Runge Kutta Fehlberg Fourth-fifth Order Method Research Articles

Heat transfer analysis in magnetohydrodynamic nanofluid flow induced by a rotating rough disk with non-Fourier heat flux: aspects of modified Maxwell-Bruggeman and Krieger-Dougherty models.

Non-Newtonian fluids have unique heat transfer properties compared to Newtonian fluids. The present study examines the flow of a Maxwell nanofluid across a rotating rough disk under the effect of a magnetic field. Furthermore, the Cattaneo-Christov heat flux model is adopted to explore heat transport features. In addition, a comparison of fluid flow without and with aggregation is performed. Using similarity variables, the governing partial differential equations are transformed into a system of ordinary differential equations, and this system is then solved by employing the Runge-Kutta Fehlberg fourth-fifth order method to obtain the numerical solution. Graphical depictions are used to examine the notable effects of various parameters on velocity and thermal profiles. The results reveal that an increase in the value of Deborah number decreases the velocity profile. An increase in the thermal relaxation time parameter decreases the thermal profile. An artificial neural network is employed to calculate the rate of heat transfer and surface drag force. The R values for skin friction and Nusselt number were computed. The results demonstrate that artificial neural networks accurately predicted skin friction and Nusselt number values.

Forced convection of laminar gaseous slip flow near a stagnation point with viscous dissipation and pressure work

Forced convection problem of laminar quasi-incompressible boundary layer for the stagnation slip flow at a relatively low Mach number, considering the simultaneous effects of viscous dissipation and pressure work, has been investigated. The system of coupled partial differential equations was first transformed into a system of coupled ordinary differential equations through suitable transformations, which was then solved using Runge–Kutta–Fehlberg fourth-fifth order method. The solution obtained here is much better suited to formulating and solving the variable-property of chemically reacting flows that occur in practice, by taking into account the slip boundary conditions at the gas–wall interface. The effects of the Eckert number and the slip parameter on the heat transfer characteristics are presented graphically and discussed. The numerical results show that the pressure work, viscous dissipation play significant role on the heat transfer and could not be neglected under any circumstance for rarefied gas flows.

Second law analysis on Hall effect of natural convection flow through vertical channel in the presence of uniform heat source/sink

Purpose Microfluidics is one of the extensive elaborated technologies in thermal and engineering fields due to its wide range of applications, such as micro heat exchangers, micro mixture and microchannel heat sinks, which is used to develop a large number of microscopic devices and systems. Enhancement of thermal energy using verity of nanoliquids is one of the challenges in these applications of microfluidics. Therefore, using single wall carbon nanotubes for enhancement of thermal energy in microchannel is the main purpose of this study. Hall effect of natural convection flow in a vertical channel with slip and temperature jump condition is considered. The impacts of radiative heat flux, uniform heat source/sink, viscous dissipation and joule heating are also taken into account. Design/methodology/approach Suitable non-dimension variables are applied to the governing equations to reduce the system into ordinary differential equations. The reduced nonlinear system is then solved numerically using Runge–Kutta–Fehlberg fourth–fifth-order method along with shooting technique. The impact of different pertinent parameters on numerical solutions of primary velocity, secondary velocity, temperature, entropy generation and Bejan number is comprehensively discussed in detail. Also, the obtained numerical results are compared with existing one which perfectly found to be in good agreement. Findings It is established that, with the aspects of Joule heating, viscous dissipation, radiative heat flux and uniform heat source/sink, the production in the entropy can be improved. Further, it is found that the increasing ratio of wall ambient temperature difference and nanoparticle volume fraction leads to enhance the entropy generation. The same effect reverses with increasing values of fluid wall interaction parameter (FWIP) and rare faction. The irreversibility ratio enhances with larger values of nanoparticle volume fraction and decelerates with increment values of FWIP. Originality/value The impact of single wall carbon nanoliquid in a vertical channel flow by using radiative heat flux, heat source/sink, joule heating and viscous dissipation is first time investigated. Further, the influence of Hall current is explored in detail.

Exploration of particle shape effect on Cu-H2O nanoparticles over a moving plate

PurposeThis paper aims to explore particle shape effect on Cu-H2O nanoparticles over a moving plate in the presence of nonlinear thermal radiation. To characterize the effect, particle shape and viscous dissipation are considered. Convergent solutions for the resulting nonlinear systems are derived and the effects of embedded parameters of interest on velocity and temperature field are examined.Design/methodology/approachThe Runge–Kutta–Fehlberg fourth-fifth order method along with shooting technique is used to solve the governing equations (6) and (7) with boundary conditions (8). A suitable finite value of η∞ is considered in such a way that the boundary conditions are satisfied asymptotically.FindingsThe results show an increase in both the heat transfer and thermodynamic performance of the system. However, among the three nanoparticle shapes, disk shape exhibited better heat transfer characteristics and heat transfer rate. On the other hand, the velocity profile enhances with increasing values of ϕ in the first solution, but the opposite trend was found in the second solution.Originality/valueThe present paper deals with an exploration of particle shape effect on Cu-H2O nanoparticles over a moving plate in the presence of nonlinear thermal radiation. To characterize the effect, particle shape and viscous dissipation are considered. Convergent solutions for the resulting nonlinear systems are derived and the effects of embedded parameters of interest on velocity and temperature field are examined. The skin friction coefficient and Nusselt number are numerically tabulated and discussed. The results show an increase in both heat transfer and thermodynamic performance of the system. However, among the three nanoparticle shapes, disk shape exhibited better heat-transfer characteristics and heat-transfer rate. On the other hand, the velocity profile enhances with increasing values of ϕ in the first solution, but the opposite trend was found in the second solution.

Biot number effect on MHD flow and heat transfer of nanofluid with suspended dust particles in the presence of nonlinear thermal radiation and non-uniform heat source/sink

This paper considers the problem of steady, boundary layer flow and heat transfer of dusty nanofluid over a stretching surface in the presence of non-uniform heat source/sink and nonlinear thermal radiation with Biot number effect. The base fluid (water) is considered with silver (Ag) nanoparticles along with suspended dust particles. The governing equations in partial form are reduced to a system of non-linear ordinary differential equations using suitable similarity transformations. An effective Runge–Kutta–Fehlberg fourth-fifth order method along with shooting technique is used for the solution. The effects of flow parameters such as nanofluid interaction parameter, magnetic parameter, solid volume fraction parameter, Prandtl number, heat source/sink parameters, radiation parameter, temperature ratio parameter and Biot number on the flow field and heat-transfer characteristics were obtained and are tabulated. Useful discussions were carried out with the help of plotted graphs and tables. Under the limiting cases, comparison with the existing results was made and found to be in good agreement.

Nonlinear thermal radiation effect on magneto Casson nanofluid flow with Joule heating effect over an inclined porous stretching sheet

In this paper, mixed convection on MHD flow of casson nanofluid over a non-linearly permeable stretching sheet has been investigated and analyzed numerically. The effects of thermal radiation, chemical reaction, heat generation/absorption, viscous dissipation, suction and Joule heating are considered. The Brownian motion and thermophoresis phenomenon are used to model nanoparticles (Buongiorno's model). After converting PDEs governing the problem to ODEs, they have been solved by Runge-Kutta Fehlberg fourth-fifth order method. Obtained results of investigating the effects of different parameters changes on velocity, temperature, and concentration profiles are reported as diagrams. Fluid flow velocity reduction by increase in Hartman number (magnetic field parameter) is due to existence of Lorentz drag force against flow, flow velocity reduction due to increase in casson fluid parameter, increase in temperature profile due to increase in radiation parameter, and nanoparticle concentration profile reduction due to increase in chemical reaction parameter are some of valuable obtained results. Also, in final section of this paper effects of different parameters on skin friction coefficient, local Nusselt and Sherwood numbers are investigated that positive and ascending behavior for all three are reported.

On multiple solutions of non-Newtonian Carreau fluid flow over an inclined shrinking sheet

This paper presents the multiple solutions of a non-Newtonian Carreau fluid flow over a nonlinear inclined shrinking surface in presence of infinite shear rate viscosity. The governing boundary layer equations are derived for the Carreau fluid with infinite shear rate viscosity. The suitable transformations are employed to alter the leading partial differential equations to a set of ordinary differential equations. The consequential non-linear ODEs are solved numerically by an active numerical approach namely Runge–Kutta Fehlberg fourth-fifth order method accompanied by shooting technique. Multiple solutions are presented graphically and results are shown for various physical parameters. It is important to state that the velocity and momentum boundary layer thickness reduce with increasing viscosity ratio parameter in shear thickening fluid while opposite trend is observed for shear thinning fluid. Another important observation is that the wall shear stress is significantly decreased by the viscosity ratio parameter β∗ for the first solution and opposite trend is observed for the second solution.

Unsteady magnetohydrodynamics flow about a stagnation point on a stretching plate embedded in porous medium

This paper presents an unsteady two-dimensional laminar boundary layer flow of an incompressible, viscous and electrically conducting fluid in a vicinity of a stagnation point towards a stretching plate embedded in porous medium with variable free stream velocity. The governing nonlinear partial differential equations of mass, momentum and energy are reduced to a system of nonlinear ordinary differential equations by using suitable similarity variables. The ordinary differential equations are tackled numerically by using Runge-Kutta Fehlberg fourth-fifth order method with shooting technique. The effect of the significant parameters on the fluid temperature and velocity are graphically presented and discussed. The numerically computed results are in good agreement with the available published results in the literature. It is found that the velocity increases with the increasing unsteadiness parameter where the velocity ratio parameter λ = 0.5 but it decreases for λ = 1.5.

Slip effects on MHD Hiemenz stagnation point nanofluid flow and heat transfer along a nonlinearly shrinking sheet with induced magnetic field: multiple solutions

The present investigation deals with two-dimensional boundary layer flow of an incompressible electrically conducting nanofluid induced by a nonlinearly (power-law) shrinking flat surface in the presence of passively controlled nanoparticle boundary condition along with induced magnetic field effect. The similarity transformations developed by Lie group method are used which transforms the set of governing equations into a set of coupled similarity equations. This system of nonlinear ordinary differential equation is solved to obtain multiple solutions using a Runge–Kutta–Fehlberg fourth–fifth-order method (RKF45) with shooting method. This study claims the existence of multiple solutions of velocity and temperature profiles as function of suction (\(s\)) and shrinking parameter (\(\chi\)). The critical points (turning points) have also been reported for suction (\(0 < s_{c} < s\)) and shrinking parameter (\(\chi_{c} < \chi < 0\)) for the default set of other parameters. The temporal stability analysis has been performed to confirm the uniqueness of stable solution.

Mathematical Model for Cyber Attack in Computer Network

This investigation focuses to develop an e-SEIRS (susceptible, exposed, infectious, recovered) epidemic computer network model to study the transmission of malicious code in a computer network and derive the approximate threshold condition (basic reproduction number) to examine the equilibrium and stability of the model. The authors have simulated the results for various parameters used in the model and Runge-Kutta Fehlberg fourth-fifth order method is employed to solve system of equations developed. They have studied the stability of crime level to equilibrium and found the critical value of threshold value determining whether or not the infectious free equilibrium is globally asymptotically stable and endemic equilibrium is locally asymptotically stable. The simulation results using MATLAB agree with the real life situations.

MHD mixed convection slip flow near a stagnation-point on a non-linearly vertical stretching sheet in the presence of viscous dissipation

In this study, MHD mixed convection stagnation-point flow toward a non-linearly stretching vertical sheet in the presence of thermal radiation and viscous dissipation is numerically analyzed. The partial momentum and heat transfer equation are transformed into a set of ordinary differential equations by employing suitable similarity transformations. Using the Runge-Kutta Fehlberg fourth-fifth order method, numerical calculations to the desired level of accuracy are obtained for different values of dimensionless parameters. The results are presented graphically and in tabular form. The results for special cases are also compared to those obtained by other investigators and excellent agreements were observed. The effect of injection on the MHD mixed slip flow near a stagnation-point on a non-linearly vertical stretching sheet is to enhance the velocity field which results from the suppression of the skin friction on the wall surface. The heat transfer rate at the surface increases with increasing values of the non-linearity parameter. The velocity and thermal boundary-layer thicknesses are found to be decreasing with increasing values of the non-linearity parameter.

STAGNATION POINT FLOW OF MHD DUSTY FLUID TOWARD STRETCHING SHEET WITH CONVECTIVE SURFACE

This paper presents the study of stagnation point of hydromagnetic flow of dusty fluid over a stretching sheet with the bottom surface of sheet heated by convection from hot fluid. The governing partial differential equations are transformed into a system of nonlinear ordinary differential equations using similarity transformation. These resulting nonlinear ordinary differential equations are solved numerically by using Runge-Kutta Fehlberg fourth-fifth order method (RKF45 Method). The characteristics of velocity and temperature profiles of hydromagnetic fluid flow of dusty fluid are analyzed and discussed for different parameters of interest such as convective Biot number, fluid-particle interaction parameter, magnetic parameter, ratio of free stream velocity parameter and Prandtl number on the flow. The numerical results are compared with previous published results for validation

Stagnation Point Flow of Nanofluid over a Moving Plate with Convective Boundary Condition and Magnetohydrodynamics

A theoretical investigation is carried out to examine the effects of volume fraction of nanoparticles, suction/injection, and convective heat and mass transfer parameters on MHD stagnation point flow of water-based nanofluids (Cu and Ag). The governing partial differential equations for the fluid flow, temperature, and concentration are reduced to a system of nonlinear ordinary differential equations. The derived similarity equations and corresponding boundary conditions are solved numerically using Runge-Kutta Fehlberg fourth-fifth order method. To exhibit the effect of the controlling parameters on the dimensionless velocity, temperature, nanoparticle volume fraction, skin friction factor, and local Nusselt and local Sherwood numbers, numerical results are presented in graphical and tabular forms. It is found that the friction factor and heat and mass transfer rates increase with magnetic field and suction/injection parameters.

Effect of Radiation on Flow and Heat Transfer of MHD Dusty Fluid Over a Stretching Cylinder Embedded in a Porous Medium in Presence of Heat Source

In the framework of the two-phase model, flow of a conducting dusty fluid due to linearly stretching cylinder immersed in a porous media with the effect of radiation is considered. The flow is described in terms of a ‘dusty gas’ model proposed by Saffman, which treats the discrete phase (particles) and the continuous phase (fluid) as two continua occupying the same space. Similarity transformations are used to convert the governing partial differential equations corresponding to momentum and energy equations into non-linear coupled ordinary differential equations. Numerical solutions of these equations are obtained using Runge–Kutta Fehlberg fourth-fifth order method and results are illustrated graphically. Comparisons with previously published work are performed and the results are found to be in good agreement. The numerical results for the local skin-friction coefficient and local Nusselt number are also presented.

Boundary layer flow past a stretching sheet with fluid-particle suspension and convective boundary condition

The steady two-dimensional boundary layer flow of a viscous dusty fluid over a stretching sheet with the bottom surface of the sheet heated by convection from a hot fluid is considered. The governing partial differential equations are transformed into ordinary differential equations using a similarity transformation, before being solved numerically by a Runge–Kutta–Fehlberg fourth-fifth order method (RKF45 Method) with the help of MAPLE. The effects of convective Biot number, fluid particle interaction parameter, and Prandtl number on the heat transfer characteristics are discussed. It is found that the temperature of both fluid and dust phase increases with increasing Biot number. A comparative study between the previous published and present results in a limiting sense is found in an excellent agreement.

Suspended Particle Effect on Nanofluid Boundary Layer Flow Past a Stretching Surface

This article presents a numerical investigation on effect of suspended particles on boundary layer flow and heat transfer characteristics of a viscous nanofluid immersed in a porous medium over a stretching sheet. Copper Cu as nanoparticle and water with suspended particles as its base fluid have been considered. Two different temperature conditions are studied, namely (i) the sheet with prescribed surface temperature (PST) and (ii) the sheet with prescribed wall heat flux (PHF). Similarity transformations are used to reduce the governing nonlinear boundary layer equations to coupled higher order non-linear ordinary differential equations. These equations were numerically solved using Runge–Kutta–Fehlberg fourth-fifth order method. The effect of governing physical parameters on velocity and temperature profiles for the both nanofluid phase and dust phase are computed, analyzed and discussed. Besides, the local skin friction coefficient and local Nusselt number are computed and presented in tabular form. It is found that permeability parameter and volume fraction of nanoparticle has a significant influence on increasing rate of heat transfer. And momentum and thermal boundary layer are thinner due to the influence of suspended particles. Comparisons with previously published work are performed and excellent agreement is obtained.