Fehlberg Fourth-fifth Order Research Articles

Physics-informed neural networks approach for pollutant dispersion flow caused by magnetic dipole

Abstract Research on the liquid flow and distribution of waste discharge concentration over cylindrical surfaces might lead to the growth of efficient waste management and pollution controlling strategies. In view of this, the pollutant concentration and magnetic dipole impact on the liquid flow through the cylinder with a porous medium are explored in the present analysis. Moreover, Fourier’s and Fick’s laws are considered to explore heat transport features. The governing partial differential equations (PDEs) of the present study are converted into non-dimensional ordinary differential equations (ODEs) utilizing similarity variables. Also, the current study proposes a novel implementation of neural networks based on physics knowledge for the liquid flow past a cylinder. The self-updating weights and bias facilitate the network to satisfy the minimization objective function and provide accurate results. Additionally, the solution obtained by Runge Kutta Fehlberg's fourth-fifth order (RKF-45) method is utilized to validate the physics-informed neural network (PINN) results, presenting mean squared errors from 10-9 to 10-11. Furthermore, it is found that the increase in temperature relaxation time parameter decreases the Nusselt number gradually at the rate of 1 to 5%. An upsurge in the values of the pollutant external source parameter and the pollutant external source variation parameter decreases the concentration profile.

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.

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.

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.

Nonlinear radiation effects on water-based nanofluid containing CNTs subject to heat source/ sink past a wedge

In this paper, a two-dimensional and incompressible laminar flow comprised of water-based carbon nanotubes over convectively heated moving wedge under the magnetic field and nonlinear radiation and heat production/ absorption is investigated. The base nanofluid (water) contains single wall carbon nanotubes (SWCNTs) and multiple walls carbon nanotubes (MWCNTs). In order to convert the dimensional nonlinear partial differential equations in nondimensional nonlinear ordinary differential form, an adequate set of similarity variables had been used. These set of equations and boundary conditions are evaluated by the implementation of RKF-45 (Runge–Kutta–Fehlberg fourth-fifth) order scheme. The influence of several physical parameters on particular nanoparticle’s volume friction, temperature and velocity ratio parameter, heat source/ sink parameter, nonlinear radiative constraint, exponent constant, magnetic factor, Eckert and Biot numbers is studied. An opposite behavior of volume fraction and velocity ratio parameters on velocity and energy profiles is achieved.

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.

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.

Effect of inclined magnetic field on flow, heat and mass transfer of Williamson nanofluid over a stretching sheet

This research explores numerically the impact of the aligned magnetic field on Williamson's nanofluid on a stretching surface with convective boundary conditions. The ordinary differential equations (ODE's) are obtained from the partial differential equations (PDE's) employing the transformation of similarity. Such equations are computed with the respective limits with Runge-Kutta – Fehlberg's fourth-fifth order method along with a shooting procedure. The functions of the many parameter values on the flow field profiles are graphically presented and examined. The current results agree well with previous works. The local skin friction, the local Nusselt number, and the local Sherwood number are shown with graphs and tables for changed values of the flow constraints.

Features of entropy optimization on MHD couple stress nanofluid slip flow with melting heat transfer and nonlinear thermal radiation

Numerical analysis is performed for magnetohydrodynamics (MHD) couple stress nanofluid flow over a stretching sheet with melting and nonlinear radiation. The second law of thermodynamics is also incorporated with first-order slip. Nanofluid characteristics for thermophoresis and Brownian moments are encountered. The system that comprises differential equations of partial derivatives is remodeled into the system of differential equations via similarity transformations and then solved numerically through the Runge–Kutta–Fehlberg fourth-fifth (RKF-45) order technique. The physical parameters, which emerges from the derived system are discussed in graphical format. The significant outcomes of the current investigation are that the velocity field decays for a higher magnetic parameter. Another, important outcome of the study is both temperature and concentration are increasing functions of the first-order slip. Nusselt and Sherwood numbers are decreasing with an increase in magnetic strength. Further, Bejan number augment due to enhancement in the first-order slip and couple stress fluid parameters whereas a differing tendency is shown for magnetic and radiation parameters.

Comparison of some numerical methods for ozone decomposition of the second order in aqueous solutions

A comparative study between optimal homotopy asymptotic method (OHAM) and some other well-known methods such as the Homotopy perturbation, Adomian decomposition and Runge-Kutta fehlberg fourth-fifth order methods has been presented in the current paper. These methods have been applied in solving the system of the second order decomposition of ozone in aqueous solutions. The obtained results indicate these methods are appropriate to solve the equations whereas applying OHAM is more simpler with the same accuracy in comparison to other methods.

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.