It has been shown experimentally that when a drop is deposited at the center of a substrate with an axial temperature gradient (hotter in the center), thermocapillarity effects makes an outward flow ...to appear so that the drop evolves towards a ring whose radius increases with time. Upon reaching a critical radius, the contact line becomes unstable, showing gentle undulations whose amplitudes grow with time. Using the lubrication approximation and adopting appropriate dimensionless variables, a parameter-free differential equation is obtained that governs this type of thermocapillary flow. Numerical solutions of this equation are presented to study the unstable stage. Experimental results are compared with those obtained from the numerical solutions.
Resumen Se ha mostrado experimentalmente que cuando se deposita una gota en el centro de un sustrato con un gradiente axial de temperatura (más caliente en el centro), por efectos termocapilares se ...genera un flujo hacia afuera de modo que la gota evoluciona hacia un anillo cuyo radio crece con el tiempo. Al alcanzar un radio crítico, la línea de contacto se inestabiliza, mostrando suaves ondulaciones cuyas amplitudes crecen con el tiempo. Utilizando la aproximación de lubricación y adoptando adecuadas variables adimensionales, se obtiene una ecuación diferencial libre de parámetros que gobierna este tipo de flujo termocapilar. En este trabajo se presentan soluciones numéricas de dicha ecuación para estudiar en particular la etapa inestable. Se comparan los resultados experimentales con los obtenidos con las soluciones numéricas.
Jarosite KFe
(SO
)
(OH)
minerals are effective scavengers of potentially toxic elements (PTEs) and are abundant, for example, in acid rock/mine drainage scenarios. The retention process is highly ...relevant for environmental attenuation of heavy metals and metalloids since these are usually highly soluble and thus mobile under acidic conditions. We investigated both macroscopically and at the molecular scale the extent and the effects of concomitant incorporation of As(v) and Pb(ii) into synthetic jarosite at different As/Pb starting molar ratios, using XRD-Rietveld, SEM, ATR-FTIR spectroscopy and wet chemistry. The amount of arsenate substituted in the jarosite structure was larger when Pb(ii) was also incorporated, the former filling up to approximately 33% of the tetrahedral sites normally occupied by SO
, as compared to 21% when Pb(ii) was absent. Similarly, the amount of Pb(ii) incorporated in the structure was larger when As(v) was also taken up. The jarosite unit cell volume increased as higher amounts of As(v) incorporated into its structure, but simultaneous Pb(ii) incorporation seemed to limit this increase due to its smaller size as compared to K
. The extent to which As and Pb can accommodate in the jarosite structure was found to be limited by concentration maxima under the imposed synthesis conditions. At As/Pb ratios up to 1, Pb-As-jarosites were the only crystalline products. Above this ratio, a mixture of Pb-As-jarosite, anglesite (PbSO
) and poorly-crystalline ferric arsenate (AFA) phases was observed. At the highest As/Pb ratio investigated of 1.80 Pb-As-jarosite was no longer formed. Infrared spectroscopy analysis was applied for the first time here to substituted jarosites with both cations and anions, showing spectral changes in the solids as the As/Pb ratio increased: a characteristic As-O doublet at ∼810 and ∼855 cm
was observed upon Pb incorporation, showing an indirect effect of Pb(ii) on the As-O bonds in the jarosite structure. Thus, structural incorporation of Pb plays a pivotal role in the unit cell environment of jarosite to balance the distortion caused by AsO
-for-SO
substitution. The retention processes found in this work have important environmental implications and impacts: through the synergistic incorporation encountered, remediation enhancement of cationic pollutants such as Pb(ii) is possible in a concomitant fashion with As(v) attenuation in acidic mining and metallurgical environments.
In this study, the classical two-dimensional potential VN=12mω2r2+1NrNsin(Nθ), N∈Z+, is considered. At N=1,2, the system is superintegrable and integrable, respectively, whereas for N>2 it exhibits a ...richer chaotic dynamics. For instance, at N=3 it coincides with the Hénon–Heiles system. The periodic, quasi-periodic and chaotic motions are systematically characterized employing time series, Poincaré sections, symmetry lines and the largest Lyapunov exponent as a function of the energy E and the parameter N. Concrete results for the lowest cases N=3,4 are presented in complete detail. This model is used as a benchmark system to estimate the accuracy of the Sparse Identification of Nonlinear Dynamical Systems (SINDy) method, a data-driven algorithm which reconstructs the underlying governing dynamical equations. We pay special attention at the transition from regular motion to chaos and how this influences the precision of the algorithm. In particular, it is shown that SINDy is a robust and stable tool possessing the ability to generate non-trivial approximate analytical expressions for periodic trajectories as well.
•Hénon–Heiles type system is proposed as a benchmark model to estimate SINDy’s accuracy.•Analysis of periodic and chaotic motions is done using extensive analytical tools.•SINDy’s accuracy drops when time series data has finite expansions in the basis.•SINDy offers analytical expressions for periodic trajectories in Hamiltonian system.
In this study, the classical two-dimensional potential
$V_N=\frac{1}{2}\,m\,\omega^2\,r^2 + \frac{1}{N}\,r^N\,\sin(N\,\theta)$, $N \in
{\mathbb Z}^+$, is considered. At $N=1,2$, the system is ...superintegrable and
integrable, respectively, whereas for $N>2$ it exhibits a richer chaotic
dynamics. For instance, at $N=3$ it coincides with the H\'enon-Heiles system.
The periodic, quasi-periodic and chaotic motions are systematically
characterized employing time series, Poincar\'e sections, symmetry lines and
the largest Lyapunov exponent as a function of the energy $E$ and the parameter
$N$. Concrete results for the lowest cases $N=3,4$ are presented in complete
detail. This model is used as a benchmark system to estimate the accuracy of
the Sparse Identification of Nonlinear Dynamical Systems (SINDy) method, a
data-driven algorithm which reconstructs the underlying governing dynamical
equations. We pay special attention at the transition from regular motion to
chaos and how this influences the precision of the algorithm. In particular, it
is shown that SINDy is a robust and stable tool possessing the ability to
generate non-trivial approximate analytical expressions for periodic
trajectories as well.
In this study, the classical two-dimensional potential \(V_N=\frac{1}{2}\,m\,\omega^2\,r^2 + \frac{1}{N}\,r^N\,\sin(N\,\theta)\), \(N \in {\mathbb Z}^+\), is considered. At \(N=1,2\), the system is ...superintegrable and integrable, respectively, whereas for \(N>2\) it exhibits a richer chaotic dynamics. For instance, at \(N=3\) it coincides with the Hénon-Heiles system. The periodic, quasi-periodic and chaotic motions are systematically characterized employing time series, Poincaré sections, symmetry lines and the largest Lyapunov exponent as a function of the energy \(E\) and the parameter \(N\). Concrete results for the lowest cases \(N=3,4\) are presented in complete detail. This model is used as a benchmark system to estimate the accuracy of the Sparse Identification of Nonlinear Dynamical Systems (SINDy) method, a data-driven algorithm which reconstructs the underlying governing dynamical equations. We pay special attention at the transition from regular motion to chaos and how this influences the precision of the algorithm. In particular, it is shown that SINDy is a robust and stable tool possessing the ability to generate non-trivial approximate analytical expressions for periodic trajectories as well.
