Mixture Theory Research Articles

Nonequilibrium thermodynamics of non-ideal reaction-diffusion systems: Implications for active self-organization.

We develop a framework describing the dynamics and thermodynamics of open non-ideal reaction-diffusion systems, which embodies Flory-Huggins theories of mixtures and chemical reaction network theories. Our theory elucidates the mechanisms underpinning the emergence of self-organized dissipative structures in these systems. It evaluates the dissipation needed to sustain and control them, discriminating the contributions from each reaction and diffusion process with spatial resolution. It also reveals the role of the reaction network in powering and shaping these structures. We identify particular classes of networks in which diffusion processes always equilibrate within the structures, while dissipation occurs solely due to chemical reactions. The spatial configurations resulting from these processes can be derived by minimizing a kinetic potential, contrasting with the minimization of the thermodynamic free energy in passive systems. This framework opens the way to investigating the energetic cost of phenomena, such as liquid-liquid phase separation, coacervation, and the formation of biomolecular condensates.

Energy balance during Bragg wave resonance by submerged porous breakwaters through a mixture theory-based δ-LES-SPH model

This paper presents a numerical analysis of the time behaviors of mechanical and internal fluid energies during the Bragg wave resonance induced by two-arrayed trapezoidal submerged porous breakwaters based on a reformatted δ-LES-SPH model (Di Mascio et al., 2017). In the present work, a mixture theory is introduced into the δ-LES-SPH model by reformulating the governing equations with the incorporation of a volume fraction. In this approach, the viscous and diffusive terms are also modified by the volume fraction. The energy equation is then written for the presented model highlighting the presence of two additional components compared with the classical δ-LES-SPH formulation: one coming from the fluid compression and another one due to the dissipation both induced by the interaction of the porous structure with the fluid phase. The numerical results are validated by available experimental data for a gravity-driven mass flow passing through a porous dam case and two Bragg wave resonance by two-arrayed submerged trapezoidal porous breakwaters cases. A numerical analysis is then conducted on Bragg wave resonance by two-arrayed trapezoidal porous breakwaters, by investigating the effects of the distance between the two breakwaters and their porosity. Interesting insights about the type and magnitude of dissipation occurring during the wave-structure interaction are captured by analyzing the time evolutions of each energy component.

Development and Validation of Asphalt Pavement Rutting Prediction Model with Transient Temperature Field Considered

This study developed a transient rutting analysis program based on the principles of heat transfer and the viscoelastic–viscoplastic theory of asphalt mixtures. Using the finite element model (FEM) model corresponding to the multi-layer pavement structure and the subprogram interface of ABAQUS software, along with Python programming, the program was designed to estimate rutting under transient temperature field conditions with climate conditions, traffic flow, and structure layer constitutive parameters of pavement considered. Subsequently, the program was used to validate the efficiency of the solution and analyze the influencing factors of internal temperature field and rutting depth within the asphalt pavement structure. Results showed that, based on meteorological data and recommended values of material thermal properties, the FEM of transient temperature field can accurately simulate the temperature field distribution of pavement structure. Based on loading information and dynamic modulus test, the transient temperature variable rutting with FEM proposed in this paper can simulate and calculate the rutting of pavement structure layer with some errors for different pavement conditions from 5% to 40%. Through parametric analysis of the influence factors of temperature field and rutting, the modified transient temperature field rutting prediction program can effectively simulate rutting of pavement structure, and the minor errors were less than 20%.

A numerical study on the oscillatory dynamics of tip vortex cavitation

In this paper, we numerically study the mechanism of the oscillatory flow dynamics associated with the tip vortex cavitation (TVC) over an elliptical hydrofoil section. Using our recently developed three-dimensional variational multiphase flow solver, we investigate the TVC phenomenon at Reynolds number $Re = 8.95 \times 10^5$ via dynamic subgrid-scale modelling and the homogeneous mixture theory. To begin, we examine the grid resolution requirements and introduce a length scale considering both the tip vortex strength and the core radius. This length scale is then employed to non-dimensionalize the spatial resolution in the tip vortex region, the results of which serve as a basis for estimation of the required mesh resolution in large eddy simulations of TVC. We next perform simulations to analyse the dynamical modes of tip vortex cavity oscillation at different cavitation numbers, and compare them with the semi-analytical solution. The breathing mode of cavity surface oscillation is extracted from the spatial-temporal evolution of the cavity's effective radius. The temporally averaged effective radius demonstrates that the columnar cavity experiences a growth region followed by decay as it progresses away from the tip. Further examination of the characteristics of local breathing mode oscillations in the growth and decay regions indicates the alteration of the cavity's oscillatory behaviour as it travels from the growth region to the decay region, with the oscillations within the growth region being characterized by lower frequencies. For representative cavitation numbers $\sigma \in [1.2,2.6]$ , we find that pressure fluctuations exhibit a shift of the spectrum towards lower frequencies as the cavitation number decreases, similar to its influence on breathing mode oscillations. The results indicate the existence of correlations between the breathing mode oscillations and the pressure fluctuations. While the low-frequency pressure fluctuations are found to be correlated with the growth region, the breathing mode oscillations within the decay region are related to higher-frequency pressure fluctuations. The proposed mechanism can play an important role in developing mitigation strategies for TVC, which can reduce the underwater radiated noise by marine propellers.

Density functional theory for responsive hard-sphere fluids

We investigate the equilibrium properties of responsive hard-sphere fluids, where the particle size is a coarse-grained property that fluctuates within an internal free-energy landscape, responding to changes in concentration or the application of external fields. For this purpose, we employ a generalised density functional theory for responsive colloids based on the fundamental measure theory for polydisperse hard-sphere mixtures. We find that increasing particle softness yields a substantial reduction in mean particle size, volume fraction and pressure. We also examine the density profiles and size segregation of responsive hard-sphere fluids subjected to three representative external potentials: a planar hard wall, gravitational and Archimedes buoyant fields. We observe that increasing stiffness or particle concentration enhances density oscillations, converging to the behaviour of a monodisperse fluid. In a gravitational field, size segregation occurs, with smaller particles accumulating at the bottom due to sedimentation. This effect is more pronounced when the Archimedes force is included, causing larger colloids to accumulate at the top, leading to density oscillations in this region that intensify with particle softness, an effect not observed in monodisperse systems. Finally, we demonstrate that responsiveness leads to distinct behaviour compared to conventional polydisperse non-responsive fluids, in which particle compression is not allowed.

Two-dimensional nonlinear Brinkman and steady-state Navier–Stokes equations for fluid flow in PCL

indent To remove mucus from the human body, periciliary layer (PCL) is an important region found in the human respiratory system. When a human inhales strange particles along with air into the body, goblet cells inside the epithelial cells secrete mucus to catch those particles and form a mucus layer on the top of the PCL. Since the velocity of the fluid in the PCL and cilia residing in the PCL affect the movement of mucus, in this work, we apply two-dimensional nonlinear Brinkman and steady-state Navier–Stokes equations to find the velocity of the fluid in the PCL. In the equations, the velocity of cilia is also contributed in the mathematical model which perturbs the fluid movement instead of the pressure gradient. Because bundles of cilia are considered in this work rather than a single cilium, the governing equations are derived from the hybrid mixture theory (HMT) which are the equations in a macroscopic scale. The numerical solutions are obtained by using a mixed finite element method of Taylor-Hood type and Newton’s method. We focus on five different angles of cilia that make with the horizontal plane. The velocity of the PCL fluid is presented for each angle. The numerical solutions obtained in this study can be useful in finding the mucus velocity that can help physicians to treat patients who have massive mucus in their lungs.

Development of a smoothed particle hydrodynamics model for porous media flows with enhanced volume conservation and the revisit of the mass conservation equation

Porous media exist extensively in hydraulic and coastal engineering structures, while the modeling of wave/flow interaction with porous media remains challenging. This work develops a smoothed particle hydrodynamics (SPH) model for accurately simulating wave/flow interaction with porous media. The mass and momentum conservation equations incorporating the mixture theory are adopted. The resistant forces of the solid skeleton of porous media on fluid flows are described by the nonlinear empirical formula. The research contributions of the work lie in two aspects. First, two categories of mass conservation equations for porous media flow are revisited and analyzed to examine the influences of the local time derivative term of fluid volume fraction on simulation results. Second, the Volume Conservation Shifting scheme is, for the first time, introduced into SPH to enhance volume conservation for simulating porous media flows. The developed SPH model is validated by an analytical case of seepage flows in a U-tube with porous media and then applied to study four benchmark examples involving both saturated and unsaturated porous media, i.e., dam-break flow through a crushed stone dam, rapid seepage flow through a rockfill dam, solitary wave propagation over a porous seabed, and solitary wave propagation over a submerged porous breakwater. The morphological features and dynamic pressure heads of the porous media flows have been satisfactorily predicted, demonstrating the good accuracy and enhanced volume conservation of the developed SPH model.

Fatigue delamination damage analysis in composite materials through a rule of mixtures approach

The present study investigates delamination damage initiation and propagation within a homogenization theory of mixtures, using the concept of virtual layers and virtual interfaces. It eliminates spatial discretization of layers, introducing a resultant damage variable to capture structure’s bulk response under both monotonic and cyclic loads. Fatigue-induced deterioration is classified into sub-critical, critical, and over-critical stages based on interfacial stresses. Calibration is conducted employing the widely-available Wöhler curves for each loading mode independently. An advance-in-time strategy is included in the model to enhance the simulation speed. The reliability of the approach is assessed for crack initiation and propagation separately through standard test coupons, showing good correlation with experimental data in mode I, mode II, and mixed-mode loading conditions. Depending on the calibration procedure adopted, the model is applicable to a wide range of stress ratios. In addition, it could be integrated into any standard finite element framework using the desired number of elements through the thickness regardless of the physical amount of layers. This allows easy modification of stacking sequences or the number of layers within the constitutive law without mesh structure changes, facilitating simulation of large-scale composite laminates with minimal accuracy loss and reduced computational costs.

Adaptive integration of history variables in constrained mixture models for organ-scale growth and remodeling.

In the last decades, many computational models have been developed to predict soft tissue growth and remodeling (G&R). The constrained mixture theory describes fundamental mechanobiological processes in soft tissue G&R and has been widely adopted in cardiovascular models of G&R. However, even after two decades of work, large organ-scale models are rare, mainly due to high computational costs (model evaluation and memory consumption), especially in long-range simulations. We propose two strategies to adaptively integrate history variables in constrained mixture models to enable large organ-scale simulations of G&R. Both strategies exploit that the influence of deposited tissue on the current mixture decreases over time through degradation. One strategy is independent of external loading, allowing the estimation of the computational resources ahead of the simulation. The other adapts the history snapshots based on the local mechanobiological environment so that the additional integration errors can be controlled and kept negligibly small, even in G&R scenarios with severe perturbations. We analyze the adaptively integrated constrained mixture model on a tissue patch for a parameter study and show the performance under different G&R scenarios. To confirm that adaptive strategies enable large organ-scale examples, we show simulations of different hypertension conditions with a real-world example of a biventricular heart discretized with a finite element mesh. In our example, adaptive integrations sped up simulations by a factor of three and reduced memory requirements to one-sixth. The reduction of the computational costs gets even more pronounced for simulations over longer periods. Adaptive integration of the history variables allows studying more finely resolved models and longer G&R periods while computational costs are drastically reduced and largely constant in time.

Non-linear biphasic mixture model: Existence and uniqueness results

Abstract This paper is concerned with the development and analysis of a mathematical model that is motivated by interstitial hydrodynamics and tissue deformation mechanics (poro-elasto-hydrodynamics) within an in-vitro solid tumour. The classical mixture theory is adopted for mass and momentum balance equations for a two-phase system. A main contribution of this study is we treat the physiological transport parameter (i.e., hydraulic resistivity) as anisotropic and heterogeneous, thus the governing system is strongly coupled and non-linear. We derived a weak formulation and then formulated the equivalent fixed-point problem. This enabled us to use the Galerkin method, and the classical results on monotone operators combined with the well-known Schauder and Banach fixed-point theorems to prove the existence and uniqueness of results.

Experimental study on the effects of temperature on mechanical properties of 3D printed continuous carbon fiber reinforced polymer (C[sbnd]CFRP) composites

Design for safety of 3D printed continuous carbon fiber reinforced polymer (CCFRP) composites remains challenging for accommodating harsh service environments with a wide range of temperatures. To investigate thermal effects on mechanical properties and failure mechanism of CCFRP materials, this study carried out a series of experimental tests on the laminates with layups of [0]n, [90]n and [0/90]n for tension, as well as [±45]n for in-plane shearing. The application of classical laminate theory and rule of mixtures to the 3D printed CCFRP composites under varying temperatures is then evaluated. The results indicate that the transverse elastic modulus/strength and in-plane shear modulus/strength increase at a low temperature, but all the mechanical properties decrease at a high temperature. Notably, an unexpected decrease in strength of [0/90]n laminates is observed when the temperature drops from −10 °C to −40 °C. Significant strain concentrations are visualized during tensile experiments at high temperature through the digital image correlation (DIC) technique. With increasing temperature, the [0]n laminates undergo a transition from an explosive to a jagged failure mode, while the [90]n laminates shift from brittle to ductile failure. The alteration is attributed to decrease in the mechanical properties of both the matrix and the matrix fiber interface, as revealed by scanning electron microscopy (SEM) analysis. It is found that although the classical laminate theory exhibits an acceptable prediction accuracy for the 3D printed CCFRP composites under varying temperature conditions, the rule of mixtures is not applicable. For this reason, the new formulations for the rule of mixtures are then proposed to enable accurate predictions for 3D printed CCFRP composites under different temperatures. This study is anticipated to provide insightful understanding on mechanical properties and failure mechanisms for 3D printed CCFRP composites at different temperatures.

A mixture theory formulation for active suspensions

A mixture theory formulation for active suspensions

Universal phase-field mixture representation of thermodynamics and shock-wave mechanics in porous soft biologic continua.

A continuum mixture theory is formulated for large deformations, thermal effects, phase interactions, and degradation of soft biologic tissues suitable at high pressures and low to very high strain rates. Tissues consist of one or more solid and fluid phases and can demonstrate nonlinear anisotropic elastic, viscoelastic, thermoelastic, and poroelastic physics. Under extreme deformations or shock loading, tissues may fracture, tear, or rupture. Existing models do not account for all physics simultaneously, and most poromechanics and soft-tissue models assume incompressibility of some or all constituents, generally inappropriate for modeling shock waves or extreme compressions. Motivated by these prior limitations, a thermodynamically consistent formulation that combines a continuum theory of mixtures, compressible nonlinear anisotropic thermoelasticity, viscoelasticity, and phase-field mechanics of fracture is constructed to resolve the pertinent physics. A metric tensor of generalized Finsler space supplies geometric insight on effects of rearrangements of microstructure, for example degradation, growth, and remodeling. Shocks are modeled as singular surfaces. Hugoniot states and shock decay are analyzed: Solutions account for concurrent viscoelasticity, fracture, and interphase momentum and energy exchange not all contained in previous analyses. Suitability of the framework for representing blood, skeletal muscle, and liver is demonstrated by agreement with experimental data and observations across a range of loading rates and pressures. Insight into previously unresolved physics is obtained, for example importance of rate sensitivity of damage and quantification of effects of dissipation from viscoelasticity and phase interactions on shock decay.

Artificial Intelligence Using FFNN Models for Computing Soil Complex Permittivity and Diesel Pollution Content

Soil pollution caused by hydrocarbons, such as diesel, poses significant risks to both human health and the ecosystem. The evaluation of soil pollution and various soil engineering applications often relies on the analysis of complex permittivity, encompassing parameters such as dielectric constant and dielectric loss. Various computational models, including theoretical physics-based models, mixture theory models, statistical empirical models, and artificial neural network (ANN) models, have been explored for computing soil complex permittivity and predicting water and pollutant content. Theoretical models require detailed data that is often unavailable, and thus have limited applicability. Mixture models tend to underestimate soil characteristics due to inaccuracies in permittivity estimation of soil phases. While empirical models are widely used, their applicability is restricted to specific soil types, datasets, and locations. ANN models offer promising predictions, accommodating nonlinear phenomena and allowing for missing information and variables. In this study, capacitive electromagnetic electrode sensors were utilized to determine the complex permittivity of soil contaminated with varying levels of diesel at different moisture levels. Theoretical mixture, empirical, and Feed Forward Neural Network (FFNN) models were employed to compute the permittivity of polluted soil based on its phases and to predict the level of diesel pollution. A comparison of these modeling approaches revealed that the FFNN model exhibited the best performance. The ANN model demonstrated superior performance metrics, including a high correlation coefficient and lower mean square error. Specifically, the correlation coefficients for the FFNN model were 0.9942 for training samples, 0.9967 for validation samples, and 0.9977 for test samples. Additionally, the ANN model yielded the lowest mean square error compared to the other three models. Doi: 10.28991/CEJ-2024-010-09-018 Full Text: PDF

Equilibrium solution of two – dimensional non-homogeneous equations in the theory of elastic mixtures

Equilibrium solution of two – dimensional non-homogeneous equations in the theory of elastic mixtures

A poroelastic [formula omitted]-SPH model for modeling biofilm deformation and sloughing in microfluidic channels

We present a poroelastic fluid–structure interaction δ-SPH model for simulating biofilm deformation and sloughing in microfluidic channels. Based on the mixture theory, this model accurately handles large deformations of solid structures and simultaneously simulates seepage flow within the porous biofilm and external flow. A key advantage of the model is its reliance on the solid phase material properties rather than the averaged properties of the biofilm, enabling the modeling of changes in porosity and biofilm material properties induced by loads. Model verification is achieved through two benchmark problems, followed by application to biofilm deformation and sloughing studies. The Young’s modulus of biofilms is a key parameter of interest in many studies. Calibration of the biofilm’s solid material elasticity modulus by using the presented model yields a value of 200 Pa, aligning with previous reports in literature. Additionally, the model is capable of simulating the sloughing process by evaluating local von Mises equivalent stress, reproducing different detachment patterns corresponding to various material strengths. This δ-SPH model offers an efficient numerical tool for biofilm analysis and is extendable to simulate other fluid–structure interaction problems involving porous media with large deformations, such as soil, debris flow, and biological tissues.

Equilibrium Solution of Two – Dimensional Non-Homogeneous Equations in the Theory of Elastic Mixtures

Abstract: The problem of plane elasticity for a doubly connected body with inner and outer boundaries in a regular polygonal form with common centre and parallel sides has been studied. The sides of the polygon were exposed to external forces. The nature of the force term was determined by application of complex variable theory. Kolosov’s method of solution was applied to obtain the biharmonic equation of the forcing term. The forces on the particle were studied under 2-dimensions from which the compatibility and equilibrium equations were derived. The compatibility and equilibrium equations were used to derive the force – stress relations. The results shows that there is a significant relationship between the angle of the force term on the plane of the particle and the stress state of the particle, which is in conformity with existing experimental results.

Entropy Analysis of Implicit Heat Fluxes in Multi-Temperature Mixtures.

We propose new implicit constitutive relations for the heat fluxes of a two-temperature mixture of fluids. These relations are frame-indifferent forms. However, classical explicit forms of the stress tensors and the interaction forces (specified as explicit forms of constitutive relations) as given in mixture theory are used. The focus here is to establish constraints imposed on the implicit terms in the heat fluxes due to the Second Law of Thermodynamics. Our analysis establishes that the magnitude of the explicit entropy production is equal to or greater than that of the implicit entropy production.

On the comparison between phenomenological and kinetic theories of gas mixtures with applications to flocking

We study the comparison between the phenomenological and kinetic models for a mixture of gases from the viewpoint of collective dynamics. In the case in which constituents are Eulerian gases, balance equations for mass, momentum, and energy are the same in the main differential part, but production terms due to the interchanges between constituents are different. They coincide only when the thermal and mechanical diffusion are sufficiently small. In this paper, we first verify that both models satisfy the universal requirements of conservation laws of total mass, momentum, and energy, Galilean invariance and entropy principle. Following the work of Ha and Ruggeri (ARMA 2017), we consider spatially homogeneous models which correspond to the generalizations of the Cucker Smale model with thermal effect. In these circumstances, we provide analytical results for the comparison between two resulting models and also present several numerical simulations to complement analytical results.

Mixture theory for diffuse interface models of two-phase flows

Mixture theory for diffuse interface models of two-phase flows