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This work explores the heat transfer flow characteristics of an incompressible non-Newtonian Jeffrey fluid over a stretching/shrinking surface with thermal radiation and heat source. The sheet is linearly stretched in the presence of a transverse magnetic field with convective boundary conditions. Appropriate similarity variables are used to transform the basic governing equations (PDEs) into ODEs. The resulting equations are solved by utilizing MATLAB bvp4c. The impact of distinctive physical parameters and dimensionless numbers on the flow field and heat transfer is analysed graphically. It is noticed that the measure of heat raised with increasing the Biot number and opposite effect with the rise of the suction parameter.

The current research work is devoted to study the exploration of dissipation and chemical reaction on MHD radiative Casson nanoliquid thin film flow generated due to the sheet stretching. Brownian motion, viscous dissipation, radiation and chemical reactions are considered, respectively, in energy and concentration equations. The popularly used similarity variables are utilized to remodel the nonlinear partial differential equations to a system of ordinary differential equations. The computational results are obtained for this system by adopting shooting technique. The impressions of significant flow variables on liquid stream are obtained through diagrams and tables. The outcomes of current work conclude that the thickness of thin film decreases with the rise of unsteady, magnetic and Casson parameters. Also, for higher estimates of non-Newtonian parameter a reduction in the coefficient of wall friction is noticed. The outcomes are matched with the viscous fluid flow and the results are in better agreement as a limiting case.

This work examines the effects of non-linear thermal radiation and Joule heating on MHD three-dimensional visco-elastic nanofluid flow due to a surface stretching in lateral directions. A coupled nonlinear differential system is generated from the boundary layer equations by using self-similarity variables and is then solved numerically by using most powerful shooting technique with Runge Kutta method of fourth order. The computational results for the flow variables are plotted graphically and are discussed in detail for various governing parameters that emerged in the analysis. It is observed that the momentum of the visco elastic nanofluid is better than that of a viscous fluid. Thicker thermal and concentration boundary layers are formed for increasing nonlinear thermal radiation and temperature ratio parameters. Also the results are in very good agreement with the outcomes available in the literature as a particular case. This model may play a significant role in the field of manufacturing and engineering applications.

The importance of thermophoresis and its essential role in particle migration have led to many published reports (i.e. aim and objectives). However, there exists no report on thermo-migration of tiny/nano-sized particles in the motion of various fluids. A meta-analysis on the significance of either nano or tiny particles exposed to thermophoretic force owing to temperature gradient during the dynamics of liquid substances is deliberated upon in this report. The method of slope linear regression through the data point was adopted to scrutinize sixty (60) published reports in which the effects of thermophoresis (thermodiffusion) is deliberated upon. The outcome of the study shows that different responses to the force of a temperature gradient are sufficient enough to enhance the temperature distribution and the concentration of non-Newtonian fluid due to an increase in thermophoresis. Thermophoretic effect increases the concentration of fluids in which the relationship between the shear stress and shear strain is non-linear. Skin friction coefficients is a decreasing function of thermophoresis. Increase in thermophoretic deposition is achievable due to an increase in thermophoresis. The effect of haphazard motion of nanoparticles should be investigated when it increases negligibly and considerably large. Thermal radiation strongly influences the significance of thermo-migration of tiny particles on fluid flow.

The present research explores the features of thermal and solutal transport of a 3D micropolar liquid stream on an elongated convectively heated inclined sheet taking Soret effect. Mathematical modelling is designed with the aid of suitable scaling analysis on the governing PDEs conceiving the small magnetic Reynolds number. The resultant set of coupled nonlinear ODEs are derived with MATLAB to obtain computational solutions. Impression of the emerged flow parameters on the three boundary layers is graphically traced and deliberated. The parameters of magnetic field and stretching ratio and power law index diminished frictional drag. Hike in rate of thermal diffusion is prevailed with stronger surface heat convective and Prandtl numbers. Outcomes are collated with the data available in the literature and found to agree very closely as a limiting case.

In this report, the effects of homogeneous-heterogeneous autocatalytic chemical reaction together with the variable thermal conductivity in the Maxwell fluid flow due to nonlinear surface of variable thickness are investigated. Thermal radiation and heat generation / absorption effects are also incorporated in the analysis. Appropriate scaling analysis is implemented to reduce the mathematical model describing the physics of the problem in to a set of nonlinear differential equations and are subsequently solved computationally. Graphical illustrations indicating the effect of pertinent parameters on momentum, thermal and solutal boundary layers are presented and discussed. The study reveals that velocity distribution shows a decreasing (increasing) tendency for larger values of wall thickness parameter when the velocity power law index is less (greater) than unity. The concentration of the homogeneous bulk fluid with catalyst at the surface decreases with increasing chemical reaction rate parameters.

In the present work, dynamical aspects of boundary layer flow of hydromagnetic fluid (suspended with microorganisms) are investigated near the stagnation region over a stretchable heated permeable sheet. Viscosity is taken as a linear function of temperature where the flow field accommodates diffusion processes due to temperature and concentration gradients (Soret/Dufour numbers). A system of partial differential equations is set up for the mathematical description of the related bio-physical phenomenon. Unit free conversions and analysis of symmetry are implemented to obtain nonlinear dimensional free differential equations setup which can be solved numerically via the Runge–Kutta–Fehlberg scheme. Results in the form of pictorial and tabulation representation reveal that velocity profile increases when viscosity is a function of temperature difference, but the velocity profile influences the fluid temperature in the opposite direction.

The candid objective of this study is to simulate stagnation point flow of the MHD chemically reactive Maxwell based nanofluid under the action of Arrhenius activation energy function with radiative thermal energy and frictional heating. Application of scaling analysis followed by the RKF-45 method to the governing flow equations yield the computational solution. Maxwell parameter results in abatement of flow. Higher the Eckert number greater is the temperature with a overshoot in the vicinity of the stretched surface Nanoparticle concentration enhances proportionally with activation energy and temperature difference parameter.

In this paper we discuss the MHD unsteady flow of a micropolar nanofluid through a vertical channel with expanding/contracting walls under the influence of nonlinear thermal radiation considering weak permeability. The PDEs of the governing problem are reduced into a set of ODEs by using self-similar transformations and are then solved numerically. Influence of flow parameters such as wall dilation ratio, permeability Reynolds number, micropolar parameters, thermal radiation parameter, nanofluid parameters etc. on the flow, thermal energy and volume fraction of nanoparticles has been discussed in detail. We trust that the results of this study are not only useful for industrial applications but also present a basic understanding of the physical model.

A mathematical model is established to examine the influence of viscous dissipation and joule heating on magnetohydrodynamic (MHD) flow of an incompressible viscoelastic nanofluid over a convectively heated stretching sheet. Brownian motion and thermophoresis effects have been introduced in this nanofluid model. The governing equations are transformed into ODE’s by using suitable similarity conversions and are then solved numerically by the most robust shooting technique. The significance of numerous physical flow constraints is performed for, and distributions through graphs. It is noticed that, the increases for higher values of and reduces for rising values of heat source and Biot numbers. An outstanding contract was found between our numerical results and previously publicised results.

The present study discusses two dimensional Darcy-Forchheimer steady flow of a doubly stratified Maxwell nanofluid over a sheet of continuous stretching. Analysis of thermal energy and species concentration is carried out incorporating radiative heat, thermal and solutal stratifications, Brownian motion and thermophoresis. By introducing suitable transformations the system of equations of the flow are recast into a set of nonlinear ODEs which are then solved numerically by using the RKF-45 method. Flow characteristics are deliberated for different variations of governing parameters. Surface drag force, thermal energy and mass transfer rates are computed and discussed. Favourable comparisons with published work in the literature for different special cases of the problem are examined.

An analysis illustrating the flow of an Ostwald-de-Waele liquid film on an unsteady stretching sheet under the influence of thermocapillary force, magnetic field and viscous dissipation is carried out. In this study, thermal conductivity is assumed to be a function of fluid temperature. Numerical solutions for the partial differential equations governing the flow are obtained by employing the elegant Runge-Kutta-Fehlberg method for certain representative values of controlling parameters, such as thermocapillarity number, magnetic field parameter, etc. Film thickness is calculated for various values of flow parameters. Film thickness of shear thinning fluids is found to be smaller than that of a Newtonian fluid and a converse trend holds true for shear thickening fluids. Thicker films are noticed for increasing values of thermocapillarity number. In the presence of thermocapillary force, an initial decrease in the velocity of a shear thinning fluid occurs before fluid velocity experiences a significant increase towards the free surface. Stronger magnetic field strengths are seen to increase the free surface velocity. Themocapillary force on temperature in a shear thinning fluid is more prominent.

In this study we analyze the effects of thermal radiation and surface mass transfer on the flow generated by a shrinking sheet. Homotopy analysis method (HAM) is employed to obtain analytical solutions for the flow, temperature and mass concentration fields. The plots for the velocity, temperature and concentration fields are presented and their behavior is discussed for several sets of values of the governing parameters. Dual solutions are observed to exist for the shrinking sheet problem. Surface drag coefficient of the stable solution is enhanced for increasing values of the magnetic parameter and suction parameter. Also, thermal radiation parameter and Eckert number increase the rate of heat transfer in the case of stable solution. However, mass transfer rate of the stable solution is noticed to be a decreasing function of Schmidt number and chemical reaction parameter. We expect that the results obtained will not only provide useful information for industrial applications but also complement the existing literature.

Entropy generation is inherently affiliated with transport of thermal energy. However, not much attention has been paid to the study of entropy generation in liquid thin film flows. Hence, in this paper an analysis is carried out to study the entropy generation in a thin viscous fluid film on a stretching sheet embedded in a porous medium subject to a magnetic field and thermocapillarity force taking thermal radiation and internal heating/absorption into account. The numerical solution of the equations of momentum and energy governing the flow is obtained. Influence of various parameters emerged in the analysis on the velocity, temperature, surface drag, Nusselt number, entropy generation number and the Bejan number are graphically illustrated and discussed. Thermocapillarity number shows an enhancement of velocity at the free surface. Film thickness is found to increase with increasing thermocapillary force. Thinner films are noticed for stronger magnetic field strengths. It is found that in the absence of thermocapillary force, the effect of magnetic field on entropy generation number near the stretching surface is prominent. Thermocapillarity is seen to have a stronger effect on the entropy generation number near the stretching surface as well as at the free surface.

The partial derivative DOW is missing in Eqs. (3–5) in the original publication. The corrected equations are shown below.

Study of thermal radiative heat transport is relevant in several industrial applications. Analysis of radiative heat transfer in Newtonian fluids under different conditions has been studied extensively. However, not much attention has been paid to analyze the non-linear thermal radiative heat transfer in non-Darcy Casson fluid flows over stretching surfaces. Hence, in this paper an attempt is made to study the combined effect of non-linear thermal radiation and Navier slip in a magnetohydrodynamically and chemically reactive Casson fluid flow on a stretching surface immersed in a non-Darcy porous medium in the presence of thermo-diffusion. The physics of the problem is delineated by a set of partial differential equations. To solve these equations numerically scaling analysis and Runge–Kutta–Fehlberg method are sequentially applied. Computational results of the investigation are analyzed through graphs and tables. It is noted that the non-Newtonian rheology parameter, inertial and Lorentz forces are seen to suppress the flow while thermal and solutal buoyancy forces have an opposite effect. Parameters of temperature ratio and thermal radiation enhanced the temperature.

The aim of this investigation is to analyze the effectiveness of Lorentz force, viscous dissipation and internal heating on the heat and flow characteristics of a non-Newtonian Casson fluid thin film resting on a stretching surface under the influence of a magnetic field. Employing suitable similarity variables and shooting technique and integrating scheme numerical solutions for velocity and temperature are obtained. The results of this analysis are compared with the published work and are found to be in good agreement. The thickness of the thin film is evaluated and is observed that Lorentz force and the non-Newtonian nature of the fluid have a thinning influence on the film. Velocity and temperature distributions in the thin film are discussed for various flow parameters.

The unsteady dispersion of a solute in blood flow modeling blood as a Newtonian fluid under the influence of a body acceleration is studied using the generalized dispersion model proposed by Gill and Sankarasubramanian [13]. As a result, the total process of dispersion can be described in terms of a simple diffusion process with the effective diffusion coefficient as a function of time. The model brings out mainly the effect of body acceleration and radius of the artery on the overall dispersion process. In the absence of body acceleration, the dispersion coefficient is found to increase rapidly and maintains a steady value in aorta while in other arteries it oscillates about a mean value after reaching it. Body acceleration is observed to enhance the value of the dispersion coefficient in all arteries. The effective diffusivity is found to depend on body acceleration. In the presence of body acceleration, it is noticed that there is a decrease in the effective diffusivity in aorta, femoral, and carotid while in coronary it is increased.

The combined effect of annular gap, yield stress and irreversible boundary reaction on the dispersion process in a Casson fluid flow is studied using generalized dispersion model. The study describes the development of dispersive transport following the injection of a tracer in terms of the three effective transport coefficients viz. absorption, convection and dispersion coefficients. The combined effect of annular gap, yield stress and wall absorption parameter on the above three effective transport coefficients is discussed. It is observed that the absorption coefficient is independent of the yield stress of the fluid and depends on the annular gap and wall absorption parameter. It is also observed that the asymptotic convection, dispersion coefficients are dependent on the yield stress of the fluid, annular gap and wall absorption parameter. The effect of the flow parameters on the mean concentration is studied. Application of this model for understanding the dispersion of solute in blood in a catheterized artery is discussed.

The dispersion of a solute in the flow of a Casson fluid in an annulus is studied. The generalized dispersion model is employed to study the dispersion process. The effective diffusion coefficient, which describes the whole dispersion process in terms of a simple diffusion process, is obtained as a function of time, in addition to its dependence on the yield stress of the fluid and on the annular gap between the two cylinders. It is observed that the dispersion coefficient changes very rapidly for small values of time and becomes essentially constant as time takes large values. In non–Newtonian fluids the steady state is reached at earlier instants of time when compared to the Newtonian case and the time taken to reach the steady state is seen to depend on the values of the yield stress. It is observed that a decrease in the annular gap inhibits the dispersion process for all times both in Newtonian as well as in non–Newtonian fluids. When the yield stress is 0.05, depending upon the size of the annular gap (0.9–0.7) the reduction factor in the dispersion coefficient varies in the range 0.58–0.08. The application of this study for understanding the dispersion of an indicator in a catheterized artery is discussed.

In this paper, diffusion of solute in a Newtonian fluid flowing in a pipe/channel under the influence of transverse magnetic field with interphase mass transfer at the boundaries is analysed using the generalized dispersion model. The study describes the development of dispersive transport following the injection of a tracer in terms of the three effective transport coefficients viz. the absorption, the convection and the dispersion coefficients. The absorption coefficient does not depend on magnetic field but the convection and dispersion coefficients are influenced by the magnetic field. The effect of magnetic field on the mean concentration has also been discussed

We discuss the combined free and forced convection flow of a viscous incompressible fluid in a vertical pipe of variable cross-section. The governing nonlinear equations are solved in terms of the stream function and temperature. The behaviour of the velocity, temperature and heat transfer coefficient is discussed numerically in detail.