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In many energy and combustion applications, particles experience large temperature gradients, which can affect the coagulation process due to thermophoresis. This study presents a rigorous theory of thermophoretically modified Brownian coagulation in the entire particle size regime. The theoretical derivations are based on the kinetic theory for the free-molecular regime and the harmonic mean method for the transition regime. The coagulation kernels in different size regimes can be expressed as the basic Brownian coagulation kernel times an enhancement factor. The enhancement factor represents the coagulation rate enhancement induced by thermophoresis and is a function of specific dimensionless numbers. Based on the enhancement factor, the thermophoretic enhancement effects on particle coagulation are further analyzed under a wide range of gas and particle conditions. The results show that thermophoretic enhancement effects are ignorable in the free-molecular regime, but need to be considered in the continuum regime and the transition regime. In addition, the enhancement effects increase significantly with increase of gas temperature and temperature gradient while decrease with increase of gas pressure. The present study can improve understanding of thermophoretic effects on Brownian coagulation in the entire size regime and provide a useful tool to calculate the coagulation rates in presence of thermophoresis.

Aerosol particles suspended in a diluted gas with non-uniform temperature distribution are expected to experience a thermophoretic force. In theoretical treatment of thermophoresis, it is usually assumed that the particle temperature is equal to the surrounding gas temperature. However, this might not always be the case. In some particular applications, the particle temperature can significantly differ from the gas temperature. In the present paper, we theoretically investigate the effect of the particle temperature on the thermophoresis of nanoparticles in the free molecule regime. Theoretical formulas for the thermophoretic force and thermophoretic velocity are obtained based on the gas kinetic theory. As examples, a spherical Ag nanoparticle suspended in a dilute He gas is considered, and the Rudyak–Krasnolutski potential is employed to model the gas–particle interaction. It is found that the influence of the particle temperature on the thermophoresis of nanoparticles can be significant. With increasing particle size, the error due to the equal gas–particle temperature assumption can be neglected.

The viscoelastic fluid flow over a non linear stretching porous sheet is considered in this analysis. The case of suction and injection is also discuused. The effects of thermophoresis and brownain motion with viscous dissipation is taken into account. Under the flow assumptions, boundary layer approximation applied on the mathematical model and developed the partial differential equations. The similarity variable is applied on the partial differential equations which converted into ordinary differential equations. The dimensionless system further solved through numerical technique bvp4c method. The results of the current problem achieved after solving the problem which presented through graphs and table. Influence of numerous physical parameters on velocity function, concentration function and temperature function are manipulated through graphs. The tabular results are schemed to display skin friction and local Nusselt number. The higher values of N b which enhanced the velocity profile but declines velocity profile due to higher values of Ec . The thermal thickness enhances due larger values of γ and Pr but thermal thickness declines due to higher values of Ec and N t . The concentration profile enhances due to enhancing the values of Ec and Pr but concentration profile declines due to increasing the values of N t . Nusselt number enhaced due to higher values of Ec , S c and N t but declines of higher values of Pr .

Carbonaceous deposits formed in fuel-film cooling act as a thermally insulating barrier, but the mechanisms for their formation at the high enthalpies and pressures of rocket engines are not well studied. Various spectroscopic techniques were used to investigate the microscopic chemical and physical structure of carbonaceous deposits formed in a small-scale H2O2-kerosene rocket combustor. Where possible, spectra were measured as a function of axial distance from the fuel-film injector to identify changes in deposit structure as the fuel-film boundary layer vaporized and mixed with the hot core flow. Two carbon layers were found, a tenacious dense layer beneath a porous soot layer. Gas chromatography mass spectrometry was used to determine condensable polycyclic aromatic hydrocarbon (PAH) concentrations in the dense and soot deposits. Raman spectroscopy measured planar coherence length and other indicators of structural order while x-ray photoelectron spectroscopy determined carbon hybridization, surface oxidation, and deposit impurities. All three of these spectroscopy techniques showed independent signs of the dense layer forming via heterogeneous condensation of heavy PAHs, whereas the soot layer deposited via thermophoresis.

Nanofluids have already proven great potential in the thermal amplification of several manufacturing industries and have been widely employed in energy technologies in recent times. Therefore, the current framework investigates the characteristics of the Jeffrey nanofluid flow with the influence of activation energy and motile microorganisms over a sheet is considered. The influence of the magnetic field is another important physical parameter in the flow analysis and has been regarded in this review. The remarkable properties of nanofluid are demonstrated by thermophoresis and Brownian motion characteristics. Thermophoresis has relevance in mass transport processes in many higher temperature gradient operating systems. An appropriate similarities transformation is utilized make convenient to partial differential equation into ordinary differential equations. The well-known shooting tactic is utilized to estimate numerical outcomes of obtained ordinary system of flow. The governing dimensionless equations are integrated subject to the aid of the bvp4c scheme in the built-in software of MATLAB to find out the solution. The procedure assumed employing MATLAB. The characteristics of physical parameters against the velocity of fluid, temperature, concentration, and motile microorganisms are elaborated through graphs and tables. The dynamic physical declaration of attained results reveals that buoyancy ratio parameter and bioconvection Rayleigh number plays a vibrant role in the declining flow of Jeffery nanofluid while contrasting nature is analyzed for mixed convection parameter. The increasing behavior of temperature is observed for larger variations of heat sink/source parameters and thermophoresis parameter while it shows conflicting nature for increasing values of Prandtl number. The concentration of nanoparticle is reduced with the advanced values of Prandtl number, Lewis number and Brownian motion, perceived from the results. The mounting valuation of the Peclet number and Bioconvection Lewis number caused a reduction in motile microorganism concentration. The envisaged hypotheses could be beneficial for modifications to extrusion systems, bio-mimetic systems, efficient energy generation, bio-molecules and improved production systems.

A numerical study is accomplished to analyze the Brownian motion of micro-polar hydromagnetic nano-fluid over a stretched surface with thermophoresis and thermal radiation effect. The dimensional equations of the proposed model are reformed by acceptable similarity transformations and the numerical approach using MATLAB in connection with bvp4c is implemented to solve the final equations. The convergence, salient features of the pertinent parameters on micro-polar nano-fluid flow for velocity, angular velocity, temperature and concentration are emphasized on the plots. We also discuss the effect on Skin friction, Nusselt number and Sherwood number and displayed via Tables. The influence of Grashof number is remarkable throughout the study. Outcomes obtained indicate that for enlarged values of micro-polar parameter, the angular velocity is raised. The validity and accuracy of the present model have been checked and found adequate agreement. The importance of such analysis over stretched sheet have numerous manufacturing, industrial and engineering applications in plastic sheets extrusion, polymer extraction, blowing of glass, manufacture of paper and rubber sheets.

In this manuscript a study of the unsteady magneto-hydrodynamic mixed convection Jeffrey fluid flow over an inclined permeable moving plate in presence of thermal radiation, heat generation, radiation, aligned magnetic field, thermophoresis effect and homogenous chemical reaction, subjected to variable suction has been undertaken. The governing equations are solved by using a regular perturbation technique. The expressions for the distributions of velocity, temperature and species concentration are obtained. With the aid of these, the expressions for skin friction, Nusselt number and Sherwood number also have been derived. The influences of various physical Parameters involved in the problem on the above mentioned quantities are discussed with the help of graphs and tables. From the significant findings, it has been found that the velocity increases with an increase in Jeffrey fluid in the presence of permeability. It shows reverse effects in the case of magnetic parameter, radiation parameter and chemical reaction parameter. Comparisons with previously published work performed and the results are found to be in the excellent agreement.

In this paper, the effect of viscous dissipation and Joule heating on magneto-hydrodynamic (MHD) nanofluid flow towards a vertical sinusoidal wavy surface has been investigated numerically. The wavy surface is considered to be heated with uniform flux. Effects of radiation and nanoparticle volume concentration are also considered. The fluid flow over the surface geometry is designed through the highly non-linear partial differential equations, and governing equations must be transformed into a dimensionless non-similar system of equations to get a better solution by applying the highly efficient well-known Keller-box scheme. In this regard, the numerical solutions are obtained for a concentration of mass fluid, velocity profile, temperature profile, local skin friction number coefficient, local Nusselt number, and Sherwood number shown graphically. Connected parameters involve in the modelled problem such as heat generation parameter, radiative conduction flow parameter, the amplitude of the waviness of the surface, Prandtl number, Brownian motion parameter, Buoyancy ratio, magnetic parameter, Thermophoresis parameter and the Lewis number for momentum, temperature profile, and concentration profile of the fluid through different graphs are explained. In this process, certain iterations were performed for different step sizes in the direction of η = 0 . 01 and x = 0 . 005 , respectively. So, keeping this in view to get the more accurate results with all parameters considered, 0 ≤ M ≤ 1 , 0 ≤ R d ≤ 1 , 0 ≤ Q ≤ 1 , 0 . 01 ≤ α ≤ 0 . 04 , 0 ≤ L e ≤ 40 , 0 . 1 ≤ N b ≤ 0 . 7 , 0 . 1 ≤ N t ≤ 0 . 7 , 0 . 1 ≤ N r ≤ 0 . 7 , 0 . 1 ≤ E c ≤ 0 . 4 , 0 . 1 ≤ J ≤ 0 . 4 , . The obtained results illustrate that Nusselt number, Sherwood number, and velocity profile decreases respectively with the increase of waviness amplitude, but skin friction changes its behaviour to wavier. Also, temperature and concentration profiles increase respectively with enhancing the value of the amplitude of waviness. Finally, it is noted that Nusselt number and Sherwood number decrease for higher values of the Brownian parameter. But after a specific value of x, the trend of Sherwood number increases. However, the behaviour of skin friction decreases, but in a precise manner, it remains almost unchanged.

Numeric simulations are performed for comparative study of MHD rotational flow of hybrid nanofluids (MoS2–Ag/ethylene glycol–water (50%–50%)) and nanofluids (Ag/ethylene glycol–water (50%–50%)) over a horizontally elongated plane sheet. The principal objective is concerned with the enhancement of thermal transportation. The three dimensional formulation governing the conservation of momentum, mass, energy, and concentration is transmuted into two-dimensional partial differentiation by employing similarity transforms. The resulting set of equations (PDEs) is then solved by variational finite element procedure coded in Matlab script. The intensive computational run is carried out for suitable ranges of the particular quantities of influence. Both the primary and secondary velocities and temperature are greater in values for the hybrid phase than that of nano phase but nanoparticle concentration is smaller for the hybrid phase. The increased values of parameters for thermophoresis, Brownian motion, shape factor, and volume fraction of ϕ 2 made significant improvement in temperature of the two phases of nano liquids. Results are also computed for the coefficients of skin friction (x, y-directions), Nusselt number, and Sherwood number. The present findings manifest reasonable comparison to their existing counterparts.

The developing interest in thermal engineering and nano-technology presented the idea of nanoparticles with enhanced thermal mechanisms and attained convinced novel applications in heat transportation systems, solar energy, micromechanics, air conditioning systems, cooling and heating facilities, bio-medical applications like cancer treatment, artificial heart surgery etc. Beside this, the phenomenon of entropy generation is another fascinating research area which used to improve the assessment of heat transportation with optimized procedure. The entropy generation in bio-convective flow of Eyring-Powell nanofluid over permeable stretched surface of cylinder is scrutinized in this research. The effects of magnetic field, thermal radiation and viscous dissipation are considered. Concentration communication is obtained in view of activation energy associated with binary chemical reaction. Furthermore, properties of Brownian diffusion and thermophoresis of nanoparticles are accounted. Suspended nanoparticles are made stable through mutual effects of bioconvection and buoyancy forces. Total entropy generation rate is modeled through second thermodynamics law. The governing flow model is obtained with the help of boundary layer approximations. Dimensional model is transformed into non-dimensional one by transformations and then tackled by Newton built-in shooting procedure. Impact of different influential variables on entropy generation, velocity, temperature, concentration, Bejan number and motile density of microorganisms are analyzed via graphs. Numerical values of skin friction coefficient, density, Sherwood and Nusselt numbers are tabulated and examined. Major findings are listed at the end. Since this analysis is based on theoretical flow assumptions, therefore the flow parameters specify the constant range like 0.0 ⩽ Γ ⩽ 6.0 , 1.0 ⩽ γ ⩽ 4.0 , 0.1 ⩽ H a ⩽ 1.2 , 0.2 ⩽ λ ⩽ 0.6 , 0.2 ⩽ R d ⩽ 1.7 , 0.1 ⩽ K ⩽ 3.0 , 0.1 ⩽ Pr ⩽ 4.5 , 0.3 ⩽ N b ⩽ 1.8 , 0.2 ⩽ N t ⩽ 1.7 , 0.0 ⩽ E 1 ⩽ 1.0 , 1.0 ⩽ E c ⩽ 4.0 , 0.2 ⩽ σ ⩽ 1.4 , 0.4 ⩽ S c ⩽ 1.7 , 1.0 ⩽ δ ⩽ 2.5 , 0.5 ⩽ P e ⩽ 2.0 , 0.5 ⩽ Ω ° ⩽ 2.0 , 0.1 ⩽ B r ⩽ 1.0 , 0.1 ⩽ L b ⩽ 1.5 . The results show that fluid velocity decreases with curvature parameter and porosity constant. A decreasing variation in nanofluid temperature due to curvature parameter is examined. The entropy generation and Bejan number decline Brinkman number, motile density diffusion parameter and mass concentration diffusion variable. Furthermore, it is observed that skin friction coefficient increases with curvature parameter and porosity constant.

Here magnetohydrodynamic (MHD) bioconvection Walter-B nanofluid flow due to stretched sheet is considered. Significance of motile gyrotactic microorganisms and melting phenomena are scrutinized. Energy equation is assisted with Brownian motion, thermal radiation, Joule heating and thermophoresis diffusion. Diffusion-thermo and thermo diffusion impacts are also considered. Suitable transforms are used to reduce related expressions into ordinary differential systems. Non-linear ODEs are tackled through homotopy analysis method (HAM). Convergent series solutions are obtained. Impacts of melting parameter, Dufour number, bioconvection Lewis number, radiation parameter, Soret number, Brownian motion parameter, magnetic parameter, Prandtl number and Peclet number are addressed. Velocity intensifies with increase in melting and viscoelastic parameters while it decay against magnetic parameter. Decay in temperature in noted against melting and radiation parameters while it enhances for higher Brownian motion parameter and Prandtl number. Concentration increases against Brownian motion parameter while it decays against higher melting parameter. Microorganism field increases with Peclet number while it decays with bioconvection Lewis number.

Numerical investigation of the three-dimensional flow and heat transfer of 36 nm Al2O3–H2O nanoliquid over a wedge surface is carried out by utilizing the modified Buongiorno model (MBM). The boundary layer approximation is assumed to be valid. The thermophysical properties of Al2O3–H2O nanoliquid are deliberated in the study by modeling them through the use of correlations based on experimental data. The thermal boundary layer equation comprises the Brownian motion and thermo-migration effects caused by nanoparticles. Zero mass flux boundary condition is also accounted. Optimization of the heat transfer rate of nanoliquid is made using the Response surface methodology (RSM). Two stream functions are used to derive the similarity transformations and they are employed to arrive nonlinear ordinary differential system from the governing partial differential system, and then the subsequent nonlinear problem is treated numerically using the Finite Element Method (FEM). The heat flow features are scrutinized using two-dimensional, surface, and streamline plots. The results of flow due to a moving wedge in the same direction of free stream velocity are compared with those of a moving wedge in the opposite direction of free stream velocity. The thermal layer and nanoparticles volume fraction layers are enlarged due to the chaotic movement of nanoparticles. The thermophoresis mechanism improves the thermal layer at the wedge surface. The high level of pressure gradient factor, high level of shear-to-strain rate and low level of Lewis number, were found to be the optimal operating condition that would maximize the heat transfer rate.

This theoretical and computational analysis reveals the thermal characteristics for the flow of Williamson Sutterby nanofluid through the Darcy–Forchheimer sponge medium over an extending surface. The Cattaneo-Christov heat flux, radiation heat flux, and convective boundary are taken into account. Moreover the impacts of bio-convection of self-propelled micro-organisms and electromagnetic fields are considered. The amalgamation of Sutterby–Williamson with slight homogeneous diffusion of nano-particles and micro-organism exposes the novelty of this work. The principal objective pertains to avoid the possible settling of nano-entities to given effective heat transportation. The physical problem is formulated in the form of partial differential equations which are then altered to a system of ordinary differential equations. The influences of emerged parameters on the physical quantities of interest are evaluated by utilizing the Runge–Kutta method. The parameters of Sutterby fluid, inertia friction and porosity decelerate the flow but the parameters of electric field and mixed convection are directly responsible to promote the velocity of fluids. It is noted that fluid accelerates for Williamson model λ ≠ 0 . The parameter of the electric field, Brownian motion, radiations, thermophoresis, and Eckert number raises the temperature of the fluid. The heat transfer rate at the surface recedes in magnitude for larger b (Cattaneo-Christov diffusion). The motile density number increases with the upsurge values of L b , P e and Ω . Two sets of graphical outputs are presented for Williamson fluid parameter λ ( λ = 0 , λ = 0 . 5 ) .

Bioconvection for rotational flow is conceived to provide stability to improved thermal transportation for nanofluid flow over Riga plate design. Nanoparticles are considered due to their unusual characteristics like extraordinary thermal conductivity, which are significant in heat exchangers, advanced nanotechnology, electronics, and material sciences. Cattaneo–Christov theory and activation energy are incorporated. The unsteady three dimensional partially differentiate formulation is simplified in the form of two independent coordinates ( ζ , η ) . For steady-state solution ( ζ = 1 ) , Galerkin discretization in used to employ finite element simulation in MATLAB environment. The unsteady parameter rotating parameter, thermophoresis, and Brownian motion parameter escalated the nanofluid temperature field. Modified electromagnetic parameter M H accelerated the primary flow velocity and activation energy augmented the volume fraction of nanoparticles in the boundary layer region. The larger modified Hartmaan number M H reduces the coefficient of skin friction in primary direction but the magnitude of coefficient of skin friction in secondary direction is augmented. The local Nusselt number R e x 1 / 2 N u x is directly proportional to M H but it is inversely related to β . The higher values of Brownian motion and thermophoresis boosted the temperature. An excellent accord among the present and previously existing solutions is establishes the validity of the current findings.

This research work highlight the newly developed concept of Rosseland approximation and gyrotactic microorganisms in steady, two-dimensional, incompressible flow of Walter's-B nanofluid (non-Newtonian) over a stretchable surface of sheet. Buongiorno nanofluid model, which represents seven important slip mechanisms (i.e., Brownian motion, inertia, Magnus impact, thermophoresis, diffusion-phoresis, gravity and fluid drainage) is utilized in the mathematical modeling of governing expressions. In this research work, only two important factors of seven slip mechanisms (Brownian diffusion, thermophoresis) are studied and the rest of neglected. Furthermore, the Rosseland approximation and heat generation/absorption effects are used in the modeling of the energy equation. The behavior of thermal and solutal stratification effects are addressed at the stretched boundary of the sheet. The nonlinear dimensional flow expressions lead to dimensionless ordinary equations through appropriate similarity transformations. The total residual error is calculated through Homotopy Analysis Method (HAM) for the momentum, temperature, concentration and motile density. The influences of important flow parameters of the governing flow equations are discussed and plotted graphically. The obtained results are compared with fruitful and valuable research in the literature and found very good agreement with them. Over obtained outcomes highlight that the velocity field, declined versus higher estimations of Weissenberg number. It is also remarked that the temperature and concentration fields have contrast impact subject to thermophoresis parameter. The physical quantities like skin friction coefficient, motile density, concentration and Nusselt number are discussed physically via various flow parameters.

This problem considers the instabilities, which can occur when a horizontal sparsely filled nanofluid porous layer is saturated with a binary mixture with fast chemical reaction. The upper and lower plates are heated isothermally. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. The modified Darcy equation that includes the time derivative term is used to model the momentum equation. The energy equation includes cross and regular diffusion conditions. In conjunction with the Brownian motion, the nanoparticle fraction becomes stratified; hence the viscosity and the conductivity are stratified. The nanofluid is assumed to be diluted and this enables the porous medium to be treated as a weakly heterogeneous medium with variation, in the vertical direction, of conductivity and viscosity. The oscillatory and stationary convection are evaluated analytically. The reaction rate is supposed to be greater than the diffusion rate. We find that both a stationary instability and an oscillatory instability can occur as the first bifurcation, depending on the sign and the value of the heat of reaction. The execution of the porous parameter, Prandtl number, Soret parameter, Dufour parameter, Lewis number, solutal Rayleigh number, chemical reaction and the modified diffusivity ratio parameters is implemented pictorially. It is found that the presence of nanoparticles the critical Rayleigh number is increased and helps to stabilize the system. The presence of chemical reaction parameter destabilizes the system. The results of previously-published work are also obtained as special cases of the general solution.

The current research article explores the bio-convection magnetized Carreau-nanofluid flow utilizing activation energy and magnetic dipole. Magnetic nanomaterials have significant effect on the magnetized behavior of ferrofluids which leads to robust the changes in fluid viscosity and thermophysical characteristics. Darcy-Forchheimer flow model is also present. Thermal conductivity of nanofluid is dependent on the temperature. Brownian motion and thermophoresis aspects are also accounted. The PDEs are reduced to set of ODEs via appropriate functions. Numerical solutions are achieved with the help of the famous bvp4c solver a built-in function in MATLAB. Numerous flow parameters are graphically and numerically presented with physical significance. The proposed findings could be useful for applications of extrusion systems, organic compounds, improved energy generation and improved manufacturing techniques.

Natural convection flow and heat transfer characteristics of Buongiorno's mathematical model nanofluid flow inside square cavity with isothermal conditions on both side walls and adiabatic conditions on top and bottom walls is studied numerically in this analysis. Finite difference method is employed to solve the governing partial differential equations formulated in stream function, nanoparticle volume fraction and temperature numerically. The results are presented in the form of streamlines, isotherms, isoconcentrations, local Nusselt number and Sherwood number for various values of influenced parameters, such as, Rayleigh number (100 ≤ Ra ≤ 300), buoyancy ratio parameter (0.1 ≤ Nr ≤ 0.9), Lewis number (1.0 ≤ Le ≤ 10), thermophoresis number (0.1 ≤ Nt ≤ 0.5), Brownian motion number (0.1 ≤ Nb ≤ 0.9) and radiation number (0.1 ≤ R ≤ 0.9) and are represented through graphs. It is detected that the values of rate of heat transfer elevates with rising values of Rayleigh number (Ra).

In this paper, a theoretical model is developed for the deliberation of time-dependent 3D magnetohydrodynamics second-grade (SG) nanofluid with radiation effects past an unsteady stretched surface. The work is carried out by involving the physical effects of thermophoresis and Brownian movement. To find the desired solutions, more practical boundary conditions are introduced familiar as the Robin and zero-mass diffusion conditions. The derived model of partial differential form is reframed into the structure of ordinary differential expressions by implementing the dimensionless variables which are reported numerically through the Runge-Kutta-Fehlberg (RKF) scheme with shooting procedure. Numerically determined skin-friction data and energy transportation rate are also addressed with appropriate analysis. Solutions are performed for nanofluid velocity, concentration of nanoparticles and nanofluid temperature. The analysis of dissimilar parametric values on liquid velocity, concentration of nanoparticles and temperature is reported using plots and tabulated data. The presence of Hartman and Biot numbers demonstrated the higher temperature curves. The nanoparticle concentration and temperature distributions reduced for larger unsteady parameter values. A comparison is also provided for limiting cases to authenticate our obtained results.

Background: Emerging applications in nanomaterials processing are increasingly featuring\ud multiple physical phenomena including magnetic body forces, chemical reactions and high\ud temperature behavior. Stimulated by developing a deeper insight of nanoscale fluid dynamics in\ud such manufacturing systems, in the current article, we study the magnetic nanofluid dynamics\ud along a nonlinear porous stretching sheet with Arrhenius chemical kinetics and wall transpiration.\ud Appropriate similarity transformations are employed to simplify the governing flow problem.\ud Methods: The emerging momentum, thermal energy and nanoparticle concentration ordinary\ud differential conservation equations are solved numerically with a hybrid technique combining\ud Successive Linearization and Chebyshev Spectral Collocation. A parametric study of the impacts\ud of magnetic parameter, porous media parameter, Brownian motion parameter, parameters for\ud thermophoresis, radiation, Arrhenius function, suction/injection (transpiration) and nonlinear\ud stretching in addition to Schmidt number on velocity, temperature and nanoparticle (concentration)\ud distribution is conducted. A detail numerical comparison is presented with different numerical and \ud 2\ud analytical techniques as a specific case of the current investigation.\ud Findings: Increasing chemical reaction constant parameter significantly decreases nanoparticle\ud concentration magnitudes and results in a thickening of the nanoparticle concentration boundary\ud layer. Enhancing the values of activation energy parameter significantly increases the nanoparticle\ud concentration magnitudes. Increasing thermophoresis parameter elevates both temperature and\ud nanoparticle concentration. Increasing radiation parameter increases temperature and thermal\ud boundary layer thickness. Enlarging Brownian motion parameter (smaller nanoparticles) and\ud Schmidt number both depress the nanoparticle concentration.

This paper presents the bioconvection flow of Casson nanofluid by a rotating disk with gyrotactic microorganisms, multiple slips, and thermal radiation. The non-Newtonian fluid (Casson fluid) in the existence of thermophoretic and Brownian diffusion effects is incorporated. The research of rotatable disk has practical applications in numerous processes like rotating machinery, lubrication, computer storage devices, oceanography, crystal growth, etc. Using appropriate variables, a system of strongly non-linear PDEs is translated into a system of ODEs. To construct a numerical solution of nonlinear flow equations, the bvp4c technique is used. The impacts of various numbers on velocities, thermal field, the volumetric concentration of nanoparticles, and microorganism field are investigated. From the results, it can be noticed that radial velocity rises by varying values of mixed convection parameter. An increase in bioconvection Rayleigh number causes decays in velocity fields. A larger Prandtl number declines the temperature distribution. Furthermore, an increment in the estimations of thermophoresis parameter increases the thermal and solutal field of species. Significant heat transfer is acquired with higher radiation effects. Slip parameter for temperature and concentration decreased the thermal and solutal profiles. The microorganisms field is declined with greater microorganisms slip parameter.

The cornerstone of a new method to modeling nanofluids is particle movement. There are three primary forms of nanofluid simulation. The single-phase modeling with nanofluid properties (SPU), the multiphase fluid with discrete particle (MDP) and the discrete element method with particle (DEMP). In this study, by comparing three approaches to modeling the natural convection heat transfer in nanofluid, this method's (discrete element method with particle) ability was examined to predict properties and processes, such as heat transfer and friction factor. With a high range of Rayleigh numbers (0 ≤ Ra ≤ 100, 000), this method tries to model the natural convection process in a conical cavity filled with Al2O3 nanofluid and nanoparticles. For modeling the MDP and thermophoresis effect (DEMP), a unique code with a novel grid generation algorithm was developed. The results demonstrated that the initial conditions determined the accuracy of the three methods for modeling the final parameters of natural convection within the conical cavity, the accuracy of the used equations, the procedure through which the boundary conditions were applied and the Rayleigh number. Accordingly, the DEMP and MDP methods were recommended for low Rayleigh numbers and those beyond 5500, respectively.

The current study is concerned with the investigation of the combined effects of thermophoretic motion and thermal radiation on steady, viscous, incompressible and two-dimensional mixed convection flow of optically dense grey fluid. The governing equations are made dimensionless by using suitable dimensionless variables and further are transformed into a convenient form for a numerical algorithm. The transformed flow model is integrated by using an efficient finite difference methd. Effects of the assorted parameters involved in the flow model such as Prandtl number, mixed convection parameter, modified mixed convection parameter, Schmidt number, radiation parameter, thermophoresis parameter and thermophoretic coefficient on velocity field, temperature distribution, and mass concentration are demonstrated graphically. The numerical results of skin friction, the heat transfer rate and the mass transfer rate under the effect of pertinent parameters are displayed in tabular form.

The nanoparticles proved a motivating research area in the fourth generation of the world due to their extensive use in science and infrastructure, such as vehicle cooling, higher heat transfer rates in microchips, food manufacturing, biotechnology, biochemistry, transportation, metrology and nuclear reactors. Dispersing the nanoparticles within base fluid is a newly approach for implementations of heat transfer and biomedicine/bioengineering. The current determination is committed to explore the features of bioconvection in Carreau nanofluid flow under the influence of various thermal consequences. The flow is originated by a stretched cylinder. The characteristics of Cattaneo-Christov heat and mass flux are applied to examine the heat/mass transportation of nanofluid. The effects of thermal radiation and activation energy are also considered. The consequences of Brownian movement and thermophoresis features are analyzed by incorporating Buongiorno’s nanofluid model. The governing partial differential equations are transmuted into the structure of nonlinear ordinary differential equations by introducing suitable transformation. The shooting technique is used to achieve the numerical simulations of nonlinear system. The physical impacts of prominent parameters on velocity, temperature distribution, concentration field and microorganisms profile are examined and captured graphically. The numerical outcomes against various flow quantities are also presented in tabular form. The results convey that a higher temperature profile is observed with larger values of thermal Biot number, exponential base sink parameter and thermal relaxation parameter while a decrement in temperature is noticed with increasing mixed convection parameter. The concentration profile shows an increasing trend with mass concentration parameter and concentration relaxation parameter. Moreover, the microorganism field decline with Peclet number and bioconvection Lewis number.

Here we scrutinize magnetohydrodynamic mixed convection flow of viscous nanoliquid by a stretching cylinder. Heat flux, Joule heating and dissipation effect are examined in heat equation. Thermophoresis and Brownian diffusion behaviors are accounted. Entropy generation investigation is addressed. A first order chemical reaction is also accounted. Nonlinear PDEs are converted to ordinary differential system employing similarity variables. To develop a convergent solution we implemented ND-solve method. Variation of various sundry variables on Bejan number, velocity, entropy rate, concentration and temperature are graphically deliberated. Variation of various influential variables on Nusselt and Sherwood numbers are graphically examined. Larger slip parameter reduces the velocity field. Temperature is improved against the larger radiation variable. Concentration has a reverse trend against thermophoresis and Brownian motion variables. Radiation and magnetic variables have similar behavior on entropy optimization. Bejan number is reduced versus larger Brinkman number.

International audience; This work studies the influence of phoretic motion (thermophoresis and diffusiophoresis) of fine alumina particles ("smoke") produced during the combustion of aluminum. Direct numerical simulations on a single aluminum droplet burning in a quiescent environment suggest that thermophoresis is the main mechanism driving smoke back to the aluminum surface, hence a major contributor to the oxide lobe development. The presence of this lobe is found to distort the flowfield, which favors hot and smoke-rich regions closer to the lobe, thereby enhancing thermophoresis. This combination of aerodynamic and thermophoretic effects leads to a mass rate of deposited smoke which is consistent with experimental data. A simplified model, deduced from simulation results, is able to predict the size of the final oxide residue in good agreement with measurements. This study supports that aluminum oxide present on the burning aluminum particle is largely due to material formed in the flame, subsequently desposited by thermophoresis.

Linear stability analysis was applied to the onset of penetrative convective motion under the changeable gravity field in an asymmetric arrangement of porous bed saturated by a nanofluid. It also integrates, along with thermophoresis, the influence of Brownian motion, and using the Galerkin processes, the eigenvalue problem is solved. Four different kinds of linear and nonlinear gravity force variation functions are considered. The expression of the eigenvalue number has been derived, and the effects of changeable linear and nonlinear gravity, heat source parameter, and parameters arises from the asymmetric arrangement of a porous bed on the Rayleigh number have been presented graphically.

This article provides insight into the study of hydromagnetic flow of a chemically reacting water based nanofluid of Copper (Cu), Silver (Ag) and Ferrous Ferric Oxide (Fe 3 O 4 ) nanoparticles over a stretching permeable sheet with heat generation, nanoparticle volume fraction, Soret number, Eckert number and porosity. The governing system of PDEs is reduced to nonlinear ODEs by the tool of similarity transformations, and is solved by fourth order Runge–Kutta scheme with the shooting method via MATLAB. Output of the nanofluid velocity, temperature and concentration for the involved material parameters as well as the correlated engineering physical quantities like coefficient of skin friction and rate of heat transfer are demonstrated via single plots of three nanopartciles and Tables. Heat generation and Soret number for three different nanoparticles play a significant role throughout the investigation. Greater nanoparticle volume fraction decreases the liquid velocity, while a non-decline situation has occurred for liquid temperature. Furthermore, the velocity always overshoots for Fe 3 O 4 -water nanofluid, followed by Cu and Ag–water nanofluid, whereas the reverse performance is observed for the temperature profiles. An excellent validation of this model is accomplished. The adopted numerical scheme of RK-Shooting is stable and convergent in the same domain throughout the numerical calculations. The proposed investigation shows the importance of three nanoparticles in medical industry and biocompatibility engineering. Also, these nanoparticles can be used in physics, engineering, space technology, high temperature and cooling process, medicines, biosensors, paints, cosmetics, conductive coatings and medical devices.

In this paper, the groundwork of some thermophysical properties of higher-order chemical processing and dissipation of viscous on nanofluid along with a continuously stretching porous sheet is taken. The porous medium is considered with two space coordinates, laminar, time-invariant, MHD incompressible Newtonian nanofluid. The equations are framed to govern the fluid flow as coupled equations involving nonlinear partial derivatives. The impacts of electric and magnetic fields on nanofluid with viscous dissipation in the presence of higher-order chemical reaction, analyzing conservation of momentum and energy, is the novelty of the problem. The level of raising thermal conductivity and the output of transferring the heat on nanofluid is observed. Finally, the governing equations involving partial derivatives have complied with nonlinear ordinary differential equations. The transformations are subjected to the similarity variable used to solve these equations. Approximate solutions are obtained using a numerical method of the Runge-Kutta-Felburg method with shooting technique. The effects of emerging parameters K r , E r , λ , N t , δ , N b are porous, electric, mixed convection, thermophoresis, chemical process and, Brownian motion, and non-dimensional numbers such as Hartmann, Prandtl, Schmidt, and Eckert are extensively explained. The electrically conducting nanofluid flow for velocity fluid, temperature fluid and, nanoparticles concentration volume fraction fluid with transferring heat, Nusselt, and transferring mass, Sherwood number are examined with graphical representation. The Lorentz resistive force due to the applied strength of electric develops the thickness of boundary layers of momentum and thermal regions. This helps to cool the electronic systems and radiators. The dimensionless Nusselt number diminishes with various values of thermophoresis and Brownian motion parameters as a dependent function of Hartmann, electric number, and homogeneous chemical reaction parameter.

In this study, we have discussed thermophoresis particle deposition effects under the action of both pressure and buoyant forces flow of magneto-Walter’s B nanofluid induced by a stretched surface. The Buongiorno nanofluid model is employed to analyze the dynamic impact of thermophoretic dispersion and Brownian motion. The effects of chemical reaction, Joule heating and non-linear radiation relations are also incorporated. The analysis has been performed in view of solutal and heat convective boundary constraints. The analytical technique namely homotopy analysis scheme followed to solve the resulting non-linear governing equations. The behavior of velocity, temperature and concentration profiles are observed graphically. The physical consequences for all physical parameters are justified. It is noted that heat thermal Biot number, thermophoretic constant and viscoelastic parameter increases the nanofluid temperature and concentration. A decaying concentration profile is noted for Schmidt number.

In the present work, a two-phase mathematical model of intravenous magnetic ( Fe 3 O 4 , Fe 2 O 3 or FePt) nanoparticle hyperthermia is considered for the treatment of muscle and prostate tumours. The blood-nanoparticle suspension is described as a non-Newtonian (Quemada) nanofluid, and the effects of thermophoresis and Brownian motion are considered. The blood vessel region is surrounded by a cancerous (muscle or prostate) tumour region, a non-cancerous (muscle or prostate) region and a fat layer. Heat is generated within these tissue regions using an external alternating magnetic field. A numerical solution of the problem is obtained using the mixed finite element method with P 2 - P 1 Taylor-Hood elements. Using this numerical solution, the influence of tumour size, blood vessel radius and haematocrit on blood flow, convective heat transfer, nanoparticle mass transport and blood pressure is examined.

This article explains the effect of magnetite nanofluid taking into account water as the base fluid, over a rotating disk in the presence of external magnetic field. The governing boundary layer equations of laminar and incompressible flow over rotating disk have been formulated under the simultaneous influence of thermophoresis and Brownian motion. Additionally, influence of chemical reaction between species, prompted by Arrhenius activation energy is considered into the model. The Von Karman’s transformations of the Navier-Stokes equations are used to obtain the non-dimensional forms of the governing equations, and then fourth order finite difference scheme is used to find the numerical solutions using Noumerov’s discretization. This technique has strong impact on the accuracy of finite difference scheme. The physical descriptions of fluid velocity, temperature, concentration profiles and quantities of practical interest such as skin friction coefficient, Nusselt number and Sherwood number are presented with the help of graphs. One of the significant findings of this analysis include that an intensification in the thermophoresis parameter causes a downfall in the rate of heat transfer at the surface of the disk whereas another significant outcomes of the present investigation is that there is a downfall in the behaviours of the species concentration with the strengthening of Arrhenius activation energy. The fluid flow due to rotating disks has application in mechanisms of thermal transformations for nuclear propulsion devices.

A description for magnetohydrodynamic effects on the transient rotational flow of Oldroyd-B nanofluids is considered. The temperature and concentration distributions are associated with Cattaneo-Christove double diffusion, Brownian motion, thermophoresis, Soret, and Dufour. The governing equations in the three-dimensional form are transmuted into dimensionless two-dimensional form with the implementation of suitable scaling transformations. The variational finite element procedure is harnessed and coded in Matlab script to obtain the numerical solution of the coupled non-linear partial differential problem. The varying patterns of velocities, skin friction coefficients, Nusselt number, Sherwood number, fluid temperature, and concentration functions are computed to reveal the physical nature of this study. It is observed that higher inputs of the parameters for magnetic force, Deborah number, rotational fluid, and cause to slow the primary as well as secondary velocities but they raise the temperature like thermophoresis and Brownian motion does. However, thermal relaxation parameter reduces the nanofluid temperature. The local heat transfer rate reduces against Nt, rotational, and Nb parameters, and it is higher for Prandtl number. The current FEM (finite element method) solutions have been approved widely with the recent published results, showing an excellent correlation. The examination has significant applications in the food industry and relevance to energy systems, biomedical, and modern technologies of aerospace systems.

This paper reports an analytical and numerical study of thermosolutal convection in a Brinkman horizontal porous layer saturated with a binary mixture, destabilized by submitting it either to both heat and mass fluxes ( a = 0 , with Soret effect) or only to a heat flux to recover the case of pure Soret effect ( a = 1 ). The analytical solution is derived based on the parallel flow approximation and validated numerically using a finite difference method. A good agreement is found between the predictions of the parallel flow approximation and the numerical results obtained by solving the full governing equations. The governing parameters of this study are the Lewis number, Le , the Rayleigh thermal, R T , the Soret number, Sr , and the buoyancy ratio, φ . The thresholds for the onset of convection and the generated heat and mass transfer rates are examined both in the presence ( a = 0 ) and in the absence ( a = 1 ) of an external mass flux. The influence of the governing parameters on the onset of motion and the resulting fluid flow, temperature and concentration fields is discussed. It was found that the onset of convection for a = 0 and positive values of Sr is delayed compared to the case a = 1 (pure Soret effect) and, for the latter case, the onset of convection is delayed compared to a = 0 and negative values of the Soret number.

This paper reports on the numerical investigation of assisting and opposing mixed convection flows and heat transfer performance of nanofluids in a cubical enclosure under turbulent conditions. The enclosure side wall consists of a partially heated heat source and the nanofluid is pumped into the enclosure through the inlet port. A three-dimensional, turbulent two-phase mixture model is developed to investigate the thermal performance of nanofluids by considering the effects of Brownian motion and thermophoresis. The base fluid is considered as water into which different nanoparticles such as aluminum oxide (Al2O3), copper (Cu) and silver (Ag) are dispersed for different particle volume fractions (ϕ) varied from 0% to 4%. The Richardson number (Ri) and Reynolds number (Re) are varied between 0.01 ≤ Ri ≤ 10 and 3.1 × 103 ≤ Re ≤ 105 and the Grashof number (Gr) is maintained at a constant value of 108. The nanoparticles are considered as monodispersed spherical particles with a uniform diameter of 100 nm. The results indicate that the temperature and turbulent kinetic energy distributions for opposing flows are higher than assisting flows. It also found that the average heat transfer rate increases with decrease in Richardson number and increase in volume fraction for both assisting and opposing flows. The effects of Brownian motion and thermophoresis are more significant in assisting flows than opposing flows.

In this present article, the influence of MHD flow of a non-Newtonian Casson nanoliquid conveying gyrotactic microorganisms over a moving wedge with thermal radiation, thermophoresis and Brownian motion are reported. The basis boundary layer equations of momentum, energy, nanoparticle concentration and density of motile microorganisms, which are nonlinear PDE’s, are transformed into a nonlinear ODE’s through a set of similarity variables. The renovated equations are solved analytically using the Homotopy Perturbation Method (HPM). The analytical results are validated with the numerical results using shooting method also compared with available literature as a limiting case. The impact of salient parameters of the Casson nanoliquid are illustrated with several graphs and tables.

A rotating MHD flow of electrically conducting Oldroyd-B fluid through non-Darcy porous medium across a stretching/shrinking surface is investigated in purview of bioconvection effects. The fluid velocity field, temperature field, concentration of nano materials and that of bio microorganisms have been simultaneously formulated in the form of coupled nonlinear partial differential equations. The quest for improved thermal performance in practicable situations has motivated to utilize some real fluids like Oldroyd-B nanofluids with support from bioconvection effect to stabilize nanoparticle fraction. The zero normal flux of the nanoparticles and convective boundary condition are imposed. The substitution of appropriate similarity functions has resulted a set of ordinary differential equations (ODE). The sparse system of transformed ODEs was simulated numerically with bvp4c solver facilitated by MATLAB commercial software. Computations were executed for specific parameters to perceive the response of velocity, temperature, concentration of nanoparticles and concentration of microorganisms, skin friction coefficient and local heat transfer rate with variation of influential parameters. It is detected that fluid temperature profile rises when Deborah number, Biot number, parameters for buoyancy ratio, porosity, rotation, magnetic field strength, thermophoresis diffusion are enhanced.

A numerical investigation is presented to show the effects of thermo-diffusion on the hydromagnetic stagnation-point flow of Sisko nanofluid over a linearly stretching sheet by considering viscous dissipation and Joule heating with suction/injection. A suitable set of similarity transformations is considered in order to convert the basic partial differential equations into a set of coupled nonlinear ordinary differential equations. An efficient numerical method along with shooting technique is involved in order to solve the reduced governing basic equations. The influences of several emerging physical parameters of Sisko nanofluid on the profiles of velocity, temperature, solutal concentration, nanoparticle volume fraction, skin-friction coefficient, Nusselt number, and Sherwood number have been studied and analyzed in detail through graphs and tables. It is found that the skin friction coefficient decays more rapidly against the material parameter of Sisko fluid. Also, it is noticed that the Brownian motion and thermophoresis parameter have the reverse effects on Sisko nanofluid Sherwood number. It is analyzed that the Nusselt number decreases with an increase in the values of thermophoresis parameter, Brownian motion parameter, and Eckert number, while its value increases with material parameter and Dufour solutal Lewis number. It is observed that the Sherwood number has ascending behavior for thermophoresis and, Brownian motion parameters, whereas nanofluid Sherwood number gets amplified with a hike in the nano Lewis number parameter for all the values of Brownian motion parameter.

In a suspension of tangent hyperbolic bionanofluid keeping both nanoparticles and motile microorganisms, the thermobioconvective boundary layer flow was studied through an exponentially stretching surface utilizing response surface methodology (RSM). The constructed model of a tangent hyperbolic nanofluid in boundary layer flow is studied with implications of thermophoresis and Brownian motion. Condition of zero normal flux of nanomaterials is added at the surface to scatter the nanomaterials from the plate surface. The rate of heat transfer is analyzed using convective boundary condition. Numerical shooting strategy with Runge-Kutta scheme is to follow intently behind the similarity transformation to solve the system of governing equations. It is assumed that the output variables of interest are dependent on the governing input parameters. The sensitivity analysis is additionally introduced. It is discovered that the sensitivity of local Nusselt number increments by expanding Lewis and thermophoresis number while the highest non-dimensional Nusselt number appears close to the significant level for the thermophoresis and low level for the Brownian motion variable. Additionally, it is demonstrated that the average maximum mean thickness of motile microorganism appears at the highest level of Brownian motion and thermophoresis number and thermophoresis and Lewis numbers. The results would provide initial guidance for potential manufacture of devices.

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 dust accumulation on a solar panel may be one of the most serious losses in the energy yield of photovoltaic systems, especially in the desert environment. The transport of airborne dust particles onto solar panels is governed by various physical processes and described by particle dynamics in a fluid. Here we investigate how the temperature gradient between the solar panel and the ambient air affects the dust accumulation. This so-called thermophoresis, which drives particles moving from the hot solar panel to the cold air, is examined using the boundary layer approximation for non-isothermal laminar airflow along the solar panel. We find that a strong thermophoretic force is built up at the entry region of the solar panel. With the velocity and temperature profiles of non-isothermal laminar airflow, we simulate Lagrangian particle trajectories. We demonstrate that for submicrometer-size particles, the thermophoresis builds a region free of the dust accumulation on the solar panel. It is shown that the thermophoretic effect may decrease the accumulation of submicrometer-size particles on the solar panel whose temperature is a few tens of degrees higher than the ambient air in the daytime.

The cooling channel can capture both the droplets and particles in the industrial waste gas. The flow field and particle motion in the corrugated cooling channel were numerically simulated and the capture mechanism was investigated. Submicron and micron particles are captured under the actions of thermophoresis and inertia, coupling with turbulence, respectively. The capture probability shows a high distribution for particles near the wall, and a low distribution for particles in the central area. The deposition probability density distribution of particles on the channel wall changes periodically along the flow direction. The overall capture efficiency decreases first and then increases with increasing particle size for submicron particles, in the range of 40% to 50%, and it keeps increasing with particle size for micron particles, reaching 77% for 10 μm particles. A corrugated cooling channel performs well in capturing particulate matter with the help of thermophoresis and inertia coupled with turbulence.

Current investigation elaborates the impacts of Brownian motion and thermophoretic force on electrically conducting Prandtl-Eyring nanofluid flow yielded by stretched surface. Buongiorno nano-model is used to trace the heat and mass transfer characteristics in the flow regime. The mathematical formulation concerning to the adopted physical parameters is modeled in the form of complex partial differential structure. Boundary layer theory is obliged to reduce non-linearity of subsequent equations by truncating higher order terms. To facilitate the computation process, the governing problem in partial differential form is converted into dimensionless ordinary differential expressions. Numerical solution for attained boundary value problem is procured by R-K-Fehlberg methodology. The consequences of flow governing parameters on interested physical quantities (momentum, heat, concentration) are depicted in graphical manner while tabular representation is used to demonstrate the variations in wall drag coefficient, wall thermal flux and particles concentration flux. The computed results show that the presence of magnetic field is not favorable for fluid momentum, albeit, both Brownian motion and thermophoresis phenomenon surges the thermal energy of fluid. Besides these, concentration profile increases versus Brownian motion while thermophoresis phenomenon has reverse impacts on it. Surface drag force enriched by magnifying the magnetic field intensity, furthermore, the surface heat flux shows reduction versus Brownian motion and thermophoresis parameters. In addition, the surface mass flux shows increasing trend versus all governing parameters.

Linear and weakly nonlinear stability analyses of Brinkman–Benard convection in water with a dilute concentration of single-walled carbon nanotubes ( S W C N T s ) is studied analytically in the paper. The thermophysical properties of water- S W C N T s nanoliquid and water- S W C N T s nanoliquid-saturated high-porosity medium are calculated using phenomenological relations and the mixture theory. The five-mode Lorenz model is derived under the assumptions of Boussinesq approximation, small-scale convective motion, weak thermophoresis and porous friction. Using the method of multiscales the five-mode Lorenz model is reduced to a cubic Ginzburg–Landau equation the solution of which helps in quantifying the unsteady heat transport. The effects of S W C N T s , porous parameter and viscosity ratio on onset of convection and the heat transport are documented. Results of the unicellular-Brinkman–Benard convection problem is extracted from those of the multicellular-Brinkman–Benard convection problem. Three different enclosures are considered in the study and their influence on the onset of convection and the heat transport are compared. The single-phase model which incorporates nanoliquid thermophysical properties is shown to be a limiting case of the model proposed in the present paper.

In this present communication, we proposed the mathematical model for non-Newtonian liquid (Ree-Eyring model) towards a stretched surface. The Buongiorno model is used in the modelling of flow problem subject to magnetohydrodynamics, entropy generation, nonlinear thermal radiation, homogeneous-heterogeneous reactions, Brownian motion, Joule heating and thermophoresis diffusion. Convective boundary conditions are implemented at the stretched surface. With the assistance of rheological expression of Ree-Eyring model, we construct the governing flow expressions. The energy equation is developed via Brownian diffusion, nonlinear thermal radiation, Joule heating and thermophoresis. Two different types of species (homogeneous-heterogeneous reactions) are considered for the analysis of mass transport. The flow is solely induced due to a nonlinear stretched surface. The electrical conducting and chemically reactive fluid is considered for the analysis towards a stretched surface. The appropriate transformations are implemented to transform governing PDEs into a set of coupled ODEs. Shooting method has been applied to get the solutions of obtained highly non-linear ODEs. The thermodynamics second law is employed to model and calculate total entropy rate. Impact of influential variables on the flow field (velocity), gradient of velocity, temperature, gradient of temperature, concentration and entropy generation rate are studied. The engineering quantities like velocity, temperature gradients are numerical discussed in tabular form. Furthermore, the entropy rate and Bejan number show contrast impact versus larger Brinkman number.

Here flow behavior of stratified Williamson nanofluid over porous surface of stretching cylinder is examined. The concept of gyrotactic miscroorganisms is implemented to control the random movement of suspended nanoparticles. Effects of magnetic field is accounted. Further chemical reaction with Arrhenius activation energy is considered for the modeling of concentration equation. Brownian motion and thermophoresis effects are further considered. The flow model is obtained by employing the boundary layer assumptions. Appropriate transformations are used to reduced the dimensional system into non-dimensional ones. NDSolve code in MATHEMATICA software is used to tackle the obtained non-dimensional flow expressions. Behavior of velocity, mass concentration, temperature and motile microorganisms versus involved variables is examined graphically. The engineering curiosity like skin friction coefficient, heat and mass transfer rates (Nusselt and Sherwood numbers) and density number are computed and analyzed. Important observations are highlighted at the end.

There are still debates regarding effects driven by temperature gradient (∇T) during unconventional sintering, such as field assistant sintering technology and selective laser sintering, with existent analytical tools insufficient to reflect in-process features. In this work we propose a brand-new approach to investigate ∇T-driven effects during unconventional sintering via non-isothermal phase-field simulations. 2D and 3D simulations on sintering of yttria-stabilized zirconia micro-particles not only reproduce well features investigated by analytical models, but also discover new in-process phenomena, e.g. ∇T-induced coalescence of identical particles, and reveal potential competitions between thermophoresis and Ficktian diffusion.

The present study aims to investigate the impact of nanoparticle migration due to Brownian motion and thermophoresis on Ag-MgO/Water hybrid nanofluid natural convection. An enclosure with sinusoidal wavy walls is considered for this investigation; right and cold walls of this enclosure are in constant temperature while the upper and bottom walls are insulated. This simulation follows Buongiorno's mathematical model; Brownian and thermophoresis diffusion of Ag occurs in MgO-Water nanofluid while the diffusion of MgO happens in Ag-water nanofluid. The result indicates that Nu number increments up to 11% by increasing thermophoresis diffusion for both nanoparticles. Also, increasing Brownian diffusion of Ag augments nanoparticle concentration on hot wall, while the accumulation of nanoparticles unifies by incrementing Brownian diffusion of MgO.

In high-temperature gas-cooled reactors, graphite dust particles within the reactor core and the heat transfer equipment experience large temperature gradients. Under such conditions, thermophoresis may play an important role in determining aerosol evolution. This study presents a theoretical and numerical analysis of the thermophoretic effects on aerosol coagulation within these reactors. The coagulation rates for Brownian versus thermophoretic coagulation are calculated and compared for various temperature gradients. Our results show that thermophoretic coagulation dominates over Brownian coagulation for large temperature gradients. We defined an enhancement factor to evaluate the role of thermophoretic coagulation under various reactor conditions. The enhancement factor increased dramatically with increasing temperature gradient, decreasing pressure and increasing particle diameter, but was not very sensitive to temperature change. The time evolution of the particle size distribution related to combined Brownian and thermophoretic coagulation was simulated using a log-skew-normal method of moments. The simulation results indicate that aerosol evolution can be strongly accelerated by thermophoretic coagulation under large temperature gradients.

Peristaltic flow of non-Newtonian nano fluid through a non-uniform surface has been investigated in this paper. The fluid motion along the wall of the surface is caused by the sinusoidal wave traveling with constant speed. The governing equations are converted into cylindrical coordinate system and assuming low Reynolds number and long wave length partial differential equations are simplified. Analytically solutions of the problem are obtained by utilizing the homotopy perturbation method (HPM). In order to insight the impact of embedded parameters on temperature, concentration and velocity some graphs are plotted for different peristaltic waves. At the end, some observations were made from the graphical presentation that velocity, pressure rise and nano particle concentration are increasing function of thermophoresis parameter Nt while temperature and frictional forces show opposite trend.