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The fibers are loaded by the cosine component of the external load, when a fiber fails, and due to the local load-sharing nature, its force is shared by surviving neighboring fibers. The results show that the system presents a greater resistance and toughness toward the applied load compared to the classical one.

In this paper, the authors adopt the dynamics of a local load-sharing fiber bundle model in two dimensions under an external load to study scaling law in failure process of composite materials with randomly oriented fibers. The model is based on the fiber bundle model where the fibers are randomly oriented. The system is different to the classical one where the fibers are arranged in parallel with the applied load direction.

The evolution time of the fraction of broken fiber is described by an exponential law with two characteristic times. The latter decrease linearly and exponentially respectively with both applied load and temperature.

Scaling behavior of the broken fiber numbers with the size system shows that the system exhibits a scaling law of Family–Vicsek model with universal exponents.

The model incorporates key factors of membrane such as permeability and resistance, feed concentration, fluid viscosity and pressure differentials. Special emphasis is placed on the fouling mechanisms, including pore blockage and cake layer formation, which significantly impact the efficiency of the filtration process.

In this study, we present a numerical analysis of permeate flux through a membrane, focusing on the intricate dynamics of fouling phenomena. Utilizing the Langevin model, we simulate the permeation process to understand how various operational parameters affect the flux rates.

Our results demonstrate that fouling not only reduces the permeate flux but also alters the membrane’s hydraulic resistance over time. The results show that the increasing of the diffusion process on membrane reduces the clogging phenomenon. Hence, the increases of the transmembrane pressure reduce exponentially blocking pore process.

By analyzing these changes, we provide insights into optimizing membrane performance and developing strategies to mitigate clogging membrane. This research contributes to the field of membrane technology by enhancing our understanding of permeate flux behavior under fouling conditions and offering potential pathways for improving long-term operational sustainability.

The aim of the present paper is to investigate the behavior of collective motion of living biological organisms in the two-dimensional (2D) plane by adopting a new approach based on the use of Langevin dynamics. Langevin dynamics is a powerful tool to study these systems because they present a stochastic process due to collisions between their constituents.

In this paper, the dynamical properties and scaling behavior of self-propelled particles were studied numerically by using Langevin dynamics. These dynamics have been affected by the use of only the alignment zone of radius R.

The results indicated that the system’s velocity increases with time and reaches to finite value at the equilibrium phase.

This result is more consistent with that of Vicsek’s model. However, the system’s velocity decreases exponentially with the applied noise without taking the zero value for the highest noise value.

As well as, the crossover time of the growth kinetic system decreases exponentially with noise.

Scaling behavior has been checked for this system and the corresponding results prove that behavior scales with the same law of the one in Vicsek’s model but with different scaling exponents.

The phase transition observed in Vicsek’s model cannot be reproduced by the Langevin dynamics model, which describes more about the dynamical properties of self-propelled systems.