A Generalized Finite Volume Method for Density Driven Flows in Porous Media

In this article, we consider a time evolution equation for solute transport, coupled with a pressure equation in space dimension 2. For the numerical discretization, we combine the generalized finite volume method SUSHI on adaptive meshes with a time semi-implicit scheme. In the first part of this article, we present numerical simulations for two problems: a rotating interface between fresh and salt water and a well-known test case proposed by Henry. In the second part, we also introduce heat transfer and perform simulations for a system from the documentation of the software SEAWAT.

Chemical Potential-Induced Wall State Transitions in Plant Cell Growth

The pH/T duality of acidic pH and temperature (T) action for the growth of grass shoots was examined in order to derive the phenomenological equation of wall properties for living plants. By considering non-meristematic growth as a dynamic series of state transitions (STs) in the extending primary wall, the critical exponents were identified, which exhibit a singular behaviour at a critical temperature, critical pH and critical chemical potential (μ) in the form of four power laws: $$f_{\pi } \left( \tau \right) \propto \left| \tau \right|^{\beta - 1}$$ f π τ ∝ τ β - 1 , $$f_{\tau } (\pi ) \propto \left| \pi \right|^{1 - \alpha }$$ f τ ( π ) ∝ π 1 - α , $$g_{\mu } (\tau ) \propto \left| \tau \right|^{ - 2 - \alpha + 2\beta }$$ g μ ( τ ) ∝ τ - 2 - α + 2 β and $$g_{\tau } (\mu ) \propto \left| \mu \right|^{2 - \alpha }$$ g τ ( μ ) ∝ μ 2 - α . The indices α and β are constants, while π and τ represent a reduced pH and reduced temperature, respectively. The convexity relation α + β ≥ 2 for practical pH-based analysis and β ≡ 2 “mean-field” value in microscopic (μ) representation were derived. In this scenario, the magnitude that is decisive is the chemical potential of the H+ ions, which force subsequent STs and growth. Furthermore, observation that the growth rate is generally proportional to the product of the Euler beta functions of T and pH, allowed to determine the hidden content of the Lockhart constant Ф. It turned out that the pH-dependent time evolution equation explains either the monotonic growth or periodic extension that is usually observed—like the one detected in pollen tubes—in a unified account.

Derivation and soliton dynamics of a new non-isospectral and variable-coefficient system

Under investigation in this paper is a new and more general non-isospectral and variable-coefficient non-linear integrodifferential system. Such a system is Lax integrable because of its derivation from the compatibility condition of a generalized linear non-isospectral problem and its accompanied time evolution equation which is generalized in this paper by embedding four arbitrary smooth enough functions. Soliton solutions of the derived system are obtained in the framework of the inverse scattering transform method with a time-varying spectral parameter. It is graphically shown the dynamical evolutions of the obtained soliton solutions possess time-varying amplitudes and that the inelastic collisions can happen between two-soliton solutions.

Lax Integrability and Exact Solutions of a Variable-Coefficient and Nonisospectral AKNS Hierarchy

In this paper, a variable-coefficient and nonisospectral Ablowitz–Kaup–Newell–Segur (vcniAKNS) hierarchy with Lax integrability is constructed by embedding a finite number of differentiable and time-dependent functions into the well-known AKNS spectral problem and its time evolution equation. In the framework of inverse scattering transform method with time-varying spectral parameter, the constructed vcniAKNS hierarchy is solved exactly. As a result, exact solutions and their reduced n-soliton solutions of the vcniAKNS hierarchy are obtained. It is graphically shown that the parity of an embedded time-dependent function has connection with the symmetrical characteristics of the spatial structures and singular points of the obtained one-soliton solutions.

Shear reversal in dense suspensions: the challenge to fabric evolution models from simulation data

Dense suspensions of hard particles are important as industrial or environmental materials (e.g. fresh concrete, food, paint or mud). To date, most constitutive models developed to describe them are, explicitly or effectively, ‘fabric evolution models’ based on: (i) a stress rule connecting the macroscopic stress to a second-rank microstructural fabric tensor $\unicode[STIX]{x1D64C}$; and (ii) a closed time-evolution equation for $\unicode[STIX]{x1D64C}$. In dense suspensions, most of the stress comes from short-ranged pairwise steric or lubrication interactions at near-contacts (suitably defined), so a natural choice for $\unicode[STIX]{x1D64C}$ is the deviatoric second moment of the distribution $P(\boldsymbol{p})$ of the near-contact orientations $\boldsymbol{p}$. Here we test directly whether a closed time-evolution equation for such a $\unicode[STIX]{x1D64C}$ can exist, for the case of inertialess non-Brownian hard spheres in a Newtonian solvent. We perform extensive numerical simulations accessing high levels of detail for the evolution of $P(\boldsymbol{p})$ under shear reversal, providing a stringent test for fabric evolution models. We consider a generic class of these models as defined by Hand (J. Fluid Mech., vol. 13, 1962, pp. 33–46) that assumes little as to the micromechanical behaviour of the suspension and is only constrained by frame indifference. Motivated by the smallness of microstructural anisotropies in the dense regime, we start with linear models in this class and successively consider those increasingly nonlinear in $\unicode[STIX]{x1D64C}$. Based on these results, we suggest that no closed fabric evolution model properly describes the dynamics of the fabric tensor under reversal. We attribute this to the fact that, while a second-rank tensor captures reasonably well the microstructure in steady flows, it gives a poor description during significant parts of the microstructural evolution following shear reversal. Specifically, the truncation of $P(\boldsymbol{p})$ at second spherical harmonic (or second-rank tensor) level describes ellipsoidal distributions of near-contact orientations, whereas on reversal we observe distributions that are markedly four-lobed; moreover, ${\dot{P}}(\boldsymbol{p})$ has oblique axes, not collinear with those of $\unicode[STIX]{x1D64C}$ in the shear plane. This structure probably precludes any adequate closure at second-rank level. Instead, our numerical data suggest that closures involving the coupled evolution of both a fabric tensor and a fourth-rank tensor might be reasonably accurate.

Fractional Effects on the Light Scattering Properties of a Simple Binary Mixture

A fractional generalized hydrodynamic (GH) model of the concentration fluctuations correlation function is analyzed. Our analysis is based on a previously proposed irreversible thermodynamics non-fractional model that describes the first GH deviations in a finite frequency and wavelength regime where the local equilibrium assumption is not valid. Our basic purpose is to generalize this theory to investigate the time fractional effects on the intermediate scattering function (ISF) of a model of a binary mixture. We discuss two different forms of introducing the fractional derivatives into the hydrodynamic time evolution equation of the ISF and examine their consistency in the non-fractional limit. We calculate analytically the fractional intermediate scattering function (FISF) and compare it with the non-fractional one. We find that although the FISFs are in general less than ISF, the FISF may be a significant fraction of the ISF (∼36%) and might be measurable by light or neutron scattering techniques. We show that fractional time derivatives provide a consistent description of this correlation function in the GH regime and reduce to its well-known behavior in the Navier–Stokes limit (NS) where local equilibrium is restored. We also suggest an experimental verification of our results for near equilibrium states. Finally, we summarize the main results of our work and make some further physical remarks.