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In this paper, a universal computational algorithm is constructed by using F and G series, which can be applied for any of conic orbit. In particular, to find the solution of the two–body problem. In this context, the solution of geocentric system motion of the Mercury planet in the solar system is found using the obtained computational algorithm.

This work aims to construct a new regularisation version for the perturbed two-body problem using bipolar coordinates. We mainly use two transformation techniques to deal with this problem when the distance between two bodies tends to be zero. The system exhibits a well-stabilised periodic orbit over a long time span through numerical simulations. With the increase in the oblate parameter value of the body, the periodic orbit will gradually evolve into a quasi-periodic form.

Abstract Using a thermally stratified water-based nanofluid and a permeable stretching sheet as a simulation environment, this research examines the impact of nanoparticle aggregation on MHD mixed convective stagnation point flow. Nanoparticle aggregation is studied using two modified models: the Krieger–Dougherty and the Maxwell–Bruggeman. The present problem's governing equations were transformed into a solvable mathematical model utilizing legitimate similarity transformations, and numerical solutions were then achieved using shooting with Runge–Kutta Fehlberg (RKF) technique in Mathematica. Equilibrium point flow toward permeable stretching surface is important for the extrusion process because it produces required heat and mass transfer patterns and identifies and clarifies fragmented flow phenomena using diagrams. Nanoparticle volume fraction was shown to have an impact on the solutions' existence range, as well. Alumina and copper nanofluids have better heat transfer properties than regular fluids. The skin friction coefficients and Nusselt number, velocity, temperature profiles for many values of the different parameters were obtained. In addition, the solutions were shown in graphs and tables, and they were explained in detail. A comparison of the current study's results with previous results for a specific instance is undertaken to verify the findings, and excellent agreement between them is observed.

Abstract A magneto couple stress nanofluid flow along with double diffusive convection is presented for peristaltic induce flow through symmetric nonuniform channel. A comprehensive mathematical model is scrutinized for couple stress nanofluid magneto nanofluids and corresponding equations of motions are tackled by applying small Reynolds and long wavelength approximation in viewing the scenario of the biological flow. Computational solution is exhibited with the help of graphical illustration for nanoparticle volume fraction, solutal concentration and temperature profiles in MATHEMTICA software. Stream function is also computed numerically by utilizing the analytical expression for nanoparticle volume fraction, solutal concentration and temperature profiles. Whereas pressure gradient profiles are investigated analytically. Impact of various crucial flow parameter on the pressure gradient, pressure rise per wavelength, nanoparticle volume fraction, solutal concentration, temperature and the velocity distribution are exhibited graphically. It has been deduced that temperature profile is significantly rise with Brownian motion, thermophoresis, Dufour effect, also it is revealed that velocity distribution really effected with strong magnetic field and with increasing non-uniformity of the micro channel. The information of current investigation will be instrumental in the development of smart magneto-peristaltic pumps in certain thermal and drug delivery phenomenon.

Abstract Ethylene glycol is commonly used as a cooling agent in the engine, therefore the study associated with EG has great importance in engineering and mechanical fields. The hybrid nanofluid has been synthesized by adding copper and graphene nanoparticles into the Ethylene glycol, which obeys the power-law rheological model and exhibits shear rate-dependent viscosity. As a result of these features, the power-law model is utilized in conjunction with thermophysical characteristics and basic rules of heat transport in the fluid to simulate the physical situations under consideration. The Darcy Forchhemier hybrid nanofluid flow has been studied under the influence of heat source and magnetic field over a two-dimensionally stretchable moving permeable surface. The phenomena are characterized as a nonlinear system of PDEs. Using resemblance replacement, the modeled equations are simplified to a nondimensional set of ODEs. The Parametric Continuation Method has been used to simulate the resulting sets of nonlinear differential equations. Figures and tables depict the effects of physical constraints on energy, velocity and concentration profiles. It has been noted that the dispersion of copper and graphene nanoparticulate to the base fluid ethylene glycol significantly improves velocity and heat conduction rate over a stretching surface.

This work is divided to two parts; the first part analyzes the features of Hénon–Heiles’s potential and finding the energy levels for bounded and unbounded motions. The critical points are explored in different phase spaces from the classical potential to the generalized one. In the second part, the possible solutions of the generalized (fifth-degree) Hénon–Heiles system are analyzed using the averaging theory. Two consequent transformations are used to set the Hamiltonian of this system in standard form for applying the averaging theory. In this context, eight solutions are found, where one of them is not convenient for the proposed assumptions, and the other seven solutions are proper and adequate to represent seven periodic orbits for the generalized Hénon–Heiles dynamical system, which has at least seven periodic orbits.

All previously published data on the dynamics of ethylene glycol conveying copper and gold nanoparticles over a convective surface, nothing is known about the importance of dual branch solutions. Hybrid nanofluids improve the thermal conductivity of the fluid. The nanoparticles copper and gold having ethylene glycol as a base fluid are used here. The flow problem is described over a stretching/shrinking surface with the influence of Ohmic heating, non-linear radiation, and a convectively heated surface. Furthermore, the magnetic field strength is applied perpendicular to the direction of the flow. To control the fluid, flow-governing equations are numerically solved by using bvp4c, a built-in approach in MATLAB. For hybrid nanomaterials, the consequence of different physical parameters is discussed graphically and with tabular data. A comparison with previous findings reveals that the present findings are in good agreement. The results revealed that the coefficient of skin friction for the physically stable branch declines over a certain range of shrinking parameters; nonetheless, for the unstable branch, the reverse pattern is discovered. The magnetic force diminishes the flow field and energy dispersion in the upper branch but improves it in the lower branch.

This study analyzed thermal and mass transport in magnetohydrodynamic Maxwell nanofluids over a cylinder stretched along the z-direction. The Cattaneo-Christov diffusion theory and Buongiorno’s model were employed to model the problem. The influences of Joule heating, chemical reaction rate, and heat generation were also considered. Appropriate similar variables were utilized to transform the constitutive equations. A semi-analytical method, namely the homotopy analysis method (HAM) in Wolfram Mathematica, was used to compute the problem solution. The results demonstrated the inverse variation in flow behavior with increased Maxwell parameter values; however, thermal and solutal transport displays the opposite trend. Additionally, the flow field showed resistance due to the presence of the magnetic field, while Joule heating enhanced the energy and mass transport phenomena. The results regarding the coefficient of skin friction along the radial direction are consistent with values reported in the literature.

The analysis of nanofluids under various physical scenarios convinced the researchers and scientists because of their broad range of applications in potential area of the current time like chemical engineering, biomedical engineering and applied thermal engineering etc. To give the final shape of many industrial and engineering processes, enhanced heat transfer desired, therefore, the study of Al 2 O 3 -H 2 O, γAl 2 O 3 -H 2 O, Al 2 O 3 -C 2 H 6 O 2 , and γAl 2 O 3 - C 2 H 6 O 2 nanofluids is reported. The model successfully achieved after mathematical operations and by appealing similarity transforms. To examine the behavior of heat transfer, numerical tools utilized and performed the results. It is observed that enhanced heat transfer in Al 2 O 3 -H 2 O, γAl 2 O 3 -H 2 O, Al 2 O 3 -C 2 H 6 O 2 , and γAl 2 O 3 -C 2 H 6 O 2 could be attained by setting nanoparticles concentration up to 20%. For Al 2 O 3 -H 2 O, γAl 2 O 3 -H 2 O, optimum heat transfer trends noticed due to their prominent thermophysical values. Also, fewer effects of combined convection on θ ( η ) examined.

In this work, the restricted three-body system is studied in the framework of the continuation fractional potential with its application on the Earth–Moon system. With the help of a numerical technique, we obtained thirteen equilibrium points, such that nine of them are collinear while the remaining four are non-collinear points. We found that the collinear points near the smaller primary were shifted outward from the Moon, whereas the points near the bigger primary were shifted towards the Earth as the value of the continuation fractional parameter increased. We analyzed the zero-velocity curves and discussed the perturbation of the continuation fractional potential effect on the possible regions of the motion. We also discussed the linear stability of all the equilibrium points and found that out of thirteen only two were stable. Due to such a prevalence, the continuation fractional potential is a source of significant perturbation, which embodies the lack of sphericity of the body in the restricted three-body problem

In this work, the quantized Hill problem is considered in order for us to study the existence and stability of equilibrium points. In particular, we have studied three different cases which give all whole possible locations in which two points are emerging from the first case and four points from the second case, while the third case does not provide a realistic locations. Hence, we have obtained four new equilibrium points related to the quantum perturbations. Furthermore, the allowed and forbidden regions of motion of the first case are investigated numerically. We demonstrate that the obtained result in the first case is a generalization to the classical one and it can be reduced to the classical result in the absence of quantum perturbation, while the four new points will disappear. The regions of allowed motions decrease as the value of the Jacobian constant increases, and these regions will form three separate areas. Thus, the infinitesimal body can never move from one allowed region to another, and it will be trapped inside one of the possible regions of motion with the relative large values of the Jacobian constant.

In this work, we investigated the differences and similarities among some perturbation approaches, such as the classical perturbation theory, Poincaré–Lindstedt technique, multiple scales method, the KB averaging method, and averaging theory. The necessary conditions to construct the periodic solutions for the spatial quantized Hill problem—in this context, the periodic solutions emerging from the equilibrium points for the spatial Hill problem—were evaluated by using the averaging theory, under the perturbation effect of quantum corrections. This model can be used to develop a Lunar theory and the families of periodic orbits in the frame work for the spatial quantized Hill problem. Thereby, these applications serve to reinforce the obtained results on these periodic solutions and gain its own significance.

In this paper, we study the existence and stability of collinear and noncollinear equilibrium points within the frame of the perturbed restricted problem of 2 + 2 bodies by a planetesimal belt. We compare and investigate the corresponding results of the perturbed and unperturbed models. The impact of the planetesimal belt is observed on collinear and noncollinear equilibrium points. We demonstrate that all equilibrium points are unstable, and we numerically investigate the noncollinear equilibrium points. Finally, we emphasize that the proposed problem is a credible model for describing the capture of small bodies by a planet.

AbstractThe nanofluids has attractive applications in the industrials and engineering field because nanofluids offer the improvement in the thermal efficiencies of flow, which is very fruitful in the energy sector. This study emphases on the numerical exploration of heat and mass transfer efficiencies on a mixed convective flow of tangent hyperbolic nanofluid under the consideration of gyrotactic microorganisms. To analyze the magnetized boundary layer flow, the stratification boundary conditions are considered on the surface of the stretching cylinder. The thermal analysis involves the important features such as activation energy, nonuniform heat source/sink, and nonlinear thermal radiation. The flow model is altered into system of nonlinear ordinary differential equations by utilization of suitable similarity variables and Bvp4c solver on the MATLAB to find numerical solutions. The graphical and tabulated investigation is presented for various developing parameters thoroughly. The results indicate that by the increment of mixed convection parameters the fluid velocity accelerate consequently, whereas raising values of the Weissenberg number reduce the fluid velocity. TThe greater values of the Prandtl number lead to a decrease in fluid temperature, while the concentration of nanoparticles rises with the increase in activation energy parameters. Additionally, the heat transfer rate reduces due to stronger values of thermophoresis parameter, although reverse tendency is observed for the greater values of Brownian motion parameter. [ABSTRACT FROM AUTHOR]

The objective of current investigation is to examine Maxwell fluid flow generated by cylindrical stretching pipe. Thermophoresis and Brownian moment phenomena will also incorporate on the said fluid. Melting heat shifting phenomenon incorporated at the surface of the elongating cylinder. Mathematical formulation is supported by boundary layer assumption. Suitable similarity transformations diminishes model of nonlinear partial differential equations into a model of nonlinear ordinary differential equations that ultimately solved via numerical technique named Finite Element Technique. The properties of numerous effective variables are graphically displayed and examined. Further the physical quantities like velocity gradient, heat and mass fluxes are tabulated numerically and analyzed. The dimensionless temperature distribution rises for rising values of curvature parameter and Brownian motion parameter however an opposite behavior is observed for melting parameter and Prandtl number. Moreover, a reduction occurs in heat and mass fluxes for sufficiently high values of curvature parameter. [ABSTRACT FROM AUTHOR]

After applying a magnetic field, the behavior of the partly ionized liquids is completely different from that of the ordinary fluids. In this study, we concentrated on the Cattaneo–Christov heat flux model-based three-dimensional partly ionized bio-convective flow of a second-order fluid on a bidirectional permeable stretching surface. The development of the thermal and solutal flow models takes into account the impacts of non-uniform sources and sinks, Ohmic viscous dissipation, and chemical reactions. In addition, the surface boundary effects of electron and ion collisions with convective boundary conditions are seen. The mathematical flow model is transformed appropriately to create an ordinary differential equations, which is then numerically solved with MATLAB's BVP4C approach. To demonstrate the physical relevance of the flow field along various developing parameters, graphical and tabular results are created. It is noteworthy to note that while fluid temperature decreases with stronger values of the second-order fluid parameter, fluid velocity improves in both directions. In addition, it is shown that raising the thermal and concentration relaxation parameters, respectively, causes a drop in the fluid temperature and nanoparticle concentration. [ABSTRACT FROM AUTHOR]

Particle dissolution is an important process that occurs during heat treatment in the transportation process. We present a one-dimensional model of spherical wax particle dissolution in a fluid. The model includes a nonlinear equation at the free boundary, which was established to describe the dissolution rate of a wax particle assuming a diffusion process. At the moving boundary, we compute the concentration distributions of partial solutions. To estimate wax dissolution in solution and related expressions for the dissolved radius, an analytical solution of a mathematical model for spherical particle dissolution in flow is used. [ABSTRACT FROM AUTHOR]

This article presents an analysis of the radiative flow of nanofluid past a convected vertical Riga plate with activation energy and nonlinear heat generation. A Riga plate is a surface-mounted electromagnetic actuator with a spanwise aligned array of alternating electrodes and permanent magnets. The aforementioned rheological model is formulated in the way of the nonlinear partial differential equations (PDEs). The system of nonlinear coupled ordinary differential equations (ODEs) is obtained from the modeled nonlinear PDEs by incorporating valid similarity transformations. The system of ODEs is then figured out by the reliable shooting method. The effects of mixed convection, modified Hartmann number, Biot number, thermophoresis, Eckert number, reaction rate, the width of magnets and electrodes, and Lewis and Prandtl numbers on the velocity, temperature, and concentration profiles are investigated graphically and numerically. It is found that the speed of the fluid can be managed with the separation of magnets and electrodes. Further, the concentration of nanofluid is significantly enhanced in the presence of activation energy. Also, the nonlinear exponential index parameter of heat generation declines the temperature of the fluid. Further, it is found that in the presence of thermal radiation, viscous dissipation, and nonlinear heat generation, the Nusselt number increases averagely by about 11.3%. [ABSTRACT FROM AUTHOR]