Interfacial pattern selection in defiance of linear growth

2017 ◽  
Vol 829 ◽  
pp. 345-363 ◽  
Author(s):  
Jason R. Picardo ◽  
R. Narayanan
Keyword(s):  
Linear Growth ◽  
Evolution Equations ◽  
Equation Model ◽  
Galerkin Projection ◽  
Mode Interaction ◽  
Pattern Selection ◽  
Physical Situation ◽  
Interfacial Instabilities ◽  
Linear Mode ◽  
Counter Example

In this study, we revisit Rayleigh’s visionary hypothesis (Rayleigh, Proc. R. Soc. Lond., vol. 29, 1879a, pp. 71–97), that patterns resulting from interfacial instabilities are dominated by the fastest-growing linear mode, as we study nonlinear pattern selection in the context of a linear growth (dispersion) curve that has two peaks of equal height. Such a system is obtained in a physical situation consisting of two liquid layers suspended from a heated ceiling, and exposed to a passive gas. Both interfaces are then susceptible to thermocapillary and Rayleigh–Taylor instabilities, which lead to rupture/pinch off via a subcritical bifurcation. The corresponding mathematical model consists of long-wavelength evolution equations which are amenable to extensive numerical exploration. We find that, despite having equal linear growth rates, either one of the peak-modes can completely dominate the other as a result of nonlinear interactions. Importantly, the dominant peak continues to dictate the pattern even when its growth rate is made slightly smaller, thereby providing a definite counter-example to Rayleigh’s conjecture. Although quite complex, the qualitative features of the peak-mode interaction are successfully captured by a low-order three-mode ordinary differential equation model based on truncated Galerkin projection. Far from being governed by simple linear theory, the final pattern is sensitive even to the phase difference between peak-mode perturbations. For sufficiently long domains, this phase effect is shown to result in the emergence of coexisting patterns, wherein each peak-mode dominates in a different region of the domain.

scholarly journals Diffusion Processes Satisfying a Conservation Law Constraint

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
J. Bakosi ◽  
J. R. Ristorcelli
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Conservation Law ◽  
Evolution Equations ◽  
Diffusion Processes ◽  
Random Variables ◽  
Planck Equation ◽  
Equation Model ◽  
First Four Moments ◽  
And Diffusion

We investigate coupled stochastic differential equations governing N nonnegative continuous random variables that satisfy a conservation principle. In various fields a conservation law requires a set of fluctuating variables to be nonnegative and (if appropriately normalized) sum to one. As a result, any stochastic differential equation model to be realizable must not produce events outside of the allowed sample space. We develop a set of constraints on the drift and diffusion terms of such stochastic models to ensure that both the nonnegativity and the unit-sum conservation law constraints are satisfied as the variables evolve in time. We investigate the consequences of the developed constraints on the Fokker-Planck equation, the associated system of stochastic differential equations, and the evolution equations of the first four moments of the probability density function. We show that random variables, satisfying a conservation law constraint, represented by stochastic diffusion processes, must have diffusion terms that are coupled and nonlinear. The set of constraints developed enables the development of statistical representations of fluctuating variables satisfying a conservation law. We exemplify the results with the bivariate beta process and the multivariate Wright-Fisher, Dirichlet, and Lochner’s generalized Dirichlet processes.

Non-linear mode interaction between spin torque driven and damped modes in spin torque nano-oscillators

2015 ◽  
Vol 106 (19) ◽  
pp. 192405 ◽  
Author(s):  
M. Romera ◽  
E. Monteblanco ◽  
F. Garcia-Sanchez ◽  
B. Delaët ◽  
L. D. Buda-Prejbeanu ◽  
...  
Keyword(s):  
Mode Interaction ◽  
Spin Torque ◽  
Linear Mode ◽  
Non Linear

scholarly journals The spreading and stability of a surfactant-laden drop on an inclined prewetted substrate

2015 ◽  
Vol 772 ◽  
pp. 535-568 ◽  
Author(s):  
J. V. Goddard ◽  
S. Naire
Keyword(s):  
Numerical Simulations ◽  
Evolution Equations ◽  
Leading Edge ◽  
Equation Model ◽  
Nonlinear Evolution Equations ◽  
Late Time ◽  
Differential Algebraic Equation ◽  
Scaling Properties ◽  
Insoluble Surfactant ◽  
Free Liquid Film

We consider a viscous drop, loaded with an insoluble surfactant, spreading over an inclined plane that is covered initially with a thin surfactant-free liquid film. Lubrication theory is employed to model the flow using coupled nonlinear evolution equations for the film thickness and surfactant concentration. Exploiting high-resolution numerical simulations, we describe the late-time multi-region asymptotic structure of the spatially one-dimensional spreading flow. A simplified differential–algebraic equation model is derived for key variables characterising the spreading process, using which the late-time spreading and thinning rates are determined. Focusing on the neighbourhood of the drop’s leading-edge effective contact line, we then examine the stability of this region to small-amplitude disturbances with transverse variation. A dispersion relationship is described using long-wavelength asymptotics and numerical simulations, which reveals physical mechanisms and new scaling properties of the instability.

scholarly journals Multimode Analysis of the Dynamics and Integrity of Electrically Actuated MEMS Resonators

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Serge Bruno Yamgoué ◽  
Alain Juvenal Tchiegang
Keyword(s):  
Time Integration ◽  
Free Vibrations ◽  
Galerkin Projection ◽  
Mode Shapes ◽  
Galerkin Procedure ◽  
Linear Mode ◽  
Mems Resonators ◽  
Numerical Time Integration ◽  
The Right ◽  
Subharmonic Resonances

We present a theoretical investigation of the dynamic behavior of a microelectromechanical system (in brief, MEMS) device modelled as a clamped-clamped microbeam subjected to electrostatic and electrodynamic actuation. We use the Galerkin projection technique to reduce the partial integro-differential equation governing the dynamics of the microbeam to a system of coupled ordinary differential equations which describe the interactions of the linear mode shapes of the microbeam. Analytical solutions are derived and their stability is studied for the simplest reduced-order model which takes into account only the first linear mode in the Galerkin procedure. We further investigate the influence of the first few higher modes on the Galerkin procedure, and hence its convergence, by analysing the boundaries between pull-in and pull-in-free vibrations domains in the space of actuation parameters. These are determined for the various multimode combinations using direct numerical time integration. Our results show that unsafe domains form V-like shapes for actuation frequencies close to the superharmonic, fundamental, and subharmonic resonances. They also reveal that the single first-mode reduced model usually considered underestimates the left branches and overestimates the right branches of these boundaries.

A kinetic theory based thermal lattice Boltzmann equation model

2017 ◽  
Vol 28 (04) ◽  
pp. 1750047 ◽  
Author(s):  
Qinglan Zhai ◽  
Song Zheng ◽  
Lin Zheng
Keyword(s):  
Boltzmann Equation ◽  
Lattice Boltzmann ◽  
Evolution Equations ◽  
Analytic Solutions ◽  
Equation Model ◽  
Distribution Functions ◽  
Convection Flow ◽  
Lattice Boltzmann Equation ◽  
Thermal Fluid Flow ◽  
Thermal Lattice Boltzmann

A thermal lattice Boltzmann equation (LBE) model within the framework of double distribution function (DDF) method is proposed from the continuous DDF Boltzmann equation, which has a clear physical significance. Since the discrete velocity set in present LBE model is not space filled, a Lax–Wendroff scheme is applied to solve the evolution equations by which the spatial interpolation of two distribution functions is overcome. To validate the model, some classical numerical tests include thermal Couette flow and natural convection flow are simulated, and the results agree well with the analytic solutions and other numerical results, which showed that the present model had the ability to describe the thermal fluid flow phenomena.

scholarly journals REDUCED-ORDER MODELLING OF PARAMETERIZED TRANSIENT FLOWS IN CLOSED-LOOP SYSTEMS

2021 ◽  
Vol 247 ◽  
pp. 06055
Author(s):  
Péter German ◽  
Mauricio Tano ◽  
Jean C. Ragusa ◽  
Carlo Fiorina
Keyword(s):  
Stokes Equations ◽  
Momentum Equation ◽  
Equation Model ◽  
Reduced Order Modeling ◽  
Galerkin Projection ◽  
Reduced Basis ◽  
Reduced Order ◽  
Transient Flows ◽  
Continuity Equations ◽  
The One

In this paper, two Galerkin projection based reduced basis approaches are investigated for the reduced-order modeling of parameterized incompressible Navier-Stokes equations for laminar transient flows. The first approach solves only the reduced momentum equation with additional, physics-based approximations for the dynamics of the pressure field. On the other hand, the second approach solves both the reduced momentum and continuity equations. The reduced bases for the velocity and pressure fields are generated using the method of snapshots combined with Proper Orthogonal Decomposition (POD) for data compression. To remedy the stability issues of the two-equation model, the reduced basis of the velocity is enriched with supremizer functions. Both reduced-order modeling approaches have been implemented in GeN-Foam, an OpenFOAM-based multi-physics solver. A numerical example is presented using a two-dimensional axisymmetric model of the Molten Salt Fast Reactor (MSFR) and the dynamic viscosity as the uncertain parameter. The results indicate that the one-equation model is slightly more accurate in terms of velocity, while the two-equation model, built with the same amount of modes for the velocity, is far more accurate in terms of pressure. The speed-up factors for the reduced-order models are 3060 for the one-equation model and 2410 for the two-equation model.

Idiographic Dynamics of Positive Affect in GAD

2020 ◽  
Vol 36 (3) ◽  
pp. 500-509
Author(s):  
Hannah G. Bosley ◽  
Devon B. Sandel ◽  
Aaron J. Fisher
Keyword(s):  
Positive Affect ◽  
Structural Equation ◽  
Alternative Hypothesis ◽  
Equation Model ◽  
Aggregated Data ◽  
Regulatory Strategy ◽  
Constant State ◽  
Automatic Search ◽  
Time Lagged ◽  
Exploratory Procedure

Abstract. Generalized anxiety disorder (GAD) is associated with worry and emotion regulation difficulties. The contrast-avoidance model suggests that individuals with GAD use worry to regulate emotion: by worrying, they maintain a constant state of negative affect (NA), avoiding a feared sudden shift into NA. We tested an extension of this model to positive affect (PA). During a week-long ecological momentary assessment (EMA) period, 96 undergraduates with a GAD analog provided four daily measurements of worry, dampening (i.e., PA suppression), and PA. We hypothesized a time-lagged mediation relationship in which higher worry predicts later dampening, and dampening predicts subsequently lower PA. A lag-2 structural equation model was fit to the group-aggregated data and to each individual time-series to test this hypothesis. Although worry and PA were negatively correlated in 87 participants, our model was not supported at the nomothetic level. However, idiographically, our model was well-fit for about a third (38.5%) of participants. We then used automatic search as an idiographic exploratory procedure to detect other time-lagged relationships between these constructs. While 46 individuals exhibited some cross-lagged relationships, no clear pattern emerged across participants. An alternative hypothesis about the speed of the relationship between variables is discussed using contemporaneous correlations of worry, dampening, and PA. Findings suggest heterogeneity in the function of worry as a regulatory strategy, and the importance of temporal scale for detection of time-lagged effects.

Sign in / Sign up

Export Citation Format

Share Document