Similar Equations Research Articles

A revised model on three-dimensional hydromagnetic convective flow of nanofluids over a nonlinearly inclined stretching sheet

Theoretical research has been done on the hydromagnetic three-dimensional nanofluid flow across an inclined stretching sheet in the presence of Brownian motion and thermophoresis. It is taken into account that there are two different kinds of water-based nanofluids that contain copper and alumina. Three-dimensional nonlinear-type similarity transformations are used to convert the governing boundary-layer equations into a collection of similarity equations, which are then numerically solved by MATLAB. Further, results obtained in some limiting cases are compared to some results that have already been published, and it is found that there is good agreement. The problem is controlled by a number of physical factors, and the impact of these factors on different flow distributions is thoroughly examined with the help of graphs and tables. It has been observed that the nanofluid has better heat transfer conductivity than the ordinary base fluid.

New Similarity Solutions of Magnetohydrodynamic Flow Over Horizontal Plate by Lie Group with Nonlinear Hydrodynamic and Linear Thermal and Mass Slips

The viscous laminar magnetohydrodynamic convective boundary layer flow with the combined effects of chemical reaction and nonlinear velocity slip and linear thermal and concentration slips have been considered across a flat plate in motion. Using a non-dimensional transformation attained by the single parameter continuous group method, the governing equations are transformed into a system of nonlinear ordinary similarity equations, then, the solutions of the coupled system of equations are constructed for velocity, temperature, and concentration functions by using the numerical methods. Among the parameters that have been looked at are the buoyancy parameter N, the nonlinear slip parameter {mathrm{n}}_{1}, the order of chemical reaction n, the Prandtl number Pr, and the Schmidt number Sc. An investigation was made on the profiles with respect to mixed convection parameter uplambda , order of chemical reaction n, arbitrary index parameter {mathrm{n}}_{1}, velocity slip parameter a, thermal slip parameter b, mass slip parameter c, suction parameter fw, magnetic parameter M. Verification of the results were possible due to comparison of two numerical methods to obtain the solution to the differential equations. The present study indicates that, for a range of values of the magnetic parameter, the wall shear stress decreases with increasing mixed convection. Moreover, for a variety of mixed convection parameter instances, the wall heat transfer decreases with increasing perpendicular magnetic effect.

Reilly’s Law and Logistic Type Equation in a Model for Studying Group Selection and Intercity Relations

This paper presents a model based on logistic type equations, but taking Reilly's Law into account. It consists of differential equations, each of which describes the dynamics of a population located at the center of a territory whose resources it exploits. Similar differential equations, but without elements that take into account the Reilly's Law, have already been presented for studying the evolution of altruism from binary altruism (between two individuals) to Benthamian altruism (toward the group). The model presented here aims to study the outcome of competition for resources at least partially shared by populations concentrated at different points. It shows that a group's degree of cohesion is crucial to its survival if it competes with other groups for resources, even without the use of violence. It also shows that cooperation can replace competition between different populations with mutual benefits (even if the resulting equilibrium can be rather precarious). Finally, the differential equations used, while containing elements similar to those found in Newton's gravitational law (as well as Reilly's), predict the existence, alongside the force of attraction between economic actors, of its opposite (that is a sort of social antigravity).

Similarity solution of fractional viscoelastic magnetohydrodynamic fluid flow over a permeable stretching sheet with suction/injection

A novel easy‐to‐use similarity technique for spatial fractional partial differential equations is introduced to solve the flow and heat transfer of fractional viscoelastic magnetohydrodynamic (MHD) fluid over a permeable stretching sheet. The proposed method represents a new technique for deriving similarity transformations, which allows the conversion of the fractional partial differential equations to fractional ordinary differential equations. Moreover, using the Grünwald–Letnikov formula, the fourth‐order Runge–Kutta method is adopted here to solve the obtained similarity equations, using the shooting method. The transformed governing equations were found to be dependent on six physical parameters, namely, fractional parameter, magnetic field parameter, generalized local Reynolds number, generalized Prandtl number, the wall suction/injection parameter, and the wall stretching exponential. The results reveal that maximums of the skin friction coefficient and local Nusselt number are obtained for fractional‐order derivative of about 0.5. Moreover, application of a magnetic field on electrically conducting fractional fluid thins the velocity boundary layer and thickens the thermal boundary layer; therefore, the skin friction coefficient increases whereas the local Nusselt number decreases. Furthermore, an increase in any parameters of the local Reynolds number, suction parameter, or the wall stretching exponential leads to thinner boundary layers and higher values of skin friction coefficient and local Nusselt number, whereas imposing injection has opposite effects. Finally, increasing the generalized Prandtl number leads to a thinner thermal boundary layer and higher local Nusselt number for the fractional viscoelastic fluid. It would be of interest to extend the proposed method to other practical problems in the same field.

Viscous dissipation effects on hybrid nanofluid flow over a non-linearly shrinking sheet with power-law velocity

This research intends to investigate the effect of the nonlinearity of the surface velocity on the hybrid nanofluid flow behavior. Here, the total composition of Al2O3 (alumina) as well as Cu (copper) volume fractions, are implemented in a one-to-one ratio and then dispersed in water. The similarity equations are gained employing a similarity transformation, which is programmed in MATLAB software. The dual solutions are attainable for certain ranges with respect to the mass flux parameter S and the power-law index n. Also, the turning point occurs in the region of S<0 and n>1. Besides, the rise of n led to reduce the skin friction as well as the heat transfer coefficients with 39.44 % and 11.71 % reduction, respectively. Moreover, 14.39 % reduction of the heat transfer rate is observed in the presence of viscous dissipation (Eckert number). It is found that only the first solution is stable as time progresses. Generally, this study gives scientists and engineers a starting point for predicting how to control the parameters to achieve the best results for relevant practical applications.

A consistent phase field model for brittle fracture with new crack driving force

Phase field model has been widely used in fracture due to its excellent properties, but diverse simulation results are obtained from different phase field models. To address the variability of the results of different models, a consistent phase field model for brittle fracture is constructed in this paper. Firstly, by re-expressing the evolution equation of the phase field model, it is found that the differences between different models mainly come from the crack driving force. Therefore, a new crack driving force is presented in this paper, and a similar phase field model evolution equation is established to make a closer connection between the AT1 and AT2 models. Subsequently, the effect of phase field gradient is analyzed theoretically. When there is no gradient effect, the new AT1 and AT2 models have the same damage evolution characteristics; when there is a gradient influence, they also have good consistency and reduce the inter-model variability. Finally, some numerical examples are used to study the mechanical properties of the AT1 and AT2 models in the absence and presence of phase field gradient. The results indicate that the two models have similar damage evolution properties, verifying the validity of the models.

Arbitrarily high-order explicit energy-conserving methods for the generalized nonlinear fractional Schrödinger wave equations

A novel category of explicit conservative numerical methods with arbitrarily high-order is introduced for solving the nonlinear fractional Schrödinger wave equations in one and two dimensions. The proposed method is based on the scalar auxiliary variable approach. The equations studied is first transformed into an equivalent system by introducing a scalar auxiliary variable, and the energy is then reformulated as a sum of three quadratic terms. Applying the explicit relaxation Runge–Kutta methods in temporal and the Fourier pseudo-spectral discretization in spatial, the resulting time–space full discrete scheme is proved to preserve the reformulated energy in the discrete level to machine accuracy. The proposed methods improve the numerical stability during long-term computations, as demonstrated through numerical experiments. Also this idea can be easily extended to other similar equations, such as the nonlinear fractional wave equation and the fractional Klein–Gordon–Schrödinger equation.

Wood Diaphragm Deflections. I: Generalizing Standard Equations Using Mechanics-Based Derivations for Panel Construction

Horizontal wood diaphragm systems, whether decked with conventional or mass timber panels, transfer wind and seismic loads to vertical elements of a lateral force-resisting system (LFRS), in flexible, rigid, or semirigid fashion. Characterizing and calculating the resulting diaphragm deflections determines the distribution of forces to critically loaded components and a significant portion of lateral building translations and rotations. Deflection equations for sheathed wood structural panel (WSP) diaphragms are well established in US design standards in a four-term expression that models flexural, shear, and fastener-slip deformations, and its full derivation using principles of mechanics is provided herein. Derivations of similar equations for cross-laminated timber (CLT) diaphragms have yet to unfold, despite growing industry consensus that CLT panels make efficient slabs and decks. In this first of two companion papers, the corrected full derivation of the current four-term WSP diaphragm deflection expression is provided and assessed, and two ways to quantify the cumulative contribution of fastener slip are presented to expand its usage to a wider variety of WSP and CLT configurations in current use. Building on this generalized mechanics-based derivation, the authors are able to propose and assess in the companion paper a unified diaphragm deflection model to compute both WSP and CLT diaphragm deflections as implemented under current practice and guide further development.

Математичне моделювання пристінної оливної плівки в камері підшипника ГТД

The most preferable result of modeling the flow in the BC is the determination of the heat transfer coefficient to the inner wall. The complexity of solving this problem is due to both the complex geometry of the BC and the presence of a two-phase flow, the structure of which changes from air-droplet in the core to a nearly liquid in the wall oil film. Available research results show that even three-dimensional CFD modeling of such a flow does not completely solve the problem. At the same time, the calculation time is long, and the results require, at a minimum, selective experimental validation. At the same time, it can be considered proven that the main mechanism of heat transfer from the core to the wall region of the BC is related to the radial flow of droplets, and the thermal resistance of the wall oil film has a decisive effect on the value of the internal heat transfer coefficient. It is advisable to model these media on the basis of a two-dimensional problem with the averaging of phase parameters along the axis. Considering the small volume fraction of droplets, the Lagrangian approach can be used to model the two-phase flow in the core of the BC. This allows consideration of not only the polydispersity of the droplets, but also the creation and movement of secondary droplets during the formation of the wall oil film. One of the main problems in the modeling of the wall film is the definition of its flow regime and the corresponding criterion equations for calculating the coefficients of friction and heat transfer. Most of the equations use the longitudinal coordinate of the plate as a geometric parameter and cannot be applied to the bearing chamber. In this study, the possibility of converting the film flow into a flow similar to that occurring in a flat pipe is substantiated. This allows not only to consider the geometric features of the BC, but also to use the corresponding Reynolds numbers and similarity equations for the equivalent flow. Along with the use of a two-layer model for the boundary region and the concept of the analogy of transfer processes, this made it possible to form a mathematical model of the film, which considers all the components that determine the formation, movement, and heat transfer of the wall oil film. In addition, the model does not contain restrictions on the appearance of the parameters transverse profile, which are, for example, for the EWF model of the oil film in ANSYS FLUENT. The obtained results along with the previously developed air-droplet flow model in the core provide a complete two-dimensional model of the gas-liquid flow in the GTE bearing chamber, which allows the main geometric and all mode parameters to be used to determine the heat transfer coefficient to the inner wall of the chamber. Given the short calculation time, the proposed model allows for a detailed investigation of each factor’s contribution and element-by-element identification of the model based on the results of more detailed modeling and by comparing calculated and experimental data.

Finite-Time Suboptimal Control Design for Aerobatic Maneuver of Variable-Pitch Quadrotor

The variable-pitch quadrotors can perform aggressive maneuvers such as aerobatic flip by varying the thrust in both positive and negative vertical directions via the blade pitch, which can provide an additional degree of freedom as opposed to the fixed-pitch quadrotor that only changes the rotational rate of motors. However, an inevitable challenge exists when designing an operative controller for the aerobatic flip due to the resultant singularity of the rotation matrix. In this paper, a finite-time suboptimal control design strategy named finite-time <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\theta -D$</tex-math></inline-formula> algorithm combined with the geometric control technique is proposed for the aerobatic maneuver control of the variable-pitch quadrotor. Based on this scheme, the singularity issue can be circumvented. The <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\theta -D$</tex-math></inline-formula> control design provides a closed-form suboptimal control law by offline solving a differential Riccati equation and a set of linear Lyapunov equations analytically, which does not require complex online numerical and iterative solutions compared with the similar finite-time state dependent Riccati equation technique. Simulation results demonstrate that the proposed geometric finite-time <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\theta -D$</tex-math></inline-formula> method provides an effective and efficient control of the variable-pitch quadrotor.

Increasing the efficiency and environmental safety of ship power plants cooling systems with jet heat removal devices

There are numerous cases when during the ship operation it is advisable to exclude for some time the intake of sea water used to cool the ship’s power plant. The issues of the possibility of such operation of widely used open double-circuit cooling systems are considered. To this end, the systems can be switched to the sea water tanks available on board. Work according to this scheme is limited by the time of heating the available water to the maximum allowable temperature. The issues of increasing this time by rising the heat removal through the ship’s plating are considered. To do this, the heated water is not simply discharged into the tank, but is jet supplied through the pipeline system directly to the inner surface of the ship’s plating. Conducted visual and thermal engineering studies have shown that the best results are achieved with the use of a jet supply, supplemented by a screen one. As a result, a channel for hot water movement is formed between the screen and the ship’s plating. Water radially spreads outward from the center of the screen, forming a fan jet. At the same time, the maximum flow velocity around the ship’s plating and the temperature difference between hot and outboard water are ensured. The advantages of this method for supplying cooled water are confirmed by the thermal engineering studies. It is shown that in this case the heat transfer increases by 25…30 %, compared with the case of a simple jet water supply. As a result of processing numerous experimental data, generalizing similarity equations have been obtained. They allow you to calculate the heat transfer for various water flow regimes. The use of these devices allows you to increase the time of systems operation in a closed loop without receiving outside water. These devices are quite simple in design and can be used both at the stage of building a new ship, and when modernizing existing ships. An example of such devices introduction on self-propelled floating cranes is given.

Thermal efficiency of environmentally friendly closed cooling systems of marine power plants

The issues of heat removal efficiency of cladding heat exchangers of closed cooling systems of ship power plants are considered. The use of such systems allows obtaining technical and environmental advantages in the operation of the vessel. The latter ensures the conservation of fish resources of the seas and continental waters. It is shown that the worst conditions for heat removal take place when the vessel is parked. In this case, heat transfer to outboard water is carried out with free convection. This mode is the main calculation mode. In the case of vessel movement, heat removal increases significantly, since it is carried out with forced convection. For the case of the ship berthing, the results of experimental thermal engineering studies on a model of 1m2 are presented. It is shown that the efficiency of heat removal significantly depends on the orientation of the ship’s plating of the temperature difference in the apparatus. The transition from the bottom location of the apparatus to the side, as well as an increase in the temperature difference, provide greater thermal efficiency. Generalizing similarity equations are obtained for the case of the ship berthing. Based on them, a method for the calculation has been developed. The reliability of this technique is fully confirmed by the results of testing such a heat exchanger, with an area of 100 m2, on a self-propelled floating crane of project 15201. It is shown that the obtained generalizing similarity equations have their own characteristics compared to the known ones. This is a consequence of the peculiarities of the processes taking place in such devices. For the case of vessel movement, generalizing similarity equations are also obtained. In this case, the authors have used the results of their own tests of such devices on the floating crane of project 15201, as well as the tests results of other researchers on various ships.

On the Existence of Optimal Control in the Problem of Optimizing the Lowest Coefficient of a Semilinear Evolutionary Equation

The paper studies the problem of optimizing the lowest coefficient understood as a function with values in a Banach space, which enters linearly into an abstract semilinear pseudoparabolic evolutionary differential equation in a Banach space. For this problem, an existence theorem for an optimal control is proved. Due to the nonlinearity of the equation under study, the author uses his previous results on the total preservation of the unique global solvability (on the totally global solvability) and on the estimation of solutions for similar equations. This estimate turns out to be significant in the course of the study. As an example, the Oskolkov’s hydrodynamic system of equations is considered.

Consistent cosmological structure formation on all scales in relativistic extensions of MOND

General relativity manifests very similar equations in different regimes, notably in large scale cosmological perturbation theory, non-linear cosmological structure formation, and in weak field galactic dynamics. The same is not necessarily true in alternative gravity theories, in particular those that possess MONDian behaviour (“relativistic extensions” of MOND). In these theories different regimes are typically studied quite separately, sometimes even with the freedom in the theories chosen differently in different regimes. If we wish to properly and fully test complete cosmologies containing MOND against the ΛCDM paradigm then we need to understand cosmological structure formation on all scales, and do so in a coherent and consistent manner. We propose a method for doing so and apply it to generalised Einstein-Aether theories as a case study. We derive the equations that govern cosmological structure formation on all scales in these theories and show that the same free function (which may contain both Newtonian and MONDian branches) appears in the cosmological background, linear perturbations, and non-linear cosmological structure formation. We show that MONDian behaviour on galactic scales does not necessarily result in MONDian behaviour on cosmological scales, and for MONDian behaviour to arise cosmologically, there will be no modification to the Friedmann equations governing the evolution of the homogeneous cosmological background. We comment on how existing N-body simulations relate to complete and consistent generalised Einstein-Aether cosmologies. The equations derived in this work allow consistent cosmological N-body simulations to be run in these theories whether or not MONDian behaviour manifests on cosmological scales.

Research on Potential Equilibrium of Dynamic Combustion Performance in Electrothermal-Chemical Launching

The interior ballistic potential equilibrium theory is used to analyse the parameters at the potential equilibrium points of pressure curves of 30mm conventional launching experiment and electrothermal-chemical (ETC) launching experiments. Analysis shows that: compared to the conventional launching experiment, (1) the time for ETC launching experiment to arrive the potential equilibrium point is apparently shortened; (2) the relative burnt percentages of the propellants at the potential equilibrium points are smaller in ETC launching tests. And the curves of gas generation brisance in the three ETC launching experiments are consistent. Then based on the measured pressure curves, the actual gas generation functions in chamber for conventional launching experiment and ETC launching experiments are fitted respectively. Using the average actual gas generation functions to replace the shape functions which are derived from the hypothesis of Geometric Combustion Law, the interior similarity equations of ETC Gun are established with the potential equilibrium point as the reference state. The calculated pressure curve obtained by solving the interior ballistic similarity equations of electrothermal chemical gun is in good agreement with the experimental pressure curve. Therefore, it is expected that the interior similarity equations of ETC Gun could predict the experimental pressure curve.

An impact of radiation on laminar flow of dusty ternary nanofluid over porous stretching/shrinking sheet with mass transpiration

The goal of current research is to determine the laminar boundary layer problem for the two-dimensional flow of ternary dusty nanoparticles through a porous stretching/shrinking sheet in the presence of radiation and mass transpiration. Generalized partial differential equations are converted to nonlinear ordinary differential equations after demonstrating an appropriate similarity transformation. Exact solutions are subsequently provided for the resulting system of equations. The development of ternary dusty nanofluids has significantly improved the heat transmission process for manufacturing and industrial applications, as well as in nanotechnology research. The influence of the various interesting parameters on the flow and heat transfer is analyzed and discussed in detail through plotted graphs. It was found that the basic similarity equations admit two phases for both stretching/shrinking surfaces. Graphs are used to illustrate the findings of this work. We found that the velocity fields increase with an increase in the inverse Darcy number value for the case stretching sheet while opposite effects can be observed in the shrinking case. Moreover, the magnitude of particle interaction parameter enhances as the solution domain raises. The findings disclose that the performance of ternary nanofluid phase heat transfer is improved compared to dusty phase performance.

Unit Cell Volume as a Measure of the Framework Composition of ZSM-5 Type Zeolites

A simple X-ray powder diffraction procedure is proposed to determine the framework composition of ZSM-5 type zeolites. The procedure is based on a linear relation between the unit cell volume and the framework aluminum content found by analyzing two families of ZSM-5 zeolites with different SiO2/Al2O3 ratios. One family consisted of products synthesized in the presence of TPA+ cations while hydrogen forms were included in the second series of samples. Linear relationships between cell constants, cell volume (Vu.c.), and the framework aluminum content (NAl) were established. For the samples synthesized in the presence of an organic template and sodium cations, the relation between NAl and Vu.c. is given by NAl = (Vu.c. – 5332 Å3)/15.13. A similar equation was computed for H-ZSM-5 zeolites: NAl = (Vu.c. – 5330 Å3)/15.40. Equations derived can be used to evaluate the number of framework aluminum in the unit cell. In this way, the ratio of the framework and nonframework aluminum atoms in the samples of ZSM-5 type zeolites subjected to heat treatment can be estimated.

An experimental study on the equivalent nonlinear model for a large-sized tuned liquid damper

A fire-suppression water tank is usually required at the top of a tall building. It would be economical to design a fire-suppression tank as a tuned liquid damper (TLD) during preliminary building design. A TLD has obvious advantages in terms of economy, maintenance, and damping efficiency with small vibrations. The sloshing nonlinearity of a TLD increases design difficulty. The purpose of this study is to investigate the nonlinear dynamic response of a large-scale TLD and provide an equivalent model for the TLD under small excitation suitable for the vibration comfort control of tall buildings. The nonlinear dynamic response was investigated through shaking table experimental results. An experiment was conducted with a TLD with a rectangular tank size of 2500 mm × 1750 mm and a sloshing period of 3.2–6.2 s. A series of TLD tests were performed with different flow-damping devices, water depths, and excitations. The time history of the water sloshing wave height was obtained using a video recognition method; the base shear force of the tank was measured simultaneously. A new nonlinear stiffness and damping model (NSD-h) was proposed that simplified the TLD as a nonlinear tuned mass damper (TMD) to predict the tank base force. In the NSD-h model, the equivalent wave height was introduced to describe the sloshing amplitude of the TLD. The nonlinear frequency and damping ratio of the TLD were calculated using the equivalent wave height. Verified in wind conditions, the NSD-h model demonstrated significantly higher accuracy than the nonlinear stiffness and damping model (NSD) with similar simple equations.

Analysis of heat transfer performance for ternary nanofluid flow in radiated channel under different physical parameters using GFEM

BackgroundAdvanced nanofluids also termed as ternary nanofluids convinced the engineers and scientists due to unique molecular structure and enrich heat transfer capability. These potentially contributes in aerodynamics, biomedical engineering, aerospace sciences and electronic devices. In many industrial purposes, these can be used as coolant more specifically in electronic and chemical industries. MethodsThis study emphasized on the development of an efficient ternary nanofluid model between parallel plates channel. The model formulation is successfully completed via ternary nanofluid correlations and also exercised similarity equations for the possible domain. Moreover, model became more fascinating due to induction of thermal radiations and viscous dissipation. To investigate the heat transfer performance of ternary nanofluid the model treated via an efficient scheme based on residuals and linearly independent set of weight functions and is known as Galerkin Finite Element Method (GFEM). Significant findingsIt is examined that the ternary nanofluid [(Al2O3-CuO-Fe3O4)/EO]mbnf is a suitable heat transport fluid than common binary nanofluid [(Al2O3-CuO)/EO]bnf nanofluids. further, induction of solar thermal radiations and convection through the plate are vital sources to upshoot the internal heat transmission of the [(Al2O3-CuO-Fe3O4)/EO]mbnf and it would allow the researchers and engineers for large scale applications. The feasible parametric ranges that pride considerable heat transfer are Bi = 0.1, 0.3, 0.5, 0.7, 0.9, 1.1 and Eckert number are Ec = 0.1, 0.3, 0.5, 0.7, 0.9, 1.1 while keeping the weight concentration of nanoparticles up to 6.0%.

Mathematical analysis of heat and mass transport under thermal radiations and dissipation for electrically conducting second-grade fluid

This research deals with heat and mass transport in 2D, steady laminar unidirectional flow of second-grade fluid inside converging/diverging stretchable channel with no slip effects. The similarity equations were engaged for the model development and acquired a model in the form of a joint system of ODEs comprising the innovative effects of thermal radiations, magnetic field and dissipation function. Then, mathematical treatment of the model was done via HAM Package and the results were furnished under the inspiration of physical constraints. It is pointed out that the fluid moves quite slowly in diverging cases than that of converging ones under multiple Re values, whereas magnetic field effects are almost negligible for the velocity profile. The temperature of the fluid can be enlarged by increasing Ec and Pr and Rd is not proven good in this study. Further, as far as the mass transport is concerned, the dominant contribution of Sc is observed for both stretching/shrinking walls.