Runge Kutta Fehlberg Fourth-fifth Order Research Articles

Thermophoretic diffusion deposition velocity effect in the flow-induced due to inner stretched and outer stationary coaxial cylinders

This study's primary goal is to examine the mass and heat transfer in the fluid flow between two cylinders while taking the thermophoretic particle deposition impact into account. Only the stretching effect of the inside cylinder under the stationary outside cylinder causes the flow. The governing partial differential equations (PDEs) of the flow dynamics are converted into ordinary differential equations (ODEs) by utilizing similarity transformations. Then, using the shooting approach and the Runge-Kutta Fehlberg fourth-fifth order (RKF-45) method, these simplified equations are numerically solved. Graphical depictions are utilized to analyze the physical behaviour of key factors in detail. The comparison of the published findings with the obtained outcome is given here to check the accuracy and obtained a good agreement with the published results. Results reveal that the gap size strongly influences the momentum, heat and mass transfer fields. Velocity and thermal profiles increase as the curvature parameter upsurges, but the concentration profile declines. It is also observed that when gap width rise, the rate of heat transmission reduces. The findings demonstrate that the mass transportation rate declines as the thermophoretic parameter and thermophoretic coefficient value increase.

Sensitive analysis of heat transfer enhancement in ternary Casson nanofluid flow between a conical surface and disk

This study explores the ternary nanofluid flow within the canonical gap between a cone and a disk with particle deposition and magnetic field effects. Reduced titanium dioxide, magnetite, and graphene oxide are used as nanoparticles in the base fluid ethylene glycol. The governing equations of the problem are in the form of partial differential equations, which are converted to nonlinear ordinary differential equations by using appropriate similarity transformations, and they are solved numerically by using Runge–Kutta–Fehlberg fourth fifth-order (RKF 45) technique. The main agenda of this work is to discuss the impacts of parameters on three cases. The effects of essential aspects on fluid flow, heat and mass transfer rates were studied and analyzed using a graphical representation. Additionally, the response surface methodology and sensitivity analysis are carried out to enhance the importance of the heat transfer rate. The results reveal that the flow field increases significantly with increased Reynolds numbers for both cone and disk rotations. It is observed that the sensitivity analysis of the Nusselt number toward the Eckert number is more for all the radiation parameter values and the Eckert number’s middle level.

Nanoparticle aggregation kinematics and nanofluid flow in convectively heated outer stationary and inner stretched coaxial cylinders: Influenced by linear, nonlinear, and quadratic thermal radiation

The consequence of nanoparticle aggregation and convective boundary condition on the nanofluid stream past the co-axial cylinder with radiation impact is investigated in the present examination. The influence of linear, nonlinear, and quadratic thermal radiation on the nanofluid flow is analyzed. The outer cylinder stays stable, while the inner cylinder deforms horizontally in the axial direction, allowing fluid to flow. By using similarity variables, the governing equations are transformed into ordinary differential equations (ODEs). Subsequently, the Runge–Kutta–Fehlberg fourth-fifth order (RKF-45) method is employed to solve the reduced ODEs. The upshot of several nondimensional terms on the temperature and velocity profiles is displayed with graphical representation. The comparison of linear, quadratic, and nonlinear thermal radiation on the thermal profile is illustrated. The upsurge in curvature parameter increases velocity and thermal profile. The increase in radiation parameter intensifies the temperature profile. The thermal profile improves with a rise in the values of radiation parameter. The radiation parameter generates thermal energy in the flow zone, which is why the temperature field has improved. The thermal Biot number exhibits an increasing response with temperature and thermal boundary layer thickness. The linear thermal radiation shows better heat transfer compared to quadratic and nonlinear thermal radiation.

Irreversibility scrutiny of SWCNT and MWCNT nanofluid convective flow using Darcy-Forchheimer rule in an upright microchannel with variable thermal conductivity and Brownian motion and thermophoresis

Carbon nanotubes are used to attain greater thermal transfer rates in a various of enormous applications in the field of microsystem, emission, conductive polymers, fibers, and fabrics. Hence the aim of this article is to explore the entropy generation analysis of single wall carbon nanotubes and multiwall carbon nanotubes flow in an upright micro channel with variable thermal conductivity. The exponential heat source and non-linear thermal radiation were considered. The micro channel retains the no slip and convective boundary conditions. The Buongiorno model was used in this study, which emphasizes the light on Brownian motion and thermophoresis phenomena that occur during the fluid flow. Darcy- Forchheimer model is also taken into account. The governing equation numerically solved by employing Runge-Kutta Fehlberg's fourth-fifth order. The repercussion of this study upholds that the inflates of variable thermal conductivity magnify the thermal field. Entropy production inflates at right wall of the channel with augmented Brownian parameter and exhibits exactly opposite behavior at left wall of the channel. One of the important tasks of the investigation was to compare SWCNT and MWCNT. The findings confirmed that multiwall carbon nanotubes exhibit more magnificent heat and mass transfer than single wall carbon nanotubes.

Impact of waste discharge concentration on fluid flow in inner stretched and outer stationary co-axial cylinders

The effect of Stefan blowing and pollutant concentration distribution on the flow of fluid through the co-axial cylinder is investigated in this study. Investigating the fluid flow and dispersion of waste materials in co-axial cylinders can lead to the progress of effective waste disposal and pollution prevention methods. The outer cylinder remains stationary and the inner cylinder deforms horizontally along the axial direction which initiating the liquid flow. The governing equations are converted into ordinary differential equations (ODEs) utilizing similarity variables. Then, the reduced ODEs are solved numerically using the Runge Kutta Fehlberg fourth-fifth order (RKF-45) approach. The consequence of various parameters on the velocity, thermal, and concentration profiles are displayed in graphical representation. The rise in values of the curvature parameter upsurges the temperature profile. The concentration decreases as the pollutant external source variation parameter and pollutant source parameter increases. The rise in pollutant external source variation parameter values declines the mass transfer rate, but a converse trend is seen for increasing pollutant external source parameter values.

Impact of solid–liquid interfacial layer in the nanofluid flow between stretching stationary disk and a rotating cone

The present study explores the flow of nanofluid between a cone and a disk with the thermophoresis and Brownian motion impact. The flow is due to an expanding disk and a rotating cone. In the current study, radiative heat flux is considered. Using proper similarity transformations, the governing differential equations of momentum, temperature and concentration are transformed into a set of nonlinear ordinary differential equations (ODEs). The Runge-Kutta Fehlberg fourth fifth-order (RKF-45) technique and the shooting strategy are employed to solve the reduced ODEs. The impact of various parameters is illustrated through graphs and used to comprehend flow profile behaviour easily. The obtained results demonstrate that the velocity profiles enhance as improve in values of the Reynolds number. The rise in radiation, thermophoresis and Brownian motion parameters increases the thermal profile due to the potency of Brownian movement which results in a compelling nanoparticle interaction and boosts the fluid's thermal efficiency. Increasing values of the thermophoresis parameter reduces the heat transmission rate at the disk. As the Brownian motion parameter elevates, the concentration profile reduces, but the concentration profile increases as the thermophoresis parameter is raised.

Framing the features of nanolayer and diameter of carbon nanotubes in the flow of blood over a stretching cylinder in presence of magnetic induction

This study explores the slip boundary layer electrically conducting flow of nanofluid over a stretching cylinder. Flow type in present study is considered to be steady and incompressible flow with mixture of SWCNTs/MWCNTs and human blood. Thermal conductivity of present flow model is measured by merging the consequence of diameter of SWCNTs/MWCNTs nanoparticles and solid–liquid interfacial-layer. Velocity slip, non-linear radiation, and induced magnetic-flux are considered in flow. The mode of heat transmission in flow is presented in appearance of dual heat flux, that is, Cattaneo-Christov heat flux and prescribed heat flux. By executing similarity exploration, we modernize the leading PDEs of the acknowledged flow model into ODEs. The consequential equations are solved numerically by recalling Runge–Kutta–Fehlberg fourth-fifth order scheme through shooting system. Deliberations on consequence of flow features on flow specific are organized accurately via graphs and charts. The professed numerical consequences are novel and distinctive. These consequences can contribute to other scholars on electing the suitable factors to enlarge the heat conduction technique in medical science, society, and the industry.

Effect of thermophoretic particle deposition on the three-dimensional flow of nanofluid induced by a nonlinear stretched surface

The current investigation is carried out to study a heat source/sink in a porous medium generated by a nonlinear stretched surface with the impact of thermophoretic particle deposition (TPD) on three-dimensional (3D) nanofluid flow due to the vast range of industrial applications like a condensation of aerosol particles on walls, extraction of oil, coated steel cooling, and radial reflectors. The mathematical model was subjected to a boundary layer approximation, which resulted in the development of partial differential equations (PDEs) then converting these equations to ordinary differential equations (ODEs) the similarity variable is used. The system of reduced ODEs is solved with Runge–Kutta-Fehlberg fourth fifth-order (RKF-45) and shooting techniques with the help of MATLAB software. Discussions are made with the help of graphs obtained for various dimensionless constraints. The results show that nanoparticle addition will improve the thermal profile, but contrary behavior is seen in the velocity and concentration profiles. Three-dimensional figures are drawn to show the behavior of different constraints over Nusselt, Sherwood, and Skin friction factors along with the numerical tabulation presented.

The effects of nanoparticle aggregation and radiation on the flow of nanofluid between the gap of a disk and cone

The heat transfer in three-dimensional coordinates is the main focus of the present article. The cone-disk (CND) apparatus is presumed to include a stationary cone (SCN) and a rotating disk (RTD), or counter-rotating, or both of them co-rotating, along with Titanium dioxide-ethylene glycol-based nanofluid. The accommodative impact of nanopaprticles aggregation and thermal radiation is considered in the modelling equations for the flow pattern available in the literature. The modelling equations are converted into ordinary differential equations (ODEs) using similarity transformations. The behaviour of flow profiles is readily understood by employing the Runge Kutta Fehlberg fourth fifth-order (RKF45) method and the shooting approach, which is strategized and explained using graphs. The outcomes reveal that the dynamics of flow with nanoparticles aggregation case shows better-quality heat transport for increased values of radiation parameter. Furthermore, the fluid flow with aggregation shows the developed heat transfer rate at cone surface for increased values of radiation parameter.

Impact of nanoparticle aggregation on heat transfer phenomena of second grade nanofluid flow over melting surface subject to homogeneous-heterogeneous reactions

It is well known that the inclusion of a certain quantity of nanoparticles boosts the thermal conductivity of the nanofluid. The reason for this tremendous improvement is yet unknown. Consequently, finding the proper thermal effect of nanoscale particles requires an understanding of nanoparticle aggregation kinematics. The utilization of nanomaterials may be seen in a variety of technological and industrial applications. The influence of homogeneous and heterogeneous chemical reactions on an incompressible flow of second-grade nanofluid through a stretched cylinder with NP aggregation is investigated in this work. Similarity transformations are used to change partial differential equations (PDEs) into a system of ordinary differential equations (ODEs). The Runge Kutta Fehlberg fourth fifth-order (RKF 45) technique and shooting approach are used to numerically solve these ODEs. The influence of major elements on flow fields and heat transfer rates is investigated and addressed using graphical representations. The results suggest that the fluid flow without NPs aggregation has better heat transmission than when the melting parameter increases. Furthermore, the higher mass transfer for fluid flow with aggregation condition is detected for increased values of strength of heterogeneous and homogeneous reaction parameters.

Effect of thermal radiation on heat transfer in plane wall jet flow of Casson nanofluid with suction subject to a slip boundary condition

A Glauert type laminar wall jet issuing into a stationary liquid medium lying above a wall has technical uses in wall cooling and flow control. It plays a vital role in industrial applications like cooling/heating by impingement of jet, turbine blades, film cooling, mass and heat transfer phenomena. In this regard, a steady incompressible two-dimensional laminar Glauert kind wall jet is scrutinized in this study by considering nanoparticles suspension in the base liquid sodium alginate with suction and wall slip boundary conditions. Further, a comparative study is done by considering aluminum alloy and single-walled carbon nanotube as nanoparticles. The reduced ordinary differential equations (ODEs) are numerically solved by applying Runge–Kutta–Fehlberg fourth fifth-order (RKF-45) technique along with the shooting method. Results reveal that Casson nanofluid shows enhanced heat transfer than Casson nanoliquid for increased values of radiation parameter. The rising values of the Casson parameter deteriorate the heat transfer rate of both nanoliquids but an inverse trend is seen for improved values of radiation parameter.

Forced Convection of MHD Radiative Jeffrey Nanofluid Over a Moving Plate

Convectively heated Jeffrey nanofluid flow in the presence of magnetic field and thermal radiation is investigated from a moving plate. Parameter of Brownian motion from Boungiorno model is the imperative mechanism that contributes to the heat transfer enhancement. Governing equations, consisting of the continuity, momentum, energy and nanoparticle concentrations equations are transformed into dimensionless form by means of the appropriate similarity transformation variables. Numerical results via Runge-Kutta Fehlberg Fourth-Fifth order (RKF45) method are specifically acquired on the impact of physical parameters such as Brownian motion, magnetic parameter, ratio of relaxation to retardation and radiation parameters over the temperature and nanoparticles concentration profiles. Comparison of the present results with existing published studies has validated the accuracy of the numerical solutions. Graphical representation of different magnetic parameters has caused the increment in both temperature and nanoparticles concentration profiles. On the other hand, enhancement of Brownian motion has intensified the temperature but declined the nanoparticles concentration.

Thermal behaviour of annular hyperbolic fin with temperature dependent thermal conductivity by differential transformation method and Pade approximant

The current paper explores the steady-state heat transfer and thermal stress analysis of a hyperbolic annular fin with temperature dependent thermal conductivity. The framed equations are articulated in terms of non-linear ordinary differential equations. The domination of non-dimensional parameters on the thermal gradient of the fin has been analysed graphically by using Runge–Kutta Fehlberg fourth-fifth order (RKF-45) process and DTM-Pade approximant. Here, differential transformation method (DTM)-Pade approximant has been implemented to find the analytical solution for the non-linear ordinary differential equations. The RKF-45 method is used to obtain numerical outcomes for different non-dimensional parameters. Further, the obtained numerical results are compared with the results achieved from DTM-Pade approximant solution. This relation demonstrates that numerical outcome obtained by DTM-Pade approximant are nearer to that of numerical results. Outcome results exposes that, the escalating values of thermal conductivity parameter upsurges the thermal distribution. The growing values of the radius ratio elevate the thermal distribution in the fin. Further, the radial stress and tangential stress distribution varies for larger values of dimensionless parameters.

Impact of Binary Chemical Reaction and Activation Energy on Heat and Mass Transfer of Marangoni Driven Boundary Layer Flow of a Non-Newtonian Nanofluid

The flow and heat transfer of non-Newtonian nanofluids has an extensive range of applications in oceanography, the cooling of metallic plates, melt-spinning, the movement of biological fluids, heat exchangers technology, coating and suspensions. In view of these applications, we studied the steady Marangoni driven boundary layer flow, heat and mass transfer characteristics of a nanofluid. A non-Newtonian second-grade liquid model is used to deliberate the effect of activation energy on the chemically reactive non-Newtonian nanofluid. By applying suitable similarity transformations, the system of governing equations is transformed into a set of ordinary differential equations. These reduced equations are tackled numerically using the Runge–Kutta–Fehlberg fourth-fifth order (RKF-45) method. The velocity, concentration, thermal fields and rate of heat transfer are explored for the embedded non-dimensional parameters graphically. Our results revealed that the escalating values of the Marangoni number improve the velocity gradient and reduce the heat transfer. As the values of the porosity parameter increase, the velocity gradient is reduced and the heat transfer is improved. Finally, the Nusselt number is found to decline as the porosity parameter increases.

Impact of magnetic dipole on ferromagnetic hybrid nanofluid flow over a stretching cylinder

Nanofluids manage heat in the internal combustion of the engines or machines by avoiding corrosion in the cooling system as well as assist in eradicating the engine’s waste heat. Hence, they are used as coolants in many automotive industries. Inspired by these applications, the thermal and mass transfer in hybrid nanoliquid flow over a stretching cylinder on taking account of magnetic dipole is studied in this investigation. Here, we have done a comparative study on flow of two diverse combinations of hybrid nanofluids, namely and The modelled equation for the assumed flow is converted to ODEs by opting appropriate similarity variables. These ODEs are solved by utilizing the Runge–Kutta Fehlberg fourth-fifth order (RKF-45) method by adopting shooting technique. Physical clarification of relevant parameters for non-dimensional discrete flow fields are discussed briefly by using graphs. Also, skin friction, Sherwood and Nusselt numbers are deliberated with the assistance of graphs. Results reveal that, the upsurge in ferromagnetic interaction parameter declines the velocity in both fluids but converse trend is detected in temperature and concentration of the liquids. The heightening of ferromagnetic interaction parameter declines the rate of heat and mass transfer.

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.

Thermal radiation effect on boundary layer over a flat plate having convective surface boundary condition

This paper presents the effect of thermal radiation on the boundary layer over a flat plate. The convective boundary condition is applied at the surface of the flat plate. The solution to the coupled non-linear transport equations is obtained using the Runge–Kutta–Fehlberg fourth-fifth order (RKF45) method. The impact of thermal radiation on mercury, air, sulphur oxide and water whose Prandtl numbers ( $$\Pr$$ ) are 0.044, 0.72, 2, and 7 respectively are depicted using line graphs and tables. The impact of Prandtl number ( $$\Pr$$ ), local convective heat transfer (a) and temperature difference ( $$C_{T}$$ ) on temperature distribution are also presented. The results indicated that the boundary layer thickness decreases with $$\Pr$$ augment but increases with increasing values $$R$$ . Furthermore, R augment is inversely proportional to the temperature gradient near the plate while an opposite trend is observed away from the plate. The results also indicated that $$R$$ augment reduces heat transfer but an opposite trend is observed with $$\Pr$$ augment. $$\Pr$$ augment has a decreasing effect on boundary layer thickness and a augment has an increasing effect on temperature.

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.

STAGNATION POINT FLOW OF DUSTY CASSON FLUID WITH THERMAL RADIATION AND BUOYANCY EFFECTS

The aim of the study is to examine the stagnation point flow of a dusty Casson fluid over a stretching sheet with thermal radiation and buoyancy effects. The governing boundary layer equations are represented by a system of partial differential equation. After applying suitable similarity transformations, the resulting boundary layer equations are solved numerically using the Runge Kutta Fehlberg fourth-fifth order method (RKF-45 method). The behaviors of velocity, temperature and concentration profiles of fluid and dusty particles with respect to change in fluid particle interaction parameter, Casson paramter, Grashof number, radiation parameter, Prandtl number, number density, thermal equilibrium time, relaxation time, specific heat of fluid and dusty particles, ratio of diffusion coefficients, Schmidt number and Eckert number are analysed graphically and discussed. Our computed results interpret that velocity distribution decays for higher estimation of Casson parameter while temperature distribution shows increasing behavior for larger radiation parameter.

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.