Effect Of Convection Research Articles

Convection-effect-enhanced micro thermoelectric module directly heated by catalytic combustion

A novel micro thermoelectric power generation module directly heated with localized catalytic combustion has been proposed. Arrays of high-temperature plate-type thermoelectric generator (TEG) elements, of which the hot junction is directly heated by exothermic catalytic oxidation of hydrocarbon fuel, are designed. Heat losses caused by heat conduction through the TEG element are recovered by the convection effect with unburned mixture flows. Catalyst layers were prepared by impregnating Pd or Pt catalyst on porous Al2O3 supports coated by using the suspension plasma spray. A series of one-dimensional numerical analyses considering the catalytic combustion in addition to the conductive and convective heat transfer were made to examine the temperature field of the TEG module and to investigate how various parameters affect the convection effect. A model combustor using quartz plates was developed, which mimics the temperature fields of the SiGe TEG module. It is found that, with the aid of the convection effect, temperature differences over 700 °C between the hot/cold junctions can be achieved, while the unreacted fuel concentration is less than 10%. Finally, 40 mm-long Ni-based alloy TEG elements have been fabricated using laser cutting and spot welding techniques, which ensure high operating temperatures and reliability. In a power generation demonstration using butane as a fuel, an open circuit voltage of 0.64 V has been obtained from a single TEG module with a peak output power and a power generation density as high as 10.1 mW and 10.8 kW/m3, respectively.

Influence of chemical reaction and thermal convective condition on the heat and mass transport in boundary layer flow over a magneto-radiated wedge with cross diffusion

The analysis of Boundary Layer Flows (BLFs) is of huge significance due to its large-scale applications most likely in aerodynamics, manufacturing of vehicles front faces and many other advanced research areas. Therefore, this study is organized by exercising significant important engineering parameters like surface convection, radiative flux, MHD and thermal and mass diffusion. The developed model embedded the concrete effects of velocity slip, diffusion convection, thermal radiations, magnetic field, chemical reaction and thermo-mass gradients. Later, the numerical treatment is made via RK scheme and the model efficaciously is tackled with desired accuracy. Then the results are decorated over the region (wedge and sheet) and deeply examined. It is observed that slippery surface of wedge and sheet [Formula: see text] enlarges the fluid movement rapidly. The directed magnetic field ([Formula: see text] controlled the fluid motion which can be beneficial from industrial view point. Further, temperature and concentration distributions upsurge by strengthening convective heat ([Formula: see text] and diffusion convection processes. Moreover, skin friction and Nusselt number improved against the parameters under consideration.

Lattice Boltzmann finite-difference-based model for fully nonlinear electrohydrodynamic deformation of a liquid droplet.

Electric field-induced flows involving multiple fluid components with a range of different electrical properties are described by the coupled Taylor-Melcher leaky-dielectric model. We present a lattice Boltzmann (LB)-finite difference (FD) method-based hybrid framework to solve the complete Taylor-Melcher leaky-dielectric model considering the nonlinear surface charge convection effects. Unlike the existing LB-based models, we treat the interfacial discontinuities using direction-specific continuous gradients, which prevents the miscalculation arising due to volumetric gradients without directional derivatives, simultaneously maintaining the electroneutrality of the bulk. While fluid transport is recovered through the LB method using a multiple relaxation time (MRT) scheme, the FD method with a central difference scheme is applied to discretize the charge transport equationat the interface, in addition to the electric field governing equationsin the bulk and at the interface. We apply the developed numerical model to study the different regimes of droplet deformation due to an external electric field. Similar to the existing analytical and other numerical models, excluding the surface charge convection (SCC) term from the charge transport equation, the present methodology has shown excellent agreement with the existing literature. In addition, the effect of SCC in each of the regimes is analyzed. With the present numerical model, we observe a strong presence of SCC in the oblate deformation regime, contrary to the weak effect on prolate deformations. We further discuss the reason behind such differences in the magnitude of nonlinearity induced by the SCC in all the regimes of deformation.

Mixed convection and thermal radiation effects on non-Newtonian nanofluid flow with peristalsis and Ohmic heating

Introduction: This investigation explores the heat and mass transfer properties of a non-Newtonian nanofluid containing graphene nano-powder and ethylene glycol during peristalsis. The rheological characteristics of the nanofluid are determined using the Carreau-Yasuda model, and various factors such as viscous dissipation, Lorentz force, Ohmic heating, and Hall effects are taken into account. Mixed convection and thermal radiation effects are also considered in the analysis, and the problem is mathematically described using the long wavelength and low Reynolds number approximations.Methods: The resulting nonlinear system is solved using numerical methods to obtain the solutions. The dominant effects of mixed convection and thermal radiation are given particular attention, while the influences of other parameters are discussed in relation to these dominant effects.Results and Discussion: The results demonstrate that increasing the Brinkman number, heat source, and thermal slip parameter leads to higher nanofluid temperatures. However, the heat transfer rate decreases with a higher Hall parameter. The velocity near the center of the channel increases for higher values of the concentration Grashof and Hall parameters. Furthermore, an increase in the Hall and Brownian motion parameters results in a higher concentration of nanoparticles. These findings have practical implications in various fields, including materials science, chemical engineering, and biomedical engineering.

Comparative thermal analysis of Nickel and Tantalum based hybrid nanofluid using constant proportional Caputo and Atangana–Baleanu operators with time-controlled condition

The main purpose of this research work is to examine how the immersion of nickel and tantalum nanoparticles in water affects the flow and thermal properties of water, with the aim of anticipating potential enhancements. The investigation pertains to the flow behavior of a hybrid nanofluid over an extensively elongated vertical surface under the influence of a condition that varies with time. The phenomenon of heat transfer transpires due to the combined effects of natural convection and thermal radiation. An in-depth comprehension of flow patterns and heat transfer over a vertical surface is essential for optimizing heat exchange processes, designing efficient cooling systems, and enhancing overall energy efficiency. It also helps in climate modeling, weather prediction, and the improvement of the performance of solar thermal systems. The basic model is generalized using Atangana–Baleanu (AB) and constant proportional Caputo (CPC) operators. This generalization leads to the development of two different fractional settings for which separate mathematical analyses are performed. With the help of some new unit-independent quantities, exact solutions for both fractional frameworks are computed using the Laplace transform. For the elucidation of changes in velocity and thermal fields due to different physical effects, several graphical and tabular representations are produced. The outcomes illustrate that the incorporation of uniform concentrations of nickel and tantalum nanoparticles in water leads to raising its thermal efficacy up to 40.39%. The nanoparticles consisting of lamina shapes are observed to produce the maximum amelioration in thermal features. The CPC model based profiles of flow and temperature fields are lower than those procured via the AB model. As a result of nanoparticles’ intrinsic characteristics, the insignificant heat transfer capability of water is greatly amplified, leading to a notable boost in its thermal performance. Thus, the examined hybrid nanofluid can be used as a viable alternative to water in various engineering applications.

Synergistic improvement of Seebeck coefficient and power density of an aqueous thermocell using natural convection for low-grade heat utilization

As a cutting-edge heat-to-electricity technology, thermocells possess the advantages of large Seebeck coefficient, low cost and high scalability, indicating a great prospect in low-grade heat recovery. Previous studies about thermocells focus on the electrode materials and electrolytes, but the internal heat and mass transfer mechanisms are less considered, especially the effects of natural convection on the performance of aqueous thermocells. The purpose of this study is to explore the influence of natural convection on the performance of aqueous thermocells. Firstly, a fundamental test bench for aqueous ferricyanide/ferrocyanide ([Fe(CN)6]3−/4-) thermocells using Pt as electrodes is designed and established. Based on the self-developed test bench, then the performance of thermocells related to electrode orientation θ are experimentally investigated under typical thermal boundary conditions, including the constant electrode temperature, and constant heating and cooling temperature. The experimental results under constant electrode temperature show that natural convection can significantly reduce the internal resistance of thermocells. The thermocell obtains the largest |Se| of 1.75 mV/K at θ = 90°, which is 19.9% higher than that of θ = 0°. Meanwhile, the thermocell achieves the maximum power density of 0.211 W/m2 at θ = 135°, which is 744% higher than that of θ = 0°. In addition, by combining the experimental results under constant heating and cooling temperature, it can be observed that natural convection can also sharply reduce the thermal resistance and then lead to a smaller temperature difference of two electrodes. Overall, natural convection has trade-off effects on internal resistance and thermal resistance, and the proper use of natural convection can succeed in prominent improvement of Seebeck coefficient and power density for thermocells.

Entrainment Makes Pollution More Likely to Weaken Deep Convective Updrafts Than Invigorate Them

AbstractAre the results of aerosol invigoration studies that neglect entrainment valid for diluted deep convective clouds? We address this question by applying an entraining parcel model to soundings from tropical and midlatitude convective environments, wherein pollution is assumed to increase parcel condensate retention. Invigoration of 5%–10% and <2% is possible in undiluted tropical and midlatitude parcels respectively when freezing is rapid. This occurs because the positive buoyancy contribution from freezing is larger than the negative buoyancy contribution from condensate loading, leading to positive net condensate contribution to buoyancy. However, aerosol‐induced weakening is more likely when realistic entrainment rates occur because water losses from entrainment more substantially reduce the latent heating relative to the loading contribution. This leads to larger net negative buoyancy contribution from condensates in polluted than in clean entraining parcels. Our results demonstrate that accounting for entrainment is critical in conceptual models of aerosol indirect effects in deep convection.

Convection of snow: when and why does it happen?

Convection of water vapor in snowpacks is supposed to have a major impact on snow density and microstructure profiles with strong implications for the thermal regime and snow stability. However, the process has never been directly measured and only recently been simulated for idealized conditions. The analysis suggests that natural convection is not likely to happen in typical horizontally homogeneous polar or Alpine snow covers. This paper studies the potential impact of heterogeneity induced, e.g., by shrubs on convection of water vapor. We find that natural convection triggered by buoyancy occurs even with sub-critical Rayleigh number as low as 5 due to heterogeneity in snow density. This leads to complementing contributions of diffusive and convective flux divergence on snow density changes. The combined effect of diffusion and convection helps to generate the often-observed low density foot and high-density top of, e.g., Arctic snowpacks. The strongest effect of convection is not for very thin or thick snow covers but for snow covers with thickness in the order of 0.5 m. This scale facilitates the development of convection cells. Further work should address the additional effects of sub-snow lateral temperature variations and assess the effect of convective vapor fluxes on snow microstructure.

Impact of innovative fin design on phase change material melting for thermal energy storage system

Among the main types of thermal energy storage, Latent Heat Storage System has the advantage of storing heat due to the small size temperature changes and the high stored energy. These benefits are attributed to the use of phase change material. However, the most fantastic drawback of phase change material is its low thermal conductivity. To further improve the weak thermal conductivity of phase change materials in these systems, it is imperative to look at different thermal enhancement techniques. This paper discusses two primary thermal improvement approaches for a rectangular thermal energy storage system: fins and a metal matrix. The rectangular enclosure is filled with n-eicosane and the bottom surface is subjected to constant heat flux. A comparison of different structures (conventional fin, Y-shaped fin, and honeycomb structure) with a reference configuration that contains only pure phase change material was discussed. The comparison between these structures is based on the charging time, which is one of the main objectives of this work. Outcomes reveal that adding a hexagonal Y-shaped fin and honeycomb structure can help improve the thermal performance of the latent heat thermal energy storage system and uniformize the temperature compared to the reference configuration. However, the effect of natural convection is not significant for the honeycomb structure but is crucial for the reference configuration. An optimization study was done to study the effect of varying the number of the hexagonal Y-shaped fin, PCM types, and heat flux. Results show that deploying four Y-shaped hexagonal fins is the best convincing argument in favor of improving the performance of the latent storage system. The RT44HC is the predominant coolant at lower heat fluxes, whereas the RT28HC was the best choice at higher heat fluxes. These results could be beneficial in applying phase change material to the thermal management of electronic components.

Revisiting Forced Convective Enhanced Heat Transfer in Offset-Strip-Fin Cores and Rationalized Performance Prediction Correlations

Abstract The development of rationalized correlations for Fanning friction factor f and Colburn j factor for forced air-flow convection in rectangular offset-strip-fin cores is presented in this study with 100 ≤ Re ≤ 8000. New experimental data and three-dimensional conjugate heat transfer computational simulations were acquired to understand the flow physics and heat transfer phenomena. The offset arrangement of the fins disrupts the fully-developed condition prevalent in plain fins to promote secondary flow and enhanced heat transfer, and this effect is found to be fundamentally scaled by offset length ratio λ(=ℓ/dh). Furthermore, because of the blunt surface edge or finite thickness of the offset fin, the flow stagnation and wake effects are integral parts to the secondary flow. The influence is found to be characterized by the thickness ratio ζ(=tRe/dh) as well as the rectangular flow cross section aspect ratio α(=s/hf). New models are proposed for f and j in both laminar and turbulent regimes based on the enhanced convection effects, which are represented by these scaling parameters, and are superimposed on the fully-developed condition in a rectangular channel. The correlations are thereby devised from the new sets of experimental data as well as that given in the open literature and thus cover a wide range of λ, ζ, and α. Because the transition from laminar to turbulent regimes is smooth and continuous, the general correlations of f and j are developed by asymptotic matching as single expressions and are shown to predict the extended dataset to within ±20%.

Effects of thermophoresis and mixed convection on Carreau fluid flow with gold nanoparticles

Effects of thermophoresis and mixed convection on Carreau fluid flow with gold nanoparticles

Analysis of nonlinear radiative peristaltic transport of hyperbolic tangent nanomaterial with chemical reaction

In this research, we examine the mixed convection effects for peristaltic motion of tangent hyperbolic nanoliquid. On flexible channel walls, partial slip characteristics are applied. Influences of Ohmic heating and viscous dissipation are analyzed in the modeling. We evaluated the transportation of heat under nonlinear thermal radiation. The governing problem is first reduced into dimensionless form by using suitable transformations and then numerical technique is employed for the solution after utilizing small Reynolds number and large wavelength assumption. The problem is simulated employing the so-called Buongiorno’s formulation which includes the features of thermophoresis and random motion. In addition, a mass transport scheme is developed with first-order chemical reactions. The roles of sundry variables on velocity, coefficient of heat transfer, temperature and concentration are examined graphically. Velocity and temperature show similar behavior against thermal Grashof parameter. Concentration decreases for larger thermophoresis parameter. Heat transfer coefficient and temperature have decreasing effects against radiation parameter.

Study of Stable Stratification in HiRJET Facility With Direct Numerical Simulation

Stable density stratification in a large enclosure could significantly hamper the effectiveness of natural convection cooling in pool-type liquid metal or gas-cooled advanced reactors. In addition, accurate prediction of stratified front behavior remains to be a challenging task for turbulence modeling. With the rapid growth of high-performance-computing capabilities in recent years, conducting high-fidelity simulations for a large-timescale transient has become more affordable and hence a valuable data source to support turbulence modeling as well as to gain further physical insights. In this work, direct numerical simulation is performed at experiment-consistent conditions to simulate the density stratification transient High-Resolution Jet (HiRJET) facility. Specifically, we focus on the case where an injected aqueous sugar solution has 1.5% density higher than that in the enclosure. In the early stage of the transient, the impingement of the denser jet to the bottom surface of the enclosure promoted turbulent mixing locally. This rendered the establishment of the mixture layer, formation and swift upward propagation of the stratified front, and elevation with (locally) the highest vertical concentration gradient. As the front rose, the diminishing turbulent mass flux slowed down the propagation, and a larger vertical concentration gradient was established. In this stage, both the velocity and the concentration scalar showed large-timescale fluctuation behavior around the stratified front. For the concentration time signal, the characteristic frequency in the power spectral density was found to agree well with the Brunt-Väisällä frequency. The preliminary validation endeavor showed that the stratified front location and the corresponding concentration gradient magnitude in the simulation agreed well with the experiment data. Further validation will mainly revolve around benchmarking against high-resolution density measurement and high-order flow statistics.

Importance, influence and limits of CFD radiation modeling for containment atmosphere simulations

In the appearance of steam, the CFD modeling of thermal-hydraulic phenomena in reactor containments during various phases of an accident necessarily requires consideration of radiative heat transfer. These radiation phenomena involve (i) energy transfer within the gas mixture and (ii) between the gas and the surrounding structures which have a much higher thermal inertia. Preliminary calculations performed for the present experiments in the OECD/NEA HYMERES-2 project with the CFD code containmentFoam using a Monte Carlo thermal radiation solver with non-gray gas modeling of steam absorption, showed that radiant heat transfer is important even with a very low amount of steam (≈ 2%). Therefore, the test matrix was tailored to the two opposite extremes: Either gas compositions with low steam content, where radiative heat transfer can be neglected, or gas mixtures containing larger amounts of steam, so that radiative heat transfer plays a dominant role. For the two selected experiments within the H2P2 series and the corresponding CFD calculations, a vessel with a diameter of 4 m and a height of 8 m was preconditioned with different mixtures of air/steam at room and elevated temperatures. This was followed by the build-up of a stable helium stratification in the upper part of the vessel. Helium was then injected from the top at a higher mass flow rate, which (a) compresses the gas atmosphere, (b) resembles piston compression and (c) leads to a height-dependent and transient increase in gas temperature below this helium stratification, influenced by thermal radiation effects — or their absence. These experiments and the associated CFD calculations were developed to isolate the thermal radiation phenomena as much as possible from convective and diffusive effects. They are the first of their kind to be conducted in a large-scale thermal-hydraulic facility. It is demonstrated (a) that neglecting thermal radiation for the high steam content case, the temperature rise is significantly over-predicted by a factor of two compared to the application of the suggested Monte Carlo thermal radiation solver with non-gray gas modeling, (b) the computational effort considering CFD and non-gray gas radiation is feasible with the suggested tailored Monte Carlo solver and (c) that the suggested radiation model outperforms the widespread used P1 radiation model.

Variable thermo-physical properties of fluid flow along with exponential stretching; A chemical reaction study

Heat and mass transfer having variable fluid properties has wide range of applications in industrial and engineering processes.The transference of species arises in several processes like oil transport phenomenon, food processing and diffusion of assured medicines in blood (drug delivery). Current research deals with the mixed convection and radiation effects on flow of a non-Newtonian tangent hyperbolic fluid with variable thermo-physical properties and activation energy. Arrhenius kind of chemical reaction is taken along an exponentially stretchable surface. The main focus of the current exploration is to execute shear thinning tangent hyperbolic fluid past an exponentially stretchable surface under the influence of variable viscosity, mixed convection, thermal radiation and activation energy. Transformation approach is opted to attain the non-linear ODE's based on standard conservation laws of mass, linear momentum and energy. Numerical solutions of the governing problem are computed via the shooting technique. Characteristics of the parameters appearing in modeling like radiation parameter, variable viscosity parameter, mixed convection parameter, Williamson parameter, variable conductivity and diffusivity are comprehensively analyzed through graphical behavior.

Acoustic Modeling Using the Aeroacoustic Wave Equation Based on Pierce’s Operator

The capabilities of an aeroacoustic wave equation based on Pierce’s operator (AWE-PO) for modeling subsonic flow-induced sound and for sound prediction are investigated. The wave equation is applied to an isothermal two-dimensional mixing layer computed by direct numerical simulation. In contrast to a direct numerical simulation, providing the acoustic fluctuations directly, the simulations based on Lighthill’s wave equation and the AWE-PO rely on a hybrid workflow to predict the generated sound field. Special attention is put on the interpretation of the right-hand side of both wave equations. Comparing the terms on the right-hand side in Lighthill’s theory and AWE-PO suggests a source amplitude for AWE-PO that is 90% smaller. This reduction is attributed to the filtering property of the material derivative. Finally, the results of the acoustic far-field pressure are compared. It is shown that the radiated sound field’s directivity, propagation, and convection effects are well captured for both wave equations. The computations using Lighthill’s equation and AWE-PO are found to provide acoustic intensities within 1.8 dB from the reference direct numerical simulation. This error is comparable with the errors reported for Lighthill’s equation in previous studies.

Mixed convective-quadratic radiative MoS 2–SiO 2/H 2 O hybrid nanofluid flow over an exponentially shrinking permeable Riga surface with slip velocity and convective boundary conditions: Entropy and stability analysis

In this work, the flow of a hybrid Mo S 2 − Si O 2 /water nanofluid across a porous exponentially diminishing Riga surface with the effects of quadratic heat radiation is taken into account. We also considered the effects of mixed convection, slip velocity, and convective thermal boundary conditions. Before numerically solving the governing partial differential equations, the MATLAB function bvp4c was used to get a comprehensive description of all relevant flow characteristics. The ramifications of physical characteristics concerning fluid flow and thermal profile are investigated using graphs and tables. The primary objective of this study is to investigate the Nusselt number, skin friction coefficient, and entropy production. Using the second law of thermodynamics, the irreversibility factor is computed. For a given value of the mass suction parameter, the findings also indicate the occurrence of dual-nature solutions in the shrinking sheet area, with a stable upper solution branch and unstable bottom solution branch. In addition, the first solution produces a positive minimum eigenvalue, whereas the second solution produces a negative eigenvalue, showing the stability of the first solution. For heat transfer enhancement, the existence of quadratic thermal radiative parameter impacts is more advantageous. With a larger value for the modified Hartman number, the velocity field has expanded. Also, recent research demonstrates that raising the thermal Biot number enhances the thermal gradient. Increases in nanoparticle volume fraction and thermal radiation parameters lead to a commensurate rise in entropy production. Because of these findings, we can better regulate the temperature of various environments by controlling the flow of heat.

Experimental and simulated thermal resistance of thermogalvanic cells with triply periodic minimal surface structures

Buildings release an abundance of waste heat that is left unused. Thermogalvanic cells (TGCs) can take advantage of waste heat to generate electricity with a low temperature gradient. Application of Triply Periodic Minimal Surface (TPMS) structures allow TGCs to potentially be used as building components for construction. TPMS in TGCs increase thermal resistance and the magnitude of increase varies by the type of TPMS used. Natural convection effects are one of the driving forces in TGCs, and those effects can be controlled with the use of TPMS structures while at the same time allowing the electrochemical process to occur. In this study, we ran multiple experiments for the TGCs with the Split P, Schwarz P, Gyroid, and IWP TPMS structures. Split P consistently gave relatively higher thermal resistances (0.012m−2KW) because of its higher surface area/volume ratio which effectively increased the temperature difference across the cell. The TGC containing the IWP structure had the least thermal resistance because of a low surface area as well as having large void spaces which increased heat transfer across the TGC. Furthermore, we simulated the thermal transport of the TGC containing the Schwarz P structure. By using the Boussinesq approximation, we were able to capture natural convection effects in our CFD simulations. The deviation of the ΔT from the experiment and simulation was due to the placement of the thermocouples at the top side of the electrode. This means that the measured electrode temperatures were overestimated since the warmer fluid was always present at the top while the simulated electrode temperature was calculated by taking the mean temperature of the electrode. The simulated thermal resistance matched the experimental values when the temperature was taken from the top sides of the electrodes, and was found to be 0.007 ±0.001m−2KW.

The Atlas of Morocco: A Plume‐Assisted Orogeny

AbstractWe explore the connections between crustal shortening, volcanism, and mantle dynamics in the Atlas of Morocco. In response to compressional forces and strain localization, this intraplate orogen has evolved far from convergent plate margins. Convective effects, such as lithospheric weakening and plume‐related volcanism, contributed in important ways to the building of high topography. We seek to better understand how crustal and mantle processes interacted during the Atlas' orogeny by combining multiple strands of observations, including new and published data. Constraints on crustal and thermal evolution are combined with new analyses of topographic evolution, petrological, and geochemical data from the Anti‐Atlas volcanic fields, and a simple numerical model of the interactions among crustal deformation, a mantle plume, and volcanism. Our findings substantiate that: (a) crustal deformation and exhumation accelerated during the middle/late Miocene, contemporaneous with the onset of volcanism; (b) volcanism has an anorogenic signature with a deep source; (c) a dynamic mantle upwelling supports the high topography. We propose that a mantle plume and the related volcanism weakened the lithosphere beneath the Atlas and that this favored the localization of crustal shortening along pre‐existing structures during plate convergence. This convective‐tectonic sequence may represent a general mechanism for the modification of continental plates throughout the thermo‐chemical evolution of the supercontinental cycle.

Investigation of Williamson nanofluid in a convectively heated peristaltic channel and magnetic field via method of moments

The study of Williamson nanofluid under peristaltic pumping is conducted in this work. The model equations are developed using the magnetic field and convection effects, and consequently, a nonlinear system of ordinary differential equations is achieved. Then, the residual method based on a linearly independent set of functions known as method of moments is implemented and portrayed as the results under the parameters’ variations. The model results revealed that the peristaltic pumping can be controlled by increasing the values of Gr and Gc; however, the dual effects of the directed magnetic field on the movement of Williamson fluid are examined. The heat transfer augmentation is observed for a stronger Brinkman number and it is higher toward the channel walls. Similarly, the thermophoretic effect and Brownian motion of the particles highly affect the concentration of Williamson nanofluid.