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The aim of this paper is to formulate and analyze thermophoresis effects on mixed convection heat and mass transfer from vertical surfaces embedded in a saturated porous media with variable wall temperature and concentration.

The governing partial differential equations (continuity, momentum, energy, and mass transfer) are written for the vertical surface with variable temperature and mass concentration. Then they are transformed using a set of non‐similarity parameters into dimensionless form and solved using Keller‐box method.

Many results are obtained and a representative set is displaced graphically to illustrate the influence of the various physical parameters. It is found that the increasing of thermophoresis constant or temperature differences enhances heat transfer rates from vertical surfaces and increases wall thermophoresis velocities; this is due to favorable temperature gradients or buoyancy forces. It is also found that the effect of thermophoresis phenomena is more pronounced near pure natural convection heat transfer limit, because this phenomenon is directly temperature gradient‐ or buoyancy forces‐dependent.

The predicted results are restricted only to porous media with small pores due to the adoption of Darcy law as a force balance.

The paper explains the different effect of thermophoresis on forced, natural and mixed convection heat, and mass transfer problems. It is one of the first works that formulates and describes this phenomenon in a porous media. The results of this research are important for scientific researches and design engineers.

The purpose of this paper is to study thermophoresis particle deposition and thermal radiation interaction on natural convection heat and mass transfer by steady boundary layer flow over an isothermal vertical flat plate embedded in a fluid saturated porous medium.

The governing partial differential equations are transformed into non‐similar form by using special transformation and then the resulting partial differential equations are solved numerically by using an implicit finite difference method.

Different results are obtained and displaced graphically to explain the effect of various physical parameters on the wall thermophoresis deposition velocity and concentration profiles. It is found that the increasing of thermal radiation parameter or dimensionless temperature ratio heats the fluid and decreases temperature gradients near permeable wall, which increases local Nusselt numbers and decreases wall thermophoresis velocities. It is also found that the effect of power indices of either temperatures or concentration enhances both local Nusselt numbers and wall thermophoresis velocities. Comparison with previously published work in the limits shows excellent agreement.

The paper presents useful conclusions based on graphical results obtained from studying numerical solutions for thermophoresis‐thermal radiation heat and mass transfer interaction by steady, laminar boundary layer over a vertical flat plate embedded in a porous medium.

Nanofluids are widely used in heat transfer phenomena owing to the higher rate of heat removal as compared to their base fluids. Nanoparticle’s motion in nanofluids is analysed by slip mechanisms that consider physical properties, which can be found in literature. It is assumed that among few, only Brownian motion and thermophoresis affect the slip mechanism to produce a relative velocity between the nanoparticles and the base fluid. The purpose of this paper is to study the effects of Brownian motion and thermophoresis in a square cavity by considering it pure fluid as well as porous cavity.

A finite element method is used to solve the flow porous equations together with the heat transfer equation and the mass transfer equation numerically. The heat and mass transfer equations were modified to take into consideration the Brownian motion as well as the thermophoresis effect.

A negligible amount of Brownian motion and thermophoresis effect has been found by considering 1 to 3 Vol.% of aluminium oxide as nanoparticles suspended in base fluid of water.

This study has provided an interesting insight into the importance of Brownian motion as well as the thermophoresis effect in heat enhancement.

The present study is believed to be an interesting and original contribution on nanofluid thermal behaviours.

The purpose of this paper is to do a comprehensive study on the unsteady general three-dimensional stagnation-point flow and heat transfer of a nanofluid by Buongiorno’s model.

In this study, the convective transport equations include the effects of Brownian motion and thermophoresis. By introducing new similarity transformations for velocity, temperature and nanoparticle volume fraction, the basic equations governing the flow, heat and mass transfer are reduced into highly non-linear ordinary differential equations. The resulting non-linear system has been solved both analytically and numerically.

The analysis shows that velocity, temperature and nanoparticle concentration profiles in the respective boundary layers depend on five parameters, namely unsteadiness parameter A, Brownian motion parameter Nb, thermophoresis parameter Nt, Prandtl number Pr and Lewis number Le. It is found that the thermal boundary layer thickens with a rise in both of the Brownian motion and the thermophoresis effects. Therefore, similar to the earlier reported results, the Nusselt number decreases as the Brownian motion and thermophoresis effects become stronger. A correlation for the Nusselt number has been developed based on a regression analysis of the data. This correlation predicts the numerical results with a maximum error of 9 percent for a usual domain of the physical parameters.

The stagnation point flow toward a wavy cylinder (with nodal and saddle stagnation points) that a little attention has been given to it up to now. The examination of unsteadiness effect on the general three-dimensional stagnation-point flow. The application of an interesting and global model (Boungiorno’s model) for the nanofluid that incorporates the effects of Brownian motion and thermophoresis. The study of the effects of Brownian motion and thermophoresis on the nanofluid flow, heat and mass transfer characteristics. The prediction of correlation for the Nusselt number based on a regression analysis of the data. General speaking, we can tell the problem with this geometry, characteristics, the applied model, and comprehensive results, was Not studied and analyzed in literature up to now.

The purpose of this paper is to study natural convection heat transfer and fluid flow in a square cavity filled with CuO-water nanofluid.

The entire length of the bottom wall of the cavity is covered by two pairs of heat source-sink, whereas the other walls are insulated. The governing equations of fluid flow are discretized using a finite volume method with a collocated grid arrangement. The coupling between velocity and pressure is solved using the SIMPLEC and the Rhie and Chow interpolation is used to avoid the checker-board solutions for the pressure.

The numerical results are reported for the effect of Rayleigh number, solid volume fraction and both presence and absence of thermophoresis and Brownian motion effects. The numerical results show an improvement in heat transfer rate for the whole range of Rayleigh numbers when Brownian and thermophoresis effects are considered. Furthermore, an increase in the Rayleigh number and nanoparticle volume fraction in both cases – when Brownian and thermophoresis effects are neglected or considered – has an excellent influence on heat transfer of nanofluids.

The area of nanofluids is very original.

The purpose of this paper is to study about the natural convection of water-based nanofluid in a partially open trapezoidal cavity under the influence of Brownian diffusion and thermophoresis.

Governing equations formulated in dimensionless stream function – vorticity variables – have been solved by finite difference method with a homemade code C++. Effects of Rayleigh number (Ra = 50-1,000), Lewis number (Le = 10-1,000), buoyancy-ratio parameter (Nr = 0.1-5.0), Brownian motion parameter (Nb = 0.1, 1.0) and thermophoresis parameter (Nt = 0.1, 1.0) on nanofluid flow and heat transfer have been studied.

It is found that high values of Rayleigh and Lewis numbers lead to the homogenization of nanoparticles distributions. For high values of Nt and Nb, heating is more essential and the cavity average temperature rises.

The originality of this work is to analyze natural convection in an open-sided trapezoidal cavity with Brownian diffusion and thermophoresis.

The purpose of this paper is to analyze heat and mass transport mechanism of unsteady MHD thin film flow of aluminium–copper/water hybrid nanofluid influenced by thermophoresis, Brownian motion and radiation.

The authors initially altered the time dependent set of mathematical equations into dimensionless form of equations by using apposite transmutations. These equations are further solved numerically by deploying Runge–Kutta method along with shooting technique.

Plots and tables for skin friction coefficient, Nusselt number, Sherwood number along with velocity, temperature and concentration profiles against pertinent non-dimensional parameters are revealed. The study imparts that aluminium–copper hybrid nanoparticles facilitate higher heat transfer rate compared to mono nanoparticles. It is noteworthy to disclose that an uplift in thermophoresis and Brownian parameter depreciates heat transfer rate, while concentration profiles boost with an increase in thermophoretic parameter.

The current study targets to investigate heat transfer characteristics of an unsteady thin film radiative flow of water-based aluminium and copper hybrid nanofluid. The high thermal and electrical conductivities, low density and corrosion resistant features of aluminium and copper with their wide range of industrial applications like power generation, telecommunication, automobile manufacturing, mordants in leather tanning, etc., have prompted us to instil these particles in the present study.

The present study has many practical implications in the industrial and manufacturing processes working on the phenomena like heat transfer, magnetohydrodynamics, thermal radiation, nanofluids, hybrid nanofluids with special reference to aluminium and copper particles.

To the best extent of the authors’ belief so far no attempt is made to inspect the flow, thermal and mass transfer of water-based hybridized aluminium and copper nanoparticles with Brownian motion and thermophoresis.

The purpose of present communication is to analyze Darcy–Forchheimer flow of viscous nanofluid by curved stretchable surface. Flow in porous medium is characterized by Darcy–Forchheimer relation. Brownian diffusion and thermophoresis are considered. Convective heat and mass boundary conditions are also used at the curved stretchable surface.

The resulting nonlinear system is solved through shooting technique.

Skin friction coefficient is enhanced for larger porosity parameter and inertia coefficient while reverse trend is noticed for curvature parameter. Local Nusselt number is enhanced for higher Prandtl number and thermal Biot number, whereas the opposite trend is seen via curvature parameter, porosity parameter, inertia coefficient, thermophoresis parameter and Brownian motion parameter. Local Sherwood number is enhanced for Schmidt number, Brownian motion parameter and concentration Biot number, while reverse trend is noticed for curvature parameter, porosity parameter, inertia coefficient and thermophoresis parameter.

To the best of author’s knowledge, no such consideration has been given in the literature yet.

This study aims to investigate the micropolar fluid flow through a moving flat plate containing CoFe₂O₄-TiO₂ hybrid nanoparticles with the substantial influence of thermophoresis particle deposition and viscous dissipation.

The partial differential equations are converted to the similarity equations of a particular form through the similarity variables. Numerical outcomes are computed by applying the built-in program bvp4c in MATLAB. The process of flow, heat and mass transfers phenomena are examined for several physical aspects such as the hybrid nanoparticles, micropolar parameter, the thermophoresis particle deposition and the viscous dissipation.

The friction factor, heat and mass transfer rates are higher with an increment of 1.4%, 2.2% and 1.4%, respectively, in the presence of the hybrid nanoparticles (with 2% volume fraction). However, they are declined because of the rise of the micropolar parameter. The imposition of viscous dissipation reduces the heat transfer rate, significantly. Meanwhile, thermophoresis particle deposition boosts the mass transfer. Multiple solutions are developed for a certain range of physical parameters. Lastly, the first solution is shown to be stable and reliable physically.

As far as the authors have concerned, no work on thermophoresis particle deposition of hybrid nanoparticles on micropolar flow through a moving flat plate with viscous dissipation effect has been reported in the literature. Most importantly, this current study reported the stability analysis of the non-unique solutions and, therefore, fills the gap of the study and contributes to new outcomes in this particular problem.

The purpose of this paper is to present a numerical investigation of the transient two-dimensional natural convective boundary-layer flow of a nanofluid past an isothermal vertical plate by incorporating the effects of Brownian motion and thermophoresis in the mathematical model.

The problem is formulated using the Oberbeck–Boussinesq and the standard boundary-layer approximations. The governing coupled non-linear partial differential equations for conservation of mass, momentum, thermal energy and nanoparticle volume fraction have been solved by using an efficient implicit finite-difference scheme of the Crank–Nicolson type, which is stable and convergent. Numerical computations are performed and the results for velocity, temperature and nanoparticle volume fraction are presented in graphs at different values of system parameters such as Brownian motion parameter, thermophoresis parameter, buoyancy ratio parameter, Prandtl number, Lewis number and dimensionless time. The results for local and average skin-friction and Nusselt number are also presented graphically and discussed thoroughly.

It is found that the velocity, temperature and nanoparticle volume fraction profiles enhance with respect to time and attain steady-state values as time progresses. The local Nusselt number is found to decrease with increasing thermophoresis parameter, while it increases slightly with increasing Brownian motion parameter. To validate the present numerical results, the steady-state local Nusselt number results for the limiting case of a regular fluid have been compared with the existing well-known results at different Prandtl numbers, and the results are found to be in an excellent agreement.

The present analysis is limited to the transient laminar natural convection flow of a nanofluid past an isothermal semi-infinite vertical plate in the absence of viscous dissipation and thermal radiation. The unsteady natural convection flow of a nanofluid will be investigated for various physical conditions in a future work.

Unsteady flow devices offer potential performance improvements as compared with their steady-state counterparts, and the flow fields in the unsteady flow devices are typically transient in nature. The present study provides very useful information for heat transfer engineers to understand the heat transfer enhancement with the nanofluid flows. The present results have immediate relevance in cooling technologies.

The present research work is relatively original and illustrates the transient nature of the natural convective nanofluid boundary-layer flow in the presence of Brownian motion and thermophoresis.