Numerical Simulations of a Dispersive Model Approximating Free-Surface Euler Equations

In some configurations, dispersion effects must be taken into account to improve the simulation of complex fluid flows. A family of free-surface dispersive models has been derived in Fernández-Nieto et al. (Commun Math Sci 16(05):1169–1202, 2018). The hierarchy of models is based on a Galerkin approach and parameterised by the number of discrete layers along the vertical axis. In this paper we propose some numerical schemes designed for these models in a 1D open channel. The cornerstone of this family of models is the Serre – Green-Naghdi model which has been extensively studied in the literature from both theoretical and numerical points of view. More precisely, the goal is to propose a numerical method for the $$LDNH_2$$ L D N H 2 model that is based on a projection method extended from the one-layer case to any number of layers. To do so, the one-layer case is addressed by means of a projection-correction method applied to a non-standard differential operator. A special attention is paid to boundary conditions. This case is extended to several layers thanks to an original relabelling of the unknowns. In the numerical tests we show the convergence of the method and its accuracy compared to the $$LDNH_0$$ L D N H 0 model.

Analysis of equilibrium dispersive model of liquid chromatography considering a quadratic-type adsorption isotherm

A single-component equilibrium dispersive model (EDM) of liquid chromatography is solved analytically for a quadratic-type adsorption isotherm. The consideration of quadratic isotherm leads to a nonlinear advection-diffusion partial differential equation (PDE) that hinders the derivation of analytical solution. To over come this difficulty, the Hopf-Cole and exponential transformation techniques are applied one after another to convert the given advection-diffusion PDE to a second order linear diffusion equation. These transformations are applied under the assumption of small nonlinearity, or small volumes of injected concentrations, or both. Afterwards, the Fourier transform technique is applied to obtain the analytical solution of the resulting linear diffusion equation. For detailed analysis of the process, numerical temporal moments are obtained from the actual time domain solution. These moments are useful to observe the effects of transport parameters on the shape, height and spreading of the elution peak. A second-order accurate, high resolution semi-discrete finite volume scheme is also utilized to approximate the same model for nonlinear Langmuir isotherms. Analytical and numerical results are compared for different case studies to gain knowledge about the ranges of kinetic parameters for which our analytical results are applicable. The effects of various parameters on the mechanism are analyzed under typical operating conditions available in the liquid chromatography literature.

Generalised Serre–Green–Naghdi equations for open channel and for natural river hydraulics

In this paper, we present a new non-linear dispersive model for open channel and river flows. These equations are the second-order shallow water approximation of the section-averaged (three-dimensional) incompressible and irrotational Euler system. This new asymptotic model generalises the well-known one-dimensional Serre–Green–Naghdi (SGN) equations for rectangular section on uneven bottom to arbitrary channel/river section.

Comparison of coupling constant by using momentum spectra and event shape variables in different interactions

We describe in this paper the quantum chromodynamics prediction to calculate the strong coupling constant by using event shape variables as well as momentum spectra. By fitting the dispersive model and employing our parameters on event shape distribution, we obtain the perturbative value of [Formula: see text] = 0.1305 ± 0.0474 and also the non-perturbative value of α0 = 0.5246 ± 0.0516 GeV for electron–proton interactions. Next, by using momentum spectra for the same interactions, we obtain αs = 0.1572 ± 0.029. Our values in both methods are consistent with those obtained from electron–positron annihilations measured previously. When we find coupling constant for different flavours, we observe that they do not affect our results considerably. This is in accordance with quantum chromodynamics theory. All these features will be explained in the main text.