Set Of Equations Research Articles

Analyzing the impact of thermophoresis particle deposition and magnetohydrodynamic cross‐flow of Williamson hybrid nanofluids across a permeable deformable surface

AbstractThe need to examine the thermal transport of hybrid nanofluids has arisen from the need for improved heat transfer to control the rising heat density of small‐scale and many other technical procedures. Hybrid nanofluids are made up of two different kinds of nanoparticles suspended in the primary fluid. This improves the ability of regular fluids to transport heat and makes them better heat exponents than nanofluids. In this study, we evaluate the cross‐flow and heat transport characteristics of Williamson hybrid nanofluid across an extendable surface. The impacts of thermal radiation, non‐uniform heat sources, and magnetic effects are provoked in this study. The set of partial different equations is obtained as a result of the theoretical formulation. The similarity variables are hired to get the ordinary differential equations. The bvp4c solver is availed to compute the numerical dual solutions for different physical parameters. The results indicate that the friction factor is enhanced due to the presence of magnetic, suction, and Williamson parameters. Whereas, the heat transfer rate decreases due to Williamson and magnetic parameters, whilst the opposite impact is seen due to suction. Moreover, the Williamson parameter reduces the Sherwood number, whereas the suction uplifts the Sherwood number.

Accounting for photosystem I photoinhibition sheds new light on seasonal acclimation strategies of boreal conifers.

The photosynthetic acclimation of boreal evergreen conifers is controlled by regulatory and photoprotective mechanisms that allow conifers to cope with extreme environmental changes. However, the underlying dynamics of photosystem II (PSII) and photosystem I (PSI) remain unresolved. Here, we investigated the dynamics of PSII and PSI during the spring recovery of photosynthesis in Pinus sylvestris and Picea abies using a combination of chlorophyll-a fluorescence, P700 difference absorbance measurements, and quantification of key thylakoid protein abundances. In particular, we derived a new set of PSI quantum yield equations, correcting for the effects of PSI photoinhibition. Using the corrected equations, we found that the seasonal dynamics of PSII and PSI photochemical yields remained largely in balance, despite substantial seasonal changes in the stoichiometry of PSII and PSI core complexes driven by PSI photoinhibition. Similarly, the previously reported seasonal upregulation of cyclic electron flow was no longer evident, after accounting for PSI photoinhibition. Overall, our results emphasize the importance of considering the dynamics of PSII and PSI to elucidate the seasonal acclimation of photosynthesis in overwintering evergreens. Beyond the scope of conifers, our corrected PSI quantum yields expand the toolkit for future studies aimed at elucidating the dynamic regulation of PSI.

Dynamical System Approach and Thermodynamical Perspective of Hořava‐Lifshitz Gravity

AbstractThe authors have examined a Friedmann Robertson Walker cosmological model in Hořava‐Lifshitz gravity by using a dynamical system approach. A set of autonomous equations is derived and their solutions are calculated. The critical points from these equations and find the characteristics values with the analysis of the physical interpretation of the phase space for this system are assessed. Three stable critical points are found and the values of the physical parameters and the scale factor's expressions at each critical points are displayed in Tables 1, 2, and 3. A hybrid scale factor to develop the model, which results in a phase transition from deceleration to acceleration is used. The suitable values of the parameters are governed by applying the Monte Chain Monte Carlo method technique to the Hubble 46 and joint Hubble 46 and Baryon Acoustic Oscillations 15 datasets. In contrast to the negative behavior of pressure, the positive behavior of energy density and illustrate the Universe's acceleration epoch and the model is represented by the EoS parameter . The authors investigated that the energy conditions and their model violates the strong energy condition. Utilizing diagnostic test, it is found that the model represents phantom behavior. The thermodynamical perspective for the model is also examined. The model accurately explained the Universe's propagation history and fits well with contemporary cosmic data.

Investigation of thermodynamic state evolution, phase transition and mixing of the hydrocarbon droplet in convective supercritical environments

Liquid alkane droplets transit to supercritical fluids under supercritical conditions. However, the subsequent thermodynamic state evolution, phase transition, and mixing in this case, which directly influences combustion in actual diesel engines, need further investigation. We studied the thermodynamic state evolution, phase transition, and mixing of n-heptane droplets in supercritical laminar nitrogen streams through a set of conservation equations. Non-ideal thermodynamic and transport properties were calculated to incorporate the dramatic variation of physical properties in the supercritical regime. We focus on the effects of Reynolds number, ambient pressure, and droplet size on the thermodynamic state evolution, phase transition and mixing of droplets. Results showed that increasing the Reynolds number can accelerate the evolution of thermodynamic states, which leads to the faster transition of droplets from the liquid phase and supercritical fluids to gas-phase mixtures. With the increased Reynolds number, the phase transition rate is increased and the mixing homogeneity is improved due to the enhanced mass transfer. Caused by the negative correlation between mass transfer and ambient pressure, in higher pressure, droplets after the supercritical transition require more time to transition from supercritical fluids to vapor-phase mixtures. Increasing ambient pressure inhibited the droplet phase transition process and deteriorated the mixing homogeneity, but no significant influence was shown on the droplet dynamic evolution process. Smaller droplets transition faster from liquid phase and supercritical fluids to vapor-phase mixtures. The faster thermodynamic state evolution and faster dynamic evolution process of the smaller droplet leads to its more rapid homogeneous mixing with ambient gasses, but the mode of dynamics is independent of the droplet size.

Lactic Acid Production by Lactiplantibacillus plantarum AC 11S-Kinetics and Modeling.

Lactic acid is a versatile chemical with wide application in many industries. It can be produced by the fermentation of different sugars by various lactobacilli and investigations on lactic acid production from different substrates and by different strains are still in progress. The present study aimed to study lactic acid production from lactose by Lactiplantibacillus plantarum AC 11S and to choose a mathematical model describing in the best way the experimental data obtained. The influence of initial substrate concentration was investigated, and optimal pH and temperature were determined. An unstructured mathematical model was developed comprising equations for bacterial growth, substrate consumption, and product formation. The model was solved with different terms for specific growth rates considering substrate and/or product inhibition. The best bacterial growth and lactic acid production were achieved at pH = 6.5 and 30 °C. Production of lactic acid was mainly growth-associated, and at initial substrate concentration over 15 g/L, a considerable product inhibition was observed. The parameters of different models were determined and compared. The modified Gompertz equation gave the best fit when solving only the equation for biomass growth at different initial substrate concentrations. Solving the entire set of differential equations for bacterial growth, substrate consumption, and product formation, the best results were obtained when using a variant of the logistic equation for biomass growth. This variant included a term for product inhibition and described in the best way all experimental data. Solving the model for different biomass concentrations showed that an increase in biomass led to a shorter lag phase and the stationary phase was reached faster. The results obtained, optimum conditions and the kinetic model, are good bases for studying pH-controlled fermentation, as well as a continuous process.

Exploration of thermal radiation and stagnation point in MHD micropolar nanofluid flow over a stretching sheet with Navier slip

The quest to strengthen heat conduction of thermal science base fluid for effective industrial outputs and engineering derives has recently increased. Thus, this study aims to determine how thermal radiation and slip effects affect the flow of a micropolar nanofluid near a stagnation point over an extending sheet. Using similarity transformations, the flow-controlling partial differential equations (PDEs) are turned into a set of non-linear ordinary differential equations (ODEs). The non-linear system of equations has been solved by the numerical technique Runge-Kutta-Fehlberg integration scheme implementing the shooting technique with suitable conditions to generate a numerical solution. The essential factors affecting the flow are depicted graphically and tabularly. Additionally, a comparison is conducted between the present result and previously published data on the Nusselt and Sherwood numbers; it claims that thermophoresis and Brownian motion vary under some restrictive conditions. An increase in the magnetic field parameter was found to boost the velocity of the micropolar nanofluid. In contrast, a rise in the micropolar parameter reduces the angular velocity.

Influence of thermal radiation and shape factor on a hybrid nanofluid flow over a permeable flat plate with cross-diffusion effects: An irreversibility analysis

Understanding and controlling the shape factors of nanoparticles in fluid flow problems is crucial for optimizing heat transfer, fluid dynamics, and various applications such as nanofluids, drug delivery systems, catalysis, and nanocomposites. By tailoring the shape of nanoparticles, researchers can manipulate their behavior, dispersion, and interaction with the fluid, thereby optimizing the desired outcomes in various fluid flow problems. The meticulous investigation uncovers the integration of entropy analysis in the investigation of the flow of convective magnetohydrodynamic hybrid nanofluid via a permeable flat surface, by considering the effects of Dufour, viscous dissipation, Soret, and thermal radiation. The governing equations are converted into a set of nonlinear ordinary differential equations using appropriate similarity transformations and then solved using the bvp4c solver, a MATLAB in-built function. We display the outcomes for three different shape factors: platelet, cylinder, and brick. The skin friction coefficient increases at a rate of 1.5089 for the brick shape, 1.5163 for the cylindrical shape, and 1.5212 for the platelet shape when the mixed convection parameter lies between 1 and 5. In this regard, it is important to point out that there is a direct connection between the increase in temperature transmission rate and a decline in the Dufour number ( Du ). It has been found that when 1 ≤ Du ≤ 2.4 , the Nusselt number decreases by 0.4786, 0.4618, and 0.4512 for brick, cylinder, and platelet shapes, respectively. A decline in the Sherwood number is noticed with an upsurge in the Soret number ( Sr ). There is a noticeable increase of 0.0442 in the Sherwood number for brick shape, 0.0438 for cylinder form, and 0.0435 for platelet form when 1 ≤ Du ≤ 2.4 . Furthermore, we have detected that an increase in the radiation parameter amplifies both the Bejan number and the profiles of entropy formation.

Exploring the numerical simulation of Maxwell nanofluid flow over a stretching sheet with the influence of chemical reactions and thermal radiation

The significance of the study lies in illuminating the consequences of thermophoresis and Brownian motion on heat and mass transfer, offering valuable insights crucial for practical applications. The purpose of this study is to look into how Casson–Williamson and Maxwell nanofluids flow over a stretching sheet in order to understand how heat and mass move in that situation. The mathematical structure comprises a set of partial differential equations (PDEs) converted into ordinary differential equations (ODEs) through a similarity transformation. Subsequently, the well-established MATLAB BVP4C method, integral to the finite difference approach, is applied to solve these ODEs. For the assurance of the method’s reliability and precision, the acquired outcomes are cross-referenced with existing literature, which is a crucial step in validation. The numerical results are depicted visually and in tables, specifically highlighting the impulse of different influencing elements on the Nusselt number, friction factor, and Sherwood number. Notably, an enhancement within the Brownian motion parameter (Nb) and the thermophoresis parameter (Nt) is found to correspond to higher local Nusselt numbers, indicating an enhancement in heat transfer. Conversely, elevated quantities of the Brownian motion parameter (Nb) result in a reduction in local Sherwood numbers. Physically, a higher Brownian motion parameter causes significant nanofluid particle displacement, enhancing their kinetic energy and intensifying heat generation in the boundary layer. The magnetic parameter (M) influences speed profile decline caused by the Lorentz force, which hinders fluid motion.

Polarization-sensitive optical coherence tomography for birefringence measurement of calcite nonlinear uniaxial crystal

We report on the new application, to the best of our knowledge, of a time-domain optical coherence tomography (TD-OCT) device used to measure the ordinary n o and extraordinary n e indices of calcite birefringence crystal at room temperature. A 1.25±0.05mm thick slab of calcite crystal is cut, polished, and used as a sample in the OCT arm. While the calcite slab is axially scanned, the raw carrier ordinary signals that came from its front and rear facets are received and denoised with a set of digital filters. The extraordinary signals are generated by the change of beam polarization using a 90°-rotating polarizer plate. It is found that the wavelet transform is capable of reaching the highest signal-to-noise ratio (SNR) of about 24.50 and 23.91 for denoising the ordinary and extraordinary signals, respectively. Quantitative measurement of n o and n e is carried out by extracting a desired envelope from the denoised signals using standard methods. Average values of 1.660 and 1.444 are obtained for n o and n e , respectively, using the wavelet-denoised signals. The weights of the results are finally searched with ones obtained from two sets of dispersion equations. We found a very good agreement between the wavelet-denoised OCT- and dispersion equation-based values with a very low relative differences of 0.04% and 2.8% for n o and n e , respectively, when the Ghosh equation is used and averaged ones of 1.3% and 4.2% for n o and n e , respectively, when the Zhao et al. equation is applied.

Non-similar bioconvective analysis of magnetized hybrid nanofluid (Ag + TiO2) flow over exponential stretching surface

Scientists have studied fluid flow over a stretching sheet to explore its potential applications in industries. This study investigates the exponential stretching flow of a bioconvective magnetohydrodynamic (MHD) hybrid nanofluid in porous medium taking into consideration thermal radiations, heat generation, chemical reaction, porosity, and dissipation. Moreover, microorganisms are present in the fluid, so the fluid is more stable, which is crucial in biotechnology, biomicrosystems, and bio-nano coolant systems. Silver and titanium dioxide in a water-based medium are the prototypical nanoparticles. The present study involves a transformation of the governing system into a set of dimensionless, coupled and nonlinear partial differential equations (PDEs) using nonsimilar techniques. The local non-similarity (LNS) technique is used to truncate these equations to ordinary differential equations (ODEs). This technique is also used to estimate transformed equations numerically until the second level of truncation takes place via the bvp4c algorithm, which is a built-in MATLAB solver. Furthermore, tables are provided that presents the drag coefficients, Nusselt numbers, Sherwood numbers, and densities of motile microorganisms. Results show a negative correlation between the velocity and the magnetic field parameter as well as the porosity parameter, as evidenced by a decrease in velocity corresponds to rises in these parameters. The temperature distribution exhibits a positive correlation with the rising values of both radiation parameter and Eckert number. The concentration profiles also exhibit a negative correlation with the increasing values of Lewis and bioconvection Lewis number, chemical reaction parameter, Peclet number and the differences in microbial concentration. This study will improve the future research on hybrid nanofluid regarding industrial applications. There haven't been any previous publications that have investigated the use of this model with the local non-similarity method. The main objective of this article is to enhance the heat transfer performance in a hybrid nanofluid.

Parametric energy optimization of a ventilated facade with windows in Mediterranean climates

Ventilated facades are an effective passive measure to improve the energy performance of buildings in warm and temperate climates such as the Mediterranean. However, although in the Mediterranean sociocultural context the use of windows is widespread, there is a lack of energy analysis of ventilated facades in combination with windows. This is due to the lack of accurate simulation tools to evaluate the energy performance of this combination because of its complex thermodynamics, which, among other issues, requires three-dimensional computational fluid dynamic tools. In this work, a parametric energy analysis of the design parameters of a ventilated façade with window openings has been performed using a three-dimensional numerical model developed by the authors that combines turbulent Navier–Stokes thermodynamic equations for air with a set of differential equations for the envelope temperatures. The results of the analysis showed the potential of the opaque ventilated façade (OVF) with windows when applied to the rehabilitation of energetically obsolete facades, achieving a remarkable reduction in heat flux with respect to these facades, a reduction that reaches 32% for the yearly heat flux when using the optimal configuration found.

On the spectral radius properties of a key matrix in periodic impulse control

In this work, we consider the periodic impulse control of a system modeled as a set of linear differential equations. We define a matrix that governs the qualitative behavior of the controlled system. This matrix depends on the period and effects of the control interventions. We investigate properties of the spectral radius of this matrix and in particular, how it depends on the period of the interventions. Our main result is on the convexity of the spectral radius with respect to this period. We discuss implications of this convexity on establishing an optimal and maximum period for effective control. Finally, we provide an example motivated from a real-life scenario.

A Quantum Formalism for Abstract Dynamical Systems

This paper presents a quantum formulation for classical abstract dynamical systems (ADS), defined by coupled sets of first-order differential equations. They are referred to as “abstract” because their dynamical variables can be of different interrelated natures, not necessarily corresponding to physics, such as populations, socioeconomic variables, behavioral variables, etc. A classical linear Hamiltonian can be derived for ADS by using Dirac’s dynamics for singular Hamiltonian systems. Also, a corresponding first-order Schrödinger equation (which involves the existence of a system Planck constant particular of each system) can be derived from this Hamiltonian. However, Madelung’s reinterpretation of quantum mechanics, followed by the Bohm and Hiley work, produces no further information about the mathematical formulation of ADS. However, a classical quadratic Hamiltonian can also be derived for ADS, as well as a corresponding second-order Schrödinger equation. In this case, the Madelung reinterpretation of quantum mechanics provides a quantum Hamiltonian that does provide the quantum formulation for ADS, which provides new quantum variables interrelated dynamically with the classical variables. An application case is presented: the one-dimensional autonomous system given by the logistic dynamics. The differences between the classical and the quantum trajectories are highlighted in the context of this application case.

Exploring the dynamics of collisionless spherical stellar systems using the matrix method: Insights from the dilation mode

Context. Analytical solutions to the perturbed equations that govern self-gravitating collisionless stellar systems are crucial for both code testing and theoretical insights. For spheres, a solution has been known for years that corresponds to the entire object’s shift from the origin. We recently introduced a new exact stationary solution, relevant for models with a single length parameter. This solution, referred to as the scale-invariant or dilation mode, has led to insights regarding the concept of perturbation energy within the linear theory framework. Aims. Our aim is to use Hénon’s isochrone model as an example to verify the ability of the standard matrix method to successfully predict the existence of a dilation mode, and to explore its potential application as a test disturbance. Methods. We used the standard matrix method for radial perturbations and applied Clutton-Brock potential-density pairs to determine the properties of the perturbations. Results. In this particular case of stationary radial perturbations, the typical relationship between the perturbations of the distribution function and the potential fails. This discrepancy poses a challenge when attempting to use the dilation mode as a test. When using Clutton-Brock pairs with the matrix method, a mass conservation equation as an additional equation to the ordinary set of linear equations is required. With this added equation, it’s possible to obtain the needed test: identical vanishing of the determinant of this modified set of equations with an increasing number of included basis functions.

Dual solutions of magnetized radiative flow of Casson Nanofluid over a stretching/shrinking cylinder: Stability analysis

The enhanced thermal efficiency exhibited by Casson nanofluids offers significant practical applications across various industrial and engineering sectors. This study focuses on the mathematical investigation of the steady magnetohydrodynamic (MHD) boundary layer flow of Casson nanofluid through a stretched/shrinking cylinder, taking into account the effects of suction and thermal radiation. The governing partial differential equations (PDEs) have been subjected to a similarity transformation, resulting in a set of nonlinear ordinary differential equations (ODEs). These ODEs were solved numerically utilizing the code of bvp4c in the software of Matlab which offers high accuracy (4th order). The employed nanofluid model incorporates the effects of Brownian motion and thermophoresis. The present study illustrates the graphical depictions of the impacts of different governing parameters, namely Hartmann (M) number, curvature (γ) parameter, Brownian motion (Nb) parameter, mass suction (S) parameter, thermal radiation (Rd) parameter, and thermophoresis (Nt) parameter, on heat transfer, flow, and mass transfer characteristics. Comprehensive determination and visual presentation of the coefficient of skin friction, local Nusselt number, and local Sherwood number were conducted for a range of estimates of applied parameters. Based on our examination, it has been determined that dual similarity solutions are present within a specific range of mass suction parameters. The relationship between the Casson parameter and various fluid dynamic properties, such as skin friction coefficient, heat transfer rate, and mass transfer rates, has been found to exhibit a decreasing trend. Furthermore, the stability analysis discovered that the first solution exhibits linear stability, whereas the second solution displays linear instability. Additionally, the motivation behind this study is to enhance industrial performance through the optimization of thermal power generation systems, thereby increasing their overall efficiency.

Extended relativistic kinetic model composed of the scalar and two vector distribution functions: Application to the spin-electron-acoustic waves

Detailed deterministic derivation of kinetic equations for relativistic plasmas is given. Focus is made on the dynamic of one-coordinate distribution functions of various tensor dimensions, but the closed set of kinetic equations is constructed of three functions: the scalar distribution function, the vector distribution function of dipole moment, and the vector distribution function of velocity (or the dipole moment in the momentum space). All two-coordinate distribution functions are discussed as well. They are presented together with their limits existing in the self-consistent field approximation. The dynamics of the small amplitude spin-electron-acoustic waves in the dense degenerate plasmas is studied within the kinetic model. This work presents the deterministic approach to the derivation and interpretation of the kinetic equations. So, no probability is introduced during the transition from the level of individual particles to the collective functions. The problem of thermalization is not considered, but we can see that the structure of kinetic equations is kept for the system before and after thermalization. Hence, the kinetic equations can be used to approach this item.

Bethe ansatz solutions of the 1D extended Hubbard-model

We construct an integrable 1D extended Hubbard model within the framework of the quantum inverse scattering method. With the help of the nested algebraic Bethe ansatz method, the eigenvalue Hamiltonian problem is solved by a set of Bethe ansatz equations, whose solutions are supposed to give the correct energy spectrum.

Flow Over a Thin Needle Moving in a Casson Fluid

This study examines the boundary layer flow across a thin, horizontal needle moving in a Casson fluid. The underlying equations are initially converted into a set of ordinary differential equations using similarity transformations, and thereafter successive linearization is applied. The Chebyshev collocation technique is applied to find the solution of the linearized equations. The temperature and velocity profiles, together with the skin friction coefficient and the Nusselt number, are illustrated graphically for different values of the needle size and Casson fluid parameter.

Parametric analysis of pollutant discharge concentration in non-Newtonian nanofluid flow across a permeable Riga sheet with thermal radiation

Proper wastewater disposal is crucial in various manufacturing and ecological systems. This study aims to prevent and regulate pollution in the water supply. It examines how the pollutant discharge concentration affects the flow of non-Newtonian nanofluids (NNNFs) over a porous Riga surface. Two different types of NNNFs, namely, Walter’s B and second-grade fluids, have been examined. The fluid flow is conveyed in the form of a system of partial differential equations (PDEs), which are first reduced to a non-dimensional set of ordinary differential equations (ODEs) and then to first-order differential equations. The numerical approach parametric continuation method is employed to solve these ODEs. It has been noticed that the energy curve declines with increasing numbers of TiO2-nanoparticles (NPs). The effect of the external pollutant source variation factor enriches the concentration of pollutants in both fluid cases. Furthermore, the viscoelastic parameter K1 plays a notable role in determining the behavior of the fluids. Particularly in NNNFs, the variation of K1 enhances the fluid flow, whereas the rise of second-grade fluid factor decreases the velocity of the fluid. Our findings indicate a substantial impact of the parameters under consideration on the concentration of pollutant discharge. Significantly, it was observed that an increase in the amount of NPs and the thermal radiation parameter led to an improvement in the thermal conductivity of the nanofluid, consequently decreasing the concentration of pollutants in the discharge. The nanofluid has greater efficiency in boosting the energy transfer rate of the base fluid. In the case of the second-grade fluid, the energy propagation rate increases up to 6.25%, whereas, in the case of Walter’s fluid B, it increases up to 7.85%.

An analytical solution and stability analysis of asymmetric non-actuator guided projectiles

PurposeOne of the challenges encountered in the design of guided projectiles is their prohibitive cost. To diminish it, an appropriate avenue many researchers have explored is the use of the non-actuator method for guiding the projectile to the target. In this method, biologically inspired by the flying concept of the single-winged seed, for instance, that of maple and ash trees, the projectile undergoes a helical motion to scan the region and meet the target in the descent phase. Indeed, the projectile is a decelerator device based on the autorotation flight while it attempts to resemble the seed’s motion using two wings of different spans. There exists a wealth of studies on the stability of the decelerators (e.g. the mono-wing, samara and pararotor), but all of them have assumed the body (exclusive of the wing) to be symmetric and paid no particular attention to the scanning quality of the region. In practice, however, the non-actuator-guided projectiles are asymmetric owing to the presence of detection sensors. This paper aims to present an analytical solution for stability analysis of asymmetric decelerators and apprise the effects of design parameters to improve the scanning quality.Design/methodology/approachThe approach of this study is to develop a theoretical model consisting of Euler equations and apply a set of non-dimensionalized equations to reduce the number of involved parameters. The obtained governing equations are readily applicable to other decelerator devices, such as the mono-wing, samara and pararotor.FindingsThe results show that the stability of the body can be preserved under certain conditions. Moreover, pertinent conclusions are outlined on the sensitivity of flight behavior to the variation of design parameters.Originality/valueThe analytical solution and sensitivity analysis presented here can efficiently reduce the design cost of the asymmetric decelerator.