scholarly journals Influence of viscosity contrast on buoyantly unstable miscible fluids in porous media

2015 ◽  
Vol 780 ◽  
pp. 388-406 ◽  
Author(s):  
Satyajit Pramanik ◽  
Tapan Kumar Hota ◽  
Manoranjan Mishra
Keyword(s):  
Linear Stability ◽  
Injection Velocity ◽  
External Excitation ◽  
Initial Value ◽  
Miscible Fluids ◽  
Viscosity Contrast ◽  
Stability Method ◽  
Linearized Operator ◽  
Stability Results ◽  
Dimensionless Formulation

The influence of viscosity contrast on buoyantly unstable miscible fluids in a porous medium is investigated through a linear stability analysis (LSA) as well as direct numerical simulations (DNS). The linear stability method implemented in this paper is based on an initial value approach, which helps to capture the onset of instability more accurately than the quasi-steady-state analysis. In the absence of displacement, we show that viscosity contrast delays the onset of instability in buoyantly unstable miscible fluids. Further, it is observed that by suitably choosing the viscosity contrast and injection velocity a gravitationally unstable miscible interface can be stabilized completely. Through LSA we draw a phase diagram, which shows three distinct stability regions in a parameter space spanned by the displacement velocity and the viscosity contrast. DNS are performed corresponding to parameters from each regime and the results obtained are in accordance with the linear stability results. Moreover, the conversion from one dimensionless formulation to another and its importance when comparing the different type of flow problem associated with each dimensionless formulation are discussed. We also calculate${\it\epsilon}$-pseudo-spectra of the time dependent linearized operator to investigate the response to external excitation.

Comparison of linear stability results with flight transition data

AIAA Journal ◽  
1995 ◽  
Vol 33 (1) ◽  
pp. 161-163 ◽  
Author(s):  
J. A. Masad ◽  
M. R. Malik
Keyword(s):  
Linear Stability ◽  
Stability Results

scholarly journals Nonlinear Evolution Equations with Exponentially Decaying Memory: Existence via Time Discretisation, Uniqueness, and Stability

2020 ◽  
Vol 20 (1) ◽  
pp. 89-108 ◽  
Author(s):  
André Eikmeier ◽  
Etienne Emmrich ◽  
Hans-Christian Kreusler
Keyword(s):  
Banach Space ◽  
Integral Operator ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
Inner Product ◽  
Initial Value ◽  
Time Discretisation ◽  
Volterra Integral Operator ◽  
Stability Results

AbstractThe initial value problem for an evolution equation of type {v^{\prime}+Av+BKv=f} is studied, where {A:V_{A}\to V_{A}^{\prime}} is a monotone, coercive operator and where {B:V_{B}\to V_{B}^{\prime}} induces an inner product. The Banach space {V_{A}} is not required to be embedded in {V_{B}} or vice versa. The operator K incorporates a Volterra integral operator in time of convolution type with an exponentially decaying kernel. Existence of a global-in-time solution is shown by proving convergence of a suitable time discretisation. Moreover, uniqueness as well as stability results are proved. Appropriate integration-by-parts formulae are a key ingredient for the analysis.

Stability of Vlasov equilibria. Part 1. General theory

1982 ◽  
Vol 27 (1) ◽  
pp. 13-24 ◽  
Author(s):  
K. R. Symon ◽  
C. E. Seyler ◽  
H. R. Lewis
Keyword(s):  
Distribution Function ◽  
General Theory ◽  
Linear Stability ◽  
Vlasov Equation ◽  
Stability Problem ◽  
Dispersion Matrix ◽  
Initial Value ◽  
Spectral Matrix ◽  
Concise Representation ◽  
The Stability

We present a general formulation for treating the linear stability of inhomogeneous plasmas for which at least one species is described by the Vlasov equation. Use of Poisson bracket notation and expansion of the perturbation distribution function in terms of eigenfunctions of the unperturbed Liouville operator leads to a concise representation of the stability problem in terms of a symmetric dispersion functional. A dispersion matrix is derived which characterizes the solutions of the linearized initial-value problem. The dispersion matrix is then expressed in terms of a dynamic spectral matrix which characterizes the properties of the unperturbed orbits, in so far as they are relevant to the linear stability of the system.

scholarly journals The influence of viscosity on the frozen wave instability: theory and experiment

2007 ◽  
Vol 584 ◽  
pp. 45-68 ◽  
Author(s):  
EMMA TALIB ◽  
SHREYAS V. JALIKOP ◽  
ANNE JUEL
Keyword(s):  
Linear Stability ◽  
Lower Layer ◽  
Experimental Parameter ◽  
Quantitative Agreement ◽  
Stability Study ◽  
Base Flow ◽  
Viscosity Contrast ◽  
Wide Range ◽  
Horizontal Oscillation ◽  
Theory And Experiment

We present the results of an experimental and linear stability study of the influence of viscosity on the frozen wave (FW) instability, which arises when a vessel containing stably stratified layers of immiscible liquids is oscillated horizontally. Our linear stability model consists of two superposed fluid layers of arbitrary viscosities and infinite lateral extent, subject to horizontal oscillation. The effect of the endwalls of the experimental vessel is simulated by enforcing the conservation of horizontal volume flux, so that the base flow consists of counterflowing layers.We perform experiments with four pairs of fluids, keeping the viscosity of the lower layer (ν1) constant, and increasing the viscosity of the upper layer (ν2), so that 1.02 × 102 ≤ N1 = ν2/ν1 ≤ 1.21 × 104. We find excellent quantitative agreement between theory and experiment despite the simple model geometry, for both the critical onset parameter and wavenumber of the FW. We show that the model of lyubimov:1987 (Fluid Dyn. vol. 86, 1987, p. 849), which is valid in the limit of inviscid fluids, consistently underestimates the instability threshold for fluids of equal viscosity, but generally overestimates the threshold for fluids of unequal viscosity. We extend the experimental parameter range numerically to viscosity contrasts 1 ≤ N1 ≤ 6 × 104 and identify four regions of N1 where qualitatively different dynamics occur, which are reflected in the non-monotonic dependence of the most unstable wavenumber and the critical amplitude on N1. In particular, we find that increasing the viscosity contrast between the layers leads to destabilization over a wide range of N1, 10 ≤ N1 ≤ 8 × 103. The intricate dependence of the instability on viscosity contrast is due to considerable changes in the time-averaged perturbation vorticity distribution near the interface.

Density-driven instabilities of miscible fluids in a Hele-Shaw cell: linear stability analysis of the three-dimensional Stokes equations

2002 ◽  
Vol 451 ◽  
pp. 261-282 ◽  
Author(s):  
F. GRAF ◽  
E. MEIBURG ◽  
C. HÄRTEL
Keyword(s):  
Experimental Data ◽  
Growth Rate ◽  
Linear Stability ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Interface Thickness ◽  
Gap Width ◽  
Stability Results ◽  
Hele Shaw Cell

We consider the situation of a heavier fluid placed above a lighter one in a vertically arranged Hele-Shaw cell. The two fluids are miscible in all proportions. For this configuration, experiments and nonlinear simulations recently reported by Fernandez et al. (2002) indicate the existence of a low-Rayleigh-number (Ra) ‘Hele-Shaw’ instability mode, along with a high-Ra ‘gap’ mode whose dominant wavelength is on the order of five times the gap width. These findings are in disagreement with linear stability results based on the gap-averaged Hele-Shaw approach, which predict much smaller wavelengths. Similar observations have been made for immiscible flows as well (Maxworthy 1989).In order to resolve the above discrepancy, we perform a linear stability analysis based on the full three-dimensional Stokes equations. A generalized eigenvalue problem is formulated, whose numerical solution yields both the growth rate and the two-dimensional eigenfunctions in the cross-gap plane as functions of the spanwise wavenumber, an ‘interface’ thickness parameter, and Ra. For large Ra, the dispersion relations confirm that the optimally amplified wavelength is about five times the gap width, with the exact value depending on the interface thickness. The corresponding growth rate is in very good agreement with the experimental data as well. The eigenfunctions indicate that the predominant fluid motion occurs within the plane of the Hele-Shaw cell. However, for large Ra purely two-dimensional modes are also amplified, for which there is no motion in the spanwise direction. Scaling laws are provided for the dependence of the maximum growth rate, the corresponding wavenumber, and the cutoff wavenumber on Ra and the interface thickness. Furthermore, the present results are compared both with experimental data, as well as with linear stability results obtained from the Hele-Shaw equations and a modified Brinkman equation.

Sediment-laden fresh water above salt water: linear stability analysis

2011 ◽  
Vol 691 ◽  
pp. 279-314 ◽  
Author(s):  
P. Burns ◽  
E. Meiburg
Keyword(s):  
Growth Rate ◽  
Linear Stability ◽  
Fresh Water ◽  
Settling Velocity ◽  
Interface Thickness ◽  
Particle Settling ◽  
Salinity Field ◽  
Unstable Layer ◽  
Stability Results ◽  
Double Diffusive

AbstractWhen a layer of particle-laden fresh water is placed above clear, saline water, both Rayleigh–Taylor and double diffusive fingering instabilities may arise. For quasi-steady base profiles, we obtain linear stability results for such situations by means of a rational spectral approximation method with adaptively chosen grid points, which is able to resolve multiple steep gradients in the base state density profile. In the absence of salinity and for a step-like concentration profile, the dominant parameter is the ratio of the particle settling velocity to the viscous velocity scale. As long as this ratio is small, particle settling has a negligible influence on the instability growth. However, when the particles settle more rapidly than the instability grows, the growth rate decreases inversely proportional to the settling velocity. This damping effect is a result of the smearing of the vorticity field, which in turn is caused by the deposition of vorticity onto the fluid elements passing through the interface between clear and particle-laden fluid. In the presence of a stably stratified salinity field, this picture changes dramatically. An important new parameter is the ratio of the particle settling velocity to the diffusive spreading velocity of the salinity, or alternatively the ratio of the unstable layer thickness to the diffusive interface thickness of the salinity profile. As long as this quantity does not exceed unity, the instability of the system and the most amplified wavenumber are primarily determined by double diffusive effects. In contrast to situations without salinity, particle settling can have a destabilizing effect and significantly increase the growth rate. Scaling laws obtained from the linear stability results are seen to be largely consistent with earlier experimental observations and theoretical arguments put forward by other authors. For unstable layer thicknesses much larger than the salinity interface thickness, the particle and salinity interfaces become increasingly decoupled, and the dominant instability mode becomes Rayleigh–Taylor-like, centred at the lower boundary of the particle-laden flow region.

Linear stability analysis for periodic travelling waves of the Boussinesq equation and the Klein–Gordon–Zakharov system

2014 ◽  
Vol 144 (3) ◽  
pp. 455-489 ◽  
Author(s):  
Sevdzhan Hakkaev ◽  
Milena Stanislavova ◽  
Atanas Stefanov
Keyword(s):  
Stability Analysis ◽  
Linear Stability ◽  
Wide Class ◽  
Boussinesq Equation ◽  
Travelling Waves ◽  
Periodic Waves ◽  
Zakharov System ◽  
Klein Gordon ◽  
Spatially Periodic ◽  
Stability Results

The question of the linear stability of spatially periodic waves for the Boussinesq equation (in the cases p = 2, 3) and the Klein–Gordon–Zakharov system is considered. For a wide class of solutions, we completely and explicitly characterize their linear stability (instability) when the perturbations are taken with the same period T. In particular, our results allow us to completely recover the linear stability results, in the limit T → ∞, for the whole-line case.

scholarly journals k-Generalized Ψ-Hilfer differential equations in Banach spaces

2021 ◽  
Author(s):  
Salim Abdelkrim ◽  
Mouffak Benchohra ◽  
Jamal Lazreg ◽  
Gaston NGuerekata
Keyword(s):  
Differential Equations ◽  
Banach Spaces ◽  
Fixed Point Theorems ◽  
Measure Of Noncompactness ◽  
Initial Value ◽  
Hilfer Fractional Derivative ◽  
The Subject ◽  
Fractional Differential ◽  
Stability Results ◽  
Rassias Stability

In this paper, we prove some existence and Ulam-Hyers-Rassias stability results for a class of initial value problem for implicit nonlinear fractional differential equations and generalized Ψ-Hilfer fractional derivative in Banach spaces. The results are based on fixed point theorems of Darbo and Monch associated with the technique of measure of noncompactness. Illustrative examples are the subject of the last section.

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