scholarly journals Breakdown of similarity solutions: a perturbation approach for front propagation during foam-improved oil recovery

2021 ◽  
Vol 477 (2245) ◽  
pp. 20200691
Author(s):  
Carlos Torres-Ulloa ◽  
Paul Grassia
Keyword(s):  
Oil Recovery ◽  
Similarity Solutions ◽  
Second Order ◽  
Perturbation Approach ◽  
Improved Oil Recovery ◽  
First Order ◽  
Order Solution ◽  
Concave Corner ◽  
Spatio Temporal ◽  
Material Points

The pressure-driven growth model has been employed to study a propagating foam front in the foam-improved oil recovery process. A first-order solution of the model proves the existence of a concave corner on the front, which initially migrates downwards at a well defined speed that differs from the speed of front material points. At later times, however, it remains unclear how the concave corner moves and interacts with points on the front either side of it, specifically whether material points are extracted from the corner or consumed by it. To address these questions, a second-order solution is proposed, perturbing the aforementioned first-order solution. However, the perturbation is challenging to develop, owing to the nature of the first-order solution, which is a similarity solution that exhibits strong spatio-temporal non-uniformities. The second-order solution indicates that the corner’s vertical velocity component decreases as the front migrates and that points initially extracted from the front are subsequently consumed by it. Overall, the perturbation approach developed herein demonstrates how early-time similarity solutions exhibiting strong spatio-temporal non-uniformities break down as time proceeds.

scholarly journals Foam front advance during improved oil recovery: similarity solutions at early times near the top of the front

2017 ◽  
Vol 828 ◽  
pp. 527-572 ◽  
Author(s):  
P. Grassia ◽  
L. Lue ◽  
C. Torres-Ulloa ◽  
S. Berres
Keyword(s):  
Oil Recovery ◽  
Surfactant Solution ◽  
Similarity Solutions ◽  
Simulation Data ◽  
Convex Shape ◽  
Improved Oil Recovery ◽  
Lower Region ◽  
Concave Corner ◽  
Front Advance ◽  
Material Points

The pressure-driven growth model is used to determine the shape of a foam front propagating into an oil reservoir. It is shown that the front, idealised as a curve separating surfactant solution downstream from gas upstream, can be subdivided into two regions: a lower region (approximately parabolic in shape and consisting primarily of material points which have been on the foam front continuously since time zero) and an upper region (consisting of material points which have been newly injected onto the foam front from the top boundary). Various conjectures are presented for the shape of the upper region. A formulation which assumes that the bottom of the upper region is oriented in the same direction as the top of the lower region is shown to fail, as (despite the orientations being aligned) there is a mismatch in location: the upper and lower regions fail to intersect. Alternative formulations are developed which allow the upper region to curve sufficiently so as to intersect the lower region. These formulations imply that the lower and upper regions (whilst individually being of a convex shape as seen from downstream) actually meet in a concave corner, contradicting the conventional hypothesis in the literature that the front is wholly convex. The shape of the upper region as predicted here and the presence of the concave corner are independently verified via numerical simulation data.

Radiant Heat Transfer From Isothermal Dispersions With Isotropic Scattering

1967 ◽  
Vol 89 (4) ◽  
pp. 300-308 ◽  
Author(s):  
R. H. Edwards ◽  
R. P. Bobco
Keyword(s):  
Heat Transfer ◽  
Approximate Solutions ◽  
Radiant Heat ◽  
Second Order ◽  
Isotropic Scattering ◽  
Radiant Heat Transfer ◽  
Approximate Methods ◽  
First Order ◽  
Order Solution ◽  
Radiant Transfer

Two approximate methods are presented for making radiant heat-transfer computations from gray, isothermal dispersions which absorb, emit, and scatter isotropically. The integrodifferential equation of radiant transfer is solved using moment techniques to obtain a first-order solution. A second-order solution is found by iteration. The approximate solutions are compared to exact solutions found in the literature of astrophysics for the case of a plane-parallel geometry. The exact and approximate solutions are both expressed in terms of directional and hemispherical emissivities at a boundary. The comparison for a slab, which is neither optically thin nor thick (τ = 1), indicates that the second-order solution is accurate to within 10 percent for both directional and hemispherical properties. These results suggest that relatively simple techniques may be used to make design computations for more complex geometries and boundary conditions.

scholarly journals Lie transforms and motion of a charged particle in a periodic magnetic field

1984 ◽  
Vol 7 (1) ◽  
pp. 159-169
Author(s):  
Sikha Bhattacharyya ◽  
R. K. Roy Choudhury
Keyword(s):  
Magnetic Field ◽  
Charged Particle ◽  
Averaging Method ◽  
Second Order ◽  
Lie Series ◽  
Periodic Magnetic Field ◽  
First Order ◽  
Order Solution ◽  
Lie Transforms ◽  
Spatially Periodic

We use the Lie series averaging method to obtain a complete second order solution for motion of a charged particle in a spatially periodic magnetic field. A comparison is made with the first order solution obtained previously by Coffey.

Acoustofluidic Simulations: Ultrasound Handling of Liquids and Particles in Microfluidic Systems

2008 ◽  
Author(s):  
Peder Skafte-Pedersen ◽  
Henrik Bruus
Keyword(s):  
Acoustic Radiation ◽  
Acoustic Streaming ◽  
Radiation Force ◽  
Second Order ◽  
Order Effects ◽  
Perturbation Approach ◽  
First Order ◽  
At Resonance ◽  
Liquid Flows ◽  
Second Order Equations

Within the field of lab-on-a-chip systems large efforts are devoted to the development of onchip tools for particle handling and mixing in viscosity-dominated liquid flows on the sub-mm scale. One technology involves ultrasound with frequencies in the MHz range, which leads to wavelengths of the order of 0.1–1 mm suitable for mm-sized microchambers. Due to the nonlinearity of the governing acoustofluidic equations, second-order effects will induce steady forces on fluids and suspended particles through the effects known as acoustic streaming and acoustic radiation force. We extend the basic perturbation approach for treating these effects in systems at resonance in various geometries. The first-order eigenmodes are used as source terms for the time-averaged viscous second-order equations. The theory is applied to explain experimental results on aqueous microbead solutions in silicon-glass microchips.

Stability of a Beddington–DeAngelis type predator–prey model with trichotomous noises

2016 ◽  
Vol 30 (17) ◽  
pp. 1650102 ◽  
Author(s):  
Yanfei Jin ◽  
Siyong Niu
Keyword(s):  
Correlation Time ◽  
Gaussian White Noise ◽  
Second Order ◽  
Predator Prey ◽  
First Order ◽  
Predator Prey Model ◽  
Order Solution ◽  
Prey Model ◽  
The Stability ◽  
The Moment

The stability analysis of a Beddington–DeAngelis (B–D) type predator–prey model driven by symmetric trichotomous noises is presented in this paper. Using the Shapiro–Loginov formula, the first-order and second-order solution moments of the system are obtained. The moment stability conditions of the B–D predator–prey model are given by using Routh–Hurwitz criterion. It is found that the stabilities of the first-order and second-order solution moments depend on the noise intensities and correlation time of noise. The first-order and second-order moments are stable when the correlation time of noise is increased. That is, the trichotomous noise plays a constructive role in stabilizing the solution moment with regard to Gaussian white noise. Finally, some numerical results are performed to support the theoretical analyses.

scholarly journals Lie symmetry analysis and similarity solutions for the Jimbo – Miwa equation and generalisations

2020 ◽  
Vol 21 (7-8) ◽  
pp. 767-779
Author(s):  
Amlan K. Halder ◽  
Andronikos Paliathanasis ◽  
Rajeswari Seshadri ◽  
Peter G. L. Leach
Keyword(s):  
Singularity Analysis ◽  
Travelling Wave ◽  
Similarity Solutions ◽  
Second Order ◽  
Lie Symmetry Analysis ◽  
Group Approach ◽  
First Order ◽  
Reduced Equations ◽  
Second Order Equations ◽  
Hypergeometric Solutions

AbstractWe study the Jimbo – Miwa equation and two of its extended forms, as proposed by Wazwaz et al., using Lie’s group approach. Interestingly, the travelling – wave solutions for all the three equations are similar. Moreover, we obtain certain new reductions which are completely different for each of the three equations. For example, for one of the extended forms of the Jimbo – Miwa equation, the subsequent reductions leads to a second – order equation with Hypergeometric solutions. In certain reductions, we obtain simpler first – order and linearisable second – order equations, which helps us to construct the analytic solution as a closed – form function. The variation in the nonzero Lie brackets for each of the different forms of the Jimbo – Miwa also presents a different perspective. Finally, singularity analysis is applied in order to determine the integrability of the reduced equations and of the different forms of the Jimbo – Miwa equation.

A High-Order Solution for the Added Stability Lobes in Intermittent Machining

2000 ◽  
Author(s):  
William T. Corpus ◽  
William J. Endres
Keyword(s):  
Closed Form ◽  
High Speed ◽  
Higher Order ◽  
Second Order ◽  
Machining Processes ◽  
Analytical Technique ◽  
First Order ◽  
Order Solution ◽  
Order Case ◽  
Speed Stability

Abstract An earlier work by the authors presented a solution for the added ultrahigh-speed stability lobe that has been shown to exist for intermittent and other periodically time varying machining processes. That earlier first-order solution was not clearly extendible to a higher order. A more general analytical technique presented here does permit higher-order results. The solution is developed first for the case of zero damping for which a final closed-form symbolic result can be realized up to second order. More important than improved accuracy, the higher-order nature of the result confirms that there exist multiple added lobes and permits a mathematical description of their locations along the spindle-speed axis. A solution is then derived for the structurally damped case, where the first-order case permits a final closed-form symbolic result while the second-order case requires computational evaluation. The first-order result matches perfectly the previously published one, as expected. The second-order result improves accuracy, measured relative to numerical simulation results, and, more important, permits a second added lobe to be predicted. The second added lobe tends to cut into the region of the high-speed stability peak that is predicted under traditional zero-frequency (time-averaged) analyses. The damped solutions also indicate that structural damping of the dominant mode becomes virtually unimportant at ultrahigh speeds.

Bubble Dynamics and Cavitation Inception Theory

1988 ◽  
Vol 32 (03) ◽  
pp. 155-167
Author(s):  
Blaine R. Parkin ◽  
Brian B. Baker
Keyword(s):  
Initial Conditions ◽  
Second Order ◽  
Order System ◽  
Refined Theory ◽  
Zeroth Order ◽  
Order Problem ◽  
Cavitation Inception ◽  
First Order ◽  
Order Solution ◽  
Forcing Function

In order to provide some theoretical background and to motivate the more refined theory introduced herein, some encouraging known theoretical results on bubble-ring cavitation inception are reviewed. This review is followed by the development of the theory of bubble-ring cavitation cutoff. Its outcome, when compared with experiment, shows the need for a more refined inception theory. The above comparison and the basic ideas behind the cutoff theory's formulation suggest a possible approach for a refinement based on a multiple scales expansion. This seems reasonable because the forcing function pulse in "laboratory time" f, varies slowly compared with the characteristic "bubble time,", which characterizes the response time of a typical microscopic cavitation nucleus. The ratio of these two times gives a small parameter, , appearing in the forcing function, with the result that this problem involves only a soft excitation. Expanding the forced Rayleigh-Plesset equation and its initial conditions to the second order in c, the zeroth-order problem is found to be the well-known autonomous nonlinear equation with nonhomogeneous initial conditions, giving free oscillations of a typical nucleus. The first-order system is a nonautonomous linear system with homogeneous initial conditions which governs the forced bubble growth. The second-order system consists of a linear autonomous differential equation and homogeneous initial conditions. It is needed to establish integrability conditions for the first-order solution. The first-order solution is left for future research and the zeroth-order problem is analyzed in the phase plane. Then a novel approximate integration, = t(u), is given in terms of elliptic integrals and functions. It was not possible to invert this solution and so the inverse u = u() is found numerically. These data are then used to find an analytical approximation for use in future first-order calculations.

Second Order Solutions of THz Response of Gated Two-Dimensional Electron Gas in Magnetic Field

Frequenz ◽  
2018 ◽  
Vol 72 (9-10) ◽  
pp. 471-477 ◽  
Author(s):  
Daipeng Wang ◽  
Jiuxun Sun ◽  
Chao Yang ◽  
Yan Dong ◽  
Zhenlin Yan
Keyword(s):  
Magnetic Field ◽  
Electron Gas ◽  
Second Order ◽  
Two Dimensional ◽  
Dimensional Electron ◽  
Two Dimensional Electron Gas ◽  
First Order ◽  
Order Solution ◽  
Dimensional Electron Gas ◽  
Second Order Equations

Abstract In this work, the Lifshits-Dyakonov theory for THz response of gated two-dimensional electron gas in magnetic field are analyzed and improved. Instead an approximate processing method for the response in original theory to the second order solution, the second order equations are strictly solved. The numerical results show that both first and second order solutions are damped oscillating functions of coordinate, but all amplitudes would decrease as magnetic field B increasing except for the first order solution of voltage. The variation of second order response as a function of B also shows damped oscillating variations, the agreement with experimental curves is reasonable.

Nonlinear Analysis of Visual Evoked Potentials Elicited by Color Stimulation

1997 ◽  
Vol 36 (04/05) ◽  
pp. 315-318 ◽  
Author(s):  
K. Momose ◽  
K. Komiya ◽  
A. Uchiyama
Keyword(s):  
Visual System ◽  
Evoked Potentials ◽  
Visual Evoked Potentials ◽  
Normal Subjects ◽  
Second Order ◽  
First Order ◽  
The Relationship ◽  
Perceived Color ◽  
Second Order Nonlinearity ◽  
Visual Evoked

Abstract:The relationship between chromatically modulated stimuli and visual evoked potentials (VEPs) was considered. VEPs of normal subjects elicited by chromatically modulated stimuli were measured under several color adaptations, and their binary kernels were estimated. Up to the second-order, binary kernels obtained from VEPs were so characteristic that the VEP-chromatic modulation system showed second-order nonlinearity. First-order binary kernels depended on the color of the stimulus and adaptation, whereas second-order kernels showed almost no difference. This result indicates that the waveforms of first-order binary kernels reflect perceived color (hue). This supports the suggestion that kernels of VEPs include color responses, and could be used as a probe with which to examine the color visual system.

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