A WEAKLY NONLINEAR INSTABILITY OF MISCIBLE DISPLACEMENTS IN A RECTILINEAR POROUS MEDIA FLOW

1994 ◽  
Vol 04 (05) ◽  
pp. 1319-1328 ◽  
Author(s):  
WILLIAM B. ZIMMERMAN
Keyword(s):  
Linear Theory ◽  
Landau Equation ◽  
Separation Of Variables ◽  
Finite Amplitude ◽  
Linear Stability Theory ◽  
Porous Media Flow ◽  
Perturbation Scheme ◽  
Viscous Effects ◽  
Regular Perturbation ◽  
Displacement Front

The linear stability theory of Tan & Homsy [1986] is extended to include the effects of weak nonlinear coupling between mass flux and viscous effects when the viscous fingers grow from a slowly diffusing, nearly flat displacement front. A regular perturbation scheme combined with a similarity-separation of variables technique leads to a Landau equation for the amplitude of the disturbance. The Landau constant has a simple pole for a given wavenumber within the linear theory cutoff wavenumber for growth. An argument is given that this pole leads to pairing of fingers while the instability remains small. Comparison of the length scale of the pole of the Landau constant with experimental measurements of finger scale shows good agreement where plausibly finite-amplitude effects might come into play, but with the linear theory otherwise.

Radial lift on a suspended finite-sized sphere due to fluid inertia for low-Reynolds-number flow through a cylinder

2013 ◽  
Vol 722 ◽  
pp. 159-186 ◽  
Author(s):  
Sukalyan Bhattacharya ◽  
Dil K. Gurung ◽  
Shahin Navardi
Keyword(s):  
Reynolds Number ◽  
Particle Diameter ◽  
Axial Direction ◽  
Perturbation Scheme ◽  
Fluid Inertia ◽  
Regular Perturbation ◽  
Cross Sectional ◽  
First Order ◽  
Reynolds Number Flow ◽  
Freely Suspended

AbstractThis article describes the radial drift of a suspended sphere in a cylinder-bound Poiseuille flow where the Reynolds number is small but finite. Unlike past studies, it considers a circular narrow conduit whose cross-sectional diameter is only $1. 5$–$6$ times the particle diameter. Thus, the analysis quantifies the effect of fluid inertia on the radial motion of the particle in the channel when the flow field is significantly influenced by the presence of the suspended body. To this end, the hydrodynamic fields are expanded as a series in Reynolds number, and a set of hierarchical equations for different orders of the expansion is derived. Accordingly, the zeroth-order fields in Reynolds number satisfy the Stokes equation, which is accurately solved in the presence of the spherical particle and the cylindrical conduit. Then, recognizing that in narrow vessels Stokesian scattered fields from the sphere decrease exponentially in the axial direction, a simpler regular perturbation scheme is used to quantify the first-order inertial correction to hydrodynamic quantities. Consequently, it is possible to obtain two results. First, the sphere is assumed to follow the axial motion of a freely suspended sphere in a Stokesian condition, and the radial lift force on it due to the presence of fluid inertia is evaluated. Then, the approximate motion is determined for a freely suspended body on which net hydrodynamic force including first-order inertial lift is zero. The results agree well with the available experimental results. Thus, this study along with the measured data would precisely describe particle dynamics inside narrow tubes.

Investigation of biokinetic effects on hybrid multienzyme biosensor system

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Akinbowale T. Akinshilo ◽  
Gbeminiyi M. Sobamowo ◽  
Osamudiamen Olaye ◽  
Adeleke O. Illegbusi
Keyword(s):  
Food Safety ◽  
Biomedical Applications ◽  
Substrate Inhibition ◽  
Perturbation Scheme ◽  
Regular Perturbation ◽  
Content Type ◽  
Analytical Scheme ◽  
Biosensor System ◽  
Biokinetic Parameters ◽  
Menten Constant

PurposeIn this paper, hybrid multienzyme biosensor system, which detects analyte through molecular conversion into signal response, is analyzed and presented. The biokinetic effects of pertinent parameters such as Michaelis–Menten constant, inhibitor inhibition and substrate inhibition modulus on biochemical reactions are investigated.Design/methodology/approachBiochemical reaction models are described by five nonlinear equations for bisubstrate amperometric system analyzed adopting the regular perturbation method.FindingsResults obtained reveal that increasing Michaelis–Menten constant of oxygen causes a significant decrease in hydrogen peroxide concentration while increasing Michaelis–Menten constant of glucose shows increasing effect on oxygen concentration. Hence, results obtained from this work serve as reference for further analysis of concentration models and offer useful insight to relevant applications such as food safety, environmental and biomedical applications.Practical implicationsThis work serves as reference for further analysis of concentration models and offers useful insight to relevant applications such as food safety, environmental and biomedical applications.Originality/valueThis paper examines the effect of biokinetic parameters on the concentration of the hybrid multienzyme biosensor. Here the effects of parameters such as inhibitor inhibition, substrate inhibition and Michealis–Menten were investigated on substrate, inhibition and product concentrations. It is illustrated from result that inhibitor parameter slows enzymatic catalytic reaction while substrate enhances reaction. This study applied approximate analytical scheme to investigate the biokinetic effects, adopting the regular perturbation scheme.

Finite Amplitude Waves on Aircraft Trailing Vortices

1972 ◽  
Vol 23 (4) ◽  
pp. 307-314 ◽  
Author(s):  
D W Moore
Keyword(s):  
Numerical Methods ◽  
Linear Theory ◽  
Finite Amplitude ◽  
Detailed Structure ◽  
The Core ◽  
Trailing Vortices ◽  
Finite Amplitude Waves

SummaryNumerical methods are used to study the growth of waves of finite amplitude on a pair of parallel infinite vortices. The vortices are treated as lines except in so far as the detailed structure of the core is needed to remove consistently the singularity in the line integrals for the velocities of the vortices. It is shown that the vortices eventually touch and the shape of the wave at this instant is calculated. The wave is quite distorted at this instant, but it is shown that its gross properties are given roughly by linear theory.

scholarly journals The Effect of a Magnetic Field on Stellar Pulsations as a Singular Perturbation Problem

1980 ◽  
Vol 58 ◽  
pp. 661-666
Author(s):  
M. Goossens ◽  
D. Biront
Keyword(s):  
Magnetic Field ◽  
Magnetic Pressure ◽  
Matched Asymptotic Expansion ◽  
Perturbation Scheme ◽  
Regular Perturbation ◽  
Perturbation Problem ◽  
Stellar Pulsations ◽  
Thermodynamic Pressure ◽  
The Magnetic Field ◽  
Adiabatic Oscillation

AbstractThe perturbation problem that describes the effect of a weak magnetic field on stellar adiabatic oscillation is considered. This perturbation problem is singular when the magnetic field does not vanish at the stellar surface, and a regular perturbation scheme fails where the magnetic pressure is comparable to the thermodynamic pressure. The application of the Method of Matched Asymptotic Expansion is used to obtain expressions for the eigenfunctions and the eigenfrequencies.

Spatial evolution of finite-amplitude plasma waves in collisionless plasmas

1974 ◽  
Vol 11 (2) ◽  
pp. 213-223 ◽  
Author(s):  
Shih-Tung Tsai
Keyword(s):  
Plasma Waves ◽  
Finite Amplitude ◽  
Perturbation Scheme ◽  
Spatial Evolution ◽  
Collisionless Plasmas ◽  
Electron Distributions ◽  
Resonant Region ◽  
Arbitrary Amplitude ◽  
Theoretical Results ◽  
Good Agreement

A numerical simulation concerning the spatial evolution of electron plasma waves and electron distributions in collisionless plasmas is performed. Only those particles in and around the resonant region are followed numerically, while the bulk plasma is treated analytically. Our results are in good agreement with existing theoretical results as well as experimental observations. The method introduced here is valid for waves with arbitrary amplitude. It does not depend on either a large-amplitude asymptotic expansion or a small-amplitude perturbation scheme.

Lubricated pipelining: stability of core–annular flow. Part 4. Ginzburg–Landau equations

1991 ◽  
Vol 227 ◽  
pp. 587-615 ◽  
Author(s):  
Kangping Chen ◽  
Daniel D. Joseph
Keyword(s):  
Annular Flow ◽  
Landau Equation ◽  
Finite Amplitude ◽  
Ginzburg Landau ◽  
Neutral Curves ◽  
Critical Reynolds Numbers ◽  
Ginzburg Landau Equations ◽  
Flow Part ◽  
Annular Flows ◽  
Monochromatic Waves

Nonlinear stability of core-annular flow near points of the neutral curves at which perfect core-annular flow loses stability is studied using Ginzburg-Landau equations. Most of the core-annular flows are always unstable. Therefore the set of core-annular flows having critical Reynolds numbers is small, so that the set of flows for which our analysis applies is small. An efficient and accurate algorithm for computing all the coefficients of the Ginzburg-Landau equation is implemented. The nonlinear flows seen in the experiments do not appear to be modulations of monochromatic waves, and we see no evidence for soliton-like structures. We explore the bifurcation structure of finite-amplitude monochromatic waves at criticality. The bifurcation theory is consistent with observations in some of the flow cases to which it applies and is not inconsistent in the other cases.

Effect of a stabilizing gradient of solute on thermal convection

1968 ◽  
Vol 34 (2) ◽  
pp. 315-336 ◽  
Author(s):  
George Veronis
Keyword(s):  
Rayleigh Number ◽  
Temperature Gradient ◽  
Prandtl Number ◽  
Critical Rayleigh Number ◽  
Effective Diffusion ◽  
Stratified Fluid ◽  
Finite Amplitude ◽  
Oscillatory Motion ◽  
Linear Stability Theory ◽  
Onset Of Convection

A stabilizing gradient of solute inhibits the onset of convection in a fluid which is subjected to an adverse temperature gradient. Furthermore, the onset of instability may occur as an oscillatory motion because of the stabilizing effect of the solute. These results are obtained from linear stability theory which is reviewed briefly in the following paper before finite-amplitude results for two-dimensional flows are considered. It is found that a finite-amplitude instability may occur first for fluids with a Prandtl number somewhat smaller than unity. When the Prandtl number is equal to unity or greater, instability first sets in as an oscillatory motion which subsequently becomes unstable to disturbances which lead to steady, convecting cellular motions with larger heat flux. A solute Rayleigh number, Rs, is defined with the stabilizing solute gradient replacing the destabilizing temperature gradient in the thermal Rayleigh number. When Rs is large compared with the critical Rayleigh number of ordinary Bénard convection, the value of the Rayleigh number at which instability to finite-amplitude steady modes can set in approaches the value of Rs. Hence, asymptotically this type of instability is established when the fluid is marginally stratified. Also, as Rs → ∞ an effective diffusion coefficient, Kρ, is defined as the ratio of the vertical density flux to the density gradient evaluated at the boundary and it is found that κρ = √(κκs) where κ, κs are the diffusion coefficients for temperature and solute respectively. A study is made of the oscillatory behaviour of the fluid when the oscillations have finite amplitudes; the periods of the oscillations are found to increase with amplitude. The horizontally averaged density gradients change sign with height in the oscillating flows. Stably stratified fluid exists near the boundaries and unstably stratified fluid occupies the mid-regions for most of the oscillatory cycle. Thus the step-like behaviour of the density field which has been observed experimentally for time-dependent flows is encountered here numerically.

The Faraday threshold in small cylinders and the sidewall non-ideality

2013 ◽  
Vol 729 ◽  
pp. 496-523 ◽  
Author(s):  
W. Batson ◽  
F. Zoueshtiagh ◽  
R. Narayanan
Keyword(s):  
Contact Line ◽  
Mode Selection ◽  
Single Mode ◽  
Linear Stability Theory ◽  
Viscous Effects ◽  
Low Frequencies ◽  
Dimension 2 ◽  
First Time ◽  
Wetting Fluid ◽  
Selection Of

AbstractIn this work we investigate, by way of experiments and theory, the Faraday instability threshold in cylinders at low frequencies. This implies large wavelengths where effects from mode discretization cannot be ignored. Careful selection of the working fluids has resulted in an immiscible interface whose apparent contact line with the sidewall can glide over a tiny film of the more wetting fluid, without detachment of its actual contact line. This unique behaviour has allowed for a system whose primary dissipation is defined by the bulk viscous effects, and in doing so, for the first time, close connection is seen with the viscous linear stability theory for which a stress-free condition is assumed at the sidewalls. As predicted, mode selection and co-dimension 2 points are observed in the experiment for a frequency range including subharmonic, harmonic, and superharmonic modes. While agreement with the predictions are generally excellent, there are deviations from the theory for certain modes and these are explained in the context of harmonic meniscus waves. A review of previous work on single-mode excitation in cylinders is given, along with comparison to the viscous model and analysis based upon the conclusions of the current experiments.

Liquid Sheet Jet Experiments: Comparison With Linear Theory

1981 ◽  
Vol 103 (4) ◽  
pp. 595-603 ◽  
Author(s):  
H. R. Asare ◽  
R. K. Takahashi ◽  
M. A. Hoffman
Keyword(s):  
Linear Theory ◽  
Fusion Reactor ◽  
Harmonic Excitation ◽  
Liquid Sheet ◽  
Linear Stability Theory ◽  
Liquid Lithium ◽  
Laser Fusion ◽  
Structural Walls ◽  
Water Jets ◽  
Highly Nonlinear

It has been proposed to protect the structural walls of a future laser fusion reactor with a curtain or fluid-wall of liquid lithium jets. As part of the investigation of this concept, experiments have been performed on planar sheet water jets issuing vertically downward from slit nozzles. The nozzles were subjected to transverse forced harmonic excitation to simulate the vibrational environment of the laser fusion reactor, and experiments were run at both 1 atm and at lower ambient pressures. Linear stability theory is shown to predict the onset of the unstable regime and the initial spatial growth rates quite well for cases where the amplitudes of the nozzle vibration are not too large and the waveform is nearly sinusoidal. In addition, both the linear theory and a simplified trajectory theory are shown to predict the initial wave envelope amplitudes very well. For larger amplitude nozzle excitation, the waveform becomes highly nonlinear and nonsinusoidal and can resemble a sawtooth waveform in some cases; these latter experimental results can only be partially explained by existing theories at the present time.

Nonlinear self-excited oscillations of a ducted flame

1997 ◽  
Vol 346 ◽  
pp. 271-290 ◽  
Author(s):  
A. P. DOWLING
Keyword(s):  
Linear Theory ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Acoustic Waves ◽  
Bluff Body ◽  
Nonlinear Oscillations ◽  
Function Analysis ◽  
Pressure Fluctuations ◽  
Finite Amplitude ◽  
Flow Disturbances

Self-excited oscillations of a confined flame, burning in the wake of a bluff-body flame-holder, are considered. These oscillations occur due to interaction between unsteady combustion and acoustic waves. According to linear theory, flow disturbances grow exponentially with time. A theory for nonlinear oscillations is developed, exploiting the fact that the main nonlinearity is in the heat release rate, which essentially ‘saturates’. The amplitudes of the pressure fluctuations are sufficiently small that the acoustic waves remain linear. The time evolution of the oscillations is determined by numerical integration and inclusion of nonlinear effects is found to lead to limit cycles of finite amplitude. The predicted limit cycles are compared with results from experiments and from linear theory. The amplitudes and spectra of the limit-cycle oscillations are in reasonable agreement with experiment. Linear theory is found to predict the frequency and mode shape of the nonlinear oscillations remarkably well. Moreover, we find that, for this type of nonlinearity, describing function analysis enables a good estimate of the limit-cycle amplitude to be obtained from linear theory.Active control has been successfully applied to eliminate these oscillations. We demonstrate the same effect by adding a feedback control system to our nonlinear model. This theory is used to explain why any linear controller capable of stabilizing the linear flow disturbances is also able to stabilize finite-amplitude oscillations in the nonlinear limit cycles.

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