Runge Kutta Fehlberg Fourth-fifth Order Research Articles

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.

Hydromagnetic solid–liquid pulsatile flow through concentric cylinders in a porous medium

The current research is reported about the hydromagnetic solid–liquid flow in an annulus between two concentric circular cylinders embedded in a porous media. The impact of joule heating is also accounted for. Unlike the usually applied constant pressure gradient, the pulsatile pressure gradient is employed. The flow problem is first modeled and then tackled by Runge–Kutta–Fehlberg fourth–fifth-order (RKF45) numerical scheme along with shooting algorithm. The impacts of emerging parameters namely magnetic field parameter and porosity parameter on velocity and temperature distributions are displayed through graphs and briefly addressed.

Electromagnetohydrodynamic transport of Al2O3 nanoparticles in ethylene glycol over a convectively heated stretching cylinder

In this study, the transport of Al2O3 nanoparticles in ethylene glycol conventional fluid over a linearly stretching cylinder is investigated. The current research employs a convective surface boundary condition for heat transfer exploration. Flux model proposed by Rosseland is employed to examine effect of thermal radiations. The governing flow problem comprises highly nonlinear ordinary differential equations. Similarity transformations are used to reduce the equations in similar forms, which are then solved by Runge–Kutta–Fehlberg fourth-fifth order numerical scheme with shooting algorithm in MATLAB software. In order to authenticate the accuracy of our results, we have contrasted results with those obtained by Ishak et al., Wang, and Pandey and Kumar and found that they are in better concord, as revealed in Table 2. The impact of numerous emerging parameters on velocity distribution and heat transfer distribution are argued in all aspects and depicted through graphs.

Influence of Stefan blowing on nanofluid flow submerged in microorganisms with leading edge accretion or ablation

The forced convective boundary layer flow of viscous incompressible time-dependent fluid containing water-based nanofluids and gyrotactic microorganisms simultaneously, from a flat surface with leading edge accretion (or ablation), is theoretically investigated in the present study. In doing so, the governing conservation equations are rendered into a nonlinear system of ordinary differential equations by means of utilizing appropriate coordinates transformations. MAPLE symbolic software is employed to solve these equations, which are subjected to impose boundary conditions using the Runge–Kutta–Fehlberg fourth-fifth order numerical method. It is noteworthy that the results of the present study are in an excellent agreement with previous solutions available in literature. The effect of selected parameters on velocity, temperature, nanoparticle volume fraction and motile microorganism density function is then investigated. Tabular solutions are included for the skin friction, heat transfer rate, nano-particle mass transfer rate and microorganism transfer rate. Applications of the study arise in advanced micro-flow devices to bio-modified nanomaterials processing.

Analytical solution for suction and injection flow of a viscoplastic Casson fluid past a stretching surface in the presence of viscous dissipation

In this study, steady two-dimensional flow of a viscoplastic Casson fluid past a stretching surface is considered under the effects of thermal radiation and viscous dissipation. Both suction and injection flows situations are considered. The partial differential governing equations are transformed into ordinary differential equations and solved analytical. Analytical solutions for velocity and temperature are obtained in terms of hypergeometric function and discussed graphically. Moreover, numerical results are also obtained by Runge–Kutta–Fehlberg fourth–fifth-order (RKF45) method and compared with the analytical results. The results showed that the injection and suction parameter can be used to control the direction and strength of flow. The effects of Casson parameter on the temperature and velocity are quite opposite. The effects of thermal radiation on the temperature are much more stronger in case of injection. The heat transfer coefficient shows higher value for Casson fluid while for Newtonian fluid is the lowest.

Two parameters Lie group analysis and numerical solution of unsteady free convective flow of non-Newtonian fluid

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.

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.

Effect of Newtonian Heating and Thermal Radiation on Heat and Mass Transfer of Nanofluids over a Stretching Sheet in Porous Media

Laminar boundary layer slip flow from a stretching surface in a nanofluid-saturated homogenous, isotropic porous medium is studied numerically. A Newtonian heating boundary condition in the presence of thermal radiation is incorporated and a Darcy model utilized for the porous medium. The model used for the nanofluids include the effects of Brownian motion and thermophoresis. A group theoretical analysis is conducted to generate similarity transformations. The governing transport equations are nondimensionalized and rendered into a set of coupled similarity ordinary differential equations using similarity transformations. The transformed equations are then solved using the Runge–Kutta–Fehlberg fourth-fifth order numerical method with shooting technique. It is shown that the physical quantities of interest depend on a number of parameters. The results are presented in tabular and graphical forms. Comparison of the present numerical solutions with published work shows very good agreement. The study finds applications in high-temperature nanotechnological materials processing.

Flow Over a Stretching Sheet in a Dusty Fluid With Radiation Effect

The radiation effect on a steady two-dimensional boundary layer flow of a dusty fluid over a stretching sheet is analyzed. The governing nonlinear partial differential equations have been transformed by a similarity transformation into a system of nonlinear ordinary differential equations and then solved numerically by applying Runge Kutta Fehlberg fourth-fifth order method (RKF45 method). The effect of fluid particle interaction parameter, Prandtl number, Eckert number, and radiation parameter on heat transfer characteristics in two different general cases, namely (1) the prescribed surface temperature (PST) and (2) the prescribed heat flux (PHF) are presented graphically and discussed. The rate of heat transfer is computed and tabulated for various values of the different parameters. Comparison of the obtained numerical results is made with previously published results.

Boundary Layer Flow and Heat Transfer of a Dusty Fluid over a Stretching Vertical Surface

This paper presents the study of convective heat transfer characteristics of an incompressible dusty fluid past a vertical stretching sheet. The governing partial differential equations are reduced to nonlinear ordinary differential equations by using similarity transformation. The transformed equations are solved numerically by applying Runge Kutta Fehlberg fourth-fifth order method (RKF45 Method). Here obtained non-dimensional velocity and temperature profiles has been carried out to study the effect of different physical parameters such as fluid-particle interaction parameter, Grashof number, Prandtl number, Eckert number. Comparison of the obtained numerical results is made with previously published results.