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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 37 records · Page 2
All sourcesNASA NTRSDOE OSTI
DOE OSTI2019

Concurrent coupling of atomistic simulation and mesoscopic hydrodynamics for flows over soft multi-functional surfaces

We develop an e?cient parallel multiscale method that bridges the atomistic and mesoscale regimes, from nanometers to microns and beyond, via concurrent coupling of atomistic simulation and mesoscopic dynamics. In particular, we combine an all-atom molecular dynamics (MD) description for specific atomistic details in the vicinity of the functional surface with a dissipative particle dynamics (DPD) approach that captures mesoscopic hydrodynamics in the domain away from the functional surface. In order to achieve a seamless transition in dynamic properties we endow the MD simulation with a DPD thermostat, which is validated against experimental results by modeling water at di?erent temperatures. We then validate the MD-DPD coupling method for transient Couette and Poiseuille flows, demonstrating that the concurrent MD-DPD coupling can resolve accurately the continuumbased analytical solutions. Subsequently, we simulate shear flows over grafted polydimethylsiloxane (PDMS) surfaces (polymer brushes) for various grafting densities, and investigate the slip flow as a function of the shear stress. We verify that a ‘‘universal’’ power law exists for the slip length, in agreement with published results. Having validated the MD-DPD coupling method, we simulate timedependent flows past an endothelial glycocalyx layer (EGL) in a microchannel. Coupled simulation results elucidate the dynamics of the EGL changing from an equilibrium state to a compressed state under shear by aligning the molecular structures along the shear direction. MD-DPD simulation results agree well with results of a single MD simulation, but with the former more than two orders of magnitude faster than the latter for system sizes above one micron.

Wang, Yuying↗
DOE OSTI2016

Dissipative particle dynamics of diffusion-NMR requires high Schmidt-numbers

We present an efficient mesoscale model to simulate the diffusion measurement with nuclear magnetic resonance (NMR). On the level of mesoscopic thermal motion of fluid particles, we couple the Bloch equations with dissipative particle dynamics (DPD). Thereby we establish a physically consistent scaling relation between the diffusion constant measured for DPD-particles and the diffusion constant of a real fluid. The latter is based on a splitting into a centre-of-mass contribution represented by DPD, and an internal contribution which is not resolved in the DPD-level of description. As a consequence, simulating the centre-of-mass contribution with DPD requires high Schmidt numbers. After a verification for fundamental pulse sequences, we apply the NMR-DPD method to NMR diffusion measurements of anisotropic fluids, and of fluids restricted by walls of microfluidic channels. For the latter, the free diffusion and the localisation regime are considered.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2016

Self-Assembly and Critical Aggregation Concentration Measurements of ABA Triblock Copolymers with Varying B Block Types: Model Development, Prediction, and Validation

The dissipative particle dynamics (DPD) simulation technique is a coarse-grained (CG) molecular dynamics-based approach that can effectively capture the hydrodynamics of complex systems while retaining essential information about the structural properties of the molecular species. An advantageous feature of DPD is that it utilizes soft repulsive interactions between the beads, which are CG representation of groups of atoms or molecules. In this study, we used the DPD simulation technique to study the aggregation characteristics of ABA triblock copolymers in aqueous medium. Pluronic polymers (PEG-PPO-PEG) were modeled as two segments of hydrophilic beads and one segment of hydrophobic beads. Tyrosine-derived PEG 5K -b-oligo(desaminotyrosyl tyrosine octyl ester-suberate)-b-PEG 5K (PEG5K-oligo(DTO-SA)-PEG 5K ) block copolymers possess alternate rigid and flexible components along the hydrophobic oligo(DTO-SA) chain, and were modeled as two segments of hydrophilic beads and one segment of hydrophobic, alternate soft and hard beads. The formation, structure, and morphology of the initial aggregation of the polymer molecules in aqueous medium were investigated by following the aggregation dynamics. The dimensions of the aggregates predicted by the computational approach were in good agreement with corresponding results from experiments, for the Pluronic and PEG 5K -oligo(DTO-SA)-PEG 5K block copolymers. In addition, DPD simulations were utilized to determine the critical aggregation concentration (CAC), which was compared with corresponding results from an experimental approach. For Pluronic polymers F68, F88, F108, and F127, the computational results agreed well with experimental measurements of the CAC measurements. For PEG 5K -b-oligo(DTO-SA)-b-PEG 5K block polymers, the complexity in polymer structure made it difficult to directly determine their CAC values via the CG scheme. Therefore, we determined CAC values of a series of triblock copolymers with 3–8 DTO-SA units using DPD simulations, and used these results to predict the CAC values of triblock copolymers with higher molecular weights by extrapolation. In parallel, a PEG 5K -b-oligo(DTO-SA)-b-PEG 5K block copolymer was synthesized, and the CAC value was determined experimentally using the pyrene method. The experimental CAC value agreed well with the CAC value predicted by simulation. Furthermore, these results validate our CG models, and demonstrate an avenue to simulate and predict aggregation characteristics of ABA amphiphilic triblock copolymers with complex structures.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2016

Dissipative Particle Dynamics Simulation Study of Poly(2-Oxazoline)-based Multicompartment Micelle Nanoreactor

We investigate multicompartment micelle consisting of poly(2-oxazoline)-based triblock copolymers for nanoreactor application, using DPD simulation method to characterize the internal structure of micelle and the distribution of reactant. Here, the DPD simulation parameters are determined from the Flory-Huggins interaction parameter (χ FH ). From the snapshots of the micellar structures and radial distribution function of polymer blocks, it is clearly presented that the micelle has the feature of the multicompartment. In addition, by implementing the DPD simulations in the presence of reactants, it is found that Reac-C4 and Reac-OPh are well associated with the hydrophilic shell of the micelle whereas other two reactants such as Reac-Ph and Reac-Cl are not incorporated into the micelle. From our DPD simulations, we confirm that the miscibility (solubility) of reactant with the micelle has strong correlation with the rate of hydrolysis kinetic resolution. Utilizing accurate methods evaluating accurate χ FH parameters for molecular interactions in micelle system, this DPD simulation can have a great potential to predict the micelle structures consisting of designed multiblock copolymers for useful reactions.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2014

Bottom-up derivation of conservative and dissipative interactions for coarse-grained molecular liquids with the conditional reversible work method

Molecular simulations of soft matter systems have been performed in recent years using a variety of systematically coarse-grained models. With these models, structural or thermodynamic properties can be quite accurately represented while the prediction of dynamic properties remains difficult, especially for multi-component systems. In this work, we use constraint molecular dynamics simulations for calculating dissipative pair forces which are used together with conditional reversible work (CRW) conservative forces in dissipative particle dynamics (DPD) simulations. The combined CRW-DPD approach aims to extend the representability of CRW models to dynamic properties and uses a bottom-up approach. Dissipative pair forces are derived from fluctuations of the direct atomistic forces between mapped groups. The conservative CRW potential is obtained from a similar series of constraint dynamics simulations and represents the reversible work performed to couple the direct atomistic interactions between the mapped atom groups. Neopentane, tetrachloromethane, cyclohexane, and n-hexane have been considered as model systems. These molecular liquids are simulated with atomistic molecular dynamics, coarse-grained molecular dynamics, and DPD. We find that the CRW-DPD models reproduce the liquid structure and diffusive dynamics of the liquid systems in reasonable agreement with the atomistic models when using single-site mapping schemes with beads containing five or six heavy atoms. For a two-site representation of n-hexane (3 carbons per bead), time scale separation can no longer be assumed and the DPD approach consequently fails to reproduce the atomistic dynamics.

2-2-DIMETHYLPROPANE↗
DOE OSTI2016

Coarse-grained model of water diffusion and proton conductivity in hydrated polyelectrolyte membrane

Using dissipative particle dynamics (DPD), we simulate nanoscale segregation, water diffusion, and proton conductivity in hydrated sulfonated polystyrene (sPS). We employ a novel model [Lee et al. J. Chem. Theory Comput. 11(9), 4395-4403 (2015)] that incorporates protonation/deprotonation equilibria into DPD simulations. The polymer and water are modeled by coarse-grained beads interacting via short-range soft repulsion and smeared charge electrostatic potentials. The proton is introduced as a separate charged bead that forms dissociable Morse bonds with the base beads representing water and sulfonate anions. Morse bond formation and breakup artificially mimics the Grotthuss mechanism of proton hopping between the bases. The DPD model is parameterized by matching the proton mobility in bulk water, dissociation constant of benzenesulfonic acid, and liquid-liquid equilibrium of water-ethylbenzene solutions. The DPD simulations semi-quantitatively predict nanoscale segregation in the hydrated sPS into hydrophobic and hydrophilic subphases, water self-diffusion, and proton mobility. As the hydration level increases, the hydrophilic subphase exhibits a percolation transition from isolated water clusters to a 3D network. The analysis of hydrophilic subphase connectivity and water diffusion demonstrates the importance of the dynamic percolation effect of formation and breakup of temporary junctions between water clusters. The proposed DPD model qualitatively predicts the ratio of proton to water self-diffusion and its dependence on the hydration level that is in reasonable agreement with experiments.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2019

Morphology and proton diffusion in a coarse-grained model of sulfonated poly(phenylenes)

A coarse-grained model previously used to simulate Nafion using dissipative particle dynamics (DPD) is modified to describe sulfonated Diels-Alder poly(phenylene) (SDAPP) polymers. The model includes a proton-hopping mechanism similar to the Grotthuss mechanism. The intramolecular parameters for SDAPP are derived from atomistic molecular dynamics (MD) simulation using the iterative Boltzmann inversion. The polymer radii of gyration, domain morphologies, and cluster distributions obtained from our DPD model are in good agreement with previous atomistic MD simulations. As found in the atomistic simulations, the DPD simulations predict that the SDAPP nanophase separates into hydrophobic polymer domains and hydrophilic domains that percolate through the system at sufficiently high sulfonation and hydration levels. Increasing sulfonation and/or hydration leads to larger proton and water diffusion constants, in agreement with experimental measurements in SDAPP. In the DPD simulations, the proton hopping (Grotthuss) mechanism becomes important as sulfonation and hydration increase, in qualitative agreement with experiment. The turning on of the hopping mechanism also roughly correlates with the point at which the DPD simulations exhibit clear percolated, hydrophilic domains, demonstrating the important effects of morphology on proton transport.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2014

Dissipative particle dynamics simulation of dilute polymer solutions—Inertial effects and hydrodynamic interactions

We examine the accuracy of dissipative particle dynamics (DPD) simulations of polymers in dilute solutions with hydrodynamic interaction (HI), at the theta point, modeled by setting the DPD conservative interaction between beads to zero. We compare the first normal-mode relaxation time extracted from the DPD simulations with theoretical predictions from a normal-mode analysis for theta chains. We characterize the influence of bead inertia within the coil by a ratio L{sub m}/R{sub g}, where L{sub m} is the ballistic distance over which bead inertia is lost, and R{sub g} is the radius of gyration of the polymer coil, while the HI strength per bead h* is determined by the ratio of bead hydrodynamic radius (r{sub H}) to the equilibrium spring length. We show how to adjust h* through the spring length and monomer mass, and how to optimize the accuracy of DPD for fixed h* by increasing the friction coefficient (γ ≥ 9) and by incorporating a nonlinear distance dependence into the frictional interaction. Even with this optimization, DPD simulations exhibit deviations of over 20% from the theoretical normal-mode predictions for high HI strength with h* ≥ 0.20, for chains with as many as 100 beads, which is a larger deviation than is found for Stochastic rotation dynamics simulations for similar chains lengths and values of h*.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2015

Multiscale Universal Interface: A concurrent framework for coupling heterogeneous solvers

Graphical abstract: - Abstract: Concurrently coupled numerical simulations using heterogeneous solvers are powerful tools for modeling multiscale phenomena. However, major modifications to existing codes are often required to enable such simulations, posing significant difficulties in practice. In this paper we present a C++ library, i.e. the Multiscale Universal Interface (MUI), which is capable of facilitating the coupling effort for a wide range of multiscale simulations. The library adopts a header-only form with minimal external dependency and hence can be easily dropped into existing codes. A data sampler concept is introduced, combined with a hybrid dynamic/static typing mechanism, to create an easily customizable framework for solver-independent data interpretation. The library integrates MPI MPMD support and an asynchronous communication protocol to handle inter-solver information exchange irrespective of the solvers' own MPI awareness. Template metaprogramming is heavily employed to simultaneously improve runtime performance and code flexibility. We validated the library by solving three different multiscale problems, which also serve to demonstrate the flexibility of the framework in handling heterogeneous models and solvers. In the first example, a Couette flow was simulated using two concurrently coupled Smoothed Particle Hydrodynamics (SPH) simulations of different spatial resolutions. In the second example, we coupled the deterministic SPH method with the stochastic Dissipative Particle Dynamics (DPD) method to study the effect of surface grafting on the hydrodynamics properties on the surface. In the third example, we consider conjugate heat transfer between a solid domain and a fluid domain by coupling the particle-based energy-conserving DPD (eDPD) method with the Finite Element Method (FEM)

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2014

Accelerating dissipative particle dynamics simulations for soft matter systems

Dissipative particle dynamics (DPD) is a coarse-grained particle-based simulation method that offers microscopic-scale insights into soft matter systems. We present an efficient implementation of a DPD model for graphical processing units (GPUs). As implemented in the LAMMPS molecular dynamics package, it can run effectively on current-generation supercomputers which often have hybrid nodes containing multi-core CPUs and (one or more) GPUs. Using efficient communication of information between the CPUs and GPUs, DPD interactions can be computed on the GPU while other portions of a full simulation model (boundary conditions, constraints, bonded interactions, diagnostic calculations, etc.) can be performed on the CPU. Our GPU-enhanced runs show a speedup of up to 9.5× versus many-core CPU simulations, and can run scalably across thousands of compute nodes. We briefly discuss how the new GPU implementation was validated against the CPU version for thermodynamics, diffusion, and hydrodynamic behavior. We also highlight large-scale models which the faster DPD implementation has enabled, for studies of monolayer self-assembly and thin-film instabilities.

36 MATERIALS SCIENCE↗
DOE OSTI2016

Pairwise adaptive thermostats for improved accuracy and stability in dissipative particle dynamics

We examine the formulation and numerical treatment of dissipative particle dynamics (DPD) and momentum-conserving molecular dynamics. We show that it is possible to improve both the accuracy and the stability of DPD by employing a pairwise adaptive Langevin thermostat that precisely matches the dynamical characteristics of DPD simulations (e.g., autocorrelation functions) while automatically correcting thermodynamic averages using a negative feedback loop. In the low friction regime, it is possible to replace DPD by a simpler momentum-conserving variant of the Nosé–Hoover–Langevin method based on thermostatting only pairwise interactions; we show that this method has an extra order of accuracy for an important class of observables (a superconvergence result), while also allowing larger timesteps than alternatives. All the methods mentioned in the article are easily implemented. Numerical experiments are performed in both equilibrium and nonequilibrium settings; using Lees–Edwards boundary conditions to induce shear flow.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2026

Kinetic Pathways to Diblock Copolymer Micelles via Thin Film Dissolution and Cosolvent Variation

We investigate how processing pathways control micelle formation during dissolution of lamellar diblock copolymers using dissipative particle dynamics (DPD) simulations and DPD-selfconsistent field theory (DPD-SCFT). Comparing thin film (TF) and cosolvent-assisted (CS) dissolution reveals fundamentally different micellization mechanisms and final states. In the TF pathway, dissolution proceeds through a metastable cylindrical intermediate that dictates final micelle dimensions. The lamellar melt first transforms into cylindrical micelles, which then fragment into spherical micelles. The final spherical radius is intrinsically linked to the cylinder radius by a constant ratio, Rsphere ≈ 1.4Rcylinder. DPD-SCFT confirms that both cylinders and spheres are near their equilibrium dimensions. In contrast, CS dissolution produces kinetically trapped micelles with sizes significantly below equilibrium. Micellization is hindered by high free energy barriers to single-chain exchange and fusion. Growth occurs primarily via fusion, but increasing corona repulsion suppresses further coarsening, preventing equilibration. These results demonstrate that, when starting from a bulk copolymer phase, the metastable cylindrical intermediate plays a central role in determining final micelle size.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2016

cDPD: A new dissipative particle dynamics method for modeling electrokinetic phenomena at the mesoscale

In this work, we develop a “charged” dissipative particle dynamics (cDPD) model for simulating mesoscopic electrokinetic phenomena governed by the stochastic Poisson-Nernst-Planck and the Navier-Stokes equations. In particular, the transport equations of ionic species are incorporated into the DPD framework by introducing extra degrees of freedom and corresponding evolution equations associated with each DPD particle. Diffusion of ionic species driven by the ionic concentration gradient, electrostatic potential gradient, and thermal fluctuations is captured accurately via pairwise fluxes between DPD particles. The electrostatic potential is obtained by solving the Poisson equation on the moving DPD particles iteratively at each time step. For charged surfaces in bounded systems, an effective boundary treatment methodology is developed for imposing both the correct hydrodynamic and electrokinetics boundary conditions in cDPD simulations. To validate the proposed cDPD model and the corresponding boundary conditions, we first study the electrostatic structure in the vicinity of a charged solid surface, i.e., we perform cDPD simulations of the electrostatic double layer and show that our results are in good agreement with the well-known mean-field theoretical solutions. Furthermore, we simulate the electrostatic structure and capacity densities between charged parallel plates in salt solutions with different salt concentrations. Moreover, we employ the proposed methodology to study the electro-osmotic and electro-osmotic/pressure-driven flows in a micro-channel. In the latter case, we simulate the dilute poly-electrolyte solution drifting by electro-osmotic flow in a micro-channel, hence demonstrating the flexibility and capability of this method in studying complex fluids with electrostatic interactions at the micro- and nano-scales.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2020

Direct Phase-Angle Detection for Three-Phase Inverters in Asymmetrical Power Grids

This paper introduces a signal reformation based direct phase-angle detection (DPD-SR) technique for three-phase inverters supporting asymmetrical grids. Asymmetries in three-phase systems may happen because of unbalanced three-phase loads, which can lead to the voltage asymmetry at the terminals of grid-tied inverters. The proposed DPD-SR technique can detect the voltage phase-angle under asymmetrical conditions. The DPD-SR technique also offers precise and rapid phase-angle detection via signal reformation and trigonometric functions. In essence, the phase-angle detection is directly derived from the trigonometric properties of the line-line voltages, while signal reformation is implemented to handle asymmetrical voltages. The proposed method measures two line-line voltages and detects the phase-angle of a three-phase system under both symmetrical and asymmetrical conditions. Unlike classical phase-locked loop (PLL) techniques, DPD-SR does not need any closed-loop PI controller, which significantly reduces the complexity and the delay in detecting the phase-angle. In this paper, the efficacy of the proposed technique is examined in comparison with the state-of-the-art PLLs under asymmetrical conditions via a set of laboratory experiments. Here, the performance of the phase-angle detection is also tested as part of an inverter feeding an asymmetrical grid.

24 POWER TRANSMISSION AND DISTRIBUTION↗
NASA NTRS1996

Noninvasive markers of bone metabolism in the rhesus monkey: normal effects of age and gender

Measurement of bone turnover in conditions such as osteoporosis has been limited by the need for invasive iliac bone biopsy to reliably determine parameters of bone metabolism. Recent advances in the area of serum and urinary markers of bone metabolism have raised the possibility for noninvasive measurements; however, little nonhuman primate data exist for these parameters. The purpose of this experiment was to define the normal range and variability of several of the newer noninvasive bone markers which are currently under investigation in humans. The primary intent was to determine age and gender variability, as well as provide some normative data for future experiments in nonhuman primates. Twenty-four rhesus macaques were divided into equal groups of male and female according to the following age groupings: 3 years, 5-10 years, 15-20 years, and > 25 years. Urine was collected three times daily for a four-day period and measured for several markers of bone turnoverm including pyridinoline (PYD), deoxypyrodinoline (DPD), hydroxyproline, and creatinine. Bone mineral density measurements of the lumbar spine were performed at the beginning and end of the study period. Serum was also obtained at the time of bone densitometry for measurement of osteocalcin levels by radioimmunoassay. There were no significant differences in bone mineral density, urine PYD, or urine DPD based on gender. Bone density was lowest in the youngest animals, peaked in the 15-20-year group, but again decreased in the oldest animals. The osteocalcin, PYD, and DPD levels followed an inversely related pattern to bone density. The most important result was the relative age insensitivity of the ratio of PYD:DPD in monkeys up to age 20 years. Since bone density changes take months or years to become measurable and iliac biopsies are invasive, the PYD/DPD marker ratio may have important implications for rapid noninvasive measurement of the effects of potential treatments for osteoporosis in the non-human primate model.

NASA Discipline Musculoskeletal↗
DOE OSTI2014

Targeting miR-21 enhances the sensitivity of human colon cancer HT-29 cells to chemoradiotherapy in vitro

Highlight: •MiR-21 plays a significant role in 5-FU resistance. •This role might be attributed to targeting of hMSH2 as well as TP and DPD via miR-21 targeted hMSH2. •Indirectly targeted TP and DPD to influence 5-FU chemotherapy sensitivity. -- Abstract: 5-Fluorouracil (5-FU) is a classic chemotherapeutic drug that has been widely used for colorectal cancer treatment, but colorectal cancer cells are often resistant to primary or acquired 5-FU therapy. Several studies have shown that miR-21 is significantly elevated in colorectal cancer. This suggests that this miRNA might play a role in this resistance. In this study, we investigated this possibility and the possible mechanism underlying this role. We showed that forced expression of miR-21 significantly inhibited apoptosis, enhanced cell proliferation, invasion, and colony formation ability, promoted G1/S cell cycle transition and increased the resistance of tumor cells to 5-FU and X radiation in HT-29 colon cancer cells. Furthermore, knockdown of miR-21 reversed these effects on HT-29 cells and increased the sensitivity of HT-29/5-FU to 5-FU chemotherapy. Finally, we showed that miR-21 targeted the human mutS homolog2 (hMSH2), and indirectly regulated the expression of thymidine phosphorylase (TP) and dihydropyrimidine dehydrogenase (DPD). These results demonstrate that miR-21 may play an important role in the 5-FU resistance of colon cancer cells.

62 RADIOLOGY AND NUCLEAR MEDICINE↗
DOE OSTI2015

Smoothed dissipative particle dynamics with angular momentum conservation

Smoothed dissipative particle dynamics (SDPD) combines two popular mesoscopic techniques, the smoothed particle hydrodynamics and dissipative particle dynamics (DPD) methods, and can be considered as an improved dissipative particle dynamics approach. Despite several advantages of the SDPD method over the conventional DPD model, the original formulation of SDPD by Español and Revenga (2003) [9], lacks angular momentum conservation, leading to unphysical results for problems where the conservation of angular momentum is essential. To overcome this limitation, we extend the SDPD method by introducing a particle spin variable such that local and global angular momentum conservation is restored. The new SDPD formulation (SDPD+a) is directly derived from the Navier–Stokes equation for fluids with spin, while thermal fluctuations are incorporated similarly to the DPD method. We test the new SDPD method and demonstrate that it properly reproduces fluid transport coefficients. Also, SDPD with angular momentum conservation is validated using two problems: (i) the Taylor–Couette flow with two immiscible fluids and (ii) a tank-treading vesicle in shear flow with a viscosity contrast between inner and outer fluids. For both problems, the new SDPD method leads to simulation predictions in agreement with the corresponding analytical theories, while the original SDPD method fails to capture properly physical characteristics of the systems due to violation of angular momentum conservation. In conclusion, the extended SDPD method with angular momentum conservation provides a new approach to tackle fluid problems such as multiphase flows and vesicle/cell suspensions, where the conservation of angular momentum is essential.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗