
Hopf Bifurcation ◽  
Normal Form ◽  
Reaction Diffusion ◽  
Amplitude Equation ◽  
Diffusion Reaction ◽  
Amplitude Equations ◽  
Future Studies ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Form Approach

Amplitude equation may be used to study pattern formatio. In this article, the authors establish a new mechanical algorithm AE_Hopf for calculating the amplitude equation near Hopf bifurcation based on the method of normal form approach in Maple. The normal form approach needs a large number of variables and intricate calculations. As a result, deriving the amplitude equation from diffusion-reaction is a difficult task. Making use of our mechanical algorithm, we derived the amplitude equations from several biology and physics models. The results indicate that the algorithm is easy to apply and effective. This algorithm may be useful for learning the dynamics of pattern formation of reaction-diffusion systems in future studies.

Absolute and Convective Instabilities of a Collapsible Tube

2002 ◽  
Author(s):  
Hamadiche Mahmoud
Keyword(s):  
Reynolds Number ◽  
Unstable Mode ◽  
Standing Waves ◽  
Cusp Point ◽  
Collapsible Tube ◽  
Convective Instabilities ◽  
Cusp Points ◽  
The Absolute ◽  
High Reynolds ◽  
Good Agreement

In this paper we extend a numerical method, developed previously by the author, to compute the eigen modes of collapsible viscoelastic duct convening a fowing fuid. A new technique is developed in order to eliminate the need for recurrence formula used in the old method. This gives a more powerful and tractable method to find the eigen modes of the system. The new method has allowed the identification of a new unstable modes in a collapsible tube. It is found that there is a set of standing non axisymmetric waves representing an absolute instabilities and a set of unstable upstream and downstream propagated waves representing a convective instabilities. Two standing waves have equal frequency in their cusp points. The frequency of the other standing waves, in their cusp points, are a multiple of the frequency of the first wave and that in good agreement with experimental finding available in the literature. It is found that the first absolute unstable mode becomes convective at high Reynolds number while the other standing wave remain absolutely unstable modes for Re higher than 100. The frequency ratio of the absolute unstable modes in their cusp point are preserved for all Reynolds number and that with an good agreement with the experience. The absolute unstable mode which becomes convective at high Reynolds number keeps a monochromatic wave with a frequency equal to the frequency of the second absolute unstable mode, in its cusp point, and that for all Reynolds number higher than the limit of absolute instability of the first mode in surprising agreement with the experimental results. It is founded that the viscosity of the solid stabilizes the standing wavesat different degree. The boundary separating the absolute instabilities zone from the convective instabilities zone are found.

BIFURCATION ANALYSIS OF AN AMPLITUDE EQUATION

2013 ◽  
Vol 23 (05) ◽  
pp. 1350081
Author(s):  
LEI SHI ◽  
HONGJUN GAO
Keyword(s):  
Neumann Boundary ◽  
Stationary Solutions ◽  
Amplitude Equation ◽  
Parabolic Partial Differential Equations ◽  
Bifurcation Parameter ◽  
Local Behavior ◽  
Nontrivial Solutions ◽  
Amplitude Equations ◽  
The Stability ◽  
Bifurcation And Stability

We study the bifurcation and stability of constant stationary solutions (u0, v0) of a particular system of parabolic partial differential equations as amplitude equations on a bounded domain (0, L) with Neumann boundary conditions. In this paper, the asymptotic behavior of the solutions (u0, v0) of the amplitude equations are considered. With the length L of the domain regarded as bifurcation parameter, branches of nontrivial solutions are shown by using the perturbation method. Moreover, local behavior of these branches are studied. We also analyze the stability of the bifurcated solutions.

scholarly journals Instability thresholds for thermal convection in a Kelvin–Voigt fluid of variable order

2021 ◽  
Author(s):  
B. Straughan
Keyword(s):  
Thermal Convection ◽  
Navier Stokes ◽  
Memory Term ◽  
Variable Order ◽  
Fading Memory ◽  
Oscillatory Convection ◽  
Numerical Techniques ◽  
Order Zero ◽  
Neutral Curves ◽  
Voigt Fluid

AbstractWe present numerical techniques for calculating instability thresholds in a model for thermal convection in a complex viscoelastic fluid of Kelvin–Voigt type. The theory presented is valid for various orders of an exponential fading memory term, and the strategy for obtaining the neutral curves and instability thresholds is discussed in the general case. Specific numerical results are presented for a fluid of order zero, also known as a Navier–Stokes–Voigt fluid, and fluids of order 1 and 2. For the latter cases it is shown that oscillatory convection may occur, and the nature of the stationary and oscillatory convection branches is investigated in detail, including where the transition from one to the other takes place.

scholarly journals Travelling and standing waves in magnetoconvection

1993 ◽  
Vol 441 (1913) ◽  
pp. 649-658 ◽  
Keyword(s):  
Numerical Solutions ◽  
Travelling Waves ◽  
Standing Waves ◽  
Periodic Boundary Conditions ◽  
Perturbation Techniques ◽  
Oscillatory Convection ◽  
Weakly Nonlinear ◽  
Modulated Waves ◽  
Steady Shearing ◽  
Rayleigh Numbers

The problem of Boussinesq magnetoconvection with periodic boundary conditions is studied using standard perturbation techniques. It is found that either travelling waves or standing waves can be stable at the onset of oscillatory convection, depending on the parameters of the problem. When travelling waves occur, a steady shearing flow is present that is quadratic in the amplitude of the convective flow. The weakly nonlinear predictions are confirmed by comparison with numerical solutions of the full partial differential equations at Rayleigh numbers 10% above critical. Modulated waves (through which stability is transferred between travelling and standing waves) are found near the boundary between the regions in parameter space where travelling waves and standing waves are preferred.

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