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Heat pipes are crucial in a wide range of applications, ranging from space satellites and industrial systems to electronic cooling and X-ray tube thermal management. This study introduces a method investigation into vapor flows within a flat plate heat pipe, utilizing the collocation method (CM) and the fourth-order Runge-Kutta-Fehlberg (RKF45) method. Building on previous efforts, this work explores the effects of the evaporator-to-condenser length ratio and Reynolds number on velocity and pressure distributions along the entire heat pipe. The significance of this research lies in its ability to elucidate critical parameters that directly influence heat pipe performance, offering deeper insights that are vital for optimizing design and efficiency. The primary motivation of this study is to fill existing gaps in the literature by developing a comprehensive analytical model that accurately characterizes vapor and liquid flow in asymmetrical flat plate heat pipes. The model's validity is confirmed through a satisfactory agreement with numerical results, underscoring the reliability of the methods used. Notably, the findings reveal that higher Reynolds numbers reduce pressure drop and shift the maximum velocity toward the bottom wick in the evaporation section, providing valuable guidance for future design improvements. Additionally, this research presents a powerful method for solving non-linear ordinary differential equations, offering significant time savings and enabling predictive functions. These contributions are poised to enhance the performance of thermal management systems across various engineering disciplines. [ABSTRACT FROM AUTHOR]

Heat pipes are crucial in a wide range of applications, ranging from space satellites and industrial systems to electronic cooling and X-ray tube thermal management. This study introduces a method investigation into vapor flows within a flat plate heat pipe, utilizing the collocation method (CM) and the fourth-order Runge-Kutta-Fehlberg (RKF45) method. Building on previous efforts, this work explores the effects of the evaporator-to-condenser length ratio and Reynolds number on velocity and pressure distributions along the entire heat pipe. The significance of this research lies in its ability to elucidate critical parameters that directly influence heat pipe performance, offering deeper insights that are vital for optimizing design and efficiency. The primary motivation of this study is to fill existing gaps in the literature by developing a comprehensive analytical model that accurately characterizes vapor and liquid flow in asymmetrical flat plate heat pipes. The model's validity is confirmed through a satisfactory agreement with numerical results, underscoring the reliability of the methods used. Notably, the findings reveal that higher Reynolds numbers reduce pressure drop and shift the maximum velocity toward the bottom wick in the evaporation section, providing valuable guidance for future design improvements. Additionally, this research presents a powerful method for solving non-linear ordinary differential equations, offering significant time savings and enabling predictive functions. These contributions are poised to enhance the performance of thermal management systems across various engineering disciplines.

We study the bacterial respiration through the numerical solution of the Fairen–Velarde coupled nonlinear differential equations. The instantaneous concentrations of the oxygen and the nutrients are computed. The fixed point solution and the stable limit cycle are found in different parameter ranges as predicted by the linearized differential equations. In a specified range of parameters, it is observed that the system spends some time near the stable limit cycle and eventually reaches the stable fixed point. This metastability has been investigated systematically. Interestingly, it is observed that the system exhibits two distinctly different time scales in reaching the stable fixed points. The slow time scale of the metastable lifetime near the stable limit cycle and a fast time scale (after leaving the zone of limit cycle) in rushing towards the stable fixed point. The gross residence time, near the limit cycle (described by a slow time scale), can be reduced by varying the concentrations of nutrients. This idea can be used to control the harmful metastable lifespan of active bacteria. [ABSTRACT FROM AUTHOR]

Recent advancement in nanotechnology brings the idea of hybrid nanomaterials which offer distinguish applications in thermal reservoirs, cooling systems, energy applications, chemical engineering, vehicle engines etc. The understating of shape features for hybrid nanomaterials is quite essential as such consequences highly influenced various thermal properties like viscosity, thermal conductivity, optical properties, stability etc. The objective of current work is to examine heat transfer analysis due to thin film unsteady flow of hybrid nanofluid. The properties of hybrid nanofluid are justified for entertaining the copper (Cu), aluminium oxide (Al2O3) nanoparticles with water (H2O) base fluid. Additionally, applications of viscous dissipation, heat source and nonlinear radiated effects are attributed to current flow problem. The thermal properties of nanoparticles are examined in presence of five shape features consisting of blades, platelets, cylinders, bricks and spheres. Numerical simulations of problem are performed via Runge-Kutta-Fehlberg method. Comparative heat transfer is performed for mono nanofluid (Cu/H2O) and hybrid nanofluid (Cu−Al2O3)//H2O. It has been observed that heat transfer enhancement is more stable for cylindrical particles as compared to spherical nanoparticles. The skin friction enhances due to Hartmann number for both mono nanofluid (MNF) and hybrid nanofluid (HNF). Current results claim applications in coating thin films, lubrication systems, improving the thermal efficiency in thermal and industrial systems, heat exchangers, cooling systems etc.

The main goal of this article is to perform a heat transfer analysis of unsteady magnetohydrodynamic hybrid nanofluid flow over a stretching/shrinking sheet. The hybrid nanofluid is synthesized by adding two kinds of nano-sized particles. In this examination, TiO2 and Cu nano-sized particles are taken with water as a base fluid. Using an appropriate similarity transformation approach, a set of nonlinear equations is obtained from the governing equations of the current study. To solve the changed system, the Runge–Kutta–Fehlberg approach is applied combined with the shooting method. The obtained data is presented in graphs and tables. Here, dual solutions can be seen utilizing upper and lower division solutions particularly aimed at a specific domain of unsteadiness parameter. The skin friction coefficient parameter shows a considerable improvement. Whereas in the case of the shrinking sheet, larger values of the magnetic parameter and solid volume fraction results in a significant drop in the local heat transfer rate for the upper branch case, while the lower branch case shows a different pattern. [ABSTRACT FROM AUTHOR]

The present exploration discusses the combined effect of non-linear thermal radiation along with viscous dissipation and magnetic field through a porous medium. A distinctive aspect of our work is the simultaneous use of porous wall and a porous material. The impact of thermal rays is essential in space technology and high temperature processes. At the point when the temperature variation is very high, the linear thermal radiation causes a noticeable error. To overcome such errors, nonlinear thermal radiation is taken into account. The coupled system of ordinary differential equations are derived from the partial differential equation. The dimensional model equations are transformed into non-dimensional forms using some appropriate non-dimensional transformation and the resulting nonlinear equations are solved numerically by executing persuasive numerical technique R-K integration procedure with the shooting method. Graphical analysis were used to assess the consequences of engineering factors for the momentum, angular velocity, concentration and temperature profiles. The skin friction values, local Sherwood and Nusselt number are the fascinating physical quantities whose numerical data are computed and validated against different parametric values. The vortex viscosity parameter and spin gradient viscosity parameter shows the reverse phenomenon on micro-rotation profile. The thermal radiation phenomena flattens the temperature and speeds up the heat transfer rate in the lower wall and a peak in the concentration is observed for the P e m > > 1 due to the inertial force. The Variational Iteration Method (VIM) and Adomian Decomposition Method (ADM) are the two analytical approach which have been incorporated here to decipher the non linear equations for showing better approximity. Comparisons with existing studies are scrutinized very closely and they are determined to be in good accord. [ABSTRACT FROM AUTHOR]

An efficiency of the generalized tenth order stochastic perturbation technique in determination of the basic probabilistic characteristics of up to the fourth order of dynamic response of Euler-Bernoulli beams with Gaussian uncertain damping is verified in this work. This is done on civil engineering application of a two-bay reinforced concrete beam using the Stochastic Finite Element Method implementation and its contrast with traditional Monte-Carlo simulation based Finite Element Method study and also with the semi-analytical probabilistic approach. The special purpose numerical implementation of the entire Stochastic perturbation-based Finite Element Method has been entirely programmed in computer algebra system MAPLE 2019 using Runge-Kutta-Fehlberg method. Further usage of the proposed technique to analyze stochastic reliability of the given structure subjected to dynamic oscillatory excitation is also included and discussed here because of a complete lack of the additional detailed demands in the current European designing codes.

We propose an adaptive and explicit Runge–Kutta–Fehlberg method coupled with a fourth-order compact scheme to solve the American put options problem. First, the free boundary problem is converted into a system of partial differential equations with a fixed domain by using logarithm transformation and taking additional derivatives. With the addition of an intermediate function with a fixed free boundary, a quadratic formula is derived to compute the velocity of the optimal exercise boundary analytically. Furthermore, we implement an extrapolation method to ensure that at least, a third-order accuracy in space is maintained at the boundary point when computing the optimal exercise boundary from its derivative. As such, it enables us to employ fourth-order spatial and temporal discretization with Dirichlet boundary conditions for obtaining the numerical solution of the asset option, option Greeks, and the optimal exercise boundary. The advantage of the Runge–Kutta–Fehlberg method is based on error control and the adjustment of the time step to maintain the error at a certain threshold. By comparing with some existing methods in the numerical experiment, it shows that the present method has a better performance in terms of computational speed and provides a more accurate solution. [ABSTRACT FROM AUTHOR]

Abstract On the basis of the error correction method developed recently, an algorithm, so-called error embedded error correction method, is proposed for initial value problems. Two deferred equations are used to approximate the solution and the error, respectively, at each integration step. For the solution, the deferred equation, which is based on a modified Euler’s polygon including the information of both the solution and its estimated error at the previous integration step, is solved with the classical fourth-order Runge–Kutta method. For the error, the deferred equation, which is based on a local Hermite cubic polynomial with three pieces of information—the solution, its estimated error at the previous step, and the constructed solution—is solved by the seventh-order Runge–Kutta–Fehlberg method. The constructed algorithm controls the error and possesses a good behavior of error bound in a long time simulation. Numerical experiments are presented to validate the proposed algorithm.

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An attempt has been made to examine the two dimensional free stream squeezed flow through a horizontal sensor surface of an electrically conducting Carreau-Yasuda fluid subject to transverse magnetic field. The governing nonlinear partial differential equations are modeled and then simplified with the aid of suitable similarity transformations. Well known numerical scheme Runge–Kutta–Fehlberg method is utilized to solve the system. Concrete graphical analysis is carried out to investigate the behavior of different pertinent parameters on velocity profile with inclusive discussion. Numerical influence of friction factor is also discussed.

In the present study, heat transfer and fluid flow of a pseudo-plastic non-Newtonian nanofluid over permeable surface has been solved in the presence of injection and suction. Similarity solution method is utilized to convert the governing partial differential equations into ordinary differential equations, which then is solved numerically using Runge-Kutta-Fehlberg fourth-fifth order (RKF45) method. The Cu, CuO, TiO2 and Al2O3 nanoparticles are considered in this study along with sodium carboxymethyl cellulose (CMC)/water as base fluid. Validation has been done with former numerical results. The influence of power-law index, volume fraction of nanoparticles, nanoparticles type and permeability parameter on nanofluid flow and heat transfer was investigated. The results of the study illustrated that the flow and heat transfer of non-Newtonian nanofluid in the presence of suction and injection has different behaviors. For injection and the impermeable plate, the non-Newtonian nanofluid shows a better heat transfer performance compared to Newtonian nanofluid. However, changing the type of nanoparticles has a more intense influence on heat transfer process during suction. It was also observed that in injection, contrary to the other two cases, the usage of non-Newtonian nanofluid can decrease heat transfer in all cases. [ABSTRACT FROM AUTHOR]

• The combined effects of hybrid nanoparticles and the mixture base fluid are analyzed. • The hybrid phase (Al 2 O 3 – Cu) presents lower Skin friction coefficient to that obtained for the nano-phase (Cu). • The mixture base fluid (H 2 O − C 2 H 6 O 2) predicts lower Skin friction coefficient C f than base fluid (H 2 O). • DRA analytical solutions match perfectly with numerical results and those available in literature. • This comparison shows an excellent agreement is observed between the analytical and numerical solutions. The most important objective of this research-work is to investigate the impacts of velocity-slip boundary conditions and shape factor of solid nanoparticles on the hydrodynamic behavior of the nonlinear problem of MHD Jeffery–Hamel hybrid nanofluid flow where the mixture H 2 O - C 2 H 6 O 2 (50 % - 50 %) was utilized as a base fluid. Using appropriate velocity transformations, the basic partial differential equations arising from mathematical modeling are transformed into non-linear ordinary differential equations. Afterwards, the determined nonlinear equation was numerically solved utilizing Runge-Kutta-Fehlberg 4th–5th order approach featuring shooting technique and analytically with the help of Duan–Rach Approach (DRA). The impact of active factors like Reynolds number, channel half-angle, Hartman number, base fluids nature, hybrid nanoparticles, velocity-slip boundary conditions, shape and Geometry of solid nanoparticles on hybrid nanofluid velocity and skin friction coefficient are visualized and investigated. The minimal local skin friction is found to be obtainable with the nanoparticles of Platelet form and second-order slip model where a reduction of 70% is gained compared to the local skin friction coefficient with spherical nanoparticles when the Hartmann number is higher. Results obtained also reveal that a higher reduction of 69% in local skin friction coefficient intensity is observed for both hybrid phase Al 2 O 3 - C u and mixture base fluid H 2 O - C 2 H 6 O 2 with second-order slip boundary condition model when Knudsen number Kn = 0,08. A comparison was made between the results obtained from this investigation in particular cases and the results obtained via the HAM-based Mathematica package for validation. Also, the obtained analytical DRA data are compared with numerical RKF45 data and the ones represented in the literature. The comparison revealed that the results match perfectly which justifies applicability, validity, and the higher exactness of the adopted Duan-Rach approach. [ABSTRACT FROM AUTHOR]

The two-dimensional unsteady laminar free convective heat and mass transfer fluid flow of a non-Newtonian fluid adjacent to a vertical plate has been analyzed numerically. The two parameters Lie group transformation method that transforms the three independent variables into a single variable is used to transform the continuity, the momentum, the energy and the concentration equations into a set of coupled similarity equations. The transformed equations have been solved by the Runge–Kutta–Fehlberg fourth-fifth order numerical method with shooting technique. Numerical calculations were carried out for the various parameters entering into the problem. The dimensionless velocity, temperature and concentration profiles were shown graphically and the skin friction, heat and mass transfer rates were given in tables. It is found that friction factor and heat transfer (mass transfer rate) for methanol are higher (lower) than those of hydrogen and water vapor. Friction factor decreases while heat and mass transfer rate increase as the Prandtl number increases. Friction (heat and mass transfer rate) factor of Newtonian fluid is higher (lower) than the dilatant fluid.

This paper reflects some research outcome denoting as to how Lotka–Volterra prey predator model has been solved by using the Runge–Kutta–Fehlberg method (RKF). A comparison between Runge–Kutta–Fehlberg method (RKF) and the Laplace Adomian Decomposition method (LADM) is carried out and exact solution is found out to verify the applicability, efficiency and accuracy of the method. The obtained approximate solution shows that the Runge–Kutta–Fehlberg method (RKF) is a more powerful numerical technique for solving a system of nonlinear differential equations.

Here formulation and computations are presented to introduce the novel concept of activation energy in chemically reacting stagnation point flow towards a stretching sheet. Constitutive expression for Cross liquid is taken into account. Magnetic field is utilised in the transverse direction. Application of suitable variables generates the non-linear differential systems. Numerical solution by Runge-Kutta-Fehlberg approach is presented. Characteristics for the significant variables like Weissenberg number, Hartmann number, Schmidt number, activation energy chemical reaction parameter, velocity ratio parameter and Prandtl number on the physical quantities are addressed through graphs and tables. Our computations reveal that species concentration rises via larger activation energy parameter whereas it decays when Schmidt number is incremented. The Weissenberg number has opposite characteristics for local Nusselt and Sherwood numbers when compared with surface drag force. [ABSTRACT FROM AUTHOR]

This paper deals with the thermal Marangoni convection effects in magneto-Casson liquid flow through suspension of dust particles. The transpiration cooling aspect is accounted. The surface tension is assumed to be fluctuating linearly with temperature. The fluid and dust particle’s temperature of the interface is chosen as a quadratic function of interface arc length. The governing problem is modelled by conservation laws of mass, momentum and energy for fluid and dust particle phase. Stretching transformation technique is utilized to form ordinary differential equations from the partial differential equations. Later, the numerical solutions based on Runge-Kutta-Fehlberg method are established. The momentum and heat transport distributions are focused on the outcome of distinct governing parameters. The results of Nusselt number is also presented and discussed. It is established that the heat transfer rate is higher in the case of dusty non-Newtonian fluid than dusty Newtonian fluid. The rate of heat transfer can be enhanced by suspending dust particles in a base liquid. [ABSTRACT FROM AUTHOR]

The present problem is the steady boundary layer flow and heat transfer of a hyperbolic tangent fluid flowing over a vertical exponentially stretching cylinder in its axial direction. After applying usual boundary layer with a suitable similarity transformation to the given partial differential equations and the boundary conditions, a system of coupled nonlinear ordinary differential equations is obtained. This system of ordinary differential equations subject to the boundary conditions is solved with the help of Runge–Kutta–Fehlberg method. The effects of the involved parameters such as Reynolds numbers, Prandtl numbers, Weissenberg numbers and the natural convection parameter are presented through the graphs. The associated physical properties on the flow and heat transfer characteristics that is the skin friction coefficient and Nusselt numbers are presented for different parameters.

An analysis on the subject of “induced magnetic field effect on stagnation flow of a TiO2-Cu/water hybrid nanofluid over a stretching sheet” has been carried out in this paper. It should be noted that hybrid nanofluid consists of two or more types of nanoparticles along with a base fluid and it is used to increase the heat transfer. Furthermore, the non-linear differential equations modeling this issue are included in this article. In order to solve these equations numerically, Runge-Kutta Fehlberg method is used as a numerical method in this problem. The main objective of this paper is to investigate the effects of change in parameters of stretching ratio parameter (A ∗ ), nanoparticles volumetric fractions (∅ 2 ), magnetic parameter (β) and reciprocal magnetic Prandtl number (λ) on the functions including velocity, induced magnetic field and temperature for both Cu-water nanofluid and TiO2-Cu/water hybrid nanofluid. Also Lorentz force which is derived from magnetic field is mentioned in this section. In addition, the impacts of (∅ 2 ), (β) and (λ) on the profiles of nanofluid and hybrid nanofluid temperature for three categories of nanoparticle shapes named brick, cylinders, and platelets are analyzed. At the end, the influences of (∅ 2 ), (β) and (λ) on skin friction coefficient (C f ) and Nusselt number (Nu x ) for Cu-water nanofluid and TiO2-Cu/water hybrid fluid for different nanoparticles shapes are discussed. In all of these studies it can be seen that applying platelets shaped nanoparticles is more effective. [ABSTRACT FROM AUTHOR]

An attempt has been made to examine the two dimensional free stream squeezed flow through a horizontal sensor surface of an electrically conducting Carreau-Yasuda fluid subject to transverse magnetic field. The governing nonlinear partial differential equations are modeled and then simplified with the aid of suitable similarity transformations. Well known numerical scheme Runge–Kutta–Fehlberg method is utilized to solve the system. Concrete graphical analysis is carried out to investigate the behavior of different pertinent parameters on velocity profile with inclusive discussion. Numerical influence of friction factor is also discussed. [ABSTRACT FROM AUTHOR]

A mathematical study is presented to investigate the influences of homogeneous and heterogeneous reactions in local similar flow caused by stretching sheet with a non-linear velocity and variable thickness. Porous medium effects are characterized by using Darcy-Forchheimer porous-media. A simple isothermal model of homogeneous-heterogeneous reactions is used. The multiphysical boundary value problem is dictated by ten thermophysical parameters: ratio of mass diffusion coefficients, Prandtl number, local inertia coefficient parameter, inverse Darcy number, shape parameter, surface thickness parameter, Hartman number, Homogeneous heat reaction, strength of homogeneous-heterogeneous reactions and Schmidt number. Resulting systems are computed by Runge-Kutta-Fehlberg method. Different shapes of velocity are noticed for n > 1 and n < 1. [ABSTRACT FROM AUTHOR]

The two-dimensional unsteady laminar free convective heat and mass transfer fluid flow of a non-Newtonian fluid adjacent to a vertical plate has been analyzed numerically. The two parameters Lie group transformation method that transforms the three independent variables into a single variable is used to transform the continuity, the momentum, the energy and the concentration equations into a set of coupled similarity equations. The transformed equations have been solved by the Runge–Kutta–Fehlberg fourth-fifth order numerical method with shooting technique. Numerical calculations were carried out for the various parameters entering into the problem. The dimensionless velocity, temperature and concentration profiles were shown graphically and the skin friction, heat and mass transfer rates were given in tables. It is found that friction factor and heat transfer (mass transfer rate) for methanol are higher (lower) than those of hydrogen and water vapor. Friction factor decreases while heat and mass transfer rate increase as the Prandtl number increases. Friction (heat and mass transfer rate) factor of Newtonian fluid is higher (lower) than the dilatant fluid. [ABSTRACT FROM AUTHOR]

This paper reflects some research outcome denoting as to how Lotka–Volterra prey predator model has been solved by using the Runge–Kutta–Fehlberg method (RKF). A comparison between Runge–Kutta–Fehlberg method (RKF) and the Laplace Adomian Decomposition method (LADM) is carried out and exact solution is found out to verify the applicability, efficiency and accuracy of the method. The obtained approximate solution shows that the Runge–Kutta–Fehlberg method (RKF) is a more powerful numerical technique for solving a system of nonlinear differential equations. [ABSTRACT FROM AUTHOR]

In this study, we investigate the effects of viscous dissipation and second order slip on MHD boundary layer flow of an incompressible, electrically conducting water-based nanofluid over a stretching sheet. The governing momentum boundary layer and thermal boundary layer equations with the boundary conditions are transformed into a system of nonlinear ordinary differential equations which are then solved numerically by using the Runge–Kutta–Fehlberg method. The effects of the flow parameters on the velocity, temperature, nanoparticle concentration, shearing stress, rate of heat transfer, and rate of mass transfer are analyzed, and illustrations are provided by the inclusion of figures and tables for various values of different parameters. We determine that the skin friction increases in magnitude, whereas the rate of heat transfer and rate of mass transfer decrease in magnitude as the strength of the magnetic field increases. In addition, the magnitudes of skin friction, rate of heat transfer, and rate of mass transfer decrease as the melting heat transfer and first-order slip parameter both increase. [ABSTRACT FROM AUTHOR]

The present problem is the steady boundary layer flow and heat transfer of a hyperbolic tangent fluid flowing over a vertical exponentially stretching cylinder in its axial direction. After applying usual boundary layer with a suitable similarity transformation to the given partial differential equations and the boundary conditions, a system of coupled nonlinear ordinary differential equations is obtained. This system of ordinary differential equations subject to the boundary conditions is solved with the help of Runge-Kutta-Fehlberg method. The effects of the involved parameters such as Reynolds numbers, Prandtl numbers, Weissenberg numbers and the natural convection parameter are presented through the graphs. The associated physical properties on the flow and heat transfer characteristics that is the skin friction coefficient and Nusselt numbers are presented for different parameters. [ABSTRACT FROM AUTHOR]