Onsager's Principle Research Articles

Multi-Species Compressible Solver for Non-Continuum Flow Through a Micro-Channel

Compressible multi species flow solver, developed by the authors Kumar and Maheshwari [May 15, 2020. “Viscous Multi-Species Lattice Boltzmann Solver for Simulating Shock-Wave Structure.” Computers and Fluids 203. Article ID 104539], has been extended further to non-continuum flows. The approach uses the Onsager-BGK model, which ensures Onsager's maximum entropy production principle is followed, thus avoids unphysical solutions. Equations for velocity slip and temperature jump have been derived. The viscous slip, thermal slip and diffusion slip coefficients have been successfully validated with 13-moment solver's and experimental data. The departure from equilibrium in such flows has been approximated by a collision probability function (), instead of using the complicated second order perturbation over the Knudsen number (that leads to Burnett equations). This provided the necessary reduction in (viscous, thermal, mass) dissipation and the developed approximate solver is able to predict the thermal and viscous profiles in a non-continuum regime and validates against DSMC simulations. Test cases consisting of flow through micro-channels demonstrated the ability of solver to handle monoatomic, diatomic and poly atomic gases. The developed non-continuum solver is in a finite volume framework and is easily adaptable to legacy in-house Navier–Stokes solvers.

A general framework to derive linear, decoupled and energy-stable schemes for reversible-irreversible thermodynamically consistent models

This paper presents a general numerical platform for designing accurate, efficient, and stable numerical algorithms for incompressible hydrodynamic models that obey thermodynamical laws. The obtained numerical schemes are automatically linear in time. It decouples the hydrodynamic variable and other state variables such that only small-size linear problems need to be solved at each time marching step. Furthermore, if the classical velocity projection method is utilized, the velocity field and pressure field can be decoupled. In the end, only a few elliptic-type equations shall be solved in each time step. This strategy is made possible through a sequence of model reformulations by exploring the models' thermodynamic structures. The generalized Onsager principle directly guides these reformulation procedures. In the reformulated but equivalent models, the reversible and irreversible components can be identified, guiding the numerical platform to decouple the reversible and irreversible dynamics. This eventually leads to decoupled numerical algorithms, given that the coupling terms only involve irreversible dynamics. To further demonstrate the numerical platform's power, we apply it to several specific incompressible hydrodynamic models. The energy stability of the proposed numerical schemes is shown in detail. The second-order accuracy in time is verified numerically through time step refinement tests. Several benchmark numerical examples are presented to further illustrate the proposed numerical framework's accuracy, stability, and efficiency.

Correct definition of internal energy at the phenomenological construction of model of multicomponent continuous medium by methods of nonequilibrium thermodynamics

Taking into account the methods of thermodynamics of irreversible processes using the Onsager principle, a model of a multicomponent continuous medium is constructed, the internal energy of which is "free" of the kinetic energy of diffusion. The model is designed for an imperfect continuous medium with chemical reactions in the field of conservative external forces. Generalized StefanMaxwell relations are obtained, which represent a system of hydrodynamic equations of motion of a mixture with true inertial forces. The proposed thermodynamic technique made it possible to obtain a number of algebraic relations known from the kinetic theory of gases for the transfer coefficients, relating, in particular, the coefficients of multicomponent diffusion with binary diffusivitys, thermal diffusion ratios with thermal diffusion coefficients and multicomponent diffusion coefficients, true (molecular) thermal conductivity coefficient for multicomponent mixture with partial thermal conductivity coefficient, which indicates their versatility. The results obtained are intended for modeling not only liquid imperfect solutions, but also gas-dust mixtures with a finely dispersed dust component.

Crucial Events and Biology

In the last years of the 20th century, the attention of physicists working in statistical physics moved from equilibrium processes characterized by stationary correlation functions and Poisson dynamics to biological processes exhibiting ergodicity breaking. The discovery of these processes raised a debate on whether basic properties such as the Onsager principle had to be abandoned or properly revisited. 1 The discovery of Levy processes led many researchers to replace the conventional central limit theorem with the generalized central limit theorem, responsible for a striking departure from the ordinary Gaussian statistics. The discovery of the processes of self-organization made the study of avalanches become very popular. 2 The observation of turbulent processes led to the discovery of new waiting time distribution densities, characterized by inverse power laws 3 and a new stochastic central limit theorem was invented to explain the emergence of Mittag-Leffler function, which is now widely used for the foundation of fractional derivatives. 4 The traditional Linear Response Theory of Kubo was replaced by a new form of linear response, compatible with the ergodic breakdown of complex systems, and this new form of linear response was used for the foundation of Complexity Matching (CM). 5 I plan to prove that crucial events are responsible for ergodicity breaking and that the CM phenomenon is a manifestation of crucial events. One problem still open in this field of research is the origin of [Formula: see text] noise that is traditionally interpreted as a manifestation of the Mandelbrot Fractional Brownian Motion (FBM). 6 I plan to show that the [Formula: see text] noise proposed in 5 for the foundation of the CM phenomenon has a completely different nature, involving crucial events rather than the FBM infinite memory. A recent result of my research group 7 proved that the progress of autonomic neuropathy makes the heartbeats of healthy individuals, dominated by crucial events, turn into FBM. Quite surprisingly, the same phenomenon of transition from the crucial event to the FBM regime was observed in the germination process of lentils 8 in the absence of light. The transition from Levy to Gauss statistics is supposed to be generated by environmental fluctuations and I will illustrate the experimental and theoretical research works that will shed light into their nature.

OnsagerNet: Learning stable and interpretable dynamics using a generalized Onsager principle

Machine learning is becoming an increasingly popular method for building mathematical models from observations of natural processes. Here, the central challenge is to impart structure into the model parameterization to enforce physical relevance, yet retain a degree of generality so that a large variety of dynamics can be learned. We introduce a novel methodology, based on a data-driven extension of the classical Onsager principle, that strikes a balance between these competing aspects. We demonstrate its efficacy by learning quantitatively accurate and qualitatively faithful reduced order models of the Rayleigh-B\'enard convection equations.

Second order linear decoupled energy dissipation rate preserving schemes for the Cahn-Hilliard-extended-Darcy model

The thermodynamically consistent Cahn-Hilliard-Extended-Darcy (CHED) model has been used to describe transient motion of a binary incompressible fluid flow in porous media. In this paper, we develop a series of linear, second-order, energy-dissipation-rate preserving numerical algorithms for the CHED model based on the energy quadratization strategy. We first extend the incompressible CHED model into a weakly compressible, thermodynamically consistent one using the generalized Onsager principle. Guided by the weakly compressible model, we then devise a couple of linear, second-order, decoupled, semi-discrete, temporal algorithms in the form of projection and the energy quadratization (EQ) method. The fully discrete algorithms are obtained by the use of the second-order finite difference method on staggered grids in space. We show theoretically that the obtained numerical algorithms respect the energy-dissipation-rate and the volume conservation property at the discrete level for any time steps, making them unconditionally energy stable. Mesh refinement tests, coarsening dynamics of binary fluids, and the buoyancy-driven binary fluid motion in porous media are investigated numerically. In buoyancy-driven flow simulations, a new set of inflow and outflow boundary conditions are devised using the model. The numerical results compare well with the results in the literature.

Energy stable modeling of two-phase flow in porous media with fluid–fluid friction force using a Maxwell–Stefan–Darcy approach

Two-phase incompressible flow in porous media plays an important role in various fields including subsurface flow and oil reservoir engineering. Due to the interaction between two phases flowing through the pores, the fluid–fluid friction force may have a significant effect on each phase velocity. In this paper, we propose an energy stable (thermodynamically consistent) Maxwell–Stefan–Darcy model for two-phase flow in porous media, which accounts for the fluid–fluid friction. Different from the classical models of two-phase flow in porous media, the proposed model uses the free energy to characterize the capillarity effect. This allows us to employ the Maxwell–Stefan model to describe the relationships between the driving forces and the friction forces. The driving forces include the pressure gradient and chemical potential gradients, while both fluid–solid and fluid–fluid friction forces are taken into consideration. Thermodynamical consistency is the other interesting merit of the proposed model; that is, it satisfies an energy dissipation law and also obeys the famous Onsager's reciprocal principle. A linear semi-implicit numerical method is also developed to simulate the model. Numerical simulation results are provided to show that the fluid–fluid friction force can improve the oil recovery substantially during the oil displacement process.

Birthing of a daughter vesicle in a model system for self-reproduction vesicles

Sakuma and Imai [Phys. Rev. Lett. 107, 198101 (2011)] established a temperature-controlled cyclic process for a model system of self-reproducing vesicles without feeding. The vesicle generates a smaller inclusion vesicle called “daughter vesicle” inside the original vesicle (we call this “mother vesicle”) and then the daughter vesicle is expelled through a small pore on the mother vesicle. This self-reproducing process is called birthing. In the present study, we present a theoretical model on the birthing process of a single, rigid daughter vesicle through a pore. By using a simple geometric picture, we derive the free energy constituting the material properties of the bending, stretching, and line tension moduli of the mother vesicle, as a function of the distance between the centers of the daughter and mother vesicles, and the size of the daughter vesicle. We see clearly the disappearance of the energy barrier by selecting appropriate moduli. The dynamics of the system is studied by employing the Onsager principle. The results indicate that translocation time decreases as the friction parameter decreases or the initial size of the daughter vesicle decreases.

Onsager's variational principle in active soft matter.

Onsagers variational principle (OVP) was originally proposed by Lars Onsager in 1931 [L. Onsager, Phys. Rev., 1931, 37, 405]. This fundamental principle provides a very powerful tool for formulating thermodynamically consistent models. It can also be employed to find approximate solutions, especially in the study of soft matter dynamics. In this work, OVP is extended and applied to the dynamic modeling of active soft matter such as suspensions of bacteria and aggregates of animal cells. We first extend the general formulation of OVP to active matter dynamics where active forces are included as external non-conservative forces. We then use OVP to analyze the directional motion of individual active units: a molecular motor walking on a stiff biofilament and a toy two-sphere microswimmer. Next we use OVP to formulate a diffuse-interface model for an active polar droplet on a solid substrate. In addition to the generalized hydrodynamic equations for active polar fluids in the bulk region, we have also derived thermodynamically consistent boundary conditions. Finally, we consider the dynamics of a thin active polar droplet under the lubrication approximation. We use OVP to derive a generalized thin film equation and then employ OVP as an approximation tool to find the spreading laws for the thin active polar droplet. By incorporating the activity of biological systems into OVP, we develop a general approach to construct thermodynamically consistent models for better understanding the emergent behaviors of individual animal cells and cell aggregates or tissues.

Theoretical analysis for dynamic contact angle hysteresis on chemically patterned surfaces

A dynamic wetting problem is studied for a moving thin fiber inserted in fluid and with a chemically inhomogeneous surface. A reduced model is derived for contact angle hysteresis by using the Onsager principle as an approximation tool. The model is simple and captures the essential dynamics of the contact angle. From this model, we derive an upper bound of the advancing contact angle and a lower bound of the receding angle, which are verified by numerical simulations. The results are consistent with the quasi-static results. The model can also be used to understand the asymmetric dependence of the advancing and receding contact angles on the fiber velocity, which was observed recently in the physical experiments reported in the work of Guan et al. [“Asymmetric and speed-dependent capillary force hysteresis and relaxation of a suddenly stopped moving contact line,” Phys. Rev. Lett. 116, 066102 (2016)].

Universality of capillary rising in corners

Abstract

Dynamics of active nematic defects on the surface of a sphere.

A nematic liquid crystal confined to the surface of a sphere exhibits topological defects of total charge +2 due to the topological constraint. In equilibrium, the nematic field forms four +1/2 defects, located at the corners of a regular tetrahedron inscribed within the sphere, since this minimizes the Frank elastic energy. If additionally the individual nematogens exhibit self-driven directional motion, the resulting active system creates large-scale flow that drives it out of equilibrium. In particular, the defects now follow complex dynamic trajectories which, depending on the strength of the active forcing, can be periodic (for weak forcing) or chaotic (for strong forcing). In this paper we derive an effective particle theory for this system, in which the topological defects are the degrees of freedom, whose exact equations of motion we subsequently determine. Numerical solutions of these equations confirm previously observed characteristics of their dynamics and clarify the role played by the time dependence of their global rotation. We also show that Onsager's variational principle offers an exceptionally transparent way to derive these dynamical equations, and we explain the defect mobility at the hydrodynamics level.

Formation of Deposition Patterns Induced by the Evaporation of the Restricted Liquid.

Evaporation-induced self-assembly of colloids or suspensions has received increasing attention. Given its critical applications in many fields of science and industry, we report deposition patterns constructed by the evaporation of the restricted aqueous suspension with polystyrene particles at different substrate temperatures and geometric container dimensions. With the temperature increases, the deposition patterns transition from honeycomb to multiring to island, which is attributed to the competition between the particle deposition rate UP and the contact line velocity UCL, and the dimension of the geometric container has an effect on the characteristics of patterns. In this paper, the formation of an ordered multiring pattern is mainly focused on as a result of UP keeping up with UCL such that the entire contact line can be pinned, that is, the periodic stick-slip motion of the contact line and the particle sedimentation. Moreover, based on the Onsager principle, we develop a theoretical model to reveal the physical mechanisms behind the multiring phenomena. The position and spacing of rings are measured, which shows that the theoretical prediction agrees well with experiments. We also find that the ring spacing decays exponentially from center to edge experimentally and theoretically. This may not only help us to understand the formation of the deposition patterns but also assist future design and control in practical applications.

Dynamic Crystallization and Phase Transition in Evaporating Colloidal Droplets.

Evaporating colloidal droplets are an omnipresent phenomenon in nature and engage in many scientific and commercial technologies. Despite their apparent importance, many of the fundamental aspects remain unknown, particularly the relationships between evaporation kinetics, volume fraction, crystallization, and phase transition. Here, we follow the structural evolution and drying dynamics across the liquid-to-solid transition of evaporating colloidal droplets containing polystyrene nanospheres with both spatial and temporal resolutions through the in situ small-angle X-ray scattering and ex situ electron microscopy techniques. We find the unconventional evaporation-driven heterogeneous crystallization and the sequential stacking of face-centered cubic (fcc), random hexagonal close-packed (rhcp), and random close-packed (rcp) superlattice structures. The crystallization and phase transition processes are further elucidated and coordinated with the real-time volume fraction variation, which constitutes a rich and dynamic picture of the self-assembly process. Starting with the Onsager principle, we provide quantitative analysis to the evaporation kinetics, including concentration gradient, gelation, and cavitation. Our findings impart a new mechanism of dynamic nucleation and crystallization and reveal the intimate link between structural heterogeneity and evaporation kinetics.

A second order fully-discrete linear energy stable scheme for a binary compressible viscous fluid model

We present a linear, second order fully discrete numerical scheme on a staggered grid for a thermodynamically consistent hydrodynamic phase field model of binary compressible fluid flows, derived from the generalized Onsager Principle. The hydrodynamic model possesses not only the variational structure in its constitutive equation, but also warrants the mass, linear momentum conservation as well as energy dissipation. We first reformulate the model using the energy quadratization method into an equivalent form and then discretize the reformulated model to obtain a semidiscrete partial differential equation system using the Crank-Nicolson method in time. The semi-discrete numerical scheme preserves the mass conservation and energy dissipation law in time. Then, we discretize the semi-discrete PDE system on a staggered grid in space to arrive at a fully discrete scheme using 2nd order finite difference methods, which respects a discrete energy dissipation law. We prove the unique solvability of the linear system resulting from the fully discrete scheme. Mesh refinements and numerical examples on phase separation due to spinodal decomposition in binary polymeric fluids and interface evolution in the gas-liquid mixture are presented to show the convergence property and the usefulness of the new scheme in applications, respectively.

Dynamics of Taylor Rising.

We study the dynamics of liquid climbing in a narrow and tilting corner, inspired by recent work on liquid transportation on the peristome surface of Nepenthes alata. Considering the balance of gravity, interfacial tension, and viscous force, we derive a partial differential equation for the meniscus profile and numerically study the behavior of the solution for various tilting angles β. We show that the liquid height h( t) at time t satisfies the same scaling law found for the vertical corner, i.e., h( t) ∝ t1/3 for large t, but the coefficient depends on the tilting angle β. The coefficient can be calculated approximately by the Onsager principle, and the result agrees well with that obtained by numerical calculation. Our model can be applied for a weakly curved corner and may provide guidance to the design of biomimetic surfaces for liquid transportation.

Horizontal transportation of a Maltese cross pattern in nematic liquid crystalline droplets under a temperature gradient.

Flow and director fields strongly couple with each other in liquid crystalline systems, and herein we discuss the coupling effect in cylindrical and spherical-cap droplets formed by nematic liquid crystal. Applying a temperature gradient to droplets dispersed in a liquid solvent, we observed a crosslike texture in the droplets moved toward the high-temperature side, indicating that the director field was deformed from equilibrium. Additionally, measurement of the flow field revealed that a convective flow was induced in the droplets under temperature gradient. These results suggested that the director deformation in the droplet was induced by convection. By designing a simplified model based on this, we theoretically analyzed the above phenomenon based on Onsager's variational principle. The results show that the phenomenon was well described by a balance of surface energy gradient with viscous and elastic forces.

Application of the Onsager Principle to the Polar-axial Phenomena in Crystal Physics

An approach is proposed to the determination of nature of a tensor of any rank describing the linear relationship between an external impact on a crystal and the resulting effect. It is shown that, in addition to the Onsager principle, polar nature of the tensor takes place when the external impact on a crystal and the resulting effect are axial in nature and their ranks have values from zero to two, but so that the sum of their ranks is equal to the rank of the physical property described. Analytical expressions are obtained for the considered crystal physical properties both of polar and axial nature for various ranks and various natures of the external impact and the resulting effect. Their application was demonstrated both for the description of the already known properties of crystals, for prediction of new properties of the second and third ranks, and for the establishment of the conditions for their observation. It is concluded that the proposed expansion of the Onsager principle not only increases the number of phenomena covered by it, but also predicts new properties of polar or axial nature – gradient and divergence properties, if the impact on the crystal is inhomogeneous over its volume.

An Energetic Variational Approach for the Cahn–Hilliard Equation with Dynamic Boundary Condition: Model Derivation and Mathematical Analysis

The Cahn--Hilliard equation is a fundamental model that describes phase separation processes of binary mixtures. In recent years, several types of dynamic boundary conditions have been proposed in order to account for possible short-range interactions of the material with the solid wall. Our first aim in this paper is to propose a new class of dynamic boundary conditions for the Cahn--Hilliard equation in a rather general setting. The derivation is based on an energetic variational approach that combines the least action principle and Onsager's principle of maximum energy dissipation. One feature of our model is that it naturally fulfills three important physical constraints such as conservation of mass, dissipation of energy and force balance relations. Next, we provide a comprehensive analysis of the resulting system of partial differential equations. Under suitable assumptions, we prove the existence and uniqueness of global weak/strong solutions to the initial boundary value problem with or without surface diffusion. Furthermore, we establish the uniqueness of asymptotic limit as $t\to+\infty$ and characterize the stability of local energy minimizers for the system.

Landau effective Hamiltonian and its application to magnetic systems

The Landau definition of the effective Hamiltonian (of the nonequilibrium free energy) is realized in a microscopic theory. According to Landau remark, the consideration is based on classical statistical mechanics. In his approach nonequilibrium states coinciding with equilibrium fluctuations are taken into account (the Onsager principle). The definition leads to the exact fulfillment of the Boltzmann principle written in the form with the complete free energy. The considered system is assumed to consist of two subsystems. The first subsystem is an equilibrium one. The second subsystem is a nonequilibrium one and its state is described by quantities that are considered as order parameters. The effective Hamiltonian is calculated near equilibrium in the form of a series in powers of deviations of the order parameters from their equilibrium values. The coefficients of the series are expressed through equilibrium correlation functions of the order parameters. In the final approximation correlations of six and more order parameters are neglected and correlations of four parameters are assumed to be small that leads to the corresponding perturbation theory. The developed theory is compared with the phenomenological Landau theory of phase transitions of the second kind. The obtained results are concretized for paramagnetic-ferromagnetic system. The consideration is restricted by paramagnetic phase.