Dimensionless Form Research Articles

Numerical Simulation of Natural Convection in a Chamfered Square Cavity with Fe3O4-Water Nanofluid and Magnetic Excitation

This study delves into the numerical exploration of the MagnetoHydroDynamic (MHD) characteristics of an Fe3O4-Water nanofluid contained within a chamfered square enclosure under the influence of an external magnetic field. The enclosure, characterized by distinct hot and cold imposed temperatures on its side walls, features both straight and chamfered sections. The orientation of magnetic field lines was manipulated by varying the angular placement of the magnetic source. The computational framework for nanofluid dynamics is mathematically formalized through a dimensionless formulation of the Navier-Stokes equations derived from their dimensional counterparts. A comprehensive numerical analysis was conducted employing the Finite Element (FE) method, a. The interaction between the Hartmann number and the angular placement of the magnetic source was analyzed, with a specific focus on nanofluid isotherms, temperature profiles, and velocity magnitude distributions. The results were thoroughly investigated and extensively discussed.

Simulation of sugar beet yield

Mathematical simulation of agricultural crop yields is based on the physical principle of causal interactions balance in a closed physical system. The conditions for model verification are determined, allowing to obtain objective and reliable findings. It is noted that empirical equations representing the dependence of crop yields on crop-forming factors in the form of polynomials of any degree, obtained using the multiple nonlinear regression method, are valid only for the conditions of a specific field experiment. Using them, it is impossible to carry out theoretical generalizations that would allow developing these particular solutions into a generalized mathematical model. It is shown that mathematical model of yield, presented in multiplicative form, can include an unlimited number of yield-forming factors. Atmospheric precipitation was used as a factor characterizing the moisture supply for plants in case of no irrigation. The validity of the developed solution is confirmed by 13 years of data on the yield of sugar beet (NZ-type hybrid) cultivated in Belarus at the State Agricultural Institution “Molodechno Variety Testing Station”. The mathematical model of yield is presented in dimensionless form; all blocks of factors of this model are similarity criteria. This allows to compare the results of mathematical simulation of yield of any agricultural crop on soils with any agrochemical properties. Such an analysis cannot be carried out with the results of calculating yields using the formulas of private mathematical models of agricultural crop yields in the form of algebraic polynomials.

Thermal Distribution in Fins of Different Geometries With Temperature and Wavelength Dependent Properties

ABSTRACTIn this research work, we developed a mathematical model of heat transfer for different profiles of fins. This paper investigates heat transfer dynamics in continuously moving fins (rectangular, trapezoidal and concave parabolic), focusing on temperature‐dependent thermal conductivity, heat transfer coefficients, internal heat generation and emissivity that varies with temperature and wavelength. The different values of the heat transfer coefficient capture various types of convection, nucleate boiling, condensation, and radiation effects, while treating thermal conductivity as a linear function of temperature. This problem is converted into a dimensionless form, and we adopted the Legendre wavelet collocation method (LWCM) to get the solution of the fin problem for various profiles. An exact solution in specific cases shows congruence up to seven to eight decimal places with LWCM results. Moreover, our analysis explores the effect of numerous non‐dimensional parameters such as thermal conductivity parameter , Peclet number , surface emissivity parameter B, convention‐conduction parameter , radiation‐conduction parameter , internal heat generation , D fin taper ratio on the temperature profile and fin efficiency were studied in detail. As , , and B, D increases in magnitude, the temperature inside the fin decreases, while higher values of Peclet number (), A, m, and Q cause lower heat transfer rate inside the fin. The results provide significant insights into the complex interplay of thermal processes across different fin geometries, emphasizing the importance of these dependencies for accurate modeling and optimization in thermal management systems.

The effects of heterogeneity on Darcy-Benard convection saturated with a ferrofluid

This study investigates how the vertical magnetic field influences the stability of the ferrofluid in a heterogeneous porous layer with the vertical flow. The layer, confined between two impermeable walls, is subject to different heat flux conditions on each side. The behavior of the ferrofluid governed by Darcy’s law and the Oberbeck-Boussinesq approximation, while the magnetic field is modeled using Maxwell’s equations. A dimensionless formulation and linear stability analysis are employed to derive an ordinary differential equation, which is solved using the shooting method and a Runge-Kuttasolver. The results indicate that the transition of the wave number from zero to a nonzero value is connected to the Peclet number and other magnetic parameters.

Statistical analysis and numerical simulation of wire coating process with sisko nanofluid: exploring variable thermophysical properties

The wire coating process enhances wire durability, safety, and performance, preventing corrosion and providing insulation against abrasion, temperature changes, and chemicals. To evaluate process quality, characteristics like fluid flow and heat transfer in the die are analysed. This work focuses on understanding molten polymer rheology via momentum, thermal distribution, and nanoparticle volume fraction profiles. A mathematical model is developed for a Sisko fluid using the Buongiorno model, which includes temperature-dependent thermophysical properties. The governing equation is reduced to dimensionless form using non-dimensional parameters and solved using the numerical technique. The results are visualised graphically for the parameters like Sisko fluid parameter, viscosity parameter (between 0 and 1), Brinkmann number, Hartmann number (between 2 and 10) using 2D and 3D plots. The study shows that the thermal field is enhanced by nanofluid factors, which subsequently enhances the heat transfer rate in the presence of nanoparticles. The optimization of the responses: shear stress rate ( S rz ) and rate of heat transfer ( Nur ) is carried out concurrently using the Surface Response Method. Furthermore, a sensitivity analysis has been conducted. The findings indicate that the viscosity parameter significantly impacts the sensitivity of the both shear stress rate and heat transfer rate.

Measuring economic growth factors: Dynamic trends and prospects

Subject. The article is devoted to measuring the factors of economic growth. Objectives. The purpose is to identify and measure economic growth factors in Russian regions. Methods. To analyze economic growth, the study employs the apparatus of production functions. The toolkit is based on the Cobb–Douglas production function, which, with the help of some transformations, takes a dimensionless form, namely, the resulting indicator is the growth of gross regional product, and the index of physical investment growth and employment growth in the region act as independent factors. Results. We built production functions for regions of the Central Federal District, analyzed changes in production functions for individual groups of regions. Conclusions. The findings of our research of production functions of regions of the Central Federal District for 2010–2022 showed less uniformity in parameters compared with the study of 2000–2009. Investments yield less return. The mechanisms of the classical theory of economic growth do not work as effectively as 15-20 years ago. The main economic growth depends on the amount of labor employed in the economy. The shortage of employees can be overcome only by changing the economic structure, intensification of mechanization processes, and robotization of production.

Анализ электродинамической модели приземного слоя атмосферы

The paper presents the results of analyzing the atmosphere surface layer electrodynamics model using the methods of similarity theory. In the electrode effect approximation, the mathematical model is presented in the form of a nonstationary initial-boundary value problem for a parabolic type equation. The correctness of using various types of electrodynamic models depending on meteorological conditions in the stratified surface layer is substantiated. Estimates of the characteristic scales of the electrode layer are obtained depending on the meteorological parameters of the atmosphere, which can be used in modeling problems of electrodynamics of the surface layer. For the case of neutral stratification of the atmosphere, a procedure has been implemented to reduce the system of equations of the electrode effect to a dimensionless form and obtain similarity criteria. Models of a classical, turbulent, convective-turbulent electrode layer were constructed, and approximations were obtained for the cases of intense turbulent mixing and strong convective transfer by splitting them into physical processes.

Stress analysis of a hyperbolic elastic-plastic disk under thermomechanical loading

A general solution is constructed for the distribution of stresses within a thin, hyperbolic elastic-plastic disk subjected to external and internal pressures while in a steady-state temperature field and under plane stress conditions. The pressures applied to the disk are assumed not to induce any plastic flow at the initial, uniformly distributed temperature. The disk is loaded by increasing the temperature of its inner radius. The temperature of the outer radius is kept constant. Elastic and temperature strains are related to stresses by the Duhamel–Neumann law. In the plastic region, the von Mises yield criterion is valid. The tensile yield stress is taken as constant. The boundary value problem is statically determinate. Therefore, the plastic flow rule is not required to determine the stress field. The final part of the general solution depends on the parameters classifying the boundary value problem, including whether the plastic flow initiates at the inner or outer radius of the disk. The solution is presented in dimensionless form. The loading parameter comprises the inner surface temperature of the disk, the coefficient of linear thermal expansion, the initial temperature of the disk, the dimensionless inner radius of the disk, the yield stress under uniaxial tension, and Young’s modulus. The general solution is obtained for both cases (for a plastic region appearing at the inner or outer radius of the disk). A specific numerical solution is constructed for a disk subjected to external pressure and a steady-state temperature field. The temperature of the inner radius at which the plastic region first appears is calculated. As it increases, the plastic region extends. At a certain temperature value, another plastic region appears near the outer radius of the disk. This value is determined as part of solving the boundary value problem. The influence of the parameters classifying the boundary value problem on the development of the plastic region and the stress distribution along the radius of the disk is shown. The qualitative differences between the new and existing solutions for a disk of constant thickness under the action of a uniform temperature field are discussed.

DETERMINING THE STATE VARIABLES OF A DISCRETE PNEUMATIC DRIVE BY USING THE SECANT METHOD WHEN LINEARIZING A MATHEMATICAL MODEL

The pneumatic drive as a thermodynamic system is described based on the principles of "thermodynamics of a body of variable mass". In describing thermodynamic and gas-dynamic processes in pneumatic drives, two scientific schools have developed: researchers of the first school represent processes in the cavities of the drive as polytropic processes with a variable polytropic index, and the second consider processes based on the equation of energy (heat) balance of the gas-pneumatic drive in an open space. The authors investigate a nonlinear mathematical model obtained on the concepts of the second school. In the cycles of reducing the number of independent parameters that determine the nature of the transient process in the drive, a transition to a dimensionless form of equations is carried out based on the principle of minimizing dimensionless complexes (dynamic similarity criteria) that determine the nature of the transient process. The task of linearization of the nonlinear model was set with the purpose of obtaining analytical expressions for all state variables based on the linear model, which will allow avoiding the numerical step methods of integration of the original nonlinear model in calculations. It is shown that the replacement of nonlinear dependencies by the first terms of their expansion in a Taylor series (the tangent method), which is practiced for servo hydraulic pneumatic drives, leads to large errors in relation to discrete drives. The proposed linearization by the secant method with the choice of the optimal form of the sowing allowed to significantly increase the calculated accuracy of the linear mathematical model. and obtain the state variables of the pneumatic drive in analytical form. The calculations performed using the second-order linear model convincingly indicate that it is quite adequate to the calculated accuracy of the nonlinear mathematical model. In the entire range of the most probable parameters, the calculated accuracy of the second-order mathematical model is quite sufficient for practical use. Thus, the need to use step-by-step numerical methods for integrating the equations of a nonlinear model and organizing the computational process on a computer is eliminated.

On the reconstruction of a two-dimensional density of a functionally graded elastic plate

In this article, the in-plane vibrations of a rectangular plate within the framework of a plane stress is formulated based on the general formulation of steady-state vibrations of an inhomogeneous elastic isotropic body. The left side of the plate is rigidly fixed, vibrations are forced by tensile load applied at the right side. The properties of the functionally graded material are described by two-dimensional variation laws (Young’s modulus, Poisson’s ratio and density). A dimensionless problem formulation is given. The direct problem solution of the displacement field determination is obtained using the finite element method. The effect of material characteristics on the displacement field and the value of the first resonance are shown. An analysis of the obtained results is carried out. The inverse problem of density determination from displacement field data for a fixed frequency is considered. To reduce the error in calculating two-variable table functions derivatives, an approach based on spline approximation and a locally weighted regression algorithm is proposed. Reconstruction examples of different laws are presented to demonstrate the possibility of using this approach.

Operational features of collecting pipelines in the presence of transit and groundwater level slope

The conditions of operation for pressure collecting drainage pipelines in melioration systems that function in the presence of transit flow and groundwater surface levels have been considered. The system of differential equations describing the fluid motion with variable flow rate in the drainage pipe has been analyzed, taking into account the inflow of liquid from the surrounding soil through the side walls in a filtration mode. This system of equations consists of the hydraulic equation of variable mass and a modified filtration equation. The pipeline under study is laid horizontally and operates with a groundwater level slope. A significant complication in the operation of such pipes is the presence of transit flow that enters their initial cross-section and is transported along the entire length of the drainage pipeline. By introducing new variables, the original system is transformed into a dimensionless form. The solution to this system of equations is presented. It is shown that, in this case, the solution to the original system of equations depends on the magnitude of four main factors: the resistance coefficient of the collection drainage pipeline «ζl»; the generalized parameter «A» which comprehensively considers the structural and filtration characteristics of the flow under consideration; the slope of the groundwater level «I»; and the magnitude of the transit flow «Qtr» The analysis employs the concept of an infinitely long horizontal drainage pipeline that operates with a groundwater level slope and transit flow, or, equivalently, a pipeline with infinite filtration capacity of the side wall surfaces. Based on the conducted analysis, relatively simple and convenient analytical expressions have been obtained for calculating the variation of flow rate and head loss along the length of the drainage pipeline operating with transit flow. To simplify the calculations, corresponding auxiliary graphical dependencies have been proposed.

Impact of Dual Phase Lag on Natural Convection Flow in a Porous Vertical Channel in the Presence of Periodic Boundary

ABSTRACTThe dual‐phase‐lag (DPL) heat conduction model is utilized in this research to analyze the fluid flow of viscous fluid passing through a porous vertical channel with periodic boundary conditions. Periodic heating is subjected to the channel boundary. Equations regarding the model, including the momentum and energy equations, in which the DPL term is incorporated, all in dimensional form, are stated and are being transformed to their dimensionless form, then solved analytically by undetermined coefficients and variation of parameters. The actual expressions of temperature and velocity, as well as the heat transfer rate and skin friction, are determined. The effects of the DPL parameters, suction/injection, Prandtl number, heat source/sink, and Strouhal number on the dimensionless temperature and velocity profiles are demonstrated using graphs that are constructed with the aid of MATLAB. It was found during the investigation that the introduction of the DPL model, together with suction/injection in the channel, enhances the velocity and fluid temperature within the channel. Also, the decreasing effect of temperature gradient phase lag on fluid temperature and velocity conversed with that of heat flux phase lag. As an important contribution, the discovery of the effects of the phase‐lag parameters of the DPL model and suction/injection on fluid temperature and velocity would significantly help researchers advance the design of electrical and electronic systems.

Computational study on entropy generation in Casson nanofluid flow with motile gyrotactic microorganisms using finite difference method

The present study reveals the phenomenon of flow and thermal variations of non-Newtonian nanofluids flow of Copper (Cu) nanoparticles through two orthogonal permeable porous discs. Additionally, the study delves into the significance of liquid solid interfacial layer and nanoparticle diameter is studied to optimize the thermal conductivity. Gyrotactic Microorganisms are also considered to optimize the nanoparticles stability inside the fluid, and the entropy generation caused to system disorder, so entropy analysis is also calculated. Using appropriate dimensionless variables, the controlling PDEs are converted into dimensionless forms. These dimensionless PDEs are solved using the Finite Difference technique, which yields very accurate and stable solutions. Higher values of the Rayleigh bio-convection and mixed convection terms enhance the flow velocity field in both porous discs. The less thickness of nanolayer has significance effect to optimize the temperature. Enhancing the values of bioconvection Lewis number leads to a decrease in motile microorganism concentration in both permeable discs, and entropy generation rate increases against rising power of Eckert number and thermal radiation rise. The finite difference method outcomes have been thoroughly validated against already published research, and showing an excellent correlation. This research holds significant potential for applications in the fabrication and electromagnetic control of advanced magnetic nanofluid materials, relevant to biomedical, energy, thermal management, and aerospace technologies.

Comparative analysis of transient thermal behaviour and efficiency in longitudinal metal porous fin with combined heat and mass transfer under dehumidification conditions

This investigation explores the transient thermal characteristics of longitudinal porous fins made of Aluminium (Al) and Copper (Cu), exposed to a dynamic moist airstream under dehumidification conditions. Heat and mass transfer processes are initiated when the fin surface temperature falls below the dew point temperature of the surrounding air, driven by differences in temperature and humidity ratios. Heat transfer within the porous structure is modelled using temperature-dependent convective coefficients and Darcy’s flow model, with the governing nonlinear partial differential equation transformed into dimensionless form and solved using the Finite Difference Method (FDM) to analyse the fin thermal profile and efficiency over time. These results reveal that Cu fin due to their superior thermal conductivity exhibit higher temperature profile and efficiency compared to Al fin across all pertinent parameters. A key finding is the significant impact of Relative Humidity (RH) on the thermal behaviour of the fin: as RH increases by 400%, the temperature distribution from base to tip decreases by 138% in Al due to its higher specific heat capacity, which enables it to absorb more latent heat and 85% in Cu, where the superior thermal conductivity allows for faster heat dissipation. As Darcy number (Da) decreases by 99%, the temperature distribution along the fin from base to tip increases by 0.058% in Al due to its lower thermal conductivity and 0.041% in Cu, where its higher thermal conductivity enables more efficient heat dispersion. These insights are pivotal for advancing fin design and optimization, facilitating improved thermal management in industrial applications under transient and dehumidifying conditions.

Transient gravity‐driven flow of a third‐grade fluid (TGF) on an inclined plane with non‐isothermal effects, exothermic reactions, and porous medium influence

AbstractComprehending the dynamics of reactive viscoelastic third‐grade fluids (VTGF) in gravity‐driven flows along inclined planes is crucial for numerous engineering and industrial applications. This research focuses to explore the transient behavior of such a fluid under non‐isothermal conditions, with an emphasis on the impact of exothermic reactions. The inclined plane region, filled with a porous material of constant permeability, is modeled using a modified version of Darcy's law to account for resistance to flow. The viscosity‐temperature dependence follows the Nahme‐type principle, while convective cooling at the free surface is simulated using Newton's cooling law. The exothermic chemical reaction of the material is described using Arrhenius kinetics. The resulting mathematical model for energy balanced and momentum comprises a set of non‐homogeneous and nonlinear partial differential equations (PDEs), which are transformed into dimensionless form and solved using semi‐implicit numerical techniques based on finite difference methods (FDM) implemented in Matlab. The study visually examines thermo‐dynamical phenomena such as thermal runaway due to the exothermic reaction and investigates how velocity and temperature respond to variations in system parameters. The numerical findings indicate that the interplay between porous medium, viscosity variation, and thermal effects significantly influences the flow and thermal behavior, providing valuable insights for optimizing and controlling such fluid systems in practical applications.

Catalyst pellets with Gaussian activity distribution under forced periodic operation for reactions with Langmuir-Hinshelwood kinetics

This research investigates how combining forced periodic operation with spatially distributed catalyst activity can enhance heterogeneous catalytic processes. It focuses on analyzing reaction-diffusion phenomena within porous catalyst pellets. These pellets exhibit a Gaussian distribution of active sites, and the study investigates how externally forced periodic variations in bulk reactant concentration and temperature affect the reaction process. The paper establishes a mathematical model for a non-isothermal reaction based on Langmuir-Hinshelwood kinetics. This model is then transformed into its dimensionless form for numerical analysis. Numerical simulations are employed to investigate the impact of various parameters on the concentration and temperature profiles within the pellet, as well as on the pellet productivity. These parameters include the position and width of the Gaussian distribution of active sites, the Thiele modulus, the mass and heat Biot numbers, the Arrhenius number for reaction, the energy generation function, the ratio of characteristic times for diffusion and heat conductivity, and frequencies and amplitudes of periodic variations. The simulations reveal complex relationships between the spatial and temporal profiles of concentration and temperature within the pellets. Using porous granules with a non-uniform catalyst activity profile alongside forced periodic operations for reaction-diffusion processes enables higher productivity compared to granules with a uniform activity profile and subject to the steady-state operation. This study demonstrates the potential for optimizing catalytic processes in porous pellets with non-uniform activity profiles under forced periodic operation, offering valuable insights into enhancing process efficiency.

Numerical study of an electrically conducting nanofluid flow past a vertical stretching Riga plate

The heat and mass transfer through electrically conducting and mixed convective nanofluid flow across an elongating Riga plate is studied. Riga plate is an electromagnetic sheet, in which electrodes are assembled alternatively. The arrangement of Riga plate produces electromagnetic behavior in the fluid flow. In the current analysis, the influence of Arrhenius activation energy, viscous dissipation and heat source/sink over the surface of Riga sheet is considered. The heat and mass transfer are examined by convective boundary conditions and nanoparticles (NPs) zero-mass flux. The Brownian motion and thermophoretic diffusion of fluid molecules are applied via Buongiorno's approach. Von Karman's similarity approach is implemented to reset the modeled equations into the dimensionless form of ordinary differential equations (ODEs). The system of ordinary differential equations is solved numerically via PCM (parametric continuation method). The fluid velocity, temperature field, and concentration gradients are analyzed by varying various flow parameters and graphically portrayed. The numerical results are validated by comparing them with the published studies. It can be resolved form the results that the flow rate decilnes with the effect of power index m, whereas enhances with the consequence of Hartmann number. The significance of thermophoresis term augments the temperature curve of the fluid.

Unsteady Hydromagnetic Flow Of A Nanofluid In The Presence Of Inclined Magnetic Field In A Porous Media

The unsteady hydromagnetic flow of a nanofluid in the presence of an angled magnetic field in porous media is investigated, with consideration given to the effect of the nanofluid’s viscosity variation parameter. The governing partial differential equations are obtained and transformed into dimensionless form by employing dimensionless quantities. Because of its stability, consistency, and high convergence rate, the finite difference approach is used to derive the numerical schemes of transformed partial differential equations, in line with the Crank-Nicolson method. The study subsequently examined at how dimensionless numbers, a viscosity variation parameter, and the magnetic field’s inclination angle affected velocity and temperature profiles. The results show that raising the Reynolds number and magnetic parameter increases the velocity profiles of the nanofluid, whereas increasing the permeability parameter reduces the velocity of the nanofluid. Higher Reynolds, Eckert, and Prandtl numbers lead to larger temperature profiles of the fluid flow. Moreover, study found that increasing the viscosity parameter and magnetic field inclination angle can accelerate fluid flow velocity profiles. In addition, the temperature profiles of the fluid flow grow with the viscosity parameter and the angle of inclination of the magnetic field. The current findings are consistent with previous research, indicating their correctness and validity.

Simulation of blood flow in a tapered artery with ternary hybrid nanofluid and Jeffrey model

This research simulates flow of blood in a tapered peristaltic artery integrating ternary hybrid nanofluid using the Jeffrey fluid model. The simulation examines the effects of Au, Fe3O4, and Ag nanoparticles on the flow. The artery, modeled as a peristaltic channel, is subjected to nonlinear thermal radiation, magnetic forces, and heat source. The problem is formulated using non-linear Cartesian partial differential equations, which are then transformed into dimensionless form without approximations. Adomian Decomposition Method (ADM) is employed to solve these equations, revealing how normal and shear stress, normal and axial velocities, induced magnetic fields, and temperature profiles vary with changes in relevant parameters. Additionally, biological factors are considered to graph streamlines, providing a comprehensive analysis of the flow dynamics. Some notable results include that adding ternary nanoparticles increases the Nusselt number by 26.03% in Newtonian fluids and 158.69% in Jeffrey fluids, and increasing tapering parameters and wavenumber improves axial and normal velocities, heat transfer, and flow complexity.

Analysis of Indicators of the Electrical Power Production Enterprises’ Pollution and Harm Risk Degree in Terms of the Paris Agreement

Abstract In the current decade, electricity consumption will increase by more than 16%. The main resources for electricity generation are oil – 2.53%; gas – 22.9%; coal – 35.98%; nuclear industry – 9.84%; hydraulic power regeneration – 15.01%; renewable sources – 12.85%; other – 0.89%. This causes global warming of the planet due to greenhouse gases increased emissions into the atmosphere. A resource capable to cover the thermal energy field’s balance is the nuclear energy. To compare the resources’ polluting capacity, it is necessary to introduce a general indicator that takes into account their properties. This study proposes an integrated approach based on the formation of groups of indicators reflecting greenhouse gas emissions; power consumption; economic activity, air quality, etc. For comparison needs, all indicators are normalized in dimensionless form and compiled with reference to their specific weight. Calculations of the pollution index for fossil resources and nuclear energy, carried out using this algorithm proved that the nuclear power, with a careful consideration of all possible polluting radionuclides (11 components), exceeds this indicator for gas by 22%, but is lower than ciphers for coal and oil by 36 and 26%, respectively. For two latters only 3 specific components being taken into account. Therefore, it seems advisable to use the considered complex indicator of environmental pollution to assess the resource safety level for electricity generation.