Numerical analysis was investigated for steady two-dimensional double diffusive mixed convection boundary layer flow over a semi-infinite vertical plate embedded non-Darcy porous medium filled with nanofluid, in presence of thermal dispersion and under convective boundary conditions. The Buongiorno nanofluid model is used, while the porous medium is described by the Darcy-Forchheimer extension. The governing partial differential equations are transformed into four coupled nonlinear ordinary differential equations using an appropriate similarity transformations and the resulting system of equations is then solved numerically by the finite-difference method. Numerical results are presented to illustrate how the physical parameters affect the flow field, temperature, concentration and solid volume fraction profiles. In addition, the variation of heat, mass and nanoparticle transfer rates at the plate are exhibited graphically for different values of pertinent parameters.

Present study explores stagnation point flow of nanofluid towards a nonlinear stretching sheet of variable thickness in the presence of electromagnetic field and convective heating. The effect of viscous dissipation and Joule heating are also taken into consideration. Novel concept of non-linear radiative heat flux is also considered. The nanofluid is inspired by Lorentz force which is instigated from the interaction of magnetic and electric fields. Using similarity transformation, the governing partial differential equations are transformed into a system of coupled nonlinear ordinary differential equations and then solved numerically by fourth order Runge-Kutta method along with shooting technique. The velocity, temperature and nanoparticle concentration profiles are plotted and analysed corresponding to various pertinent flow parameters. Also, the skin friction and rate of heat and mass transfers at the surface are computed and explained in detail. It is observed that higher wall thickness parameter results in the reduction of velocity, temperature and nanoparticle concentration when velocity power index is less than unity and opposite effect is observed when velocity power index is greater than unity. Due to intensification of electric field, nanofluid velocity is getting retarded and thereby resulting in enhancement of fluid temperature and nanoparticle concentration.

The problem of unsteady boundary layer flow of a nanofluid over a stretching surface is studied. Heat transfer due to melting is analyzed. Using a similarity transformation the governing coupled nonlinear partial differential equations of the model are reduced to a system of nonlinear ordinary differential equations, and then solved numerically by a Runge-Kutta method with a shooting technique. Dual solutions are observed numerically and their characteristics are analyzed. The effects of the pertinent parameters such as the acceleration parameter, the Brownian motion parameter, the thermophoresis parameter, the Prandtl number and the Lewis number on the velocity, temperature and concentration fields are discussed. Also the effects of these parameters on the skin friction coefficient, the Nusselt number and the Sherwood number are analyzed through graphs. It is observed that the melting phenomenon has a significant effect on the flow, heat and mass transfer characteristics.

Free convective flow and heat transfer of nanofluid close to the inclined plate immersed in the porous medium under the effects of uniform magnetic field and solar radiation has been studied. Boundary-layer approach, Boussinesq approximation and two-phase nanofluid model have been used for a formulation of the governing equations taking into account convective-radiative heat exchange with an environment. The local similarity method has been adopted for the analysis of the considered phenomenon. The obtained equations have been solved numerically using MATLAB software. The effects of control characteristics on profiles of velocity, temperature and nanoparticles volume fraction as well as Nusselt number have been studied in detail.

In this paper, unsteady flow and heat transfer of water flow in a backward-facing step under pulsating nanofluid blowing are studied numerically. Attention is focused to examine the impact of this type of blowing and its pertinent parameters on the heat transfer performance and to detect possible non-equilibrium between the base fluid and the nanoparticles inside the flow field. To this aim, the Eulerian-Eulerian two-phase model is adopted. This approach consists of separate equation sets for the base fluid and the nanoparticles. So, it provides details of the flow field for each of the constituents, separately. Computations are undertaken for different cases and the consequences of the frequency, amplitude, and the mean velocity of the pulsating blowing as well as the type, diameter, and the volume fraction of the nanoparticles therein on the heat transfer characteristics are analyzed. It is found that in addition to thermal conductivity of the blown nanoparticles, their penetration into the water flow is an important trait that has a momentous role on the heat transfer rate. In the current Eulerian-Eulerian simulation, temperature distributions of the base fluid and the nanoparticles are similar but the corresponding velocity fields are quite distinct. This reveals a kind of non-equilibrium between the base fluid and the nanoparticles inside the flow that invalidates equilibrium approaches (e.g., the single-phase model or the two-phase mixture model) for the description of the problem.

In this investigation, we intend to present the influence of the prominent viscous dissipation and Soret effects on mixed convection heat and mass transfer over the vertical frustum of a cone in a nanofluid. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. In addition, the uniform wall nanoparticle condition at the surface is replaced with the zero nanoparticle mass flux condition to execute physically applicable results. The governing equations of a nanofluid flow in the dimensional form are reduced to a system of partial differential equations in the non-dimensional form by using suitable non-similarity variables and then solved by using a recently introduced spectral method named as Bivariate Pseudo-Spectral Local Linearisation Method (BPSLLM). The convergence and error analysis tests are conducted to examine the accuracy of the spectral method. To validate the method, the present numerical results are compared with the existing results in some special cases and the outcomes are observed to be in very good agreement. The effects of Brownian motion, thermophoresis, Eckert number, Soret number, nanoparticle and regular buoyancy parameters on the dimensionless surface drag, heat, nanoparticle mass and regular mass transfer rates over the vertical frustum of a cone are discussed and illustrated graphically.

In this paper, the magnetic field effects on natural convection of power-law nanofluids in rectangular enclosures are investigated numerically with the lattice Boltzmann method. The fluid in the cavity is a water-based nanofluid containing Cu nanoparticles and the investigations are carried out for different governing parameters including Hartmann number (0.0≤Ha≤20.0), Rayleigh number (104≤Ra≤106), power-law index (0.5≤n≤1.0), nanopartical volume fraction (0.0≤ϕ≤0.1) and aspect ratio (0.125≤AR≤8.0). The results reveal that the flow oscillations can be suppressed effectively by imposing an external magnetic field and the augmentation of Hartmann number and power-law index generally decreases the heat transfer rate. Additionally, it is observed that the average Nusselt number is increased with the increase of Rayleigh number and nanoparticle volume fraction. Moreover, the present results also indicate that there is a critical value for aspect ratio at which the impact on heat transfer is the most pronounced.

In recent days, nanofluids have derived the attention of researchers, scientists and engineers due to their abundant applications in Engineering and technology and specific applications such as Electronics cooling, vehicle cooling, medical applications including cancer therapy and so on. Motivated by these applications of nanofluids, this work is mainly concerned with the convective heat transfer of nanofluids. MHD slip flow of nanofluids with heat absorption over a rotating disk subjected to suction has been analyzed. Two types of nanofluids such as copper-water nanofluid and silver-water nanofluid are considered for the present study. The system of axisymmetric nonlinear partial differential equations governing the hydromagnetic steady flow and heat transfer are reduced to nonlinear ordinary differential equations by introducing suitable similarity transformations. The resulting non-linear ordinary differential equations are solved numerically by most efficient Nachtsheim-Swigert shooting iteration technique for satisfaction of asymptotic boundary conditions along with Runge – Kutta Fehlberg Method. The flow field is affected by the presence of physical parameters, such as magnetic interaction parameter, suction parameter, slip parameter and solid volume fraction, whereas the temperature field is addionally affected by magnetic interaction parameter, suction parameter, internal heat absorption parameter and solid volume fraction. With the amplifying effect in magnetic interaction parameter, suction parameter, slip parameter and solid volume fraction, the radial and tangential velocities decline. Axial velocity gets decelerated for increasing magnetic interaction parameter and slip parameter whereas it gets accelerated for growing effect of suction parameter and solid volume fraction. The temperature of the fluid within the boundary layer enhances with the increasing effect of magnetic interaction parameter and solid volume fraction while it reduces for increasing values of the suction parameter and internal heat absorption parameter. Also the values of radial and tangential skin friction coefficients and Nusselt number are obtained numerically and are tabulated.

In this paper, the problem of magnetohydrodynamics (MHD) boundary layer flow of nanofluid with heat and mass transfer through a porous media in the presence of thermal radiation, viscous dissipation and chemical reaction is studied. Three types of nanofluids, namely Copper (Cu)-water, Alumina (Al 2 O 3)-water and Titanium Oxide (TiO 2)-water are considered. The governing set of partial differential equations of the problem is reduced into the coupled nonlinear system of ordinary differential equations (ODEs) by means of similarity transformations. Finite element solution of the resulting system of nonlinear differential equations is obtained using continuous Galerkin-Petrov discretization together with the well-known shooting technique. The obtained results are validated using MATLAB “bvp4c” function and with the existing results in the literature. Numerical results for the dimensionless velocity, temperature and concentration profiles are obtained and the impact of various physical parameters such as the magnetic parameter M, solid volume fraction of nanoparticles 𝜙 and type of nanofluid on the flow is discussed. The results obtained in this study confirm the idea that the finite element method (FEM) is a powerful mathematical technique which can be applied to a large class of linear and nonlinear problems arising in different fields of science and engineering.

The present study investigates the hydromagnetic pulsating nanofluid flow in a porous channel with thermal radiation. In this work, we considered water as the base fluid and silver (Ag), copper (Cu), alumina (Al2O3) and titanium dioxide (TiO2) as nanoparticles. The Maxwell-Garnetts and Brinkman models are used to evaluate the effective thermal conductivity and viscosity of the nanofluid. The governing equations are solved analytically and the influence of various parameters on velocity, temperature and heat transfer rate has been discussed through graphical results. From the results, it is found that the rate of heat transfer enhances with an increase of nanoparticle volume fraction. Further, the heat transfer rate is higher for silver nanoparticles as compared with copper, alumina and titanium dioxide.

The present study is devoted to electromagnetohydrodynamic (EMHD) flow and heat transfer characteristics of nanofluid inside a parallel plate microchannel. The nanofluid is actuated by Lorentz force which is originated from the interaction of applied electrical field and perpendicular magnetic field. A fully developed assumption with uniform flux at the surface is considered in the analysis, and the influences of viscous dissipation as well as Joule heating are also taken into account. The analytical solutions for velocity and temperature are derived. Moreover, the Nusselt number variations are examined. The results show that the Hartmann number, the dimensionless parameter S and the nanoparticls volume fraction have significant influences on flow and temperature of nanofluid. As Hartmann number increasing, the Nusselt number improves and similar trend can be observed with the augment of nanoparticls volume fraction. A diminishment of heat transfer performance can be seen with increase of the Joule parameter and Brinkman number, while an enhancement in heat transfer can be witnessed with increase of nanoparticls volume fraction.

This paper deals with studying fluid flow and heat transfer of nanofluid through a forward facing step channel which is affected by a uniform magnetic field transverse to fluid flow. All the channel walls are assumed to be in constant temperature and the fluid temperature at the channel inlet is less than that of the walls. Also, the nanofluid is considered as a single-phase Newtonian fluid and the proper correlations were utilized to determine the thermophysical properties of nanofluid. Therefore, a code has been developed and two-dimensional continuity, momentum and energy equations were solved, using CFD technique. The computations were conducted for different values of the Reynolds and Hartmann numbers, and contraction ratio and an extensive range of nanoparticles volume fraction. The results indicated that flow separation and reattachment phenomena, in vicinity of the step edge, could be influenced strongly by magnetic field and the average Nusselt number is increased significantly by increasing nanoparticles volume fraction and Hartmann number.

This work investigates mainly the double dispersion on the double diffusive convective boundary layer between a vertical plate immersed into a non-Darcy saturated porous medium with a nanofluid. The similarity transformations are involved and the governing system of nonlinear partial differential equations is converted into a set of nonlinear ordinary differential equations. Results are displayed graphically to illustrate the influence of δ, and ξ on the velocity, the temperature and concentration of the species profiles. Two interesting cases are treated, isothermal and non-isothermal wall plate. For a nanofluid, the rate of mass transfer is affected strongly by the double dispersion while the rate of heat transfer coefficient is less sensitive to it.

A numerical analysis was performed to study the effects of combined magnetohydrodynamic and thermal radiation under convective boundary condition over a semi infinite vertical plate embedded in a non-Darcy porous medium. Coupled heat and mass transfer of free convective boundary layer with viscous nanofluid are considered. The model used for the nanofluid includes the effects of Brownian motion and thermophoresis mechanisms, while the Darcy-Forchheimer model is used for the porous medium. The governing partial differential equations are transformed into the ordinary differential equations using the similarity transformations. The accuracy of the method is observed by a comparison with other works reduced to a common case. Many results are tabulated and representative set is displayed graphically to illustrate the influence of the various parameters of interest on different profiles. Extensive numerical investigations show that the flow field, temperature, concentration and nanoparticle volume fraction shapes are significantly influenced by magnetic parameter, regular Lewis number, Brownian motion parameter, thermophoresis parameter, regular buoyancy ratio parameter and Biot number. Heat and mass transfer rates are significantly affected by the level of the applied magnetic field and the convective heat coefficient.

The unsteady magnetohydrodynamic free convective flow due to an oscillating porous flat plate in a rotating frame of reference are studied when thermal radiation and Hall currents are taken into consideration. The entire system rotates with a uniform angular velocity about an axis normal to the plate. A uniform magnetic field is applied along the normal to the plate directed into the fluid region. Copper, alumina and titania water nanofluids are considered. The governing equations are solved analytically by employing the small perturbation approximation. The numerical results for fluid velocity and temperature are presented graphically for the pertinent parameters and discussed in details.

This paper investigates the laminar boundary layer flow of nanofluid induced by a radially stretching sheet. Nanofluid model exhibiting Brownian motion and thermophoresis is used. Series solutions for a reduced system of nonlinear ordinary differential equations are obtained by homotopy analysis method (HAM). Comparative study between the HAM solutions and previously published numerical results shows an excellent agreement. Velocity, temperature and mass fraction are displayed for various values of parameters. The local skin friction coefficient, the local Nusselt number and the local Sherwood number are computed. It is observed that the presence of nanoparticles enhances the thermal conductivity of base fluid. It is found that the convective heat transfer coefficient (Nusselt number) is decreased with an increase in concentration of nanoparticles whereas Sherwood number increases when concentration of nanoparticles in the base fluid is increased.

The peristaltic flow of nanofluids under the effect of slip conditions was theoretically investigated. The mathematical model was governed by a system of linear and non-linear partial differential equations with prescribed boundary conditions. Then, the exact solutions were successfully obtained and reported for the first time in the present work. These exact solutions were then used for studying the effects of the slip, thermophoresis, Brownian motion parameters and many others on the pressure rise, velocity profiles, temperature distribution, nanoparticle concentration and pressure gradient. In addition, it is proved that the obtained exact solutions are reduced to the literature results in the special cases.

In the general case, it was found that on comparing the current solutions with the approximate ones obtained using the homotopy perturbation method in literature, remarkable differences have been detected for behaviour of the pressure rise, velocity profiles, temperature distribution, nanoparticle concentration and finally the pressure gradient. An example of these differences is about effect of the Brownian motion parameter on the velocity profile; where it was shown in this paper that the small values of this parameter have not a significant effect on the velocity, while this situation was completely different in the published work. Many other significant differences have been also discussed. Therefore, these observed differences recommend the necessity of including the convergence issue when applying the homotopy perturbation method or any other series solution method to solve a physical model. In conclusion. The current results may be considered as a base for any future analysis and/or comparisons.

The unsteady laminar boundary layer flow of nanofluid caused by a linearly stretching sheet is considered. Transport equations contain the simultaneous effects of Brownian motion and thermophoretic diffusion of nanoparticles. The relevant partial differential equations are non-dimensionalized and transformed into similar forms by using appropriate similarity transformations. The uniformly valid explicit expressions of velocity, temperature and nanoparticles volume fraction are derived. Convergence of the series solutions is carefully analyzed. It is observed that an increase in the strength of Brownian motion effect rises the temperature appreciably. However rate of heat transfer and nanoparticles concentration at the sheet is reduced when Brownian motion effect intensifies. It is also found that the temperature and nanoparticles concentration are increasing functions of the unsteady parameter.

A numerical study is carried out to study the effects of variable fluid properties on the boundary layer flow and heat transfer of a nanofluid at a flat sheet. The effects of Brownian motion, thermophoresis and viscous dissipation due to frictional heating are also considered. The temperature-dependent variable fluid properties, namely, the fluid viscosity and the thermal conductivity are assumed to vary, respectively, as an inverse function and a linear function of temperature. Using a similarity transformation, the governing non-linear partial differential equations of the model problem are transformed into coupled non-linear ordinary differential equations and these equations are solved numerically by Keller-Box method. Velocity, temperature, and nanoparticles volume fraction profiles are presented and analyzed for several sets of values of the governing parameters; namely, variable fluid viscosity, variable thermal conductivity, Brownaian motion, thermophoresis and plate-velocity parameters with changes in the Prandtl and Schmidt numbers. It is observed that there is an increase in the skin friction in the upstream movement of the plate: But quite the opposite is true in the downstream movement of the plate. Also, the effect of the Schmidt number and the Brownian motion parameter is to reduce the Sherwood number, where as the effect of thermophoresis parameter is to enhance it.