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Inspired by significant physiological applications of peristaltic transport, a mathematical model for the peristaltic pumping of a Jeffrey nanofluid containing gyrotactic microorganisms through a flexible and vertical symmetric channel is investigated, along with the effects of thermal radiation, heat source/sink, chemical reaction, porous medium and Hall current. Swimming microorganisms in a model of non-Newtonian fluid has several biological and ecological benefits, including in the pharmaceutical industry, biofertilizers, biofuel technology, biosensors, etc. With these presumptions in mind, a conducting Jeffery fluid flow containing microorganisms via a porous vertical conduit is proposed. The expressions for the flow quantities are established by solving the governing equations of this study analytically using the homotopy perturbation method (HPM). A detailed examination is performed using the tables and graphical representations to comprehend the impact of significant components on this analysis. In addition, the results for Newtonian nanofluid are obtained as a special case and discovered that its velocity is lower than the Jeffrey nanofluid. It is observed that the best velocity distribution can be seen at larger thermal Grashof number and Darcy number. The size of the trapped bolus enhances with the increase of the Hall parameter. Furthermore, the density of the motile microorganism distribution declines for higher bioconvection Peclet number and bioconvection constant. [ABSTRACT FROM AUTHOR]

The novelty of this paper is examining the impacts of Hall currents on Sisko nanofluid slip flow generated by an exponentially stretching sheet in the existence of entropy generation. Entropy generation assessment, Brownian, thermophoresis and Bejan numbers during convection flow are explored in the presence of the Hall effect. The pertinent parameters are Hall parameter m, a magnetic field parameter M, Brownian motion parameter N b , thermophoresis parameter N t , generalized thermal Biot number γ 1 , generalized velocity slip parameter λ and material parameter of Sisko fluid A. The changes in pertinent parameters on principal and secondary velocities, entropy generation and temperature are discussed. The coefficients of local skin friction, Brownian, thermophoresis and Nusselt/Sherwood numbers are expressed in a tabular structure. Two different kinds of power-law nanofluid, Pseudoplastic (n = 0. 7) and dilatant fluid (n = 1. 5) are considered. The current results exposed that growth in magnetic field parameter diminishes the nanofluid axial velocity and boosts the profiles of temperature, secondary nanofluid velocity and nanoparticle volume fraction. The coefficients of local skin friction are enhanced according to a growth in the magnetic field factor. The Pseudoplastic fluid supports the velocity, temperature and nanoparticle volume fraction profiles compared to the dilatant fluid. The local entropy generation number, Bejan number, local skin friction coefficients and local Nusselt/Sherwood numbers are influenced by the pertinent parameters. [ABSTRACT FROM AUTHOR]

Due to innumerable applications of nanofluid in heat transfer phenomena as well as engineering processes, the research on this innovative fluid over conventional fluid got popularity among researchers. Here, we have considered a non-Darcy porous surface medium for generalized Eyring–Powell fluid model under thermophoretic as well as Brownian movements. No mass flux phenomena along with heat sink-source and activation energy aspects are scrutinized; characteristics of random motion of fluid and thermophoresis diffusion characteristics for generalized Eyring–Powell fluid are examined briefly. The system of nonlinear PDEs is reduced to nonlinear ODEs by using the similarity variable technique; the numerical scheme bvp4c is used along with shooting technique. The physical aspects like flow of fluid, temperature profile, concentration for variation of concerned parameters have been computed and demonstrated pictorially with the help of graphs. It is observed that the velocity of nanofluid declines by increasing values of the power law index (q) and Eyring–Powell liquid parameters. Augmented values of Brownian moment, thermophoresis parameter as well as heat source rise with increase in temperature of nanofluid. The concentration profile of generalized Eyring–Powell nanofluid deteriorates for higher values of Schmidt number. Moreover, the transportation rate of heat declines against Pr, while it rises by increasing values of N t . [ABSTRACT FROM AUTHOR]

Presently, due to its extraordinary mechanical, thermal, electrical and biomedical facets nanofluids deliver several prospects to exaggerate the propensity of isothermal systems by augmenting the conductivity features of the host fluids. In various areas of the energy partition, nanoparticles show a remarkable measure in energy storage, energy variation, and energy convertible, i.e. thermoelectric plans, petroleum cells, supercapacitors, stellar cells, rechargeable batteries, light-radiating diode and carbon-based light-radiating diode, smart coatings. In this current conversation, we anticipated an unsteady 3D flow of the Sutterby nanofluid consequence of a bidirectional extended surface. To envision the thermophoresis and Brownian motion properties in Sutterby's nanofluid, the Buongiorno association is utilized in an additional refined technique. Variable thermal conductivity with heat source/sink property occurred deliberated considering heat transmission techniques. The appropriate transformation is applied for transposing the PDEs into nonlinear ODEs. For numerical results, the bvp4c programmed is prerequisite for elucidating the subsequent Ordinary differential equations. The distinct performance of the Sutterby nanofluid temperature and the concentration field are designated and discussed in the physical parameter's aspect. It is clear that the temperature of the Sutterby fluid decreases with respect to the ratio of stretching rates parameter and similar developments are observed for the thermophoresis and Brownian motion parameters. Furthermore, the concentration profile declines for sophisticated estimates of the Lewis number and thermophoresis parameters. [ABSTRACT FROM AUTHOR]

The aim of this study is to investigate the effects of thermal radiation and chemical reactions on magnetohydrodynamic hyperbolic tangent liquid, which includes nanoparticles on a stretched surface while taking into account Brownian motion and thermophoresis. The nonlinear partial differential equations governing the system are converted into nonlinear ordinary differential equations through suitable similarity transformations. The focus of the study is to elucidate important engineering concepts such as skin friction, Sherwood number, and heat transfer, as well as to understand the effects of various expressions on the different profiles. The Keller-box approach, a sophisticated numerical tool, is used to get the numerical answers to the current enquiry. The generated findings are extensively tested for correctness and dependability. The findings of this study might have far-reaching ramifications for a variety of technical applications, including heat exchangers, chemical reactors, and thermal management systems.The results show that the rate of mass transfer rises with the increment in the factors of chemical reaction, thermal radiation, nanoparticles volume, and Brownian motion. [ABSTRACT FROM AUTHOR]

The main purpose of this work is to investigate the innovation of flow of two-dimensional (2D) ferrofluid flow passing upon stretched surface having the effect of a magnetic dipole. The said problem is selected since its use is very common in biomedical as well as engineering applications together with systems of drug targeting under magnetic effects. The impacts of magnetic dipole along with radiation parameter in flow of ferromagnetic cross fluid for a stretched region are investigated in detail by use of suitable similarity variables the nonlinear, ODEs are obtained. The desired ODEs are resolved numerically by using RK fifth-order method parallel to shooting scheme. Impacts of ferromagnetic interaction, viscous dissipation, Curie temperature, and Buongiorno parameters with convective boundary conditions are perceived for velocity, temperature and concentration fields. Furthermore, the terms like velocity, mass transfer and thermal gradients are under consideration and pictorial scrutinizing is done. Moreover, temperature of ferromagnetic fluid upsurges for enlargement in the values of ferrohydrodynamics interaction and Curie temperature parameters. Concentration of ferrofluid reveals a contrary tendency against the parameters of thermophoresis and Brownian motion. Owing to extensive application prospects of magnetized nanofluids, this area has received interest from engineers and researchers; nanofluids have a noteworthy preference over ordinary fluids. [ABSTRACT FROM AUTHOR]

This study aims to obtain the optical soliton solutions of the stochastic perturbed Radhakrishnan–Kundu–Lakshmanan equation which models the optical solitons in optical waveguides such as silica fibers with Kerr law in the presence of chromatic and third-order dispersions by multiplicative white noise in Itô calculus. The study constitutes one of the works aimed at adapting the model in accordance with the importance of the noise effect in nonlinear optics. In the first step, the nonlinear ordinary differential form of the investigated problem has been obtained by using an appropriate complex wave transformation. The enhanced Kudryashov technique, which is the combination of Kudryashov and the new Kudryashov methods, is applied to the nonlinear ordinary differential form. As the result, bright, dark and singular stochastic soliton solutions of the analyzed problem have been carried out by assigning appropriate parameter values and the impact of noise effect on the soliton behavior has been investigated. Lastly, the graphical representations and required comments have been presented in the related sections. [ABSTRACT FROM AUTHOR]

Nanofluids are a relatively new class of materials that have gained attention in recent years due to their unique physical and thermal properties. They are used as contrast agents in biomedicine, including for magnetic resonance imaging (MRI) and computed tomography (CT). Keeping the above applications in mind, this theoretical study intended to explore the consequence of transient magnetized Casson nanoliquid driven by a permeable bi-direction time-dependent stretching platform positioned inclined subject to nonlinear heat source/sink. The Brownian movement, thermophoretic force, chemical reaction, and variable internal heating impacts are incorporated into the proposed flow problem. The leading constitutive PDEs are diminished into coupled nonlinear self-similar dimensionless ODEs through pertinent non-dimensional quantities. The resulting problems are addressed utilizing "Particle Swarm Optimization (PSO)" and a hybrid shooting methodology inspired by the Runge–Kutta four (RK4) method. The novelty of this work is the investigation of the numerical solution of magnetized Casson nanoliquid enclosing thermal radiation and chemical reaction via PSO along with hybrid shooting technique over time-dependent stretching sheet, which has not been elaborated on to date. The physical appearance of relevant physical factors on different flow phenomena are evaluated via graphs. For the stability purpose of the numerical method Average Square Residue Error (ASRE) and Total Average Square Residue Error (TASRE) are computed. For the validation purpose the present work is associated with the available work and great correlation is found. [ABSTRACT FROM AUTHOR]

Modeling of physical systems must be based on their suitability to unavoidable physical laws. In this work, in the context of classical, isothermal, finite-time, and weak drivings, I demonstrate that physical systems, driven simultaneously at the same rate in two or more external parameters, must have the Fourier transform of their relaxation functions composing a positive-definite matrix to satisfy the Second Law of Thermodynamics. By evaluating them in the limit of near-to-equilibrium processes, I identify that such coefficients are the Casimir–Onsager ones. The result is verified in paradigmatic models of the overdamped and underdamped white noise Brownian motions. Finally, an extension to thermally isolated systems is made by using the time-averaged Casimir–Onsager matrix, in which the example of the harmonic oscillator is presented. [ABSTRACT FROM AUTHOR]

This paper aims to provide a consistent, finite-valued, and mathematically well-defined reformulation of Feynman’s path-integral measure for quantum fields obtained by studying the Wiener stochastic process in the infinite-dimensional Hilbert space of quantum states. This reformulation will undoubtedly have a crucial role in formulating quantum gravity within a mathematically well-defined framework. In fact, this study is fundamentally different from any previous research on the relationship between Feynman’s path-integral and the Wiener stochastic process. In this research, we focus on the fact that the classic Wiener measure is no longer applicable in infinite-dimensional Hilbert spaces due to fundamental differences between displacements in low and extremely high dimensions. Thus, an analytic norm motivated by the role of the fractal functions in the Wilsonian renormalization approach is worked out to properly characterize Brownian motion in the Hilbert space of quantum states on a compact flat manifold. This norm, the so-called fractal norm, pushes the rougher functions (physically the quantum states with higher energies) to the farther points of the Hilbert space until the fractal functions as the roughest ones are moved to infinity. Implementing the Wiener stochastic process with the fractal norm, results in a modified form of the Wiener measure called the Wiener fractal measure, which is a well-defined measure for Feynman’s path-integral formulation of quantum fields. Wiener fractal measure has a complicated formula of non-local terms but produces the Klein–Gordon action at the first order of approximation. Using complex integrals to compensate for the removal of non-local terms appearing in higher orders of approximation, the Wiener fractal measure turns into a complex measure and generates Feynman’s path-integral formulation of scalar quantum fields. This brings us to the main objective of this study. Finally, some various significant aspects of quantum field theory (such as renormalizability, RG flow, Wick rotation, regularization, etc.) are revisited by means of the analytical aspects of the Wiener fractal measure. [ABSTRACT FROM AUTHOR]

In this paper, a stochastic SIS epidemic infectious diseases model with double stochastic perturbations is proposed. First, the existence and uniqueness of the positive global solution of the model are proved. Second, the controlling conditions for the extinction and persistence of the disease are obtained. Besides, the effects of the intensity of volatility ξ 1 and the speed of reversion 1 on the dynamical behaviors of the model are discussed. Finally, some numerical examples are given to support the theoretical results. The results show that if the basic reproduction number ℛ 0 s < 1 , the disease will be extinct, that is to say that we can control the threshold ℛ 0 s to suppress the disease outbreak. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the problem of stability of linear time-varying stochastic perturbed systems. We present sufficient conditions ensuring the global practical uniform exponential stability of a different class of stochastic perturbed systems based on generalized integral inequalities of Gronwall type. Finally, we provide numerical examples to prove the usefulness of the main results of this paper. [ABSTRACT FROM AUTHOR]

The thermal radiation and Fourier flux have drawn the attention of scientists and experts due to their extensive applications in the fields of medicine, manufacturing modern airplanes, water distillation, more efficient electronic devices, more efficient batteries, radiating treatment, and the textile industry. Through careful consideration of this, the mathematical investigation of natural convection and Fourier flux on the flow of energy and diffusion transmission of a body of cone and cylinder revolution, located in a saturated medium, has been described for the dual situations (flow over a cone and a cylinder). After then, the Runge–Kutta technique was used to explain the leading mechanism. Possessions of governing physical measures of local Nusselt and Sherwood numbers as well as velocity, thermal and diffusion figures are provided with support of tables and diagrams. It is motivating to declare that mass transfer rate is advanced in cone kind of revolution matched to cylinder kind of revolution with rising N R , N t , Q S . Also, the heat transmission rate is upper in cylinder revolution equated to cone revolution. [ABSTRACT FROM AUTHOR]

This work is about the investigation of the flow of a micropolar nanoliquid over a stretching surface, taking into account the effects of thermal radiation, thermophoresis, and Brownian motion. The study focuses on the impact of these factors on heat and mass transfer rates, with the assumption that the Newtonian heat impact dominates. The homotropy approach is used to generate non-dimensional transformational parameters, which are then used to create a system of nonlinear differential equations. The study includes charts and tables that define transfer rates based on various parameters, and the results suggest that increased radiance levels and Nb parameters lead to improvements in heat and mass convection. The graphics used in the study are accurate and consistent with previous research in the field. Overall, this research provides insights into the complex dynamics of micropolar nanoliquid flow and the factors that impact heat and mass transfer in this system. [ABSTRACT FROM AUTHOR]

In this paper, we examine the Carreau nanofluid flow past a nonlinearly stretchable surface accompanied by chemical reactions, thermal slip parameter and thermal radiation. A two-phase nanofluid criterion is considered to examine the behavior of flowing fluid and temperature profile. We used the similarity transition to change the basic equations into the set of ordinary differential equations in order to obtain the results. Shooting criteria are applied to calculate the solution of momentum, energy, and concentration fields using MATLAB. The effects of physical entities on the velocity, concentration, and temperature fields are analyzed by numeric tabular data. Furthermore, it is analyzed that the temperature field and concentration profile are diminished while velocity and thermal slip parameters are increased. The increasing nature of Brownian movement N b corresponds to a lower nanoparticle profile concentration is observed. The temperature profile is a growing function of magnetic parameter. As the Eckert number rises, so does the thermal pattern. [ABSTRACT FROM AUTHOR]

In this paper, the importance of mass and heat transfer characteristics of mixed convective conditions on nonlinear free convection flow embedded with non-Fourier flux and thermal radiation is numerically investigated. A partial Casson slip and mixed impact of thermophoresis and Brownian motion is considered at the elastic surface. To change the highly nonlinear partial differential equations into ordinary differential system, we used the similarity transformations and then the shooting method is employed to solve resulting equations. The significance of various physical parameters on the velocity, concentration and temperature profiles are investigated for both negative value of ferromagnetic interaction ( β 1 = − 0. 2) and positive value of ferromagnetic interaction ( β 1 = 0. 2). The governing physical aspects of coefficient of skin friction, the Sherwood and Nusselt numbers are obtained for various embedded parameters. Our findings concluded that the ferromagnetic interaction is positive and then more mass transfer rate is identified compared to the negative value of ferromagnetic interaction, but whereas in friction and heat transfer rate, it was completely mixed sense (i.e., different effects have different results). Brownian motion has less Nusselt number values in positive ferromagnetism compared to negative ferromagnetism and quite opposite behavior is observed in the presence of thermophoresis, this happened due to nonlinear flux and buoyancy. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to provide numerical simulation for viscous nanofluid by stretching cylinder using Buongiorno's model. Dissipation, Joule heating and magnetic impact with irreversibility analysis are examined. Furthermore, Brownian movement and thermophoresis diffusion effects are studied. Through constitutive relations, nonlinear equations are obtained. By adopting suitable variables, the partial differential equations are changed to ordinary differential equations. Entropy rate is calculated using thermodynamics second law. ND-solve technique in Mathematica is employed to tackle dimensionless ordinary differential equations. The behavior of flow controlling parameters is examined graphically. Physical quantities are discussed through tables. A decrement in velocity is seen through magnetic effect, while opposite impact is seen for entropy rate. A similar scenario is seen for thermal field through random and thermophoresis variables. Reduction in concentration is seen for reaction parameter and Schmidt number. Important findings are presented. [ABSTRACT FROM AUTHOR]

This paper examines the activation energy influence in third-grade nanoparticle flow model (TG-NPFM), which is nonlinear mixed convective flow over a spinning disk under the influence of heat sink/source as well as viscous dissipation by utilizing Bayesian Regulation Method with backpropagated Artificial Neural Networks (BRM-BPANN). Nonlinear thermal radiation is also involved in the considered flow dynamics to obtain the approximated numerical solutions. The nonlinear PDEs of TG-NPFM are then transformed into nonlinear ODEs by implementing the corresponding transformation. We solved these ODEs by Optimal Homotopy Analysis Method (OHAM) to explain the dataset used as a reference for BRM-BPANN for different scenarios of TG-NPFM. This reference dataset is then exported to MATLAB to compute the results. The outcomes of TG-NPFM are figured by adopting the procedures of testing, validation and training. Moreover, approximated solution is compared with standard solution and the efficacy examination of TG-NPFM is authenticated by the studies of MSE, error histogram and regression plots. These soft computation frameworks provide incentive to use an efficient and dependable alternative paradigm built on soft computing environments to solve problems by doing a descriptive analysis to mitigate the impacts of different physical features. It is a new implementation of intelligent computational system of artificial intelligence introduced by incorporating the solver BRM-BPANN for interpreting the TG-NPFM. The absolute error values lie between 10 − 8 to 10 − 5 , 10 − 1 0 to 10 − 5 , 10 − 8 to 10 − 4 , 10 − 9 to 10 − 3 , 10 − 9 to 10 − 5 , 10 − 9 to 10 − 3 , 10 − 9 to 10 − 3 and 10 − 9 to 10 − 3 , which show the reliability and accuracy of the technique. The convergence and precision of the algorithm can easily be seen through the results of performance, training state and fitness plot, along with the regression value of R = 1. [ABSTRACT FROM AUTHOR]

Heat and mass transfer acquired the attention of investigators and experts because of massive uses in the field of medicine, manufacturing of modern aircrafts, uses in advanced water filtration plants, distillation process of water, more efficient electronic instruments, more efficient batteries, textile industry, uses as manufacturer in cosmetics industry and modern defense equipment. By viewing this, we considered the numerical study of Fourier flux and buoyancy driven forces on the flow of thermal and diffusion transmission of body revolutions (Paraboloid and cone), situated in a water-logged Darcy medium by allowing the radiation, Brownian motion, time-space dependent heat source or sink, and thermopheretic. Later on, the governing system is solved via Runge–Kutta method. Properties of convoluted governing measures of the organisms on local Nusselt and Sherwood numbers along with velocity, thermal and diffusion shapes are described with the aid of graphically and tabular form. It is stimulating to declare that the non-uniform heat source or sink is highly dominated in cone form of revolution as associated to paraboloid form of flow due to more distribution of mass transfer. From this, one can draw the conclusion that wherever there are higher mass transportation phenomena, cone-shaped body of revolution can be used. It is found that the time- and space-dependent heat source or sink performs as a regulating factor of the transmission of flow phenomena. [ABSTRACT FROM AUTHOR]

This paper investigates the stagnation point flow of a second-grade nanomaterial, considering nanofluid effects via thermophoresis and Brownian motion. The study addresses convective heat and mass transfer conditions within the flow induced by a stretching cylinder. The transformed systems are solved using the optimal homotopy solutions (OHAM) method, revealing the impact of various parameters on the quantities of interest. The results indicate that an increase in curvature, viscoelasticity, velocity ratio and injection parameters leads to an enhancement in velocity, while the opposite trend is observed due to suction. The viscoelasticity and curvature parameters also increase skin friction. Additionally, thermophoresis enhances the temperature and concentration fields, while Brownian motion reduces mass diffusion. [ABSTRACT FROM AUTHOR]

There are several uses for bio-convection mechanisms, such as in sedimentary pools, fuel models, and microbially accelerated oil restoration. These are the basis of motile microorganisms which are applied to improve the mixability of fluid and tiny nanoparticles because they are accountable for biological transmission procedure. This work examines the bio-convective nanoparticles fluid flow and electrically conducting incompressible magneto-hydrodynamic flow of nanofluid towards a chemically reactive stretchable surface. Buongiorno model is considered here, which incorporates Brownian motion and thermophoretic diffusion, is employed in this study to explain how nanofluids improve heat transfer. The controlling boundary layer nonlinear PDEs are converted into ODEs by utilizing certain suitable similarity approaches. Numerical scheme is adopted to solve the transformed ordinary differential equations. The main findings of this research are that the magnetic field slows motion of the fluid when the thermal and mass buoyancy forces speed it up, and that thermophoretic diffusion increases the temperature and concentration of dimensionless fluids, resulting in stronger concentration and thermal boundary layers. The Brownian motion, concentration exponent, and chemical reaction parameters show an inverse relationship between temperature and concentration. Additionally, the first-order chemical reaction creates a lighter concentration boundary layer while the magnetic field's presence due to impact of thermal and mass buoyancies helps to lower the heat transfer rate and shear stress. The findings show that while the Nusselt number and wall shear stress values are declining, the microorganism number is increasing. [ABSTRACT FROM AUTHOR]

C a 2 + signals have a crucial role in the immune system for cell functions such as proliferation, differentiation, apoptosis and gene transcription as well as the activation of Nuclear factor of activated T Cell (NFAT) which modulates the immune response of the cells. In this study, a fractional order spatiotemporal calcium dynamics model is formulated using the reaction-diffusion equation incorporating parameters like buffers, source influx, J SScyt and J PMCA . Grünwald estimation is employed to obtain the solution and stability analysis has been performed. The analysis of numerical results provides novel insights about the role of Brownian motion, super diffusion, source influx, buffers, etc., in the regulation of calcium and NFAT concentration levels in T cell and the conditions which might lead to disordered immune responses causing diseases such as HINI, HIV and HBV. [ABSTRACT FROM AUTHOR]

Due to recent advancements in technologies and nanoscience, research focused on nanofluid because of its extensive use in many industrial as well as manufacturing processes. Nanotechnology is widely utilized in many photovoltaic devices, mini-computers, bio-medical apparatus, electronic instruments, cooling and heating of metallic plates, etc. In view of numerous applications, the basic aim of this work is to explore the impact of ferromagnetic nanomaterial on 2-D Eyring–Powell fluid flow. Characteristics of viscous dissipation and Brownian movement are under consideration. Some suitable similarity transformations are exploited to achieve the nonlinear ODEs from PDEs. Bvp4c scheme is implemented to solve the resultant system of equation. Impacts of ferromagnetic interaction parameter, thermophoresis parameter, viscous dissipation, Curie temperature and Eckert number are perceived for velocity, temperature as well as concentration fields. Moreover, velocity, thermal and mass transfer gradients are deliberated and scrutinized graphically. It is found that upsurge in the value of magnetic parameter reduces the velocity profile while it deteriorates against Eyring–Powell fluid parameter. Additionally, greater thermophoresis parameter has augmented the temperature profile while it shows opposite behavior for concentration. [ABSTRACT FROM AUTHOR]

The present flow phenomena characterize the behavior of the Brownian motion and thermophoresis considering the Buongiorno model nanofluid through a non-linearly expanding Riga plate. The conjunction of thermal radiation with zero nanoparticle mass flux and convective boundary conditions enriches the thermal properties of the nanofluid. With the recent need in various industrial applications, the biomedical sector, for the design and development of the production process of various equipment, the role of nanofluid is vital. Further, the suitable variables leading to transformation rules are useful to get rid of the dimensional form of the governing equations. Traditional numerical technique with the help of shooting-based Runge–Kutta fourth-order technique is adopted for the solution of a transformed set of equations. The physical behavior of the characterizing components is presented and described briefly. Further, the important outcomes are the use of Hartmann number that overshoots the velocity profile, whereas the heat transfer retards significantly and enhanced thermophoresis attenuates the heat transfer rate. [ABSTRACT FROM AUTHOR]

This paper studies the mixed convective flow of a magnetohydrodynamic micropolar fluid over an extending sheet. The first-order velocity slip condition is taken to observe the slip flow of the fluid. The applications of solar radiation toward the micropolar fluid flow are analyzed in this paper. Furthermore, the Brownian motion, thermophoresis and Joule heating impacts are also studied. Also, the Cattaneo-Christov heat flux model, chemical reaction and activation energy are observed. The leading PDEs have been transformed to ODEs and then solved with the help of homotopy analysis technique. The impacts of different physical parameters have been evaluated theoretically. The outcomes exhibited that the material factors have augmented the microrotation and velocity profiles. Moreover, the velocity slip parameter has a reverse relation with velocity and microrotation profiles, while there is a direct relation of a velocity slip with the energy curve. The velocity profile has increased with higher thermal and mass Grashof numbers. With increasing Brownian motion parameter, the thermal profile is amplified while the concentration profile is declined. On the other hand, the thermal and mass profiles have been boosted with greater thermophoresis parameter. The velocity profile has decreased with higher magnetic parameter, whereas the temperature profile has augmented with higher magnetic parameter. The couple stress and skin friction have been augmented with material parameter, whereas the skin friction has been reduced with thermal and mass Grashof numbers. [ABSTRACT FROM AUTHOR]

In this paper, we consider the reflected backward stochastic differential equations driven by G -Brownian motion (reflected G -BSDEs) with time-varying Lipschitz coefficients. We obtain the uniqueness result by establishing a priori estimates. For the existence, the solution can be approximated by a family of reflected G -BSDEs with Lipschitz conditions and by penalized G -BSDEs with time-varying coefficients. The latter approximation is useful to get the comparison theorem. Finally, we study the reflected G -BSDEs with infinite time horizon. [ABSTRACT FROM AUTHOR]

In this paper, we study the rate in the Smoluchowski–Kramers approximation for the solution of the following distribution-dependent SDE driven by fractional Brownian motion X t = X 0 + ∫ 0 t b (s , X s , ℒ X s ) d s + ∫ 0 t σ (s , ℒ X s ) B s H , where ℒ X t denotes the law of X t , { B t H , t ∈ [ 0 , T ] } is a d -dimensional fractional Brownian motion with Hurst parameter H ∈ (1 / 2 , 1). Based on the techniques of multiple integrals and Malliavin calculus, we provide an explicit bound on total variation distance for the rate of convergence. [ABSTRACT FROM AUTHOR]

This paper reports the hydromagnetic mixed convective flow of Carreau nanofluid in the vertical microchannel with slip and convective mechanisms at the boundaries. The main objective of this work is to analyze the conduct of Carreau nanofluid under Buongiorno model as the nano-dimensions of the particle consequently encounters random motion. The novel impression is to retain the outcomes of the flow in regard of Brownian dispersion and thermophoresis. Parameters such as buoyancy ratio, Brownian motion and thermophoresis along with the effect of Weissenberg number are discussed on distribution of flow, temperature and concentration by using graphical demonstration. Surface drag coefficients, transport rates of mass and heat are portrayed by numeric esteems. Boundary layer approximations are made use to tackle nonlinear ordinary differential equations and solved by Runge–Kutta–Fehlberg 4-5th-order method. Results so obtained elaborate that velocity depletes with increased values of buoyancy ratio and viscosity ratio parameter along with power law index. Also, distinct behavior is noticed in the profile of velocity for varying Weissenberg number by boosting and diminishing depending on the power law index. On the other hand, temperature rises by augmentation of thermophoresis and Brownian motion parameter. Elevation in Brownian motion parameter accelerate the concentration panel of the fluid. Skin friction and Nusselt number are in direct proportion with buoyancy ratio parameter and Brownian motion parameter, respectively. [ABSTRACT FROM AUTHOR]

The exponentially stretching curved surface is considered for second-grade fluid flow. The slip effects are applied at the assumed surface. The Brownian motion, chemical reaction, and thermophoresis effects are studied to determine the influence of the second-grade fluid. The boundary layer approximation is applied to the governing equation which is reduced in terms of partial differential equations. These partial differential equations become the dimensionless form after using the appropriate transformations. Further, the numerical approach is imposed on the dimensionless system to calculate the problem-related results. Graphs and tables are used to show how involving physical parameters have an effect. The velocity profile and skin friction have been increased due to increasing values of curvature parameters. The velocity function declined due to an increment in the second-grade fluid parameter. It means that the thickness of the boundary layer shown as decline when the second-grade fluid parameter rises while the second-grade fluid parameter has an inverse relation with viscosity. The Prandtl number increased which increased the temperature function. Physically, the thickness of the thermal boundary is controlled by the Prandtl number, and temperature increases by increasing the value of the Prandtl number. [ABSTRACT FROM AUTHOR]

This work is motivated by the importance of vertical vibration in nanofluids suspension, because of its wide applications in heat exchangers from one fluid to another, controlling the mixing in microvolumes, pharmaceutical industry, and engineering. The current study analyzes the effects of high-frequency vertical vibration on a horizontal Casson nanofluid layer heated from below saturated between rigid-rigid, rigid-free, and free-free boundary conditions. Field equations are derived by using time-average method, which describes the vibrational thermo-convection. These field equations are solved by utilizing linear stability analysis and Galerkin technique to obtain the thermal stability curve. By using this curve, the impacts of vertical vibration and physical governing factors on temperature profile are discussed. High-frequency vertical vibration is observed to have a stabilizing effect on Casson nanofluid suspension, thereby reducing thermo-convective heat transfer from one fluid to another. Stability threshold of vibrated suspension is increased approximately as 5. 7 4 8 % and 8 1. 9 3 7 % when R v → 5 0 and R v → 1 0 0 , respectively. The significance of Brownian motion and thermophoretic force on convective heat transfer with vertical vibration is discussed. The resisting nature of Casson nanofluid on vibrational convective heat transfer is also tabulated. [ABSTRACT FROM AUTHOR]

This paper presented an analysis for time-varying nanofluid thin liquid layer applied over a stretched heated plate with adjustable concentration and temperature. The governing partial differential equations of problem have been reduced to similar ordinary differential equations through similarity transformation using group theoretic method those are convenient to find solutions along with numerical solution. Analysis comprises for impact of various parameters as Prandtl number Pr, Schmidt number Sc, Thermal exponent r, Concentration exponent s, Deformation rate γ , Thermophoresis parameter N t , & Brownian motion parameter N b , on auxiliary variables like Velocity f ′ (η) , Temperature θ (η) and Concentration ϕ (η). The MATHEMATICA software is used to solve and extract graphs for problem. The results have been depicted via graphs to grasp conclusive declaration. [ABSTRACT FROM AUTHOR]

This paper deals with the limit distribution for a stochastic differential equation driven by a non-symmetric cylindrical α -stable process. Under suitable conditions, it is proved that the solution of this equation converges weakly to that of a stochastic differential equation driven by a Brownian motion in the Skorohod space as α → 2. Also, the rate of weak convergence, which depends on 2 − α , for the solution towards the solution of the limit equation is obtained. For illustration, the results are applied to a simple one-dimensional stochastic differential equation, which implies the rate of weak convergence is optimal. [ABSTRACT FROM AUTHOR]

In a recent paper, Lanconelli and Scorolli10 extended to the multidimensional case a Wong–Zakai-type approximation for Itô stochastic differential equations (SDEs) proposed by Øksendal and Hu7. The aim of this paper is to extend the latter result to system of SDEs of Itô type driven by fractional Brownian motion (fBm) like those considered by Hu6. This extension is not trivial since the covariance structure of the fBm precludes us from using the same approach as that used by Lanconelli and Scorolli. Instead we employ a truncated Cameron–Martin expansion as the approximation for the fBm. We are naturally led to the investigation of a semilinear hyperbolic system of evolution equations in several space variables that we utilize for constructing a solution of the Wong–Zakai approximated systems. We show that the law of each element of the approximating sequence solves in the sense of distribution a Fokker–Planck equation and that the sequence converges to the solution of the Itô equation, as the number of terms in the expansion goes to infinite. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of a mild L p -solution for the backward stochastic evolution inclusion (BSEI for short) of the form d Y t + A Y t d t ∈ G (t , Y t , Z t) d t + Z t d W t , t ∈ [ 0 , T ] Y T = ξ , where W = (W t) t ∈ [ 0 , T ] is a standard Brownian motion, A is the generator of a C 0 -semigroup on a UMD Banach space E , ξ is a terminal condition from L p (Ω , ℱ T ; E) , with p > 1 and G is a set-valued function satisfying some suitable conditions. The case when the processes with values in spaces that have martingale type 2 has been also studied. [ABSTRACT FROM AUTHOR]

This study considered the Cattaneo–Christov heat flux for examining the nanofluid flow with heat transfer towards a stretching sheet involving thermal and solutal buoyancy, activation energy, thermal radiation, and microorganism concentration. It is suggested that the model of Buongiorno nanofluid can be used to sync the influences of thermophoresis and Brownian motion. The 2D boundary layer with non-Fourier PDEs is converted into 1D nonlinear ODEs. Using the RK-4 method, the acquired equations were analyzed numerically. In this regard, the relevant engineering quantities are extensively computed with a higher degree of precision and then tabulated. The motile concentration, temperature, nanoparticles concentration, and velocity decline for numerous values of involved parameters such as slips of microorganism concentration, thermal, solutal, and hydrodynamic, respectively. The temperature field decreased when the parameter of thermal relaxation was imposed, while the reverse behavior was shown for thermophoresis, radiation parameter, and Brownian motion. The current results manifest a reasonable assessment of their existing counterparts. [ABSTRACT FROM AUTHOR]

The study of melting heat transfer in micropolar nanofluid flow through porous media can provide insights into optimizing various industrial and engineering processes, as well as improve our understanding of fluid transport through porous media in environmental and biomedical applications. This study focuses on the impact of Brownian motion, thermophoresis, internal heat generation/absorption, chemical reaction and melting effects on two-dimensional micropolar nanofluid flow across a permeable stretchy surface with a stagnation point. Appropriate similarity transformations are chosen in order to convert the system of the nonlinear ordinary differential equation. The results for various parameters are examined by applying the shooting method with BVP5C technique. Furthermore, graphs and tables are used to explore the impact of significant factors that affected the velocity, microrotation, thermal behavior, and concentration of nanofluids. Results reveals that the increase in values of Brownian motion and thermophoresis parameters caused a growth in the Sherwood number and drop in the Nusselt number. The upshot of the stretching parameter (SP) on velocity, temperature, concentration and microrotation is studied. Results reveal that the velocity and microrotation fields remain constant while thermal activity increases for SP = 1. The higher value of the SP increases the concentration. [ABSTRACT FROM AUTHOR]

In this paper, numerical investigations have been performed on intricate double stratification impact with a radiative mixed convective nanofluid over a cylinder. Double stratifications comprise heat transfer used in many practical applications like power station engineering, ground-water reservoirs, thermal stratification of reservoirs, and rivers, density stratification of atmosphere, oceans, different heterogeneous mixtures, and manufacturing processing. Characteristics of heat and mass transfer have also been considered. The developing mathematical expression for momentum, energy transportation, and nano-concentration involving Brownian motion and thermophoresis inspiration are considered. The modeled equations are formulated into ordinary differential equations by applying a suitable similarity approach, which is then tackled numerically via MATLAB. Graphical illustrations of dimensionless velocity, energy, and volumetric concentration distribution are drawn against a few values of appropriate parameters. Results elucidate that enhancement occurs in the concentration profile, whereas decrement is noted for the temperature field for different values of thermophoresis and thermal stratification parameters. Our results elucidate good agreement as compared with the previously published one. The higher value of Hartmann number indicates a higher coefficient of friction. [ABSTRACT FROM AUTHOR]

In this paper, the complex dynamics of the pulsating regime of combustion waves propagation is numerically investigated within the Zel'dovich–Barenblatt–Dold model with two-step chain-branching reaction mechanism. It is shown that there exists a remarkable similarity in the dynamics of oscillations in this system and one-dimensional discrete-time maps such as the logistic map. In particular, both systems exhibit the period doubling route to chaos and the appearance of windows of stability as the bifurcation parameter is increased. For the first time for a model of combustion wave propagation the sequence of windows of stability is demonstrated and classified. Nevertheless, as the activation energy, which is taken as a bifurcation parameter, reaches a critical value corresponding to the dynamical quenching scenario the dynamics of the flame oscillations becomes multidimensional and therefore it cannot be described by a one-dimensional map anymore. The effect of the heat losses is also studied and it is demonstrated that the number of windows of stability decreases, the order of their appearance changes and the dynamical quenching appears for smaller values of the activation energy as the heat losses are increased. [ABSTRACT FROM AUTHOR]

In this paper, we solve a Lévy driven linear stochastic first-order partial differential equation (transport equation) understood in the canonical (Marcus) form. The solution can be obtained with the help of the method of stochastic characteristics. It has the same form as a solution of a deterministic PDE or a solution of a stochastic PDE driven by a Brownian motion studied by Kunita [First order stochastic partial differential equations, in Stochastic Analysis. Proc. Taniguchi Int. Symp. Stochastic Analysis, Katata and Kyoto, 1982, North-Holland Mathematical Library, Vol. 32, ed. K. Itô (North-Holland Publishing Company, 1984), pp. 249–269; Stochastic Flows and Stochastic Differential Equations, Vol. 24 (Cambridge University Press, 1997)]. [ABSTRACT FROM AUTHOR]

In this paper, we consider two versions of functional equations for the stochastic exponential. One equation for a function of a semimartingale and its square characteristic and the second for non-anticipative functionals. A martingale characterization of the general solutions of these equations is given. [ABSTRACT FROM AUTHOR]

Let K be an algebraic number field. With K we associate the ring of finite adeles K. In this paper we give a path integral formula for the propagator of a quantum mechanical system over the abelian group K n. Specifically, we consider matrix-valued Hamiltonian operators H K n = Δ K n ⊗ Id + V , where Δ K n is the Vladimirov operator and V is a non-negative definite potential. The free part of the Hamiltonian gives rise to a measure on the Skorokhod space of paths which allows us to prove the Feynman–Kac formula for the Schrödinger semigroup generated by − H K n . This formula is given in terms of the ordered time exponentials. [ABSTRACT FROM AUTHOR]

The applications of nanofluids (NFs) have been comprehensively explored in current years, as they have abundant potential for technical progress and more prominently offer assistances that can be associated with the applications of NFs for several determinations. Nanotechnology can be applied in various technological fields such as medicine, information technologies, food safety and novel materials. Here, novel properties of entropy generation in a mixed convective magneto flow of a Sutterby nanomaterial to an extended surface is scrutinized. Nanofluid model comprises Brownian motion and thermophoresis aspects. The expression of energy depends upon the phenomenon of viscous dissipation and thermal radiation. We formulated the Bejan number and entropy generation. To reduce PDEs into nonlinear ODEs, we use transformation of variables and then the resultant system is solved by bvp4c technique. The influence of the parameters involved, such as thermal radiation, chemical reaction parameter, diffusive variable, magnetic parameter, thermophoresis parameter and Schmidt number for temperature, concentration as well as Bejan number, entropy generation are inspected through tables and graphs. [ABSTRACT FROM AUTHOR]

In this paper, the well-posedness of a kind of backward stochastic differential equation driven by G -Brownian motions (G -BSDEs for short) with mean reflection is investigated. In particular, the generator is allowed having quadratic growth in z. Combining a representation of the solution with G -BMO martingale techniques, fixed point argument, and -method, existence and uniqueness results for such G -BSDEs are provided under bounded and unbounded terminal condition. [ABSTRACT FROM AUTHOR]

This paper explores the active and passive control implementation on the nanofluid flow over a stretching cylinder. Realistic velocity slip and thermal jump properties are clutched. Appearance of external heat source and magnetic influence is also reckoned. Concentration and temperature profiles are modified due to integration of Brownian motion together with thermophoresis. Leading equations are transfigured into ODEs by well-qualified similarity transfiguration and hence solved by RK-4 shooting technique. Entire simulation is settled by MAPLE software with proper rate of accuracy and the outcomes are portrayed by graphs and tables. Results are compared considering both slip and without slip conditions, whereas the concentration profile is described under active and passive control conditions. Mass transfer decreases for Brownian motion but reverse effect is found for thermophoresis parameter under passive control of flow. Mass transfer is changed by 10.1% in case of passive control condition for the Reynolds number parameter. Nusselt number is diminished by 6.16% under passive control provision for magnetic parameter. [ABSTRACT FROM AUTHOR]

In the recent years, nanotechnology has been widely used in several fields regarding its rapid developments which create a lot of prospects for researchers and engineers. More specifically, replacement of conventional liquid with nanofluid is considered as an innovative solution to heat transfer problems. Keeping aforesaid pragmatism of nanofluid in view, we considered a time-dependent mathematical model to formulate the heat sink-source based Sutterby nanofluid model under thermophoretic and Brownian movements. New mass flux and melting boundary conditions are used for heat/ mass transfer analyses. Moreover, Prandtl's boundary-layer idea is employed for mathematical formulation. The leading nonlinear set of partial differential equations is transformed to nonlinear set of ordinary differential equations. Numeric outcomes are acquired through bvp4c algorithm, graphical results are found via MATLAB technique. Acquired numerical data shows that temperature of nanofluid boosts for greater thermophoretic and unsteady parameters. Intensification is measured in concentration distribution. [ABSTRACT FROM AUTHOR]

Lubrication theory has attained attention lately due to its practical applications, such as the formation of thin films, adhesives, and lubrication of components of machines. Jeffrey's nanofluid flow over the stagnation region past a power-law lubricated surface is presented in this study. Buongiorno's model is employed to scrutinize the effects of thermophoresis and Brownian motion phenomena with constant wall and prescribed surface temperature (PST) and effects of heat source/sink, chemical reaction, and Joule heating. Due to the continuity of shear stress of fluid-lubricant and velocity at the interface, interfacial conditions are generated. By similarity conversions, ordinary differential equations are obtained and their solutions are computed numerically. For power-law index equaling 1 2 , local similarity solutions are calculated by adopting a finite difference scheme, viz. bvp4c in MATLAB. The energy profiles for constant and prescribed temperatures are monitored. The effects of pertinent parameters on the flow, thermal, and mass distributions are scrutinized and illustrated in graphs. Flow field decreases significantly by raising slip parameter as the aptitude of power-law lubricant to improve the velocity of the bulk fluid. The numerical comparison of wall stress and Nusselt number is also presented. The slip and Jeffrey's material parameters raise the numerical outcomes of the wall shear stress. In addition, increment in Prandtl number enhances the numerical value of the Nusselt number; however, it reduces for relaxation-to-retardation times ratio. [ABSTRACT FROM AUTHOR]

In the past few years, many technical strategies, such as molding, condenser heat exchanger, liquefied metal filtration, fusion control and nuclear reactor coolant, that involve hydromagnetic fluxes and thermal intensification in porous media have been observed. This study investigates the Carreau nanofluid of nanobiofilm through stretching/shrinking sheet with a stagnant point flow, nanoparticles and convecting microbes. The orthogonal (9 0 ∘ impinge) coating stagnant point circulation of a medium is considered, although the sheet may be stretched/shrinked as the procedure utilized in industry. The variations in the fluid (dynamic viscosity, thermal conductivity, mass permeability) and microbes are utilized. The similarity transformation factors are used to transform the system of partial differential equations into a nonlinear system of ordinary differential equations. To find the solution of a system of equations, the Runge–Kutta method with shooting technique has been used. The flow rate, temperature and concentration, as well as the heat transfer rate, and the physical quantities have been discussed. The nanoparticle volume fraction increases with the increasing effect of activating energy as well as thermophoresis parameter, but it decreases with the enhancing effect of Lewis number (Le) and Brownian motion parameter (Nb). The graphs and tables display the illustration of the influence of different parameters. [ABSTRACT FROM AUTHOR]

This investigation aims to inspect the flow and thermal characteristics of hybrid nanoparticles under the effect of thermophoresis and Brownian motion. The hybrid nanofluid is formed by dispersing the silver nanoparticles into the base fluid composed of tungsten oxide and water. The resulting hybrid nanofluid is assumed to flow over a moving wedge. The wedge is a geometry that can be commonly seen in many manufacturing industries, moulding industries, etc., where friction creates more heat and cooling becomes a necessary process. This study currently focuses on such areas of the industries. In this regard, the flow expressions in the form of Partial Differential Equations (PDEs) are obtained by incorporating the modified Buongiorno's model and using boundary layer approximations. The modified Buongiorno model helps us analyze the impact of volume fraction along with the slip mechanisms. Suitable transformations are used to achieve the nondimensional form of governing equations, and further, it transforms the PDE to Ordinary Differential Equation (ODE). The RKF-45 is used to solve the obtained ODE and the boundary conditions. Furthermore, graphic analysis of the solutions for fluid velocity, energy distributions and dimensionless concentration is provided. It was noted that the behavior of the Nusselt and Sherwood numbers was determined by analyzing numerous parameters. The conclusions show that they decrease with greater values of the stratification factors. Additionally, with higher values of the wedge parameter, the magnitude of the velocity field and the thermal boundary layer diminish. [ABSTRACT FROM AUTHOR]

In this paper, a spectral collocation method is developed to numerically approximate a class of stochastic differential equations driven by the fractional Brownian motion. The convergence of the proposed method is proved. Numerical simulations are conducted to illustrate the performance of the proposed method in different cases. [ABSTRACT FROM AUTHOR]

In this paper, we consider least squares type estimators for the drift parameters in the sub-bifractional Vasicek processes defined by d X t = (μ + X t) d t + d S t H , K , t ≥ 0 , X 0 = 0 , with unknown parameters > 0 and μ ∈ R , where S H , K is a sub-bifractional Brownian motion with indices H ∈ (0 , 1) and K ∈ (0 , 1 ]. The strong consistency results as well as the asymptotic distributions of these estimators are obtained. [ABSTRACT FROM AUTHOR]