Brownian Motion Research Articles

Stochastic soliton and rogue wave solutions of the nonlinear Schrödinger equation with white Gaussian noise

In this letter, we propose a novel integrable nonlinear Schrödinger equation and its Lax pair, influenced by Brownian motion and white Gaussian noise. We aim to construct and solve new integrable systems affected by the white Gaussian noise. Utilising the classical and generalised Darboux transformations, the stochastic soliton solutions and the stochastic rogue wave solutions of this novel integrable nonlinear Schrödinger equation are obtained and expressed in determinant form. Studies of stochastic soliton and rogue wave solutions of the NLS equation are essential for complex physical and mathematical phenomena where nonlinear interactions and randomness play crucial roles.

Natural convection and variable fluid properties of tangent hyperbolic nanofluid flow with Cattaneo-Christov theories and heat generation

The main aim behind this work is that we have to determine the enhancement that happens in thermal conductivity by using the Buongiorno model of nanofluid when the nanofluid particles are added to the tangent hyperbolic base fluid moving above the stretched surface. The viscosity of tangent hyperbolic nanofluid is assumed to be temperature-dependent. The modified form of the Fourier law of heat conduction motivated us; therefore, the Cattaneo-Christov heat and mass flux model is considered to observe its significance for heat and mass transport. The chemical reaction and heat generation are also considered to determine their influence on temperature and concentration gradients. This study is crucial because of its application in industrial manufacturing processes such as the coating of wire, the thinning of copper, the production of paper, photographic films, hot rolling, and the purification of crude oil. In electronic cooling systems, the efficient dissipation of heat in smartphones, computers, and data centers is critical to preventing overheating. Nanofluids, with their enhanced thermal properties, can significantly improve heat transfer rates, ensuring better performance, stability, and energy savings. The partial governing equations arising from fluid flow, mass, and heat transfer are transformed into ordinary differential equations via appropriate similarity variables. The obtained ordinary differential equations are solved numerically by using the Runge-Kutta Fehlberg method in the MATLAB software. The effects of the emerging parameters are represented through graphs. According to the results, the velocity profile upsurges due to the natural convection parameters and curvature parameter, while the power law index declines the velocity profile. As the Brownian motion coefficient rises, the concentration profile diminishes while the temperature profile increases. Both the concentration and temperature profiles increase for larger values of the thermophoresis parameter. The concentration profile declines for larger values of chemical reaction parameters, and enhancement happens in the temperature region.

Observation of Brownian elastohydrodynamic forces acting on confined soft colloids

Confined motions in complex environments are ubiquitous in microbiology. These situations invariably involve the intricate coupling between fluid flow, soft boundaries, surface forces, and fluctuations. In the present study, such a coupling is investigated using a method combining holographic microscopy and advanced statistical inference. Specifically, the Brownian motion of soft micrometric oil droplets near rigid walls is quantitatively analyzed. All the key statistical observables are reconstructed with high precision, allowing for nanoscale resolution of local mobilities and femtonewton inference of conservative or nonconservative forces. Strikingly, the analysis reveals the existence of a novel, transient, but large, soft Brownian force. The latter might be of crucial importance for microbiological and nanophysical transport, target finding, or chemical reactions in crowded environments, and hence the whole life machinery.

Fractal and Complex Patterns Existing in Music: Application to the Composition DIAPHONIES of Michael Paouris

This paper reports fractal patterns identified in the complex musical composition DIAPHONIES by Michael Paouris via power-law fractal analysis with sliding-windows of size 1024. From 7,647,232 analysed musical segments of DIAPHONIES, 3,222,832 (42.4%) are fractional Brownian motion (fBm) fractal segments and 4,424,400 (57.6%) are fractional Gaussian noise (fGn) stochastic ones. From the fBm fractal segments 295,294 (9.1%) exhibit strong persistency-P with power-law segments in the range of 2.3≤b≤3. These are the very strong fractal areas in DIAPHONIES. Numerous segments with strong antipersistency 1.7≤b<2 are reported together with segments with AP changes (1.7≤b<2.3). In DIAPHONIES continuous fractal fBm areas are dipped in non-fractal fGn areas of deterministic music. The results are within the fBm fractal areas reported in existing papers. Very importantly, the simple composition called Nocturnal-Angel by Michael Paouris, exhibited limited fBm areas of average b¯=1.98 (σ=0.3), namely of pure statistical, deterministic music, while simultaneously, the fractal analysis profile was completely different from the profiles of DIAPHONIES, hence reinforcing, the fractal findings of DIAPHONIES in relation to trivial music harmony.

Bounded diffusing diffusivities: Brownian yet non-Gaussian diffusion

Brownian yet non-Gaussian diffusion has been recently reported in a huge number of biological and soft matter systems. Meanwhile, an archetypal theoretical model called ‘diffusing diffusivities’ is proposed to interpret it. Based on this spirit of diffusing diffusivities, we extend the original diffusing diffusivities (with the unbounded exponential distribution) to the case that the diffusivity is constructed by a bounded stochastic process, i.e., corresponding diffusivities (with certain upper and lower bounds) obeying bounded power-law distribution. We demonstrate that Brownian yet non-Gaussian diffusion can be reproduced by this bounded diffusing diffusivities, via numerical simulations and analytic derivations. Specifically, the short-time distribution of displacement, as the indicator of the Brownian yet non-Gaussian diffusion, is derived analytically by means of superstatistical approach. This short-time distribution is distinct from the Laplace distribution that appears in the original model. The long-time Gaussian displacement distribution is obtained by utilizing the subordination concept. The bounded diffusing diffusivity here may be beneficial to further understanding the diffusive process of particles in complex and inhomogeneous environments. Our work enriches the diffusing diffusivity family and sheds new light on the concept of the Brownian yet non-Gaussian diffusion under stochastic process.

Nonthermal fluctuations accelerate biomolecular motors

AbstractIntracellular transport is essential for maintaining cellular function. This process is driven by different mechanisms in prokaryotic and eukaryotic cells. In small prokaryotic cells, diffusion is the primary means of transport, while larger eukaryotic cells also rely on active transport by molecular motors such as kinesin and dynein. Recently, it has become evident that, in addition to diffusion based on thermal fluctuations (Brownian motion), which was conventionally considered a diffusion mechanism within living cells, nonthermal fluctuations generated by metabolic activities play a crucial role in intracellular diffusion. Similarly, while molecular motors have been proposed to exploit thermal fluctuations in the environment following the direct observation and manipulation of single molecules, they have also been reported to utilize nonthermal fluctuations in recent years. This review begins with a brief overview of the historical knowledge of diffusive intracellular transport, which has been extended from the thermal fluctuations to the nonthermal fluctuations generated by metabolic activity. It then introduces recent findings on how nonthermal fluctuations accelerate the motion of molecular motors and discusses future perspectives on the general effects of these fluctuations on molecules in living cells.

Mittag–Leffler Fractional Stochastic Integrals and Processes with Applications

We study Mittag–Leffler (ML) fractional integrals involved in the solution processes of a system of coupled fractional stochastic differential equations. We introduce the ML fractional stochastic process as a ML fractional stochastic integral with respect to a standard Brownian motion. We provide some representation formulas of solution processes in terms of Mittag–Leffler fractional integrals and processes. Computable expressions of the mean functions and of the covariances of such processes are specifically given. The application in neuronal modeling is provided, and all involved functions and processes are specifically determined. Numerical evaluations are carried out and some results are shown and discussed.

Heat Transportation of 3D Chemically Reactive Flow of Jeffrey Nanofluid over a Porous Frame with Variable Thermal Conductivity

Nanofluids with variable thermal conductivity can potentially bring about a transformative impact in various industries. They offer adaptive and efficient heat transfer solutions that can adjust to changing conditions and specific requirements. The insertion of nanoparticles into the base fluid significantly changes its properties, affecting thermal conductivity and viscosity. The primary objective of this paper is to analyze the heat transfer rate and three-dimensional bio-convective flow of a non-Newtonian Jeffrey nanofluid across a porous surface with variable thermal conductivity. The investigation also considers the impacts of thermophoresis, Brownian motion, and the Lorentz force. The combine impact of thermal radiation and motile microbes also incorporated in the current study. To model these phenomena, we employ the boundary layer approximation to derive a system of partial differential equations (PDEs). These PDEs are subsequently simplified into more manageable ordinary differential equations (ODEs) using the similarity variables. The numerical analysis is performed via the finite difference approach, which consists of a three-stage Lobatto scheme using MATLAB package. Additionally, important engineering parameters under different constraints-like skin friction, Nusselt number, and Sherwood number—are given in a thorough manner using tabular and graphical representations. The results of this study demonstrate significant enhancements in various aspects, including thermophoresis, Brownian motion, and thermal boundary layer thickness are demonstrated through graphically and in the form of tables. As the thermal radiation parameter increases, the temperature profile rises accordingly. This enhancement in the temperature profile is directly attributable to the higher value of the radiation parameter, which results in a physical increase in temperature. These improvements are attributed to a reduction in viscous forces and an increase in the Brownian diffusion coefficient. This research advances the understanding of non-Newtonian thermally radiative flow with variable thermal conductivity, elucidating the complex behavior of such fluids and providing valuable insights for engineering applications.

Numerical Heat Transfer Analysis of Carreau Nanofluid Flow over Curved Stretched Surface with Nonlinear Effects and Viscous Dissipation.

The article presents a comprehensive study that thoroughly assesses the effects of Carreau fluid flow over a curved stretch surface. The study meticulously considers various factors, including nonlinear thermal radiation, convective boundary conditions, viscous dissipation, thermophoresis, Brownian motion, and nonlinear mixed convection. By using similarity variables, the problem's governing equations are transformed from nonlinear partial differential equations to nonlinear ordinary differential equations, and then the shooting method is applied to obtain the results. The study also rigorously examines the effect of modifying flow factors on velocity, temperature, and concentration in shear-thickening situations through graphical assessment. Furthermore, according to the corresponding values, a table is provided with data on skin friction, Nusselt, and Sherwood numbers. This study offers new perspectives on the complex mechanisms of Carreau fluids flowing over curved surfaces, with potential applications in materials engineering and fluid mechanics.

Real-space diffusion theory from quantum mechanics using analytic continuation

We show that a physical correspondence between Brownian motion and quantum mechanics can be established by formal analytic continuation if Wick rotation (t→it) is replaced by the mirror symmetric, complex conjugate time transformation t→±it. Invariance of the square modulus of the wave function under this transformation reveals a tight connection between Born's interpretation of the probability density and the proper time t2=(−it)(+it), as being both the result of a time symmetry breaking in the informational content of the microscopic (quantum) world, which takes place as we move to a macroscopic level. We find that under the transformation t→±it, the Schrödinger equation and its complex conjugate conforms with a stochastic-mechanical model of the Brownian motion of a particle. From the present analysis, Nelson's osmotic velocity arises naturally and the quantum phase function admits a classical interpretation in terms of a continuous (scalar) potential field that influences the motion of the particle. The maximum phase variation defines the direction of the osmotic and current velocity. Whereas the paper focuses on the correlation between quantum mechanics and Ficks's second law for a constant diffusion coefficient, a discussion is also provided on the possibility of a quantum mechanical formulation for anomalous diffusion, where the diffusion coefficient is a function of space and/or time.

Tangent hyperbolic MHD nanoliquids on non-isothermal stretched sheets: Analyzing the impact of transport parameters, variable fluid properties and convective boundary conditions

The key focus of this research is to look at how tangent hyperbolic magnetohydrodynamic (MHD) nano-liquids move in a non-isothermal coagulated stretched sheet. In this study, we intend to explore how fluid behavior responds to alternations in transport physical parameters. The coagulated sheet is subjected to two distinct boundary conditions to inspect the heat and mass transport rate of the nano liquid. The convective heat boundary condition (CBC) and mass-convective boundary condition (MCBC) are employed to specify the physical conditions at the boundaries of the problem domain. The boundary conditions, responsible for regulating heat flow and mass transfer rates, play a crucial role in shaping the liquid’s behavior. Using a similarity solution, the partial differential equation describing the flow of mass and heat within the system transforms into an interconnected network of non-linear ordinary differential equations. These mathematical equations are subsequently figured out using a numerical finite difference method. This study additionally examines the correlation between thermophoresis and Brownian motion, two fundamental concepts in the dynamics of colloidal suspensions. The study’s findings indicate that the temperature profile increases in all scenarios when the variable thermal conductivity and variable viscosity parameters are increased. In contrast, for the same parameters, the velocity profile is decreased. Further, increasing wall thickness reduces heat dissipation; consequently, the temperature profile similarly affects the velocity power index. The Biot number improves the rates of temperature transfer. These results underscore the significant influence of these parameters in predicting the behavior of MHD tangent hyperbolic nanofluids. This study elucidates the intricate interaction among different physical parameters in the dynamics of nano liquids, which holds significant implications for various industrial applications.

Dynamics of heat transfer in complex fluid systems: Comparative analysis of Jeffrey, Williamson and Maxwell fluids with chemical reactions and mixed convection

This study addresses the complex dynamics of heat transfer in magnetohydrodynamic (MHD) systems involving Jeffrey, Williamson, Maxwell, and Newtonian fluids, focusing on how chemical reactions, activation energy, porosity, and mixed convection impact fluid behavior. The problem is critical due to the significant influence these factors have on industrial processes and applications involving non-Newtonian fluids. The developed a mathematical model represented by partial differential equations (PDEs), which were solved using similarity transformations and the fourth order Runge-Kutta (R-K) method combined with shooting technique, with MATLAB software facilitating the solution process. The results reveal that variations in magnetic field strength, porosity, and buoyancy force significantly affect fluid velocities, while radiation, Brownian motion, and thermophoresis alter temperature profiles. Furthermore, chemical reaction rates, Schmidt number, relaxation constant, and activation energy influence fluid concentrations. Key findings include that increasing porosity and magnetic field strength generally decreases fluid velocity, while higher radiation and Prandtl numbers reduce temperature. Chemical reactions and activation energy decrease fluid concentrations, with non-Newtonian fluids showing more pronounced effects compared to Newtonian fluids. The novelty of this work lies in its comprehensive analysis of multiple interacting parameters and their combined effects on heat transfer in MHD systems, providing insights that extend beyond previous studies in the literature. This research offers valuable implications for optimizing fluid dynamics in various industrial applications, including food processing, ink formulation, and friction reduction.

Molecular modelling of active oil droplet propulsion: Insights from dissipative particle dynamics simulation

This study employed dissipative particle dynamics (DPD) simulations to investigate the self-propelled motion of oil droplets in water–oil–surfactant systems. It is the first attempt to replicate self-propulsion models of oil droplets at the molecular level, contrasting previous simulations focused on Brownian motion and hydrodynamic behavior of colloidal particles. The DPD model reproduced droplet propulsion and visualized internal Marangoni flow, showing that larger droplet radii and greater interfacial tension differences increase propulsion speeds. Additionally, surfactants with stronger oil–oil repulsion enhanced propulsion speed, suggesting that surfactant-induced local structures are crucial for the self-propulsion mechanism.

A computation analysis with heat and mass transfer for micropolar nanofluid in ciliated microchannel: With application in the ductus efferentes

Inherited heat enhancement capabilities and their significance in the field of medical sciences and industry make nanofluids the focus of research nowadays. Furthermore, due to the remarkable advancements in bionanotechnology and its significance in biomedical fields such as drug delivery systems, cancer tumor therapy, bioimaging, and many others, it has emerged as a key research area. Contribution of cilia for the flow in ductus efferentes of human male reproductive tract is elaborated. A novel mathematical scheme is presented for the heat and mass transfer of MHD micropolar nanofluid transport in an asymmetric channel lined with cilia. The pertinent equations of nanofluid transport are exposed to lubrication approximation theory and solution for the physical problem is examined with efficient bvp4c technique in MATLAB. Fluid rheology is explored with the variations of different transport parameters like Hartmann number, Grashof number, Brownian motion, buoyancy, thermophoresis and Darcy number. It is reported that nanofluid transport is affected with rise in the Lorentz force and show reverse behavior with rising permeability. The temperature of the nanofluid in ciliated microchannel is raised with enhanced value of Hartmann number, Grashof number, Prandtl number, and Darcy number while diffusion phenomenon of nanofluid is slowed down with these parameters. Spinning motion of the nanofluid is enhanced with Grashof number and slow down with nanoparticle Grashof number and different behavior is recorded for Darcy and viscosity parameters in different flow regime. Reported investigation presents crucial findings for ciliary transport of micropolar nanofluid and tackled with appropriate selection of micropolar parameter, Brownian motion parameter, thermophoresis and Grashof number. Moreover, this investigation will be handful for cilia-based actuators which work as micro-mixers in controlling the flow in minute bio-sensors and may prove their worth in micro-pumps employed in various drug-delivery systems.

Evaluation of thermal and concentration slip effects on heat and mass transmission of nanofluid over a moving wedge surface using Keller box scheme

The current research is based on the impact of thermal and solutal slip in the boundary layer nanofluid flow through a moving accelerating wedge. The present investigation is considered with the influence of Brownian motion and thermophoresis. Thermal insulation, geothermal engineering, crude oil extraction, and heat exchangers are very important applications of nanofluid movement over a wedge surface with thermal and concentration slip. The suggested mathematical analysis is expressed in terms of partial differential equations (PDEs). These PDEs are transformed into ordinary differential equations via similarity transformation. The Keller Box technique is used to integrate the resultant non-similar equations. The set of discretized and first order differential equations is formed with the help of central difference and the Newton–Raphson technique. The graphical and numerical results are extracted with the help of MATLAB. The numerical results with the influence of the Prandtl factor (Pr), constant moving factor (λ), thermal slip factor (S2), and concentration slip parameter (S2) are interpreted visually and numerically. Graphical representations of velocity, thermal, and mass concentration profiles are analyzed in depth. The solution for skin friction coefficient, heat transport rate, and mass transport rate is calculated. The moving velocity function increases as Pr increases. The rate of slip temperature and slip concentration rate is enhanced for a lower Prandtl factor. The maximum slip behavior in temperature function and fluid concentration slip is deduced for each value of thermal-slip and concentration-slip factors. For high Prandtl and Brownian motion factors, the rate of Nusselt number is enhanced significantly.

Exploring convective entropy optimization and activation energy impact on magneto-nanofluid flow over a vertical stretched plate with Hall current and nonlinear thermal radiation using SQLM

This paper investigates the profound impact of entropy generation and activation energy on the dynamics of hydromagnetic nanofluids, focusing on nonlinear thermal radiation, viscous and Ohmic dissipation, and Hall current effects over a linear stretching sheet. Various nanoparticles are incorporated into the nanofluid formulation using the thermophoresis and Brownian motion model. Through similarity transformation, the system of nonlinear equations is solved with high precision using the Spectral Quasilinearization Method (SQLM). Comprehensive analyses of velocity, cross-flow velocity, temperature, concentration, and entropy generation profiles are conducted for numerous physical parameters. Results indicate that velocity and cross-flow velocity increase with the Hall parameter while temperature and concentration profiles decrease. Both the velocity and cross-flow velocity profiles enhance for mixed convection parameters closer to the sheet, whereas the temperature and concentration profiles exhibit the contrary tendency. Prandtl number diminishes the profiles of horizontal velocity, cross-flow velocity and concentration distributions. Temperature distribution profile hikes for both the Brownian motion parameter and thermophoresis parameter. The activation energy parameter enhances the cross-flow velocity, with a corresponding decline in temperature distribution and a similar trend is observed in the concentration profile. Further, entropy generation profiles increase with higher Brinkman and Reynolds numbers while exhibiting mixed tendencies with the Schmidt number. These findings reveal that Hall current introduces complex interactions within the nanofluid flow, significantly influencing thermal, concentration, and entropy generation characteristics, particularly in magnetic fields and activation energy.

A comparative analysis on the dynamics of solid-liquid interfacial layer and nanoparticle diameter of magnetohydrodynamic nanofluid flow containing spherical and cylindrical-shaped alumina nanoparticles over a stretching curved surface

In this work, a comparative analysis on magnetohydrodynamic nanofluid flow containing spherical and cylindrical-shaped alumina nanoparticles over a stretching curved surface is mainly focused. The effects of Brownian motion, Joule heating, thermophoresis, chemical reaction and activation energy are taken into consideration in this work. It is important to mention that this analysis considers a 4 % volume fraction of the alumina nanoparticles. A thermal convective and zero-mass flux conditions are imposed to scrutinize the heat transfer analysis. The leading equations are transformed into dimensionless form by using suitable similarity variables. A numerical solution is obtained using the shooting technique. The acquired outcomes depict that greater magnetic factor enhances the skin friction coefficient. The greater magnetic factor, thermal Biot number, Eckert number, and interfacial layer thickness augment the heat transfer rate, while a greater nanoparticle diameter diminishes the thermal transfer rate. A higher magnetic factor has an increasing impact on thermal distribution and a reducing impact on velocity distribution. A greater curvature factor enhances both the velocity and thermal distributions. The concentration distribution is enhanced by the higher interfacial layer thickness and activation energy factor, while it is reduced by a higher nanoparticle diameter, Brownian motion factor, and chemical reaction factor. From the comparative analysis, it is found that the velocity, thermal, and concentration distributions are higher for the cylindrical-shaped nanoparticles compared to spherical-shaped nanoparticles.

Galerkin finite element analysis on radiative heat transfer in titanium dioxide-polyalphaolefin nanolubricant past a convergent/divergent channel with non-uniform heat source/sink effect

The current study inspects the radiative heat and mass transport of a polyalphaolefin (PAO) – based nanolubricant flow consisting of titanium dioxide (TiO2) nanoparticles in a convergent/divergent channel with thermophoresis, Brownian motion, and non-uniform heat source/sink effects. The mathematical modelling of the present problem results in a system of complex non-linear partial differential equations (PDEs). The Galerkin finite element method (GFEM) is adopted to solve complex equations and to obtain the solution for the field variables in convergent/divergent channels. The results revealed that as the values of Brownian motion get higher, the temperature dispersal in the channel quickens, whereas the concentration profile slows down. The rise in the value of the thermophoresis parameter increases the thermophoresis force, which causes liquid particles to be dispersed in the stream. The heat transmission rate increases with an increase in radiation parameter values.

Brownian motion in a magneto Thermo-diffusion fluid flow over a semi-circular stretching surface

The current study explores the mass and heat transport analysis of a Casson liquid stream past a curved surface. The current model considers the effect of magnetic strength brought on by the strength of the applied uniform magnetic field. The significance of thermophoresis and Brownian motion are also taken into account using the Buongiorno nano-liquid model. The study of liquid flow over stretching sheets frequently addresses practical issues that have garnered significant attention from researchers due to their importance in various domains, including microfluidics, fibreglass production, manufacturing, transportation, metal extrusion, thermal insulation, glass production, paper manufacturing, and acoustic blasting. The governing partial differential equations (PDEs) are converted into ordinary differential equations (ODEs) using the similarity variables. These equations are numerically solved using the finite difference method (FDM). The concentration, temperature, and velocity graphs were produced by varying the different physical parameters. The upsurge in the magnetic parameter reduces the velocity profile. As the magnetic parameter increases, thermal and concentration profiles upsurge. The decrease in velocity profile can be seen as the Casson parameter rises. The intensification in values of thermophoretic parameter enhances the thermal and concentration profiles. The concentration and thermal profiles reduce as the curvature parameter upsurges.

Gyrotactic micro-organism dusty fluid flow over an extended surface

The conventional magnetohydrodynamics (MHD) equations are modeled for the dusty fluid flow past a stretching sheet embedded in a porous medium. These equations are coupled with motile microorganisms, temperature and concentration equations for both fluid and dust particles dynamical equations. The two-phase dusty fluid flow is subjected to magnetic effect, porosity factor, 2nd order chemical reaction, Brownian diffusion, thermophoretic effect and Arrhenius activation energy. An appropriate transformation is used to reset the system of partial differential equations (PDEs) into non-linear system of ODEs (ordinary differential equations). The flow equations are numerically solved by using the parametric continuation method (PCM). For validity of the applied methodology, the results are compared using numerical and graphical results of Matlab built-in package “BVP4c”. Both results show accuracy up to four decimal places. Furthermore, from graphical results, it can be noticed that the fluid particles interaction parameter causes an attractive force among fluid particles which prevents the dust particles phase flow. This technique can be used for biological separations, filters, fluid dust separators and crude oil purification.