Multipseudopotential interaction: a solution for thermodynamic inconsistency in pseudopotential lattice Boltzmann models

Analysis of Microbubble-Blood cell system Oscillation/Cavitation influenced by ultrasound Forces: Conjugate applications of FEM and LBM

Sonoporation is a non-invasive method that uses ultrasound for drug and gene delivery for therapeutic purposes. Here, both Finite Element Method (FEM) and Lattice Boltzmann Method (LBM) are applied to study the interaction physics of microbubble oscillation and collapse near flexible tissue. After validating the Finite Element Method with the nonlinear excited lipid-coated microbubble as well as the Lattice Boltzmann Method with experimental results, we have studied the behavior of a three-dimensional compressible microbubble in the vicinity of tissue. In the FEM phase, the oscillation microbubble with a lipid shell interacts with the boundary. The range of pressure and ultrasound frequency have been considered in the field of therapeutic applications of sonoporation. The viscoelastic and interfacial tension as the coating properties of the microbubble shell have been investigated. The presence of an elastic boundary increases the resonance frequency of the microbubble compared to that of a free microbubble. The increase in pressure leads to an expansion in the range of the microbubble's motion, the velocity induced in the fluid, and the shear stress on the boundary walls of tissue. An enhancement in the surface tension of the microbubble can influence fluid flow and reduce the shear stress on the boundary. The multi-pseudo-potential interaction LBM is used to reduce thermodynamic inconsistency and high-density ratio in a two-phase system for modeling the cavitation process. The three-dimensional shape of the microbubble during the collapse stages and the counter of pressure are displayed. There is a time difference between the occurrence of maximum velocity and pressure. All results in detail are presented in the article bodies.

Equations of state in multiphase lattice Boltzmann method revisited

The single-component multiphase fluids can be described by a single equation of state (EOS), and various EOSs have been employed in the multiphase lattice Boltzmann (LB) method. In this work, we revisit five commonly used EOSs, including the van der Waals EOS, the Redlich-Kwong EOS, the Redlich-Kwong-Soave EOS, the Peng-Robinson EOS, and the Carnahan-Starling EOS. The recent multiphase LB model with self-tuning EOS is employed because of its thermodynamic consistency in a strict sense and clear physical picture at the microscopic level. First, the way to incorporate these multiphase EOSs is proposed. Two scaling factors are introduced to independently adjust the surface tension and interface thickness, and the lattice sound speed is EOS-dependent to ensure the numerical stability. Then, numerical tests are conducted to validate the incorporations of these EOSs and compare their numerical performances. The surface tension and interface thickness are set to the same values for different EOSs in the comparisons. The liquid and gas densities, surface tension, and interface thickness by the LB simulation agree well with the thermodynamic results. The maximum density ratios achieved with different EOSs are at the same level and could be very close to each other when the interface thickness is relatively small. The effects of multiphase EOS, density ratio, and dimensionless relaxation time on the spurious current are discussed in detail. It is interesting to find the van der Waals EOS shows the best numerical performance in reducing the spurious current.

Structure and isotropy of lattice pressure tensors for multirange potentials

We systematically analyze the tensorial structure of the lattice pressure tensors for a class of multiphase lattice Boltzmann models (LBM) with multirange interactions. Due to lattice discrete effects, we show that the built-in isotropy properties of the lattice interaction forces are not necessarily mirrored in the corresponding lattice pressure tensor. This finding opens a different perspective for constructing forcing schemes, achieving the desired isotropy in the lattice pressure tensors via a suitable choice of multirange potentials. As an immediate application, the obtained LBM forcing schemes are tested via numerical simulations of nonideal equilibrium interfaces and are shown to yield weaker and less spatially extended spurious currents with respect to forcing schemes obtained by forcing isotropy requirements only. From a general perspective, the proposed analysis yields an approach for implementing forcing symmetries, never explored so far in the framework of the Shan-Chen method for LBM. We argue this will be beneficial for future studies of nonideal interfaces.

Multipseudopotential interaction models for thermal lattice Boltzmann method simulations

In this work, in the first instance, the multipseudopotential interaction (MPI) model's capabilities are extended for hydrodynamic simulations. This is achieved by combining MPI with the multiple-relaxation-time collision operator and with surface tension modification methods. A method of approaching thermodynamic consistency is also proposed, which consists of splitting the ɛ_{j} term into separate terms. One of these terms is used in the calculation of the interparticle force, and the second one is used in the forcing scheme. Secondly, MPI is combined with thermal models in order to simulate droplet evaporation and bubble nucleation in pool boiling. Thermal coupling is implemented using a double distribution function thermal model and a hybrid thermal model. It is found that MPI thermal models obey the D^{2}-law closely for droplet evaporation. MPI is also found to correctly simulate bubble nucleation and departure from the heating element during nucleate pool boiling. It can be suggested that MPI thermal models are comparatively better suited to thermal simulations at low reduced temperatures than single pseudopotential interaction models, although such cases remain very challenging. Droplet evaporation simulations are carried out at a reduced temperature (T_{r}) of 0.6 by setting the parameters in the Peng-Robinson equation of state to a=1/6272 and b=1/168.

Chemical-potential multiphase lattice Boltzmann method with superlarge density ratios

The liquid-gas density ratio is a key property of multiphase flow methods to model real fluid systems. Here, a chemical-potential multiphase lattice Boltzmann method is constructed to realize extremely large density ratios. The simulations show that the method reaches very low temperatures, at which the liquid-gas density ratio is more than 10^{14}, while the thermodynamic consistency is still preserved. Decoupling the mesh space from the momentum space through a proportional coefficient, a smaller mesh step provides denser lattice nodes to exactly describe the transition region and the resulting dimensional transformation has no loss of accuracy. A compact finite-difference method is applied to calculate the discrete derivatives in the mesh space with high-order accuracy. These enhance the computational accuracy of the nonideal force and suppress the spurious currents to a very low level, even if the density ratio is up to tens of thousands. The simulation of drop splashing verifies that the present model is Galilean invariant for the dynamic flow field. An upper limit of the chemical potential is used to reduce the influence of nonphysical factors and improve the stability.

Central-Moments-Based Lattice Boltzmann for Associating Fluids: A New Integrated Approach

Dynamic and thermodynamic behaviors of associating fluids play a crucial role in various science and engineering disciplines. Cubic plus association equation of state (CPA EOS) is implemented in a central-moments-based lattice Boltzmann method (LBM) in order to mimic the thermodynamic behavior of associating fluids. The pseudopotential approach is selected to model the multiphase thermodynamic characteristics such as reduced density of associating fluids. The priority of central-moments-based approach over multiple-relaxation-time collision operator is highlighted by performing double shear layers. The integration of central-moments-based LBM and CPA EOS is useful to simulate the dynamic and thermodynamic characteristics of associating fluids at high flow rate conditions, which is extended to high-density ratio scenarios by increasing the anisotropy order of gradient operator. In order to increase the stability of the model, a higher anisotropy order of the gradient operator is implemented; about 34 present reduction in spurious velocities is noticed in some cases. The type of gradient operator considerably affects the model thermodynamic consistency. Finally, the model is validated by observing a straight line in the Laplace law test. Prediction of thermodynamic behaviors of associating fluids is of significance in various applications including biological processes as well as fluid flow in porous media.

Thermodynamic consistency of a pseudopotential lattice Boltzmann fluid with interface curvature

Thermodynamic consistency of pseudopotential lattice Boltzmann models is a major topic that needs comprehensive evaluations. When interface is flat, pseudopotential models can give density-pressure isotherms in excellent agreement with those from equation of state. When interface is curved, thermodynamic equilibriums are affected by interface curvature, and consistency of pseudopotential models has not been systematically evaluated. In this study, we show that the effect of Laplace pressure on phase equilibrium is quantitatively consistent with Kelvin equation at high reduced temperatures (≥0.7). At low temperatures, inconsistency that can be attributed to the effect of orientation of the interface was noted, and it can be improved by tuning of the pseudopotential. By relating interfacial tension of a simulated fluid to that of a real fluid, the lattice spacing of pseudopotential model is found to be on the order of several molecular diameters, the typical range of intermolecular interactions. Interfacial thickness at different temperatures in pseudopotential model compared well with experiments and molecular dynamics simulations, which confirms that the calculated length scale is reasonable. Evaluation of a free energy lattice Boltzmann model indicate that it is consistent with Kelvin equation at high temperatures. The free energy model, however, is not as accurate as the tested pseudopotential model, and discrepancies may come from the relative inaccuracies in the predictions of vapor densities and the thinner interfaces.

Hybridized method of pseudopotential lattice Boltzmann and cubic-plus-association equation of state assesses thermodynamic characteristics of associating fluids

It is crucial to properly describe the associating fluids in terms of phase equilibrium behaviors, which are needed for design, operation, and optimization of various chemical and energy processes. Pseudopotential lattice Boltzmann method (LBM) appears to be a reliable and efficient approach to study thermodynamic behaviors and phase transition of complex fluid systems. However, when cubic equations of state (EOSs) are incorporated into single-component multiphase LBM, simulation results are not well matched with experimental data. This study presents the utilization of cubic-plus-association (CPA) EOS in the LBM structure to obtain more accurate modeling results for associating fluids. An approach based on the global search optimization algorithm is introduced to find the optimal association parameters of CPA EOS for water and primary alcohols in the lattice units. The thermodynamic consistency is verified by the Maxwell construction and is also improved by the forcing scheme of [Q. Li, K. H. Luo, and X. J. Li, Phys. Rev. E 86, 016709 (2012)10.1103/PhysRevE.86.016709]. The spurious velocity is reduced with increasing isotropy in the gradient operator. Furthermore, an extended version of CPA EOS is introduced, which increases the system stability at low reduced temperatures. There is a very good match between the LBM results and experimental data, confirming the reliability of the model developed in the present study. The introduced approach has potential to be employed for simulating transport phenomena and interfacial characteristics of associating fluids in porous systems.

Lattice Boltzmann model with self-tuning equation of state for multiphase flows

A lattice Boltzmann (LB) model for multiphase flows is developed that complies with the thermodynamic foundations of kinetic theory. By directly devising the collision term for the LB equation at the discrete level, a self-tuning equation of state is achieved, which can be interpreted as the incorporation of short-range molecular interaction. A pairwise interaction force is introduced to mimic the long-range molecular interaction, which is responsible for interfacial dynamics. The derived pressure tensor is naturally consistent with thermodynamic theory, and surface tension and interface thickness can be independently prescribed. Numerical tests, including static and dynamic cases, are carried out to validate the present model and good results are obtained. As a further application, head-on collision of equal-sized droplets is simulated and the elusive "bouncing" regime is successfully reproduced.

Eliminating cubic terms in the pseudopotential lattice Boltzmann model for multiphase flow

It is well recognized that there exist additional cubic terms of velocity in the lattice Boltzmann (LB) model based on the standard lattice. In this work, elimination of these cubic terms in the pseudopotential LB model for multiphase flow is investigated, where the force term and density gradient are considered. By retaining high-order (≥3) Hermite terms in the equilibrium distribution function and the discrete force term, as well as introducing correction terms in the LB equation, the additional cubic terms of velocity are entirely eliminated. With this technique, the computational simplicity of the pseudopotential LB model is well maintained. Numerical tests, including stationary and moving flat and circular interface problems, are carried out to show the effects of such cubic terms on the simulation of multiphase flow. It is found that the elimination of additional cubic terms is beneficial to reduce the numerical error, especially when the velocity is relatively large. Numerical results also suggest that these cubic terms mainly take effect in the interfacial region and that the density-gradient-related cubic terms are more important than the other cubic terms for multiphase flow.

Transition point prediction in a multicomponent lattice Boltzmann model: Forcing scheme dependencies

Pseudopotential-based lattice Boltzmann models are widely used for numerical simulations of multiphase flows. In the special case of multicomponent systems, the overall dynamics are characterized by the conservation equations for mass and momentum as well as an additional advection diffusion equation for each component. In the present study, we investigate how the latter is affected by the forcing scheme, i.e., by the way the underlying interparticle forces are incorporated into the lattice Boltzmann equation. By comparing two model formulations for pure multicomponent systems, namely the standard model [X. Shan and G. D. Doolen, J. Stat. Phys. 81, 379 (1995)JSTPBS0022-471510.1007/BF02179985] and the explicit forcing model [M. L. Porter et al., Phys. Rev. E 86, 036701 (2012)PLEEE81539-375510.1103/PhysRevE.86.036701], we reveal that the diffusion characteristics drastically change. We derive a generalized, potential function-dependent expression for the transition point from the miscible to the immiscible regime and demonstrate that it is shifted between the models. The theoretical predictions for both the transition point and the mutual diffusion coefficient are validated in simulations of static droplets and decaying sinusoidal concentration waves, respectively. To show the universality of our analysis, two common and one new potential function are investigated. As the shift in the diffusion characteristics directly affects the interfacial properties, we additionally show that phenomena related to the interfacial tension such as the modeling of contact angles are influenced as well.

Chemical-potential-based lattice Boltzmann method for nonideal fluids

Chemical potential, as an important thermodynamic quantity, has been popularly used in thermodynamic modeling for complex systems, especially for those involving the phase transitions and chemical reactions. Here we present a chemical-potential-based multiphase lattice Boltzmann model, in which the nonideal force is directly evaluated by a chemical potential. The numerical computation is more efficient than the pressure-tensor-based model [Wen et al. Europhys. Lett. 112, 44002 (2015)10.1209/0295-5075/112/44002] because the calculations of the pressure tensor and its divergence are avoided. We have derived several chemical potentials of the popular equations of state from the free-energy density function. The theoretical analyses and numerical results support that the present model satisfies thermodynamics and Galilean invariance. An effective chemical-potential boundary condition is also implemented to investigate the wettability of a solid surface, and the contact angle can be linearly tuned by the surface chemical potential.

Thermohydrodynamics of an evaporating droplet studied using a multiphase lattice Boltzmann method

In this paper, the thermohydrodynamics of an evaporating droplet is investigated by using a single-component pseudopotential lattice Boltzmann model. The phase change is applied to the model by adding source terms to the thermal lattice Boltzmann equation in such a way that the macroscopic energy equation of multiphase flows is recovered. In order to gain an exhaustive understanding of the complex hydrodynamics during evaporation, a single droplet is selected as a case study. At first, some tests for a stationary (non-)evaporating droplet are carried out to validate the method. Then the model is used to study the thermohydrodynamics of a falling evaporating droplet. The results show that the model is capable of reproducing the flow dynamics and transport phenomena of a stationary evaporating droplet quite well. Of course, a moving droplet evaporates faster than a stationary one due to the convective transport. Our study shows that our single-component model for simulating a moving evaporating droplet is limited to low Reynolds numbers.

Multipseudopotential interaction: A consistent study of cubic equations of state in lattice Boltzmann models

A method is developed to analytically and consistently implement cubic equations of state into the recently proposed multipseudopotential interaction (MPI) scheme in the class of two-phase lattice Boltzmann (LB) models [S. Khajepor, J. Wen, and B. Chen, Phys. Rev. E 91, 023301 (2015)]10.1103/PhysRevE.91.023301. An MPI forcing term is applied to reduce the constraints on the mathematical shape of the thermodynamically consistent pseudopotentials; this allows the parameters of the MPI forces to be determined analytically without the need of curve fitting or trial and error methods. Attraction and repulsion parts of equations of state (EOSs), representing underlying molecular interactions, are modeled by individual pseudopotentials. Four EOSs, van der Waals, Carnahan-Starling, Peng-Robinson, and Soave-Redlich-Kwong, are investigated and the results show that the developed MPI-LB system can satisfactorily recover the thermodynamic states of interest. The phase interface is predicted analytically and controlled via EOS parameters independently and its effect on the vapor-liquid equilibrium system is studied. The scheme is highly stable to very high density ratios and the accuracy of the results can be enhanced by increasing the interface resolution. The MPI drop is evaluated with regard to surface tension, spurious velocities, isotropy, dynamic behavior, and the stability dependence on the relaxation time.

Assessment of interaction potential in simulating nonisothermal multiphase systems by means of lattice Boltzmann modeling

The pseudopotential lattice Boltzmann model (PP-LBM) is a very popular model for simulating multiphase systems. In this model, phase separation occurs via a short-range attraction between different phases when the interaction potential term is properly chosen. Therefore, the potential term is expected to play a significant role in the model and to affect the accuracy and the stability of the computations. The original PP-LBM suffers from some drawbacks such as being capable of dealing with low density ratios only, thermodynamic inconsistency, and spurious velocities. In this paper, we aim to analyze the PP-LBM with the view to simulate single-component (non-)isothermal multiphase systems at large density ratios and in spite of the presence of spurious velocities. For this purpose, the performance of two popular potential terms and of various implementation schemes for these potential terms is examined. Furthermore, the effects of different parameters (i.e., equation of state, viscosity, etc.) on the simulations are evaluated, and, finally, recommendations for a proper simulation of (non-)isothermal multiphase systems are presented.