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The purpose of this paper is to re-examine the assumptions implicit in Leveque’s approximation, and the variation of the temperature and the thickness of the boundary layer were illustrated using the developed solution. The analytical solutions are then checked against numerical solution programming by FORTRAN code obtained via using Runge–Kutta fourth-order (RK4) method. Finally, other important thermal results obtained from this analysis, such as approximate Nusselt number in the thermal entrance region, was discussed in detail. After that, the analytical results of the present paper are validated with certain previous investigations which were found in the specialized literature.

By defining a similarity variable, the governing equations are reduced to a dimensionless equation with an analytic solution in the entrance region. This paper gives justification for the similarity variable via scaling analysis, details the process of converting to a similarity form and presents a similarity solution. The calculation methodology for numerical resolution is based on the RK4 technique.

The profiles of the solutions are provided from which the authors infer that the numerical and exact solutions agreed very well. Another result that the authors obtained from this paper is the number of Nusselt in the thermal entrance region for which a parametric study was carried out and discussed well for the impact of scientific contribution.

The novelty of this paper is the application of the RK4 with a step size control, as a sequential numerical method of a ODEs system compared with the exact similarity solution of the thermal boundary layer problem.

Presents a numerical solution of the two‐dimensional laminar boundary layer problem on free and forced convection of an incompressible visco‐elastic fluid immersed in a porous medium over a stretching sheet. Here, the driving force for the flow is provided by an impermeable sheet stretched with a velocity proportional to the distance from a slit and buoyancy effects due to both temperature and concentration gradients. The resultant governing boundary layer equations are highly non‐linear and coupled form of partial differential equations, and they have been solved by employing a numerical shooting technique with fourth order Runge‐Kutta integration scheme. Numerical computations are carried out for the non‐dimensional physical parameters. The results are analyzed for the effect of different physical parameters like visco‐elasticity, permeability of the porous medium, Grashof number, Schmidt number and Prandtl number on the flow, heat and mass transfer characteristics. One of the several important observations is that the combined effect of thermal diffusion and diffusion of species is to increase the horizontal velocity profile and to decrease the temperature and concentration profiles in the boundary layer flow field.

The purpose of this paper is to study the thermal behavior of radial porous fin surrounded by water-base copper nanoparticles under the influence of radiation.

In order to optimize the response variable, the authors perform sensitivity analysis with the aid of response surface methodology (RSM). Moreover, this study enlightens the applications of artificial neural networks (ANN) for predicting the temperature gradient. The governing modeled equations are firstly non-dimensionalized and then solved with the aid of Runge–Kutta fourth order together with the shooting method in order to guess the initial conditions.

Numerical results are analyzed and presented in the form of tables and graphs. This study reveals that the temperature of the fin is decreasing as the wet porous parameter increases (m₂) and the temperature for 10% concentration of nanoparticles are higher than 5 and 1%. Physical parameters involved in the study are analyzed and processed through RSM. It is come to know that sensitivity of temperature gradient to radiative parameter (Nr) and convective parameter (Nc) is positive and negative to dimensionless ambient temperature (θ_a). Furthermore, after ANN training it can be argued that the established model can efficiently be used to predict the temperature gradient over a radial porous fin for the copper-water nanofluid flow.

To the best of our knowledge, only a few attempts have been made to analyze the thermal behavior of radial porous fin surrounded by copper-based nanofluid under the influence of radiation and convection.

The purpose of this paper is to study chemical reaction and heat and mass transfer effects on steady free convection flow in an inclined porous plate in the presence of MHD and viscous dissipation through the application of scaling group of transformation and numerical method.

The fourth-order Runge-Kutta along with the shooting method is employed in the numerical solution of the governing equations.

The magnetic field parameter, the permeability of porous medium and the viscous dissipation are demonstrated to exert a more significant effect on the flow field and, thus, on the heat transfer from the plate to the fluid.

The problem is relatively original.

In the article, analytical model of first-order shear deformation (FSDT) beams made of jute–epoxy is presented to study the low-velocity impact response.

The nonlinear Hertz contact law is applied to identify the contact between projectile and beam. The energy method, Lagrange's equations and Ritz method are applied to derive the nonlinear governing equation of the beam and impactor-associated boundary condition. The motion equations are then solved simultaneously by the Runge–Kutta fourth-order method.

Also, a comparison is performed to validate the model predictions. The contact force and beam indentation histories of the jute–epoxy simply supported beam under spherical impactor with different radius and initial velocity are investigated in detail. It is found that in response to impactor radius increase, the utilization of the contact force law has resulted in a same increasing trend of peak contact force, impact duration and beam indentation, while in response to impactor initial velocity increase, the maximum contact force and beam indentation increase while impact time has vice versa trend.

This paper fulfills an identified need to study how jute–epoxy beam behavior with simply supported boundary conditions under low-velocity impact can be enabled.

The purpose of this paper is to examine the effects of thermophoresis and magnetic field on steady two‐dimensional laminar hydrodynamic flow with heat and mass transfer over a semi‐infinite permeable flat surface in the presence of viscous dissipation and thermal radiation effects. The fluid viscosity and thermal conductivity are assumed to vary as a function of temperature.

The boundary layer equations are transformed to non‐linear ordinary differential equations using scaling group of transformations and these equations are solved numerically by using the fourth order Runge‐Kutta method with shooting technique for some values of physical parameters.

Some of the results obtained for a special case of the problem are compared to the results published in previous work and are found to be in excellent agreement. Many results are obtained and a representative set is displayed graphically to illustrate the influence of the physical parameters involved in the problem on the velocity, temperature and concentration profiles, as well as the local skin‐friction coefficient, the wall heat transfer and the particle deposition rate.

One valuable, important observation is that the effect of the variable viscosity parameter is to increase the effect of all studied parameters in the boundary‐layer's flow field. Also, the skin‐friction coefficient, wall heat transfer and wall deposition flux in a fluid of uniform viscosity are higher than in a fluid of non‐uniform viscosity when the surface is permeable.

The paper presents a numerical solution for two‐dimensional boundary‐layer flow with heat and mass transfer over a semi‐infinite permeable flat surface. Numerical results indicate that the combining effects of magnetic field and radiation strongly controls flow and mass transfer characteristics for the thermophoretic hydrodynamic flow. This problem is interesting from the physical point of view and also for its applications in engineering sciences.

Sheet processing of magnetic nanomaterials is emerging as a new branch of smart materials’ manufacturing. The efficient production of such materials combines many physical phenomena including magnetohydrodynamics (MHD), nanoscale, thermal and mass diffusion effects. To improve the understanding of complex inter-disciplinary transport phenomena in such systems, mathematical models provide a robust approach. Motivated by this, this study aims to develop a mathematical model for steady, laminar, MHD, incompressible nanofluid flow, heat and mass transfer from a stretching sheet.

This study developed a mathematical model for steady, laminar, MHD, incompressible nanofluid flow, heat and mass transfer from a stretching sheet. A uniform constant-strength magnetic field is applied transversely to the stretching flow plane. The Buongiorno nanofluid model is used to represent thermophoretic and Brownian motion effects. A non-Fourier (Cattaneo–Christov) model is used to simulate thermal conduction effects, of which the Fourier model is a special case when thermal relaxation effects are neglected.

The governing conservation equations are rendered dimensionless with suitable scaling transformations. The emerging nonlinear boundary value problem is solved with a fourth-order Runge–Kutta algorithm and also shooting quadrature. Validation is achieved with earlier non-magnetic and forced convection flow studies. The influence of key thermophysical parameters, e.g. Hartmann magnetic number, thermal Grashof number, thermal relaxation time parameter, Schmidt number, thermophoresis parameter, Prandtl number and Brownian motion number on velocity, skin friction, temperature, Nusselt number, Sherwood number and nanoparticle concentration distributions, is investigated.

A strong elevation in temperature accompanies an increase in Brownian motion parameter, whereas increasing magnetic parameter is found to reduce heat transfer rate at the wall (Nusselt number). Nanoparticle volume fraction is observed to be strongly suppressed with greater thermal Grashof number, Schmidt number and thermophoresis parameter, whereas it is elevated significantly with greater Brownian motion parameter. Higher temperatures are achieved with greater thermal relaxation time values, i.e. the non-Fourier model predicts greater values for temperature than the classical Fourier model.

The purpose of this article is to analyze the heat and mass transfer with entropy generation during magnetohydrodynamics (MHD) flow of non-Newtonian Sisko nanofluid over a linearly stretching cylinder under the influence of velocity slip, chemical reaction and thermal radiation. The Brownian motion, thermophoresis and activation energy are assimilated in this nanofluid model. Convective boundary conditions on heat and mass transfer are considered. The physical model may have diverse applications in several areas of technology underlying thermohydrodynamics including supercritical fluid extraction, refrigeration, ink-jet printing and so on.

The dimensional governing equations are nondimensionalized by using appropriate similarity variables. The resulting boundary value problem is converted into initial value problem using the method of superposition and numerically computed by employing well-known fourth-order Runge–Kutta–Fehlberg approach along with shooting technique (RKF4SM). The quantitative impacts of emerging physical parameters on the velocity, temperature, concentration, skin friction coefficient, Nusselt number, Sherwood number, entropy generation rate and Bejan number are presented graphically and in tabular form, and the salient features are comprehensively discussed.

From graphical outcomes, it is concluded that the slip parameters greatly influence the flow characteristics. Fluid temperature is elevated with rising radiation parameter and thermal Biot number. Nanoparticle concentration is reported in decreasing form with activation energy parameter. Entropy is found to be an increasing function of magnetic field, Brownian motion and material parameters. The entropy is less generated for shear-thinning fluid compared to shear-thickening as well as Newtonian fluids in the system.

Till now no study has been documented to explore the impact of binary chemical reaction with Arrhenius activation energy on entropy generation in an MHD boundary layer flow of non-Newtonian Sisko nanofluid over a linear stretching cylinder with velocity slip and convective boundary conditions.

The purpose of this study is to numerically examine the heat transfer and transport of space charges in the solid insulating materials [low density polyethylene (LDPE), flame retardant type 4 (FR4), Polytetrafluoroethylene (PTFE)] using the transmission line modeling (TLM) method. Besides, a comprehensive study is performed on the mutual influences of heat transfer and space charges transport within the solid dielectric bulk.

The obtained governing equations including continuity and circuit equations are coupled with heat transfer equations, and they are solved via fourth-order Runge–Kutta method.

The electric potential and field, current density and temperature distribution are calculated. It is shown that compared with FR4 and PTFE, the temperature increment rate in LDPE is much lower. Moreover, the heat transfer in the solid insulating materials bulk increases the homo-charges density and temperature in the vicinity of electrodes. Hence, the reduction in electric field is reflected in the potential deformations in the proximity of electrodes. Furthermore, where the electric field is maximized, the temperature is minimized.

This study is restricted to two-dimensional problems.

Interestingly, because of the lower temperature in LDPE, the current density and their increment rates in LDPE are much lower than that in FR4 and PTFE dielectric materials.

The purpose of this paper is to scrutinize the heat and mass transfer attributes of three-dimensional bio convective flow of nanofluid across a slendering surface with slip effects. The analysis is carried out subject to irregular heat sink/source, thermophoresis and Brownian motion of nanoparticles.

At first, proper transmutations are pondered to metamorphose the basic flow equations as ODEs. The solution of these ODEs is procured by the consecutive application of Shooting and Runge-Kutta fourth order numerical procedures.

The usual flow fields along with density of motile microorganisms for sundry physical parameters are divulged via plots and scrutinized. Further, the authors analyzed the impact of same parameters on skin friction, heat and mass transfer coefficients and presented in tables. It is discovered that the variable heat sink/source parameters play a decisive role in nature of the heat and mass transfer rates. The density of motile microorganisms will improve if we add Al-Cu alloy particles in regular fluids instead of Al particles solely. A change in thermophoresis and Brownian motion parameters dominates heat and mass transfer performance.

To the best of the knowledge, no author made an attempt to investigate the flow of nanofluids over a variable thickness surface with bio-convection, Brownian motion and slip effects.