Reduced Sherwood Number Research Articles

Analysis of MHD third-grade hybrid nanofluid model in Darcy-Forchheimer porous medium: Evaluation of the thermal performance of [formula omitted]-[formula omitted] cylindrical nanoparticles dispersed in ethylene glycol fluid

Researchers and scientists became interested in the newly developed idea of hybrid nanofluids because of their exceptional thermal conductivities, which resulted in enhanced thermal performance. These fluids have a wide range of uses in the industrial and technical fields. This novel class of fluids is playing a vital role in solar energy, automotive industry, cooling of the machine and manufacturing, heating and ventilation systems, electronic cooling, nuclear systems cooling, biomedical fields, space, ships, and defense systems. In light of the physical significance of the above-mentioned class of fluids, the current investigations are carried to examine the thermal performance of the mixture of cylindrical shaped aluminum oxide Al2O3 and copper Cu nanoparticles suspended in ethylene glycol using third-grade non-Newtonian fluid flow model. The fluid flow is induced by the stretching of the permeable sheet in exponential manner that is embedded in a Dacry-Forchheimer porous medium. The effects of Lorentz force applied normal to the flow and suction impacts have been incorporated in the current proposed model. The solutions of the flow model transformed to ordinary differential equations are computed using MATLAB built-in solver bvp4c. The solutions are presented in graphical representations. Graphical solutions reflect that fluid velocity slows down by growing the volume fractions of the nanoparticles. Increasing values of viscoelastic parameters reflect that there is a reduction in velocity and intensification in profiles of temperature and concentration have been noted. Rising values of cross-diffusion parameter is leading to the reduction in magnitude of velocity, and augmenting behavior of temperature and concentration has been seen. Observations indicate rising third-grade fluid parameter and porosity parameter lead to reduction in velocity. Increasing Schmidt number gives reduced Sherwood number, and magnetic field parameter improves the rate of hear transfer (Nusselt number) and skin friction coefficient for both Al2O3/ethylene glycol nanofluid and Al3O3-Cu/ethylene glycol-based hybrid nanofluid. The suggested model's validity is confirmed through comparison with previously released findings. Additionally, a grid independence test (sensitivity analysis) has been performed to verify the accuracy and verification of the numerical model.

Numerical Analysis on Stagnation Point Flow of Micropolar Nanofluid with Thermal Radiations over an Exponentially Stretching Surface

Several industrial developments such as polymer extrusion in metal spinning and continuous metal casting include energy transmission and flow over a stretchy surface. In this paper, the stagnation point flow of micropolar nanofluid over a slanted surface is presenting also considering the influence of thermal radiations. Buongiorno’s nanoliquid model is deployed to recover the thermophoretic effects. By using similarity transformations, the governing boundary layer equations are transformed into ordinary differential equations. The Keller-box approach is used to solve transformed equations numerically. The numerical outcomes are presented in tabular and graphical form. A comparison of the outcomes attained with previously published results is done after providing the entire formulation of the Keller-Box approach for the flow problem under consideration. It has been found that the reduced sherwood number grows for increasing values of radiation parameter while, reduced Nusselt number and skin friction coefficient decreases. Furthermore, the skin-friction coefficient increases as the inclination factor increases, but Nusselt and Sherwood's numbers decline.

Significance of mixed convective double diffusive MHD stagnation point flow of nanofluid over a vertical surface with heat generation

The mixed convection double-diffusive MHD flow of boundary layer nanofluids above vertical region is designed. This flow is explained near stagnation point along with heat generation. Using Buongiorno’s model, the properties of Brownian motion, thermophoresis, and diffusion of regular and cross type are included. Using appropriate similarity transformations of the local similarity technique, non-linear unstable PDEs in governing model converted to non-linear ODEs, which were then evaluated numerically by the Keller-box method (KBM) using the computational software MATLAB 2021a. Graphical analysis of possessions of parameters on the boundary layers in profiles of velocity, solute concentration, temperature and nanoparticle concentration is illustrated. The statistics of the reduced Sherwood number and the reduced Nusselt number for solute and nanoparticles in cases of assisting flow and opposing flow are added as well to the results. The fastest rate of heat transfer is achieved in a scenario with a negligible thermophoresis effect. The thermophoresis parameter and the buoyancy parameter of the regular double diffusive appear to rise rather than decrease in the nanoparticles lowered Sherwood number, while the Brownian motion parameter rises. The temperature and layer thickness for heat generation have a quite opposite effect. When the computed numerical findings are compared to earlier published work, it is discovered to be in good percentage. The need for numerous industrial applications and improvements in the efficiency and energy consumption of systems, such as cooling and heating transportation, in water heaters, nuclear reactors, optical devices, turbines, aerodynamics, and electronics, have led to the establishment of this investigation.

Study on Nanofluid Boundary Layer Flow Over A Stretching Surface by Spectral Collocation Method

The method of Spectral collocation is used to analyze the flowing Nano fluid layer in contact with a stretching surface for comprehensive information and thus to have its utility in industrial activities like the production of glass fibers, petroleum refining, hot rolling of metals, metal spinning etc. The spectral collocation model incorporates thermophoresis and Brownian motion phenomena to describe the fluid flow, fluid concentration and temperature profiles. A similarity solution has been presented for the governing equations of fluid momentum, concentration and temperature. The computational results are the function of Prandtl number (Pr), Lewis number (Le), thermophoresis and Brownian motion phenomena. The engineering quantities like thermophoresis parameter (Nt), Brownian motion parameter (Nb), buoyancy-ratio parameter (Nr) and reduced Nusselt number (Nu) and reduced Sherwood number (Sh) have tabulated corresponding to Prandtl number (Pr) and Lewis number (Le). The results of the current study thrown light on fluid velocity and heat transfer rates in the boundary layer. The numerous industrial products and manufacturing processes of superior quality can be exercised with the current studies.

Convective heating and mass transfer in Buongiorno model of nanofluid using spectral collocation method of shifted Chebyshev polynomial

In this article, the investigation is made to shed a light on the influence of convective boundary conditions on the boundary layer flow over a linearly stretching flat surface. The equations (partial differential equations) describing the model are transformed into system of nonlinear ordinary differential equations using similarity transformations. Equations contain various non-dimensional flow characterizing numbers viz. Prandtl number Pr, Lewis number Le, Biot number Bi, Brownian motion parameter Nb and thermophoresis parameter Nt. The influence of these parameters on thermal boundary layer, concentration distribution and temperature are analyzed in detail, by solving the equations using novel Shifted Chebyshev collocation method. The computed results, reduced Nusselt number, reduced Sherwood number, surface temperature and concentration profiles as functions of dimensionless numbers are validated by comparing the predicted results with available earlier findings (using other methods). To assert the convergence and stability of the scheme used (for much larger, but moderate, parameters values), predicted results are presented in various tabular forms. For presenting finer details of the computed values some results are also given graphically. The innovative semi-numerical scheme is robust and efficient compared with other conventional methods, used in previous studies and enables the analysis of the complex problem adequately.

Double-diffusive stagnation point flow over a vertical surface with thermal radiation: Assisting and opposing flows.

In numerous industrial procedures, the main concern of design engineers is ensuring adequate heat and mass transfer, such as in the heating and cooling practices of solar water heaters, geothermal systems, extrusion of metal, insulation of buildings, electronics, turbines, aerodynamics, electronics, paper manufacturing, and glass fiber production. The unsteady double-diffusive mixed convection flow of boundary layer nanofluids above a vertical region near stagnation point flow is developed and examined here. The Brownian motion and thermophoresis effects are incorporated by using Buongiorno's model. In the thermal energy equations, diffusion of regular and cross types is also used. By the use of the local similarity method along with suitable similarity transformations, nonlinear unsteady partial differential equations are converted to nonlinear ordinary differential equations and are numerically solved by the Keller-Box method. The investigation expresses that these profiles of solute concentration and nanoparticle concentration, temperature, and velocity in their boundary layers, respectively, depending on several parameters. A graphic analysis of all these parameters' possessions on nature's boundary layers is depicted. The highest rate of heat transfer is obtained with negligible thermophoresis effect. Furthermore, it is perceived that an increase in Nc and Nt results in a reduction in the reduced Sherwood number of nanoparticles, whereas addition results in an increase in the Nb number. There is a reverse effect on the temperature field and layer thickness for heat generation. In the wake of the above-mentioned potential applications, the current study of fluid flow has been found to be very interesting and innovative in the analysis of the influence of Brownian motion and thermophoresis effects near stagnation point flow, which will further make revolutions in industrial fields. Moreover, Buongiorno's model predicts the characteristics of double-diffusive fluids in enhancing heat transfers. This investigation has been established as a result of the numerous industrial applications mentioned above.

Entropy generation analysis on convective radiative stagnation point flow of a Casson nanofluid in a non-Darcy porous medium with induced magnetic field and activation energy: the multiple quadratic regression model

The present article analyzes the entropy generation on the nonlinear thermally radiating magnetohydrodynamic Casson nanofluid flow closer to the stagnation point inside a non-Darcy porous medium with viscous dissipation, magnetic dissipation, mixed convection, and heat generation. The induced magnetic field's impact and the implications of the chemical reaction and the activation energy are also included in the study. A mathematical model based on the substantive problem is developed, consisting of nonlinear partial differential equations and compatible boundary conditions. The mathematical model is converted to a set of nonlinear ordinary differential equations with appropriate boundary conditions using a suitable similarity transformation, which is then solved using the spectral quasi-linearization method. A complete examination is executed for achieving in-depth information of different varying flow parameters' consequences on the flow field and the significant physical quantities using several graphs and tables. Using the regression models on the reduced Nusselt number NuxRex−1/2 and the reduced Sherwood number ShxRex−1/2, the maximum relative errors are shown as 0.15%–0.30% and 0.016024%–0.016225%, respectively. The present results signify that the larger velocity ratio parameter boosts the velocity along the x-axis and the transverse velocity for M = 3, but the transverse velocity for M = 1, temperature, and induced magnetic field follow a reverse pattern.

Heat Transfer of Nanomaterial over an Infinite Disk with Marangoni Convection: A Modified Fourier’s Heat Flux Model for Solar Thermal System Applications

The demand for energy due to the population boom, together with the harmful consequences of fossil fuels, makes it essential to explore renewable thermal energy. Solar Thermal Systems (STS’s) are important alternatives to conventional fossil fuels, owing to their ability to convert solar thermal energy into heat and electricity. However, improving the efficiency of solar thermal systems is the biggest challenge for researchers. Nanomaterial is an effective technique for improving the efficiency of STS’s by using nanomaterials as working fluids. Therefore, the present theoretical study aims to explore the thermal energy characteristics of the flow of nanomaterials generated by the surface gradient (Marangoni convection) on a disk surface subjected to two different thermal energy modulations. Instead of the conventional Fourier heat flux law to examine heat transfer characteristics, the Cattaneo–Christov heat flux (Fourier’s heat flux model) law is accounted for. The inhomogeneous nanomaterial model is used in mathematical modeling. The exponential form of thermal energy modulations is incorporated. The finite-difference technique along with Richardson extrapolation is used to treat the governing problem. The effects of the key parameters on flow distributions were analyzed in detail. Numerical calculations were performed to obtain correlations giving the reduced Nusselt number and the reduced Sherwood number in terms of relevant key parameters. The heat transfer rate of solar collectors increases due to the Marangoni convection. The thermophoresis phenomenon and chaotic movement of nanoparticles in a working fluid of solar collectors enhance the temperature distribution of the system. Furthermore, the thermal field is enhanced due to the thermal energy modulations. The results find applications in solar thermal exchanger manufacturing processes.

MHD stagnation point flow of Casson fluid over a convective stretching sheet considering thermal radiation, slip condition, and viscous dissipation

AbstractThis study presents the problem of MHD stagnation point flow of Casson fluid over a convective stretching sheet considering thermal radiation, slip condition, and viscous dissipation. The partial differential equations with the corresponding boundary conditions that govern the fluid flow are reduced to a system of highly nonlinear ordinary differential equations using scaling group transformations. The fourth‐order method along shooting technique is applied to solve this system of boundary value problems numerically. The effects of flow parameters on the velocity, temperature, and concentration profiles are presented via graphs. The impact of the physical parameters on the skin friction coefficient reduced Nusselt numbers and reduced Sherwood numbers are investigated through tables. Comparison of the present findings with the previously published results in the literature shows an excellent agreement. It is also noted that a rise in the Eckert number results in a drop in the temperature of the fluid in the thermal boundary layer region of the fluid flow.

Unsteady rotating flow of nanofluid with prescribed thermal aspects

Motivated by the significant role of nanofluid in pollution cleaning and energy recovery, we decided to explore the unsteady three-dimensional rotating flow of nanofluid driven by the movement of a flat surface with the potencies of prescribed heat distributions. The modeling of the physical model is completed with the help of Buongiorno nanofluid model. Suitable arrangement of similarity variables is implemented to transform the model equations into strongly nonlinear ordinary differential equations. Numerical inspection of the model is made by employing Keller–Box algorithm. Influences of involved parameters on the distributions of heat and mass are discussed graphically, while the potencies of influential parameters on reduced Nusselt and reduced Sherwood numbers are physically discussed through tabular arrangements. It is deduced that increasing the values of Prandtl factor and heat controlling indices diminishes the temperature and concentration distributions, whereas intensification in the amount of rotation factor enhances the temperature as well as concentration distribution. Moreover, negative trends in the amounts of reduced Nusselt and Sherwood numbers are achieved with the escalations in the values of rotation and thermophoresis factors, whereas opposite trend is achieved with the intensification in the choice of Prandtl factor.

Entropy generation in convective radiative flow of a Casson nanofluid in non-Darcy porous medium with Hall current and activation energy: The multiple regression model

The present article’s primary goal is to analyze entropy generation on unsteady three dimensional thermally radiating magnetohydrodynamic flow of Casson nanofluid past a vertical stretching sheet embedded in a non-Darcy porous medium under the influence of mixed convection, Hall current, viscous dissipation, Ohmic heating, and heat generation. The combined effects of Brownian motion and thermophoretic diffusion are taken into consideration, and the impact of the chemical reaction initiated by activation energy is also incorporated in the flow-field. A mathematical model of the physical problem containing highly nonlinear coupled partial differential equations with convective boundary conditions is developed. Suitable similarity transformation is applied to convert the governing highly nonlinear coupled partial differential equations into nonlinear coupled ordinary differential equations, and then the resulting nonlinear coupled ordinary differential equations are numerically solved with the help of the spectral quasi-linearization method. A detailed investigation is performed to illustrate the impact of several pertinent flow parameters on the velocity, temperature, concentration, entropy generation and Bejan number profiles. In contrast, the numerical values of skin friction coefficients in x and z directions, local Nusselt number, and local Sherwood number are presented in tabular forms for varying values of pertinent flow parameters. Apart from this, linear and quadratic multiple regression analyses for the physical quantities of engineering interest have been executed to upgrade the present model’s efficiency and application in various industrial and engineering processes. With a variation in magnetic parameter, the maximum relative errors in the linear regression estimates for the skin- friction coefficients in x and z directions are found to be 2.28–2.37% and 0.85–1.72%, respectively. Besides, due to the enhancement in magnetic parameter, the maximum relative errors in the quadratic regression estimations for the reduced Nusselt number NuxRex−1/2 and the reduced Sherwood number ShxRex−1/2 are 0.56%−0.58% and 0.30%, respectively. It is observed that the error of each solution of the present physical problem approaches to be less than 10−8 in mere six iterations. Moreover, the findings of the current research work signify that the Brownian motion and thermophoretic diffusion minimize entropy generation adjacent to the sheet; on the contrary, activation energy intensifies entropy generation rate closer to the sheet. Furthermore, the appropriate resemblance is obtained through verifying the present results with previous existing results.

Effects of Chemical Reaction, Activation Energy and Thermal Energy on Magnetohydrodynamics Maxwell Fluid Flow in Rotating Frame

The purpose of this research is to see how chemical processes, activation energy, and heat radiation affect MHD flow of Maxwell fluid in a rotating frame. Using applicable similarity transformations, the partial differential equations that regulate the flow are reduced to extremely nonlinear ordinary differential equations. Graphs and tables are used to study the impact of monitoring parameters on velocity, temperature, concentration profiles, reduced Nusselt number, reduced Sherwood numbers, and skin friction coefficients. Outstanding agreement is obtained when the present findings of the study is compared with the previous related research works. In the study, it is noted that an increase of the thermal radiation parameters contributes to an increase of the flow temperature region. When a fluid is subjected to a greater rotation parameter, the thermal boundary layer thickens and the heat transfer rate decrease. Moreover, a decline of mass transfer rate is observed for a rise of Prandl number, rotational parameter or Deborah number.

Numerical investigation of effect of variable viscosity on transient heat transfer over a non-isothermal axisymmetric body: nanoparticle volume fraction-dependent viscosity

The current research paper delibrates the effect of variable viscosity of the nanofluid that saturates a porous media, on transient free convection. The viscosity term that appears in the equation of conservation of momentum depends on nanoparticle volume fraction. The effect of viscosity parameter on reduced Nusselt number and reduced Sherwood number for different combinations of Buoyancy ratio Nr, Brownian motion Nb and thermophoresis Nt is examined using the unconditionally stable Keller Box method. The conditions for existence of the numerical solution has been analytically derived. The study illustrates that the variation in viscosity has an insignificant effect on the reduced Nusselt number, but the reduced Sherwood number increases with an increase in the viscosity. Although the study can accommodate any arbitrary axisymmetric body, vertical flat plate, horizontal cylinder and sphere are three particular geometries that are considered in the paper.

MHD Boundary Layer Flow of a Power-Law Nanofluid Containing Gyrotactic Microorganisms Over an Exponentially Stretching Surface

This study focusses on the numerical investigations of boundary layer flow for magnetohydrodynamic (MHD) and a power-law nanofluid containing gyrotactic microorganisms on an exponentially stretching surface with zero nanoparticle mass flux and convective heating. The nonlinear system of the governing equations is transformed and solved by Runge-Kutta-Fehlberg method. The impacts of the transverse magnetic field, bioconvection parameters, Lewis number, nanofluid parameters, Prandtl number and power-law index on the velocity, temperature, nanoparticle volume fraction, density of motile microorganism profiles is explored. In addition, the impacts of these parameters on local skin-friction coefficient, local Nusselt, local Sherwood numbers and local density number of the motile microorganisms are discussed. The results reveal that the power law index is considered an important factor in this study. Due to neglecting the buoyancy force term, the bioconvection and nanofluid parameters have slight effects on the velocity profiles. The resultant Lorentz force, from increasing the magnetic field parameter, try to decrease the velocity profiles and increase the rescaled density of motile microorganisms, temperature and nanoparticle volume fraction profiles. Physically, an augmentation of power-law index drops the reduced local skin-friction and reduced Sherwood number, while it increases reduced Nusselt number and reduced local density number of motile microorganisms.

Numerical study of unsteady boundary layer flow and heat transmission of a nano-fluid past a continuously stretching sheet immersed in porous media

PurposeThis paper aims to incorporate a numerical investigation of unsteady flow of nano-fluid near the boundary wall and heat transmission over a flat sheet in porous media under the assumption that sheet is continuously stretching. A comprehensive study is accomplished to probe the impacts of different physical parameters on heat transmission and fluid flow.Design/methodology/approachThe problem is designed mathematically by using Navier–Stokes equations and corresponding equations are transmuted into non-dimensional form, which are reduced to local non-similarity equations. Implicit finite difference scheme is used to attain a non-similar solution for values of similarity variable ranging in [0,1]FindingsThe reduced Nusselt number is a decreasing function of each dimensionless number, while the reduced Sherwood number is an increasing function of each considered value of parameters Pr, Sc, Nb and Nt.Originality valueMotivated by the industrial, technological and scientific importance, the objective of this study is to discuss the unsteady flow generated by linear stretching of the surface in Nano-fluid in a porous medium. However, the investigation of unsteady boundary layer flow and heat transfer of a Nano-fluid past a stretching surface immersed in a porous medium has not been discussed yet. Furthermore, finite difference scheme is used to solve the problem, and authenticate results are achieved which are in good agreement with published work.

Free convection boundary layer flow on a solid sphere in a nanofluid with viscous dissipation

Present study considers the mathematical model of free convection boundary layer flow and heat transfer in a nanofluid over a solid sphere with viscous dissipation effect. The transformed partial differential equations are solved numerically using the Keller-box method. The numerical values for the reduced Nusselt number, reduced Sherwood number and the reduced local skin friction coefficient are obtained, as well as concentration profiles, temperature profiles and velocity profiles are illustrated graphically. Effects of the pertinent parameters, which are the Prandtl number, buoyancy ratio parameter, Brownian motion parameter, thermophoresis parameter, Lewis number and Eckert number are analyzed and discussed. It is found that the increase of Brownian motion parameter promoted the reduce of concentration boundary layer thickness while thermophoresis parameter did oppositely. It is worth mentioning that the results reported here are important for the researchers working in this area which can be used as a reference and comparison purposes in the future.

Exploration of flow and heat transfer of non-Newtonian nanofluid over a stretching sheet by considering slip factor

PurposeHere, the present paper characteristics flow and heat transfer of non-Newtonian nanofluid over a stretching sheet. Energy expression is modeled subject to slip factor phenomenon. Consideration of chemical reaction characterizes the mass transfer mechanism.Design/methodology/approachThe use of transformation variables reduces the PDEs into non-linear ODEs. The obtained nonlinear complex problems are computed numerically through RKF-45 technique. The effects of the different physical parameters on the temperature and concentration distribution are analyzed.FindingsThe nature of the reduced Nusselt number, reduced Sherwood number and skin friction coefficient also described as a function of different parameters arising in the problem. It is found that the rate of mass transfer enhances for enhancing values Brownian motion parameter and thermophoresis parameter.Originality/valueThe nature of the reduced Nusselt number, reduced Sherwood number and skin friction coefficient also described as a function of different parameters arising in the problem. It is found that, the rate of mass transfer enhances for enhancing values Brownian motion parameter and thermophoresis parameter.

The study of MHD Nano fluid flow with chemical reaction along with thermophoresis and Brownian motion on boundary layer flow over a linearly stretching sheet

The two-dimensional steady boundary layer flow, of a nanofluid past a stretching sheet with a convective boundary condition in the presence of chemical reaction has been studied. The equation of volume fraction concentration consists of the Brownian motion and the thermophoresis effects. The governing equations are simplified via boundary value approximation and some similarity variables and solved using the Runge-Kutta fourth order method with shooting technique and homotopy perturbation method (HPM) with Pad� approximation (HPM-Pad�). All the salient parameters on the temperature and concentration profiles have been studied and details are given graphically. Thus the results found in this study are being compared with the previously published study, where an excellent agreement is achieved. As presented on tables. It is found that the reduced Nusselt number is a decreasing function of both generative and destructive chemical reaction γ. While the reduced Sherwood number is a decreasing function when chemical reaction parameter, γ0.

Heat Transfer Analysis of a Williamson Micropolar Nanofluid with Different Flow Controls

ABSTRACTThe present model is devoted for the steady stagnation point flow of a Williamson micropolar nanofluid with magneto-hydrodynamics and thermal radiation effects passed over a horizontal porous stretching sheet. The fluid is considered to be gray, absorbing-emitting but non-scattering medium. The Cogley-Vincent-Gilles formulation is adopted to simulate the radiation component of heat transfer. By applying similarity analysis, the governing partial differential equations are transformed into a set of non-linear ordinary differential equations and they are solved by using the bvp4c package in MATLAB. Numerical computations are carried out for various values of the physical parameters. The effects of momentum, microrotation, temperature and nanoparticle volume fraction profiles together with the reduced skin friction coefficient, reduced Nusselt number and reduced Sherwood number of both active and passive controls on the wall mass flux are graphically presented. The present results are compared with previously obtained solutions and they are in good agreement. Results show that the skin friction is increasing functions of the Williamson parameter in both stretching and shrinking surfaces.

New exact and numerical solutions for the effect of suction or injection on flow of nanofluids past a stretching sheet

Abstract The flow of nanofluids past a stretching sheet has attracted much attention due to its wide applications in industry and engineering. Theoretical and numerical solutions have been discussed in this paper for studying the effect of suction or injection on flow of nanofluids past a stretching sheet. In the absence of thermophoresis the analytical exact solution of the stream function was obtained in terms of exponential function, while the exact solutions for temperature and nanoparticle volume fraction were obtained in terms of the generalized incomplete gamma function. In addition, in the presence of thermophoresis, the exact solutions are not available. Therefore, the numerical results, carried out by using Chebyshev collocation method (ChCM). It is found that a good agreement exists between the present results and with those published works. Useful results for temperature profile, concentration profile, reduced Nusselt number and reduced Sherwood number are discussed in details graphically. It was also demonstrated that both temperature and concentration profiles decrease by an increase from injection to suction. Finally, the present results showed that increase of the wall mass transfer from injection to suction decreased both reduced Nusselt number and the reduced Sherwood number when Brownian motion parameter and Lewis number increased.