Brinkman Models Research Articles

Comparative analysis of varied thermal conductivity and viscosity models in a hybrid nanofluid flow - a non-similar solution

ABSTRACT We aim to investigate the flow of an ethylene glycol-based hybrid nanofluid over a horizontal surface with a free stream velocity. To assess the viscosity and thermal conductivity of nanoparticles with various shapes, we employed the Brinkman, Timofeeva, and Yamada Ota unit cell models. The study explores heat transfer in the flow system considering the effects of viscous and ohmic dissipations, linear radiative heat flux, and heat generation under convective boundary conditions. A non-similar system is developed using appropriate non-similarity transformations up to the third-level truncation and numerically solved using the bvp4c algorithm. The novelty of the proposed model lies in obtaining the non-similar solution and incorporating the Brinkman, Timofeeva, and Yamada Ota unit cell models. The results indicated that the Brinkman viscosity model resulted in an increase in the wall heat transfer rate with a rise in particle volume fraction. It is also inferred that platelet-shaped nanoparticles exhibit a significantly higher heat transfer rate compared to spherical-shaped nanoparticles.

Modeling fluid flow in fractured porous media: a comparative analysis between Darcy–Darcy model and Stokes–Brinkman model

AbstractThere are limited comparative studies on modeling fluid transport in fractured porous media. Hence, this paper systematically compares the steady-state creeping flow Stokes–Brinkman and Darcy–Darcy models for computational efficiency and accuracy. Sensitivity analyses were also conducted on the effect of fracture orientations, fracture sizes, mesh resolution, and fractures with Local Grid Refinement (LGR) under the FEniCS computational framework. Both models were validated numerically, and the accuracy of their solution is compared using the R-squared metric and L2 norm estimates. Key results showed that both models have similar pressure and velocity field solutions for a given fracture orientation. The computational time required for solving the Stokes–Brinkman models for a single fracture case was unusually lower than that of the Darcy–Darcy model when the pressure and velocity terms in the Darcy–Darcy model were solved simultaneously using two equations, contrary to where only one equation solves for the pressure and the velocity is obtained by projecting the gradient of pressure onto a vector space. The Stokes–Brinkman model is more sensitive to mesh resolution, and as a result, the Darcy–Darcy model tends to be more accurate than the Stokes–Brinkman model at low resolutions. Local Grid Refinement (LGR) can improve the Stokes–Brinkman model's accuracy at low mesh resolution. Furthermore, both models showed similar results when compared for complex fracture systems such as multiple fracture cases: interconnecting and isolated fractured porous media systems under low-velocity and steady-state creeping flow conditions. The FEniCS code in this paper is shared for future researchers to reproduce results or extend the research work.

Dissipation induced instability in porous medium using Brinkman model: Thermal non-equilibrium effect

The convective instability that arises in a porous layer solely because of dissipative energy in the medium is studied. The thermal non-equilibrium model is considered for energy equation. The boundary at top is kept at constant temperature, and an adiabatic lower boundary is considered. The momentum equation is written according to the Brinkman model. A basic flow in the horizontal plane is considered, and the basic velocity and temperature profile have been derived. A comparison between the basic temperature profiles for solid and fluid of the porous medium is presented for Darcy and Brinkman models. Infinitesimal disturbance is introduced to the basic flow. A linear stability analysis has been carried out to study the stability of the basic flow. The parameters influencing the stability of the system are critical Rayleigh number (RaC), Gebhart number (Ge), ξ associated with the Darcy number (Da), and interphase heat transfer coefficient (H). The values of the critical Rayleigh number and wavenumber are compared by varying other variables. The flow is more stable when the Brinkman medium is considered. With the increase in interphase heat transfer coefficient, the critical Rayleigh number increases in both Darcy and Brinkman media. The longitudinal rolls appear as the most unstable rolls for all cases. The variation of convective rolls with respect to ξ and heat transfer coefficient is presented.

Study of thermodynamic, transport and volumetric properties of nanofluids containing ZrO2 nanoparticles in polypropylene glycol, polyvinyl pyrrolidone and water.

Zirconium dioxide (ZrO2) nanofluids are used in cooling systems, solar energy, and heat exchangers, offering improved heat transfer and efficiency across a wide temperature range. The aim of this work was to study the influence of polypropylene glycol (PPG) and polyvinyl pyrrolidone (PVP) and aqueous solutions of them as a base fluid on stability, volumetric properties, and viscosity of nanofluids containing Zirconium oxide (ZrO2) nanoparticles. The stability of these nanofluids has been confirmed using UV-Vis spectroscopy, and the particle size distribution of the systems using dynamic light scattering (DLS). Among these systems ZrO2-PPG and ZrO2-H2O-PVP30% have appropriate stability. The density, speed of sound and viscosity of these nanofluids have been measured at T = (293.15 to 318.15) K. From these data, the excess molar volume (V E m) and isentropic compressibility (κ s) have been determined. The effects of ZrO2 nanoparticles and temperature have also been investigated on volumetric and transport properties of aqueous solutions of PPG and PVP. The (V E m) values were fitted to the Redlich-Kister, Ott et al., and polynomial equations. Also, the isentropic compressibility (κ s) values were correlated with the polynomial equation. The Eyring-NRTL and Eyring-mNRF models have been used for correlating of the viscosity of the nanofluids with temperature dependency. The performance of the Einstein, Brinkman, Lundgren and Batchelor models in the prediction of viscosity of the nanofluids has also been analyzed.

Optimal characteristics of natural convection in a square porous-nanofluid-filled enclosure containing three tubes

A numerical model was used to simulate natural convection heat transfer in a two-dimensional square enclosure filled with nanofluid and saturated with porous media. The enclosure contained three heated tubes with Rayleigh numbers (Ra) ranging from 102 to 3 × 104. The governing equations with Boussinesq approximation for natural convection were solved iteratively using the finite volume technique through ANSYS FLUENT-CFD commercial package. The Darcy–Forchheimer–Brinkman and Local Thermal Equilibrium models were applied for water and nanofluid flow across the porous zone. The numerical analysis was executed systematically for significant parameters that have a substantial impact on natural convection heat transfer, namely, aspect ratio (AR = 1.25 − 3.75), nanoparticle volume fraction (Φ = 0.25%, 0.5%, 0.7% and 1%), and the porosity of metal foam (ε = 0.3, 0.4, 0.5, 0.6 and 0.7). The aspect ratio was found to have a considerable impact on the averaged Nusselt number (Nuave ). For AR = 3.75, the value of Nuave for all Rayleigh numbers increased by approximately 5.22 times compared to AR = 1.25. Moreover, Nuave remained approximately constant as the concentration of alumina-water nanofluid increased from 0.25% to 1%. The novelty of this study lies in the multi-objective optimal design approach adopted by adopting DX-12 with the outcomes of ANSYS FLUENT-CFD. The averaged Nusselt number and convection heat transfer rate reached their optimal values by nearly 13 and 9 times, respectively, at the lowest possible porosity (ε = 0.3), the lowest possible nanofluid concentration (Φ = 0), and the highest possible aspect ratio value (AR = 3.75).

Water thermal enhancement in a porous medium via a suspension of hybrid nanoparticles: MHD mixed convective Falkner's-Skan flow case study

Keeping in mind the whispered applications of Falkner's-Skan Flows, the present hydrothermal scrutinization intended to explore the two-dimensional flow pattern and thermal characteristics featuring the steady mixed convective motion of a radiating homogeneous hybrid nanofluidic mixture (Ag+TiO2)−H2O (i.e., a homogeneous aquatic mixture containing spherical silver and titanium dioxide nanoparticles) over a non-movable wedge surface. By combining Darcy's-Brinkman and single-phase models, the conservation equations are stated properly in the case where the hydrothermal impacts of specific variable heat and magnetic sources are taken into account effectively in the present flow problem. After many simplifications and rearrangements, the derived boundary layer equations are handled numerically with the help of a robust iterative GDQM-NRT algorithm, whose accurate results are presented graphically and tabularly. As the main findings, it is evidenced that the thermal buoyancy forces, as well as Lorentz's and Darcy's forces, act as assisting factors, which exert a cooling impact throughout the hybrid nanofluidic mixture (Ag+TiO2)−H2O due to the imposed axial pressure gradient. Energetically, it is demonstrated that the silver nanoparticles play a noteworthy role in the enhancement of the heat transfer rate at the wedge surface with a strengthening in the resulting surface viscous drag forces.

CuO–Water MHD Mixed Convection Analysis and Entropy Generation Minimization in Double-Lid–Driven U-Shaped Enclosure with Discrete Heating

Abstract The present study explores magnetic nanoliquid mixed convection in a double lid–driven U-shaped enclosure with discrete heating using the lattice Boltzmann method (LBM) numerical method. The nanoliquid thermal conductivity and viscosity are calculated using the Maxwell and Brinkman models respectively. Nanoliquid magnetohydrodynamics (MHD) and mixed convection are analyzed and entropy generation minimisation has been studied. The presented results for isotherms, stream isolines and entropy generation describe the interaction between the various physical phenomena inherent to the problem including the buoyancy, magnetic and shear forces. The operating parameters’ ranges are: Reynolds number (Re: 1–100), Hartman number (Ha: 0–80), magnetic field inclination (γ: 0°– 90°), nanoparticles volume fraction (ϕ: 0–0.04) and inclination angle (α: 0°– 90°). It was found that the N um and the total entropy generation augment by increasing Re, ϕ: and γ. conversely, an opposite effect was obtained by increasing Ha and α. The optimum magnetic field and cavity inclination angles to maximum heat transfer are γ = 90° and α = 0.

Entropy analysis in single phase nanofluid in square enclosure under effectiveness of inclined magnetic field by executing finite element simulations

One of the important phenomena concerning with dynamical and thermal attributes of heat transferring liquids is the measure of randomness generated at molecular level. This randomness in the system is generated due to presence of multiple factors which includes extensive heat transfer, strong magnetic field and viscous diffusion. To control and minimize disorder ness the concept of entropy is taken into account. Minimization in entropy is valuable in optimization of design objectivity in engineering structures. In addition, it measures and controls exergy loss to generate highly efficient systems likes heat exchangers, heat pumps, power plants, nuclear reactors, air conditioners and so many. So, the prime objective of this work is to analyze entropy in square enclosure filled with water and containing Copper (Cu) particles. Convection in the considered physical domain is generated by providing uniform temperature at upper wall while keeping rest of boundaries cold. In view of velocity field distribution is concerned all the extremities of cavity are at no slip condition. Inclined magnetic field is also employed to measure change in entropy due to magnetization. Formulation of problem expressing thermophysical relation of inducted nanoparticles in manifested in the form dimensionless representation. Patel and Brinkman models for effective thermal conductivity and viscosity are opted to visualize their effectiveness in entropy change. Finite element computations are performed to simulate results showing the impact of flow controlling parameters on associated distributions. Three different types of entropy namely, magnetic, thermal and viscous are measured against different parameters. Quantity of interest like average heat flux coefficient is also measured. It is deduced from the analysis that by increasing nanoparticle volume fraction thermal entropy increases whereas viscous and magnetic entropy uplifts. In addition, it is noticed that by increasing Hartmann number entropy and velocity of fluid decrements. Uplift in temperature magnitude is observed against elevating values of Rayleigh number. It is concluded that employment of magnetic field will assist in controlling flow, thermal and irreversibility phenomenon.

Thermal behavior of a conical antenna cooled with nanofluid saturated porous media: effects of the cavity’s inclination and aspect ratio

Purpose The aim of this work is to determine the average surface temperature of a conical antenna. Its cooling is ensured by means of a nanofluid-saturated porous structure. The volume fraction of the H2O–Cu nanofluid ranges between 0% (pure water) and 5%, whereas the ratio between the thermal conductivity of the used porous materials and that of water (fluid base) varies in the wide 4–41.2 range. The antenna is contained in a coaxial conical closed cavity with a variable distance between the cones, leading to an aspect ratio varying between 0.2 and 0.6. The axis of the assembly is also inclined with respect to the gravity field by an angle varying between 0° (a vertical axis with top of the cone oriented upwards) and 180° (a vertical axis with top of the cone oriented downwards). Design/methodology/approach Simulations have been done by means of the volume control method based on the SIMPLE algorithm. Findings Results of the numerical approach show that the cavity’s aspect ratio and inclination with respect to the gravity field significantly affect the thermal behavior of the active cone. Otherwise, the work confirms that the Maxwell and Brinkman models used to determine the nanofluid’s effective thermal conductivity and viscosity, respectively, are adapted to the considered assembly. Originality/value A new correlation is proposed, allowing the determination of the average surface temperature of the active cone and its correct thermal sizing. This correlation could be used in various engineering fields, including electronics, examined in the present study.

Experimental study on the influence of a conical cavity’s inclination angle and aspect ratio on thermal behavior of a cone cooled with nanofluid saturated porous media

ABSTRACT An experimental setup has been developed to measure the thermal state of a conical antenna cooled by means of porous media saturated with water-based copper nanofluid. Thermal conductivity of the used porous materials varies between 4 and 41.2 times that of the water which constitutes the base fluid of the nanofluid whose volume fraction ranges from 0% (pure water) to 5%. The antenna is placed in the center of a coaxial concentric cavity of the same shape. Aspect ratio of the enclosure varies between 0.2 and 0.6. Its base is inclined with respect to the horizontal plane with an angle ranging between 0° and 180° corresponding to a horizontal circular base with the top of the cone oriented upwards and downwards, respectively. Measurements validate the calculations obtained in a recent numerical approach based on the finite volume method. They confirm that the Maxwell and Brinkman models remain valid for the ranges of the Rayleigh number considered here and Water based-Copper nanofluid volume fraction (0–5%). This work also confirms the effectiveness of porous materials saturated with nanofluids to ensure the thermoregulation of electronic equipment dissipating a large amount of thermal power.

Effect of the Aspect Ratio on the heat transfer enhancement by the Al2O3-H2O Nanofluid traversing a Heated Shallow Cavity

In this work, the effects of the cavity aspect ratio (AR) on the flow of Al2O3-water nanofluid in a two-dimensional cavity subjected on its lower part to a constant and uniform temperature. The governing equations are discretized by the finite volume method based one point closure turbulence model. For nanofluid proprieties, Maxwell-Garnetts model (MG) and Brinkman models are used for the calculation respectively of the conductivity and viscosity of the nanofluid. The parameters of this study are the shape parameter of the cavity from 2 to 14, Reynolds number Re between 4.103 and 105 and volume fraction of the nanoparticles between 0 and φ=4%. The cavity aspect ratio effect on the flow structure and heat transfer was also examined. The results confirm that the flow structure and heat transfer are very sensitive to the cavity aspect ratio. The numerical results highlight the effect of the main parameters on the distribution of Nusselt number and friction coefficient.

Computation of thermo-solutal convection with soret-dufour cross diffusion in a vertical duct containing carbon/metallic nanofluids

Duct flows constitute an important category of modern thermal engineering. Optimizing efficiency has become a significant objective in the 21st century in, for example, heating ventilation and air-conditioning (HVAC), coolant or heat transfer fluid flows in a nuclear power reactor, heat exchanger design etc., and this has been achieved by either new materials (improved thermal insulation properties) constituting the duct walls, novel geometric designs, or improved working fluids. Nanotechnology has infiltrated into duct design in parallel with many other fields of mechanical, medical, and energy engineering. Motivated by the excellent potential of nanofluids, a subset of materials engineered at the nanoscale, in the present work, a new mathematical model is developed for natural convection in a rectangular vertical duct containing nanofluid. The left and right walls of the duct are maintained at constant and unequal temperatures, while the front and rear walls of the duct are insulated. Thermo-solutal (double-diffusive) natural convection of aqueous nanofluid containing various metallic nanoparticles (e. g., copper and titanium oxide) or carbon-based nanoparticles (e. g., diamond and silicon oxide) is simulated. The Tiwari–Das nanoscale volume fraction model is used in addition to the Brinkman and Maxwell models for defining the properties of the nanofluid. The partial differential conservation equations for mass, momentum and energy are non-dimensionalized via appropriate transformations and the resulting boundary value problem is solved with a second-order accurate implicit finite difference technique employing Southwell-Over-Relaxation (SOR). Mesh independence tests are conducted. Extensive visualization of the solutions for velocity, temperature, and nanoparticle concentration (volume fraction) are presented for five different nanoparticles (silicon oxide, diamond, copper, titanium oxide, and silver), thermal Grashof number, nanoparticle species (solutal) Grashof number, volume fraction of nanoparticles (i.e., percentage doping), Dufour number, Soret number, Prandtl number, Schmidt number, and duct aspect ratio. It is observed that the heat transfer rate (Nusselt number) at both the walls is maximized for diamond nanoparticles and minimized for silicon oxide nanoparticles. Further, the heat transfer rate for clear fluid is lower when compared with nanofluid, confirming that nanoparticles achieve the desired thermal enhancement at the boundaries also. The mass transfer at both walls (Sherwood number), however, is not significantly influenced by any particular type of nanoparticle, thermal and concentration Grashof number and is depleted with higher values of Dufour, Prandtl, Soret, and Schmidt numbers in addition to aspect ratio. However, Sherwood numbers at both the left and right duct walls are substantially boosted with greater solid volume fraction of nanoparticles.

Simulation of a vanadium-cerium redox flow battery incorporating graphite felt electrodes

• The reaction environment in a V-Ce redox flow battery is described by a 3D simulation. • Mass transport is calculated from known hydrodynamics in ‘flow-by’ porous electrodes. • Navier-Stokes and Brinkman models are coupled to consider two flow zones. • Mass transport is linked to electrochemical reaction rates and electrolyte potential. • The 3D simulation could predict experimental cell potential and state of charge. Computational fluid dynamics (CFD) simulations are used to predict the electrolyte dispersion, mass transport, current–potential distributions and state of charge in a vanadium-cerium redox flow battery (RFB) containing graphite felt electrodes, the half-cell flow compartments being separated by an anion exchange membrane. A polymeric mesh was placed between graphite felt and membrane; the anolyte and catholyte were pumped to separate stirred tanks in the batch recirculation mode. The simulation of single-phase flow was performed using the Brinkman and Navier-Stokes equations to describe flow dispersion within the graphite felt and polymeric mesh, respectively. At the same time, mass transport and current distribution were computed by solving mass and charge conservation equations. Fluid dynamics revealed a periodic velocity distribution within the porous electrode, which was influenced by the polymer mesh between electrode and membrane. A mean fractional conversion of 0.005 per pass was achieved. During a galvanostatic cycle, the state of charge fell from 36 to 10% over a 70 min discharge then rose from 10 to 30% during charge. A relatively uniform potential distribution was achieved along the length of the graphite felt in the presence of the low conversion per pass. The numerical model demonstrated a good agreement between the predicted and experimental cell potential and state of charge, with an average deviation below 0.12%. The unit cell RFB performance showed voltage, coulombic, and energy efficiencies of 85, 83, and 70%, respectively. A numerical model of the vanadium-cerium RFB served as the first approach to a more robust experimental study.

Unsteady thermal transport flow of Maxwell clay nanoparticles with generalized Mittag-Leffler kernel of Prabhakar's kind

The present work deals with an unsteady flow of a Maxwell fluid including clay nanoparticles via Prabhakar fractional operator. Thermal conductivity and viscosity coefficients for nanoparticles depends upon the Brinkman and Maxwell-Garnett models are computed analytically. The fractional mathematical model is developed through constitutive laws and transformed into non-dimensional form with suitable dimensionless parameters. Laplace transform is applied to obtain the generalized results for temperature and velocity distributions. The influence of volume fraction, dimensionless numbers and different fractional parameters depending upon temperature and velocity profile are justified and made up by MATHCAD software. We find several limiting solutions and compare them to the most recent literature. Furthermore, significant outcomes for clay nanoparticles with various base fluids have been reported. As a result, it is found that generalized Mittag-Leffler kernel with three parameters exhibits stronger than power law kernel and Mittag-Leffler kernel with one parameter. Water based nanoparticles shown maximum velocity with distinctive peak near the plate. For greater values of time and fractional parameters fluid properties like temperature and velocity can be enhanced.

Application of water based drilling clay-nanoparticles in heat transfer of fractional Maxwell fluid over an infinite flat surface

In the present paper, unsteady free convection flow of Maxwell fluid containing clay-nanoparticles is investigated. These particles are hanging in water, engine oil and kerosene. The values for nanofluids based on the Maxwell-Garnett and Brinkman models for effective thermal conductivity and viscosity are calculated numerically. The integer order governing equations are being extended to the novel non-integer order fractional derivative. Analytical solutions of temperature and velocity for Maxwell fluid are build using Laplace transform technique and expressed in such a way that they clearly satisfied the boundary conditions. To see the impact of different flow parameters on the velocity, we have drawn some graphs. As a result, we have seen that the fractional model is superior in narrate the decay property of field variables. Some limiting solutions are obtained and compared with the latest existing literature. Moreover, significant results can be observed for clay nanoparticles with different base fluids.

Existence of weak solutions to multiphase Cahn–Hilliard–Darcy and Cahn–Hilliard–Brinkman models for stratified tumor growth with chemotaxis and general source terms

We investigate a multiphase Cahn–Hilliard model for tumor growth with general source terms. The multiphase approach allows us to consider multiple cell types and multiple chemical species (oxygen and/or nutrients) that are consumed by the tumor. Compared to classical two-phase tumor growth models, the multiphase model can be used to describe a stratified tumor exhibiting several layers of tissue (e.g., proliferating, quiescent and necrotic tissue) more precisely. Our model consists of a convective Cahn–Hilliard type equation to describe the tumor evolution, a velocity equation for the associated volume-averaged velocity field, and a convective reaction-diffusion type equation to describe the density of the chemical species. The velocity equation is either represented by Darcy’s law or by the Brinkman equation. We first construct a global weak solution of the multiphase Cahn–Hilliard–Brinkman model. After that, we show that such weak solutions of this system converge to a weak solution of the multiphase Cahn–Hilliard–Darcy system as the viscosities tend to zero in some suitable sense. This means that the existence of a global weak solution to the Cahn–Hilliard–Darcy system is also established.

Effect of viscous shearing stresses on optimal material designs for flow of fluids through porous media

Topology optimization offers optimal material layouts, enabling automation in the design of devices. Given the recent advances in computer technology and additive manufacturing, topology optimization is increasingly being used to design complex porous structures, for example, microfluidic devices. For the flow of fluids in such miniature-sized porous structures, viscous shearing stresses will be significant. However, the Darcy model—the most popular mathematical model describing the flow of a single-phase incompressible fluid in rigid porous media—neglects the internal friction arising from viscous shearing stresses. We, therefore, develop a material design framework under the topology optimization based on the Darcy–Brinkman model—a mathematical model for the flow of fluids through porous media that accounts for internal friction besides the drag considered in the Darcy model. The proposed framework uses the total rate of mechanical dissipation—a physical quantity—for the objective function. To understand the effect of viscous shearing stresses on the design, we compare the optimal material layouts provided by the Darcy and Darcy–Brinkman models under topology optimization. In particular, we show, using analytical solutions corroborated by numerical simulations, that the optimal material layouts are alike for the class of problems exhibiting axisymmetry, for which viscous shearing stresses vanish. These analytical solutions will be valuable to check the veracity of numerical simulators. For those problems with dominant viscous shearing stresses (e.g., flow in a backward-facing step domain), we show, using numerical solutions, the optimal material layouts under the Darcy and Darcy–Brinkman models are very different. Also, the associated solution fields (i.e., velocity and pressure) are qualitatively different under their respective optimal layouts for these two models.

Effect of Nanoparticles and Base Fluid Types on Natural Convection in a Three-Dimensional Cubic Enclosure

Convective heat transfer using nanofluids play an important role in thermal applications such as heat exchangers, automotive industries, and power generation. In this work, a numerical analysis is conducted to examine the heat transfer of nanofluid in three-dimensional differentially heated cavity. The finite volume method-based SIMPLEC algorithm is used to solve the system of the mass, momentum, and energy transfer governing equations. The left and the right vertical side walls of the cube are maintained at constant temperatures T C and T H , respectively. The remaining walls of the cube are insulated. Effective thermal conductivity and viscosity of the nanofluid are determined using Brinkman and Maxwell models, respectively. Studies are carried out for three types of nanoparticles and volume fractions of nanoparticles ( 0 – 5 % ). The effects of two binary liquid mixtures as a base fluid (propylene glycol-water and ethylene glycol-water) are also examined. Results show an enhancement of 13 % for Al2O3-EG in comparison to pure ethylene glycol in the case of Ra = 10 3 . In addition, heat transfer enhancement was increased with the rise of nanoparticle volume fractions.

Numerical simulation of magnetohydrodynamics nanofluid flow in a semi-porous channel with a new approach in the least square method

Present study deals with the flow of a fluid with added nanoparticles in a semi-permeable channel with a transverse magnetic effect. We have used Maxwell-Garnett and Brinkman models for effective electrical conductivity and viscosity, respectively. Water-based nanofluids with silver and copper nanoparticles have been used in the present investigation. Firstly, governing differential equations have been nondimensionalized with appropriate similarity transformations. We have also proposed an efficient technique to handle coupled differential equations by the weighted residual method (WRM). Here the least square method has been modified to solve coupled linear/nonlinear differential equations. Residual errors are examined graphically to see the efficiency of the proposed scheme. By taking particular values of Hartmann number, Reynolds number, and volume fraction viz. 0.5, 0.5, and 0.05, respectively, it is seen that residual error almost equal to zero in 6th term approximation. Moreover, for validation, present results have been incorporated with previously published works in special cases. Finally, the nature of the solutions has been reviewed by varying the values of the involved parameters. In this regard, the values of volume fraction are taken in between 0.01 and 0.09. Further, the values of the Hartmann number are considered between 0 and 4, and finally, values of the Reynolds number are taken in between 1 and 5.

Double diffusion in a rectangular duct using metals or oxides suspended in a viscous fluid

In this study, double diffusive free convection of nanofluid within a confined rectangular duct is investigated numerically. The momentum and energy equations are placed in the form of difference equations and solved numerically. The left wall conditions for the concentration and temperature are lesser than those of the right wall and the upper and lower walls are insulated. Different nanofluids are considered such as mixtures with copper, diamond, silicon oxide and titanium oxide, suspended in water. Brinkman and Maxwell models are used to characterize the nanofluid. Tiwari and Das model is opted to define the nanofluid behavior. The simulations are conducted using different nanoparticles, thermal Grashof number 1 ≤ GrT ≤ 20, solute Grashof number 1 ≤ GrC ≤ 15, solid volume fraction 0 ≤ Φ ≤ 0.05, Dufour number 0 ≤ Df ≤ 1, Brinkman number 0 ≤ Br ≤ 2, and Soret number 0 ≤ Sr ≤ 5. Additionally, behavior of volumetric flow strength, skin friction, heat transport intensity and Sherwood number is also examined. The thermal Grashof number, Brinkman number, Dufour, Soret and Schmidt parameters accelerate the velocity and temperature and dwindle the concentration whereas the reversal effect was obtained for the solid volume fraction. The concentration Grashof number diminishes the velocity and temperature and intensifies the concentration. The silver nanoparticles produce the highest velocity whereas diamond nanoparticles cause the lowest velocity and temperature. The maximum temperature is attained with silicon oxide.