Main Equations Research Articles

The angle stabilization system of the self-balancing robot

The article is devoted to the consideration of the tilt angle stabilization system of a self-balancing robot. The object of research is self-balancing robots, and the subject is systems for stabilizing their tilt angle. A comprehensive analysis of the principles and algorithms used to ensure the stability of the robot when the tilt angle is changed is presented. The technical solutions underlying the stabilization system were studied and their effectiveness in relation to maintaining the robot in a vertical position was analyzed. A mathematical model is presented in which the self-balancing robot is represented as an inverted pendulum, i.e. a two-wheeled vehicle balances itself in a vertical position relative to the ground. It is assumed that the structure moves without friction. The main equations describing the movement of the balancing robot, taking into account the dynamics of the pendulum and the wheel, were obtained. As a regulator, the LQR regulator is proposed, which provides optimal control of a closed-loop system. The workability of the model and the proposed stabilization system was verified using simulation modeling. As a result of simulated modeling, oscillograms were obtained, which demonstrate that after an exciting effect, the system stabilizes and enters an equilibrium state. A controllability test was performed, which showed that the proposed controller can effectively control the system. Controllability ensures that a system can be subjected to a control signal in order to achieve the desired behavior. An example of the practical implementation of the robot and selected main components are presented

Development of a thermodynamic model for optimization of processes in crop production

The agricultural sector faces serious challenges related to climate change. These changes have the potential to reduce yields and food security, highlighting the importance of understanding and managing temperature dynamics. This work is the result of development a thermodynamic model that investigates the dynamics of temperature balance through heat energy transfer. A scheme of rheological heat exchange of an object with an insulated surface and graphs of irreversible rheological transformations are proposed. The main equation of heat exchange with a chemical reaction is given and the equation of the speed of heat energy transfer along the length of the object is derived. Further development of physical-mathematical models of the transformation of thermal energy into a set of states of the object is proposed. The experimental results fully correlate with the heat transfer equation. Samples with tomato seeds were irradiated with a photon irradiator with wavelengths of blue 450 nm, green 550 nm, red 650 nm with an exposure of 12/24 h. As a result, 90 % under the influence of the red spectrum of the photon irradiator for 24 h, which is 24 % more than the control sample. This will make it possible to assess the general temperature regime of agricultural objects and optimize the heating process. This study reveals the essence of temperature regulation at agricultural facilities using a thermodynamic model, which not only takes into account heat exchange, but also includes the influence of chemical reactions. The proposed thermodynamic model and associated equations provide a foundation for future research and practical applications that will ultimately benefit the agricultural industry, global food production

Non-equilibrium Onsager–Machlup theory

This paper proposes a simple mathematical model of non-stationary and non-linear stochastic dynamics, which approximates a (globally) non-stationary and non-linear stochastic process by its locally (or ‘piecewise’) stationary version. Profiting from the elegance and simplicity of both, the exact mathematical model referred to as the Ornstein–Uhlenbeck stochastic process (which is globally stationary, Markov and Gaussian) and of the Lyapunov criterion associated with the stability of stationarity, we show that the proposed non-linear non-stationary model provides a natural extension of the Onsager–Machlup theory of equilibrium thermal fluctuations, to the realm of non-stationary, non-linear and non-equilibrium processes. As an illustrative application, we then apply the extended non-equilibrium Onsager–Machlup theory, to the description of thermal fluctuations and irreversible relaxation processes in liquids, leading to the main exact equations employed to construct the non-equilibrium self-consistent generalised Langevin equation (NE-SCGLE) theory of irreversible processes in liquids. This generic theory has demonstrated that the most intriguing and long-unsolved questions of the glass and gel transitions are understood as a natural consequence of the second law of thermodynamics, enunciated in terms of the proposed piecewise stationary stochastic mathematical model.

Necessary and Sufficient Conditions for Solvability of an Inverse Problem for Higher-Order Differential Operators

We consider an inverse spectral problem that consists in the recovery of the differential expression coefficients for higher-order operators with separate boundary conditions from the spectral data (eigenvalues and weight numbers). This paper is focused on the principal issue of inverse spectral theory, namely, on the necessary and sufficient conditions for the solvability of the inverse problem. In the framework of the method of the spectral mappings, we consider the linear main equation of the inverse problem and prove the unique solvability of this equation in the self-adjoint case. The main result is obtained for the first-order system of the general form, which can be applied to higher-order differential operators with regular and distribution coefficients. From the theorem on the main equation’s solvability, we deduce the necessary and sufficient conditions for the spectral data for a class of arbitrary order differential operators with distribution coefficients. As a corollary of our general results, we obtain the characterization of the spectral data for the fourth-order differential equation in terms of asymptotics and simple structural properties.

Computing neural network to analyze heat and mass transfer in the flow of nanofluid between two disks

The model of copper nanoparticles which are suspended in the engine oil (EO) and rotated between two stretchable disks is analyzed. The flow, heat, and mass transmission phenomena of nanofluid with magnetohydrodynamics (MHD) have a vital role in many industries. A magnetic field in the vertical direction is imposed in the flow of the nanofluid and Dufour and Soret (DS) effects are discussed in the equations of energy and concentration. The main equations of motion and energy are converted into a set of nonlinear ordinary differential equations (ODEs) after applying the similarity conversions. A popular semi-analytical approach, namely the differential transform method (DTM) is used to get the solution of velocity, temperature, and concentration profiles. The effect of the various parameters on all profiles is graphically presented and explained. The present data of shear stress, the Nusselt, and the Sherwood numbers calculated by DTM are matched and verified by numerical method data and literature for the novelty of the work. The strength of the work is to analyze the validation, training, and testing by using Levenberg-Marquardt artificial neural network (ANN). This ANN is verified by mean square error, error histogram, and regression analysis.

An accurate theoretical formula for linear momentum, force and kinetic energy

The paper demonstrates that the existing mathematical formulas of linear momentum, force and kinetic energy in physics are incomplete, since such formulas have been formulated without incorporation of mass-energy equivalence relation E = mc2. Therefore, new reformulations of the main equations of linear momentum, force and kinetic energy in the realm of special relativity are proposed. The proposed formulas provide same mathematical outcomes as the old formulas, displaying same behavior of the system when velocity approaches to speed of light, but, most importantly, comprise only velocity of the light and mass of object to provide well-defined expressions. [Formulae available with the full text]. These formulas vividly reveal that every physical variable depends solely on relativistic mass. Therefore, it modifies Newton’s second law of motion and states that the force depends on rate of change of relativistic mass of object rather than its velocity. In this highly interesting topic, primary purpose here has been to present a succinct and the carefully reasoned account of a new aspect of the Newton’s second law of motion which properly allows to derive the new mathematical formulas of linear momentum, force and kinetic energy.

Electro-rheological flow of seminal fluid (K-L MODEL) induced by movement of cilia in a circular conduit

The current study deals with the flow of seminal fluid through the ductus efferentes owing to electro-osmotic force and motion of cilia. The constitutive equation of semen liquid has been assumed to be described as a K-L (Kuang and Luo) fluid model. The main equations of the ciliary flow of K-L model are solved by considering the supremacy of viscous upshot over inertia influence assuming the state of long wave-length number. The analytic expressions for velocity profile, flow rate and flow impedance are derived. This is the first time to bring out to the lime light that the flow impedance is decreased and the plug core radius is enlarged with the introduction of electric force in the ciliary flow system. The effects of pertinent physiological parameters on the time-mean flow rates of rhesus monkey and human semen are thoroughly investigated. The computational mean flow rate's value over the period of time of the current mathematical model has been well matched with the corresponding experimentation values of the time-mean flow rates of rhesus monkey and humanoid semen in the path of generative and the vas deferens of the path of generative of male. Introduction of electric field into the flow ciliary system and the rheology of semen fluid described as K-L fluid model (for the values of rheological parameters of K-L fluid are considered) reduce the flow resistance, which in turn elevates the flow velocity of seminal fluid, which activates the significant and favorable changes for Oocytes getting fertilized within hours of ovulation by means of reasonable volume of semen transport into the female reproductive tract to partake a quick contact with cervical mucus and enter the cervix.

Effect of Chemical reaction on hyperbolic tangent Williamson-Nanofluid flow past an exponentially porous stretching sheet: MHD and slip effects

In this research investigation, Runge-Kutta method solutions of dual-dimensional, steady, viscous, incompressible, chemically reacting, electrically conducting, hyperbolic tangent Williamson- flow of nanofluid past a uni-directional an while a magnetic field is present and several slip effects, an exponentially stretched sheet packed with porous material is studied numerically. Thermal radiation, Chemical reaction, the flow regulatingequations of boundary layer incorporates the impacts of thermophoresis and Brownian motion.SuitableTransformations of similarity are employed. for the main differential equations in partial to become differential equations inordinary that are nonlinear. The resultant governing equations are resolved by the application of a well utilized method in numericallycalled as the Runge-Kutta method and the shooting technique.Figures and numerical tables are used to show the representation of certain physical parameters on the flow model.. The pace of convergence for nonlinear differential systems is astonishing, according to an excellent comparison with the body of existing research, which also confirms a high degree of correctness. These tests are pertinent to the fields of plastic films, crystal growth, paper manufacture, and metal sheet cooling.

Analytical study of nonlinear models using a modified Schrödinger’s equation and logarithmic transformation

Optical solitons are special waves that maintain their shape while traveling. Hence, solitary optical waves have great significance when it comes to representing pulse propagation through nonlinear partial differential equations. In our research, we have uncovered unique solutions for the formation of solitary wave patterns in the equation of the Heisenberg ferromagnetic spin chain (HFSC). This equation aptly describes the behavior of electromagnetic waves in contemporary magnetism. By employing innovative techniques based on logarithmic transformations and analytical methods, we have obtained various solution forms expressed in a concise manner using elementary functions. We have tested the correctness of these solutions by substituting them directly into the main equation. These recent discoveries provide novel perspectives on the complex realm of solitons as depicted by this model. Moreover, since the model finds applications in fiber optic communications, fluid dynamics, and other fields, the implications of these discoveries are broad. The utilized methods stand out for being simple, reliable, and capable of creating fresh solutions for nonlinear partial differential equations in the realm of mathematical physics. The research findings presented here showcase the effectiveness of the applied methods in reliably studying nonlinear phenomena in the HFSC equation and other nonlinear problems in mathematical physics. By utilizing these tools, scholars have the opportunity to expand and enhance their understanding of the complex mathematical frameworks underlying real-world problems.

Numerical analysis of slip-enhanced flow over a curved surface with magnetized water-based hybrid nanofluid containing gyrotactic microorganisms

This article presents the two-dimensional flow of hybrid nanofluid comprising of gyrotactic microorganisms under the consequences of multiple slip conditions, magnetic field and thermal radiation across an elongating curved surface using porous media. The nanoparticles of TiO2 and Fe3O4 have dispersed in water for composition of hybrid nanofluid. Main equations of the problem are converted to ODEs by using an appropriate set of variables. Solution of the present model is determined with the help of bvp4c technique, which is explained in detail in the coming section. Validation of the current results is done versus the published work. The effects of various emerging factors on flow distributions have been considered and explained. Additionally, the slips conditions are incorporated to analyze various flow distributions. The present outcomes show that the rising magnetic factor lessens the velocity profile, whereas rises the temperature profile. The curvature factor has supported both temperature and velocity distributions. Growth in velocity, thermal, concentration, and microorganisms slip factors reduce the corresponding distributions. The greater impact of the embedded parameters is found on hybrid nanofluid flow when matched to nanofluid flow.

The IASI NH3 version 4 product: averaging kernels and improved consistency

Abstract. Satellite measurements play an increasingly important role in the study of atmospheric ammonia (NH3). Here, we present version 4 of the Artificial Neural Network for IASI (ANNI; IASI: Infrared Atmospheric Sounding Interferometer) retrieval of NH3. The main change is the introduction of total column averaging kernels (AVKs), which can be used to undo the effect of the vertical profile shape assumption of the retrieval. While the main equations can be matched term for term with analogous ones used in UV/Vis retrievals for other minor absorbers, we derive the formalism from the ground up, as its applicability to thermal infrared measurements is non-trivial. A large number of other smaller changes were introduced in ANNI v4, most of which improve the consistency of the measurements across time and across the series of IASI instruments. This includes a more robust way of calculating the hyperspectral range index (HRI), explicitly accounting for long-term changes in CO2 in the HRI calculation and the use of a reprocessed cloud product that was specifically developed for climate applications. The NH3 distributions derived with ANNI v4 are very similar to the ones derived with v3, although values are about 10 %–20 % larger due to the improved setup of the HRI. We exclude further large biases of the same nature by showing the consistency between ANNI v4 derived NH3 columns with columns obtained with an optimal estimation approach. Finally, with v4, we revised the uncertainty budget and now report systematic uncertainty estimates alongside random uncertainties, allowing realistic mean uncertainties to be estimated.

Multicomponent solutions: Combining rules for multisolute osmotic virial coefficients.

This paper presents an exploration of a specific type of a generalized multicomponent solution model, which appears to be first given by Saulov in the current explicit form. The assumptions of the underlying theory and a brief derivation of the main equation have been provided preliminarily for completeness and notational consistency. The resulting formulae for the Gibbs free energy of mixing and the chemical potentials are multivariate polynomials with physically meaningful coefficients and the mole fractions of the components as variables. With one additional assumption about the relative magnitudes of the solvent-solute and solute-solute interaction exchange energies, combining rules were obtained that express the mixed coefficients of the polynomial in terms of its pure coefficients. This was done by exploiting the mathematical structure of the asymmetric form of the solvent chemical potential equation. The combining rules allow one to calculate the thermodynamic properties of the solvent with multiple solutes from binary mixture data only (i.e., each solute with the solvent), and hence, are of practical importance. Furthermore, a connection was established between the osmotic virial coefficients derived in this work and the original osmotic virial coefficients of Hill found by employing a different procedure, illustrating the equivalency of what appears to be two different theories. A validation of the combining rules derived here has been provided in a separate paper where they were successfully used to predict the freezing points of ternary salt solutions of water.

Investigating the stagnation point and Darcy-Forchheimer flow of heat and mass transport with thermal radiation and activation energy

In this paper, the stagnation point and boundary layer flow of two-dimension Williamson fluid flow near a stretching cylinder along porous medium is examined. The transference analysis of mass and heat in the occurrence of thermal radiation, heat absorption and activation energy are investigated. The double stratification is implemented to analyze its impact on the thermal and solutal fields. The main leading equations are solved by applying the boundary layer approximation. By assuming the similarity variables, the main equations of the Williamson fluid are transformed into ordinary differential equations. In order to solve numerical solutions, a well-known numerical technique is adopted on MATLAB software namely ‘Bvp4c’. The fluid's velocity experiences an increment with changes in the buoyancy parameter, curvature parameter and the Williamson fluid coefficient, whereas the velocity distribution diminishes due to variations in the porosity parameter, stretching parameter and Forchheimer number. The temperature in a given region decreases due to the influence of both the Prandtl number and curvature parameter. The fluid temperature exhibits incrementing outputs in the presence of changes in the curvature parameter, thermal stratification parameter and heat generation. The solutal stratification parameter, temperature difference parameter and Schmidt number lead to a reduction in the fluid concentration. The concentration profile increases under the influence of curvature parameter. The validation of numerical finding is also included here which show similar trend with published work.

A comprehensive study on the free vibration of adhesive lap joint laminated conical–conical shells adherends

In this research, the free vibration analysis of lap joint laminated conical shell connected to laminated conical shell with adhesive layer for various boundary conditions are investigated. Based on the first-order shear deformation theory (FSDT), the main field equations of the analytical model are derived and the governing differential equations of laminated conical shell to laminated conical shell adherends lap joint are obtained for homogeneous and elasticity adhesive layers by using Hamilton principle method. Then, based on the generalized differential quadrature method (GDQM), the equilibrium equations of the structure adherends are investigated. The influence of the boundary conditions, the type of the composite materials and adhesive layer, the circumferential wave number, the length to radius ratio of the shell, and the length of overlap to length of shell ratio are studied. Furthermore, the thickness to radius ratio and the thickness of adhesive to thickness of shell ratio, as well as the various cone angles of conical shell on the natural frequency of laminated conical adherends shells connected to the conical shells with adhesive layer are investigated. For validation of the numerical results, the results are compared with the previous research and the comparisons show very good agreement. The numerical results show that with the increase of the internal shell and external shell of semi-angle, the non-dimensional frequency of lap joint connection of two laminated conical shells with adhesive layer decreased and increased, respectively. Furthermore, with the increase of the adhesive thickness to conical shell thickness ratio and lap joint length to length of conical shell ratio, the dimensionless natural frequency of the structure increased.

On the revealing of optical soliton solutions of the stochastic perturbed nonlinear Schrödinger equation in the presence of the Kerr law of self-phase modulation

In this study, an attempt to obtain optical soliton solutions of stochastic perturbed (1+1)-dimensional nonlinear Schrödinger equation having the Kerr law nonlinearity is presented. With the help of complex wave transform, the investigated nonlinear partial differential equation was converted into the nonlinear ordinary differential form. Analytical stochastic optical soliton solutions of the ordinary differential form were derived by applying the Kudryashov auxiliary equation method. By replacing the wave transformation and the appropriate solution set parameters, the main equation is provided. Graphic presentations were made for better interpretation of the results.

On optical solitons of the (1 1)-dimensional Biswas-Milovic equation having Kerr law and parabolic-law with weak non-local nonlinearity in the presence of spatio-temporal dispersion

In this study, an effort has been made to obtain optical soliton solutions of the (1+1)-dimensional Biswas-Milovic equation which was introduced by Biswas and Milovic in 2010, having Kerr law and parabolic-law with weak non-local nonlinearity in the presence of spatio-temporal dispersion, which is one of the important models for nonlinear optics. Although, the model is an equation that has been studied by many researchers, the fact that the form to be examined has not been studied before. As a general algorithm, first converting the model to nonlinear ordinary differential form with a complex wave transformation, then obtaining candidate optical soliton solutions by utilizing the new Kudryashov technique, determining the ones that satisfy the main equation from these solutions as the exact solution, and in order to better understand the obtained solutions by making graphical presentation and providing the necessary comments constitute the main framework of the article.

Exact solution for capillary waves on the surface of a liquid of finite depth

Using the Schwartz function method, we have obtained a new exact solution for the problem of stationary capillary waves of finite amplitude on the surface of a liquid that has a finite depth. The reliability of the solution was confirmed by the results of numerical verification of the main boundary equation. The obtained solution of the problem is general in the sense that for any Weber number one can find the corresponding wave configuration. Parametric analysis showed a nonmonotonic dependence of the wave-length and its amplitude on the Weber number. The fact that the problem has one more branch of the solution (the trivial solution) indicates the possibility of the existence of other branches. The Schwartz function method cannot guarantee finding all solutions of the problem even from the specified class of functions. Therefore, the question of reproducing the known exact solution of W. Kimmersley for this problem and its reliability remains open. Note that for the parameter ß included in the main boundary equation, W. Kimmersley preliminarily laid down assumption ß = 1. The found exact solution has the property that ß > 1 and cannot coincide with W. Kimmersley’s solution.

Magneto-Ternary Hybrid Nanofluid Flow About Stretching Cylinder in a Porous Medium with Gyrotactic Microorganism

The diluted suspension of nanoparticles in base liquids has been found in extensive applications in various industrial processes like nanomedicines, cooling of microsystems, and energy conversion. The idea of tri-hybrid nanofluid have been developed which shows the impact of three nanoparticles at the same time in a single fluid. This newly developed tri-hybrid mixture model getting more attention and performed better than hybrid and nanofluid. Owing to its important applications an attempt has made in this article to investigate the Casson ternary hybrid nanofluids flow along a stretching cylinder through porous medium subject to the influence of microorganism in the modeled equations. The strength of magnetic field has employed in upward direction of the flow system, and Activation Energy effect is addressed. The main equations of fluid motion have been converted to dimensionless format using set of suitable variables. In this work it has noticed that, growth in permeability parameter, Casson and magnetic factors result in more resistive force to fluid motion that declines the velocity characteristics. Moreover, the temperature distribution has grown up while the concentration characteristics have declined with growing values of Brownian factor. Furthermore, microorganism characteristics decay with growth in bio-convection Lewis and Peclet numbers. The impact of these parameters upon heat, mass and motile transfer rates has been evaluated in the tabular form.

A Cold Climate Wooden Home and Conflagration Danger Index: Justification and Practicability for Norwegian Conditions

The vast majority of fire-related deaths occur in residential buildings. Until recently, the fire risk for these buildings was only considered through static risk assessments or period-based assessments applying to certain periods of the year, e.g., Christmas holidays. However, for homes with indoor wooden panelling, especially in the ceiling, a dynamic fire danger indicator can be predicted for cold climate regions. Recognising the effect of fuel moisture content (FMC) of indoor wooden panelling on the enclosure fire development allows for the prediction of a wooden home fire danger indicator. In the present study, dry wood fire dynamics are analysed and experimental observations are reported to support in-home wooden panel FMC as a suitable wooden home fire danger indicator. Then, from previous work, the main equation for modelling in-home FMC is considered and a generic enclosure for FMC modelling is justified based on literature data and supported through a sensitivity study for Norwegian wooden homes. Further, ten years of weather data for three selected locations in Norway, i.e., a coastal town, an inland fjord town and a mountain town, were analysed using a three-dimensional risk matrix to assess the usability of the fire risk modelling results. Finally, a cold climate wooden home fire danger index was introduced to demonstrate how the risk concept can be communicated in an intuitive way using similar gradings as the existing national forest fire index. Based on the generic enclosure, the findings support FMC as a fire risk indicator for homes with interior wooden panelling (walls and ceiling). Large differences in the number of days with arid in-home conditions were identified for the selected towns. The number of days with combined strong wind and dry wooden homes appears to depend more on the number of days with strong wind than days of in-home drought. Thus, the coastal town was more susceptible to conflagrations than the drier inland towns. This aligns well with the most significant fire disasters in Norway since 1900. In addition, it was demonstrated that the number of high-risk periods is manageable and can be addressed by local fire departments through proactive measures. In turn, the fire risk modelling and associated index respond well to the recent changes in Norwegian regulations, requiring the fire departments to have systems for detecting increased risk levels. Testing the modelling for a severe winter fire in the USA indicates that the presented approach may be of value elsewhere as well.

Mechanistic fouling study of Hybrid imidazolium-based ionic liquid membrane during emulsified oil removal from oily wastewater; oil type effect

To overcome the strict discharge effluent regulations of oily wastewater, the treatment with membranes has a good perspective. In this work, [C12mim][Cl] ionic liquid was induced into the polyether sulfone membrane to improve membrane hydrophilicity and anti-fouling resistance. To reveal the membrane fouling type, the permeate flux of the membrane was modeled according to Hermia's equations and combined equations driven from four main fouling mechanisms equation. In addition, to develop a proper interpretation and fully new insights mechanisms, the elemental and topographical analyses, and chemical surface characterizations using FESEM, EDS MAPPING, AFM, FESEM, Zeta analyses, and contact angle and interfacial tension measurements were performed on the four different oily wastewaters and the membrane. So, the impact of interfacial tension, wettability, mean particle size, particle size distribution (polydispersity index), zeta potential, acidic and basic nature of oil, and the properties of polar fractions of oil including the asphaltene and resin concentrations and their weight percent of polar heteroatoms were thoroughly examined. The results revealed a reduction in permeate flux as the API gravity was reduced due to the particle size of the oily wastewater emulsion increased. Consequently, the wettability of the membrane altered towards a more oil-wet state leading to a reduction in the roughness of the membrane surface. The attachment and cover of the surface mainly by the acidic component of the oil with a higher affinity to attach the oxygen element of the membrane were proposed as the dominant mechanism of cake layer fouling.