Onsager's Principle Research Articles

General numerical framework to derive structure preserving reduced order models for thermodynamically consistent reversible-irreversible PDEs

In this paper, we propose a general numerical framework to derive structure-preserving reduced-order models for thermodynamically consistent PDEs. Our numerical framework has two primary features: (a) a systematic way to extract reduced-order models for thermodynamically consistent PDE systems while maintaining their inherent thermodynamic principles and (b) a general process to derive accurate, efficient, and structure-preserving numerical algorithms to solve these reduced-order models. The platform's generality extends to various PDE systems governed by embedded thermodynamic laws, offering a unique approach from several perspectives. First, it utilizes the generalized Onsager principle to transform the thermodynamically consistent PDE system into an equivalent form, where the free energy of the transformed system takes a quadratic form in terms of the state variables. This transformation is known as energy quadratization (EQ). Through EQ, we gain a novel perspective on deriving reduced-order models that continue to respect the energy dissipation law. Secondly, our proposed numerical approach automatically provides algorithms to discretize these reduced-order models. The proposed algorithms are always linear, easy to implement and solve, and uniquely solvable. Furthermore, these algorithms inherently ensure the thermodynamic laws. Our platform offers a distinctive approach for deriving structure-preserving reduced-order models for a wide range of PDE systems with underlying thermodynamic principles.

Thermodynamically consistent hybrid computational models for fluid-particle interactions

We introduce a novel computational framework designed to explore the dynamic interactions between fluid and solid particles or structures immersed in a viscous fluid medium adhering to the generalized Onsager principle. This innovative framework harnesses the power of the phase-field-embedding method, in which each solid component, whether rigid or elastic, is characterized by a volume-preserving phase field. The unified velocity within the fluid-solid ensemble governs the movement of both solid particles and the surrounding fluid, specifically for passive particles. Active particles, however, are not only influenced by this unified velocity but are also driven by their self-propelling velocities. To capture exclusive volume interactions among particles and between particles and boundaries, we employ repulsive potential forces at a coarser scale. These forces effectively model repulsion and collision effects. Rigid particles maintain structural integrity by enforcing a zero velocity gradient tensor within their spatial domains, necessitating the introduction of a constraining stress tensor. In contrast, elastic particles are governed by a quasi-linear constitutive equation describing the elastic stress within their domains, allowing for accurate modeling of their deformations. The motion of solid particles is tracked by monitoring the dynamics of their centers of mass. This approach facilitates the development of a hybrid, thermodynamically consistent hydrodynamic model applicable to both rigid and elastic particles. To numerically solve this thermodynamically consistent model for elastic particles, we present a structure-preserving numerical algorithm. Notably, in the limit of an infinite elastic modulus, this algorithm converges to the one employed for modeling rigid particles. Finally, we substantiate the effectiveness, accuracy, and stability of our proposed scheme through a series of numerical experiments. These experiments not only validate the computational framework but also showcase its capabilities, reinforcing the reliability of our approach.

Expansion Kinetics of Flexible Polymers upon Release from a Disk-Shaped Confinement.

A general theory is developed to explain the expansion kinetics of a polymer released from a confining cavity in a d-dimensional space. At beginning, the decompressed chain undergoes an explosive expansion while keeping the structure resembling a sphere. As the process continues, the chain transitions to a coil conformation, and the expansion significantly slows down. The kinetics are derived by applying Onsager's variational principle. Computer simulations are then conducted in a quasi-two-dimensional space to verify the theory. The average expansion of the chain size exhibits a distinctive sigmoidal variation on a logarithmic scale, characterized by two times and associated exponents that represent the fast and the slow dynamics, respectively. Through an analysis of the kinetic state diagrams, two important universal behaviors are discovered in the two expansion stages. The intersection of the expansion speed curves allows us to define the crossover point between the stages and study its properties. The scaling relations of the characteristic times and exponents are thoroughly investigated under different confining conditions, with the results strongly supporting the theory. Additional calculations conducted in a three-dimensional (3D) space demonstrate the robustness of the proposed theory in describing the kinetics of polymer expansion in both 2D and 3D spaces.

Translocation of a daughter vesicle in a model system of self-reproducing vesicles

Translocation of a daughter vesicle from a mother vesicle through a pore is experimentally studied by many groups using a model system of self-reproducing vesicles. However, the theoretical formulation of the problem is not fully understood. In the present study, we present a theoretical formulation of the process based on our previous work [P. Khunpetch et al., Phys. Fluids 33, 077103 (2021)]. In our previous work, we considered the daughter vesicle as a rigid body. In the present work, however, we allow the daughter vesicle to deform during the expulsion process. We thus derive the free energy constituting of the elastic moduli of both the mother and daughter vesicles, and of pressure-driven contribution. The minimum energy path of the translocation is searched by using the string method. With use of experimentally reasonable values of the elastic moduli, our improved model successfully reproduced the progress of the birthing process where there is no free energy barrier between the initial and the final states. The equations of motion of the daughter vesicle have been derived within the framework of the Onsager principle. We found that the translocation time of the daughter vesicle can be reduced when the pressure inside the mother vesicle increases, or the initial size of the daughter vesicle decreases.

A ternary mixture model with dynamic boundary conditions.

The influence of short-range interactions between a multi-phase, multi-component mixture and a solid wall in confined geometries is crucial in life sciences and engineering. In this work, we extend the Cahn-Hilliard model with dynamic boundary conditions from a binary to a ternary mixture, employing the Onsager principle, which accounts for the cross-coupling between forces and fluxes in both the bulk and surface. Moreover, we have developed a linear, second-order and unconditionally energy-stable numerical scheme for solving the governing equations by utilizing the invariant energy quadratization method. This efficient solver allows us to explore the impacts of wall-mixture interactions and dynamic boundary conditions on phenomena like spontaneous phase separation, coarsening processes and the wettability of droplets on surfaces. We observe that wall-mixture interactions influence not only surface phenomena, such as droplet contact angles, but also patterns deep within the bulk. Additionally, the relaxation rates control the droplet spreading on surfaces. Furthermore, the cross-coupling relaxation rates in the bulk significantly affect coarsening patterns. Our work establishes a comprehensive framework for studying multi-component mixtures in confined geometries.

Constructing custom thermodynamics using deep learning.

One of the most exciting applications of artificial intelligence is automated scientific discovery based on previously amassed data, coupled with restrictions provided by known physical principles, including symmetries and conservation laws. Such automated hypothesis creation and verification can assist scientists in studying complex phenomena, where traditional physical intuition may fail. Here we develop a platform based on a generalized Onsager principle to learn macroscopic dynamical descriptions of arbitrary stochastic dissipative systems directly from observations of their microscopic trajectories. Our method simultaneously constructs reduced thermodynamic coordinates and interprets the dynamics on these coordinates. We demonstrate its effectiveness by studying theoretically and validating experimentally the stretching of long polymer chains in an externally applied field. Specifically, we learn three interpretable thermodynamic coordinates and build a dynamical landscape of polymer stretching, including the identification of stable and transition states and the control of the stretching rate. Our general methodology can be used to address a wide range of scientific and technological applications.

Joint Experimental and Theoretical Study on Deposition Morphologies in Polymer Sessile Droplets.

Although past experimental and theoretical research has made substantial progress in understanding evaporation behaviors in various suspensions, the fundamental mechanism for polymer sessile droplets is still lacking. One critical effect is the molecular weight on the evaporation behaviors. Here, systematic experiments are carried out to investigate the evaporation behavior of polymer droplets under the effects of polymer concentration, evaporation rate, and especially molecular weight. We obtain polymer films with various morphologies with molecular weights ranging from 2 orders of magnitude to 4 orders of magnitude and polymer concentration across 4 orders of magnitude. We further develop a theoretical model based on the Onsager principle to explain the evaporation mechanism from a dynamic perspective. Analysis indicates that increasing molecular weight or polymer concentration enhances the contact angle hysteresis and slows down the evaporation, resulting in the transition from multiring to coffee ring and eventually to uniform films. The findings offer a guideline for achieving the desired deposition patterns via droplet processing techniques.

An energy-stable and conservative numerical method for multicomponent Maxwell–Stefan model with rock compressibility

Numerical simulation of gas flow in porous media is becoming increasingly attractive due to its importance in shale and natural gas production and carbon dioxide sequestration. In this paper, taking molar densities as the primary unknowns rather than the pressure and molar fractions, we propose an alternative formulation of multicomponent Maxwell–Stefan (MS) model with rock compressibility. Benefiting from the definitions of gas and solid free energies, this MS formulation has a distinct feature that it follows an energy dissipation law, and namely, it is consistent with the second law of thermodynamics. Additionally, the formulation obeys the famous Onsager's reciprocal principle. An efficient energy-stable numerical scheme is constructed using the stabilized energy factorization approach for the Helmholtz free energy density and certain carefully designed formulations involving explicit and implicit mixed treatments for the coupling between molar densities, pressure, and porosity. We rigorously prove that the scheme inherits the energy dissipation law at the discrete level. The fully discrete scheme has the ability to ensure the mass conservation law for each component as well as preserve the Onsager's reciprocal principle. Numerical tests are conducted to verify our theories, and in particular, to demonstrate the good performance of the proposed scheme in energy stability and mass conservation as expected from our theories.

Thermodynamically consistent hydrodynamic phase-field computational modeling for fluid-structure interaction with moving contact lines

This paper proposes a novel computational modeling approach to investigate the fluid-structure interactions with moving contact lines. By embracing the generalized Onsager principle, a coupled hydrodynamics and phase field system is introduced that can describe the fluid-structure interactions with moving contact lines in a thermodynamically consistent manner. The resulting partial differential equation (PDE) model consists of the Navier-Stokes equation and a nonlinear Allen-Cahn type equation. Volume conservation is enforced through an additional penalty term. A fully-discrete structure-preserving numerical scheme is proposed by combining several techniques to solve this coupled PDE system effectively and accurately. For the temporal discretization, we utilize the supplementary variable method for preserving the thermodynamic structure and the projection approach for reducing the problem size. Then, we use the finite difference method on the staggered grid for spatial discretization. Furthermore, we have rigorously proved that the proposed numerical scheme based on the second-order backward difference formula respects the original energy stability, i.e., the scheme is energy stable. Additionally, with the aid of the supplementary variable method, the resultant scheme can be transformed into a constrained optimization problem, where the solutions of the supplementary variables are the arguments of the objective function that reaches the optimality. Then the augmented Lagrangian method is introduced in part to bring robustness and efficiency to solving such a constrained optimization problem. Finally, various numerical simulations verify the model's capability and demonstrate the scheme's effectiveness, accuracy, and stability.

High order spatial discretization for variational time implicit schemes: Wasserstein gradient flows and reaction-diffusion systems

We design and compute first-order implicit-in-time variational schemes with high-order spatial discretization for initial value gradient flows in generalized optimal transport metric spaces. We first review some examples of gradient flows in generalized optimal transport spaces from the Onsager principle. We then use a one-step time relaxation optimization problem for time-implicit schemes, namely generalized Jordan-Kinderlehrer-Otto schemes. Their minimizing systems satisfy implicit-in-time schemes for initial value gradient flows with first-order time accuracy. We adopt the first-order optimization scheme ALG2 (Augmented Lagrangian method) and high-order finite element methods in spatial discretization to compute the one-step optimization problem. This allows us to derive the implicit-in-time update of initial value gradient flows iteratively. We remark that the iteration in ALG2 has a simple-to-implement point-wise update based on optimal transport and Onsager's activation functions. The proposed method is unconditionally stable for convex cases. Numerical examples are presented to demonstrate the effectiveness of the methods in two-dimensional PDEs, including Wasserstein gradient flows, Fisher–Kolmogorov-Petrovskii-Piskunov equation, and two and four species reversible reaction-diffusion systems.

Onsager’s Variational Principle for Nonreciprocal Systems with Odd Elasticity

Using Onsager's variational principle, we derive dynamical equations for a nonequilibrium active system with odd elasticity. The elimination of the extra variable that is coupled to the nonequilibrium driving force leads to the nonreciprocal set of equations for the material coordinates. The obtained nonreciprocal equations manifest the physical origin of the odd elastic constants that are proportional to the nonequilibrium force and the friction coefficients. Our approach offers a systematic and consistent way to derive nonreciprocal equations for active matter in which the time-reversal symmetry is broken.

Thermodynamically Consistent Models for Coupled Bulk and Surface Dynamics.

We present a constructive paradigm to derive thermodynamically consistent models coupling the bulk and surface dynamics hierarchically following the generalized Onsager principle. In the model, the bulk and surface thermodynamical variables are allowed to be different and the free energy of the model comprises the bulk, surface, and coupling energy, which can be weakly or strongly non-local. We illustrate the paradigm using a phase field model for binary materials and show that the model includes the existing thermodynamically consistent ones for the binary material system in the literature as special cases. In addition, we present a set of such phase field models for a few selected mobility operators and free energies to show how boundary dynamics impart changes to bulk dynamics and vice verse. As an example, we show numerically how reactive transport on the boundary impacts the dynamics in the bulk using a reactive transport model for binary reactive fluids by adopting a structure-preserving algorithm to solve the model equations in a rectangular domain.

An efficient and physically consistent numerical method for the Maxwell–Stefan–Darcy model of two‐phase flow in porous media

AbstractNumerical modeling of two‐phase flow in porous media has extensive applications in subsurface flow and petroleum industry. A comprehensive Maxwell–Stefan–Darcy (MSD) two‐phase flow model has been developed recently, which takes into consideration the friction between two phases by a thermodynamically consistent way. In this article, we for the first time propose an efficient energy stable numerical method for the MSD model, which can preserve multiple important physical properties of the model. First, the proposed scheme can preserve the original energy dissipation law. This is achieved through a newly‐developed energy factorization approach that leads to linear semi‐implicit discrete chemical potentials. Second, the scheme preserves the famous Onsager's reciprocal principle and the local mass conservation law for both phases by introducing different upwind strategies for two phase saturations and applying the cell‐centered finite volume method to the original formulation of the model. Third, by introducing two auxiliary phase velocities, the scheme has ability to guarantee the positivity of both saturations under proper conditions. Another distinct feature of the scheme is that the resulting discrete system is totally linear, well‐posed and unbiased for each phase. Numerical results are also provided to show the excellent performance of the proposed scheme.

Quasi-incompressible models for binary fluid flows in porous media

We derive a quasi-incompressible hydrodynamic phase field model and consistent physical boundary conditions for flows of binary incompressible fluids of distinct intrinsic densities in porous media subject to an external force using the generalized Onsager principle (GOP). When the external force is conservative, the model not only conserves mass and volume fraction but also dissipates the total mechanical energy with respect to the consistent boundary conditions in a fixed domain. In the case of gravity, an extended family of hydrodynamical phase field models parametrized by a specific volume fraction parameter, ϕˆ, is derived via the GOP in the context of the buoyancy force, which reduces to the previous model at ϕˆ=0. In the case of a constant mobility and equal intrinsic densities in the binary fluid system, the extended model is identical to the original one for any ϕˆ. While ϕˆ is chosen as the spatially averaged volume fraction in the general case, the extended model may not be dissipative and thus different from the model at ϕˆ=0. Numerical examples are presented to highlight the difference between the two models and interfacial instability with respect to distinct densities when the two models are distinct.

Derivation of Two-Fluid Model Based on Onsager Principle.

Using the Onsager variational principle, we study the dynamic coupling between the stress and the composition in a polymer solution. In the original derivation of the two-fluid model of Doi and Onuki the polymer stress was introduced a priori; therefore, a constitutive equation is required to close the equations. Based on our previous study of viscoelastic fluids with homogeneous composition, we start with a dumbbell model for the polymer, and derive all dynamic equations using the Onsager variational principle.

A fully explicit and unconditionally energy-stable scheme for Peng-Robinson VT flash calculation based on dynamic modeling

Since the Peng-Robinson (PR) equation of state (EoS) has proven itself to be one of the most reliable EoS, especially in the chemical and petroleum industries, the flash calculation based on the PR EoS is considered to be a foundation for describing complex compositional flows and for evaluating hydrocarbon reservoirs. Compared to the traditional Pressure-Temperature (PT) flash calculation, the novel Volume-Temperature (VT) flash calculation has become more appealing due to its advantages, such as less sensitivity to primary variables like pressure or volume. However, previous numerical schemes of the VT flash calculation involved many complicated nonlinear systems, which makes convergence hard to achieve. To treat this challenge, a fully explicit and unconditionally energy-stable scheme is proposed in this work. It is known that the dynamic model for VT flash calculation can preserve both the Onsager's reciprocal principle and the energy dissipation law. By combining the dynamic model and the linear semi-implicit scheme, the moles and volume can be updated, with the advantage that the energy-dissipation feature can be preserved at a discrete level unconditionally. Then, with the convex-concave splitting approach and the component-wise iteration framework, the scheme becomes fully explicit. The scheme shows promising potential not only because it inherits the original energy stability to ensure convergence, but it also reduces the implementation burden significantly in some engineering scenarios. A lot of numerical experiments are carried out. The numerical results show good agreement with benchmark data and the energy decaying trend at a very large time step demonstrates the stability and efficiency of the proposed scheme.

Capillary Rising in a Tube with Corners

We studied the dynamics of a fluid rising in a capillary tube with corners. In the cornered tube, unlike the circular tube, fluid rises with two parts, the bulk part where the entire cross-section is occupied by the fluid and the finger part where the cross-section is only partially filled. Using the Onsager principle, we derive coupled time-evolution equations for the two parts. We show the following: (a) At the early stage of rising, the dynamics is dominated by the bulk part and the fluid height h0(t) shows the same behavior as that in the circular tube. (b) At the late stage, the bulk part stops rising but the finger part continues, following the scaling law h1(t) ∼ t1/3. We also show that, due to the coupling between the two parts, the equilibrium bulk height is smaller than the Jurin's height, which ignores the effect of the finger part.

Variational approach to droplet transport via bendotaxis: Thin film dynamics and model reduction

Bendotaxis has recently been proposed as a mechanism for self-transport of droplets at small scales. When an active droplet undergoes self-transport via bendotaxis, interfacial, elastic, and active forces jointly determine the droplet motion in a deformable channel. Through simulations of thin-film dynamics and a model reduction based on Onsager's variational principle, we show that wettability and activity can jointly operate to enhance or weaken the self-transport effect of bendotaxis, depending on the sign of wettability (hydrophobic or hydrophilic) on the channel wall and the sign of activity (contractile or extensile) in the droplet.

Evolution and mechanism of dissolutive wetting between hot metal and carbon brick

The dissolutive wetting between hot metal and carbon brick is a complex problem from the viewpoint of carbon brick erosion. The current work aims to clarify the mechanism of dissolutive wetting at the interface between hot metal and carbon brick. The interfacial morphology is analyzed using SEM and EDS, and the change in wetting angle is studied using CCD camera and ImageJ software. Also, the interface shape model is established, the coupling control mechanism of dissolution and wetting is explored. The results reveal that the concave-like interface is formed after dissolutive wetting and the concave depth decreases with the increase of carbon content in hot metal. Moreover, a large amount of graphite flakes appear at the interface between hot metal and carbon brick after dissolutive wetting. The upper wetting angle initially exhibits a rapid decrease with time, followed by a gradual decrease. The lower wetting angle increases with time. The overall wetting angle initially decreases with time, followed by a stable trend. Hence, the dissolutive wetting can be divided into two stages.Furthermore, the interface shape model is established based on the scaling law and Noyes-Whitney equation to describe the dissolutive wetting. The variation of interface shape can be obtained by the interface shape model. The governing equation is obtained through the Onsager principle to clarify the coupling control mechanism of dissolutive wetting. Finally, wetting is determined by the relative situation of interfacial energy and dissolution free energy. The liquid structure of hot metal is changed due to the migration of carbon atoms from carbon brick to hot metal during dissolutive wetting, altering the free energy and facilitating the transition from the first stage to the second stage.

Projection Method for Droplet Dynamics on Groove-Textured Surface with Merging and Splitting

The geometric motion of small droplets placed on an impermeable textured substrate is mainly driven by the capillary effect, the competition among surface tensions of three phases at the moving contact lines, and the impermeable substrate obstacle. After introducing an infinite dimensional manifold with an admissible tangent space on the boundary of the manifold, by Onsager's principle for an obstacle problem, we derive the associated parabolic variational inequalities. These variational inequalities can be used to simulate the contact line dynamics with unavoidable merging and splitting of droplets due to the impermeable obstacle. To efficiently solve the parabolic variational inequality, we propose an unconditional stable explicit boundary updating scheme coupled with a projection method. The explicit boundary updating efficiently decouples the computation of the motion by mean curvature of the capillary surface and the moving contact lines. Meanwhile, the projection step efficiently splits the difficulties brought by the obstacle and the motion by mean curvature of the capillary surface. Furthermore, we prove the unconditional stability of the scheme and present an accuracy check. The convergence of the proposed scheme is also proved using a nonlinear Trotter-Kato's product formula under the pinning contact line assumption. After incorporating the phase transition information at splitting points, several challenging examples including splitting and merging of droplets are demonstrated.