The onset of convection in a nanofluid saturated porous layer using Darcy model with cross diffusion

Linear and nonlinear stability analysis for the onset of convection in a horizontal layer of a porous medium saturated by a nanofluid is studied. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. The modified Darcy equation that includes the time derivative term is used to model the momentum equation. In conjunction with the Brownian motion, the nanoparticle fraction becomes stratified, hence the viscosity and the conductivity are stratified. The nanofluid is assumed to be diluted and this enables the porous medium to be treated as a weakly heterogeneous medium with variation, in the vertical direction, of conductivity and viscosity. The critical Rayleigh number, wave number for stationary and oscillatory mode and frequency of oscillations are obtained analytically using linear theory and the non-linear analysis is made with minimal representation of the truncated Fourier series analysis involving only two terms. The effect of various parameters on the stationary and oscillatory convection is shown pictorially. We also study the effect of time on transient Nusselt number and Sherwood number which is found to be oscillatory when time is small. However, when time becomes very large both the transient Nusselt value and Sherwood value approaches to their steady state values.     

The Onset of Double-Diffusive Nanofluid Convection in a Layer of a Saturated Porous Medium

The onset of convection in a horizontal layer of a porous medium saturated by a nanofluid is studied analytically. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. For the porous medium, the Brinkman model is employed. Three cases of free–free, rigid–rigid, and rigid–free boundaries are considered. The analysis reveals that for a typical nanofluid (with large Lewis number), the prime effect of the nanofluids is via a buoyancy effect coupled with the conservation of nanoparticles, whereas the contribution of nanoparticles to the thermal energy equation is a second-order effect. It is found that the critical thermal Rayleigh number can be reduced or increased by a substantial amount, depending on whether the basic nanoparticle distribution is top-heavy or bottom-heavy, by the presence of the nanoparticles. Oscillatory instability is possible in the case of a bottom-heavy nanoparticle distribution.     

Numerical simulation of the nanoparticle diameter effect on the thermal performance of a nanofluid in a cooling chamber

The thermal performance of a nanofluid in a cooling chamber with variations of the nanoparticle diameter is numerically investigated. The chamber is filled with water and nanoparticles of alumina (Al₂O₃). Appropriate nanofluid models are used to approximate the nanofluid thermal conductivity and dynamic viscosity by incorporating the effects of the nanoparticle concentration, Brownian motion, temperature, nanoparticles diameter, and interfacial layer thickness. The horizontal boundaries of the square domain are assumed to be insulated, and the vertical boundaries are considered to be isothermal. The governing stream-vorticity equations are solved by using a secondorder central finite difference scheme coupled with the mass and energy conservation equations. The results of the present work are found to be in good agreement with the previously published data for special cases. This study is conducted for the Reynolds number being fixed at Re = 100 and different values of the nanoparticle volume fraction, Richardson number, nanofluid temperature, and nanoparticle diameter. The results show that the heat transfer rate and the Nusselt number are enhanced by increasing the nanoparticle volume fraction and decreasing the Richardson number. The Nusselt number also increases as the nanoparticle diameter decreases.     

Free Convection Boundary Layer Flow from a Heated Upward Facing Horizontal Flat Plate Embedded in a Porous Medium Filled by a Nanofluid with Convective Boundary Condition

The steady laminar incompressible free convective flow of a nanofluid over a permeable upward facing horizontal plate located in porous medium taking into account the thermal convective boundary condition is studied numerically. The nanofluid model used involves the effect of Brownian motion and the thermophoresis. Using similarity transformations the continuity, the momentum, the energy, and the nanoparticle volume fraction equations are transformed into a set of coupled similarity equations, before being solved numerically, by an implicit finite difference numerical method. Our analysis reveals that for a true similarity solution, the convective heat transfer coefficient related with the hot fluid and the mass transfer velocity must be proportional to x ^−2/3, where x is the horizontal distance along the plate from the origin. Effects of the various parameters on the dimensionless longitudinal velocity, the temperature, the nanoparticle volume fraction, as well as on the rate of heat transfer and the rate of nanoparticle volume fraction have been presented graphically and discussed. It is found that Lewis number, the Brownian motion, and the convective heat transfer parameters increase the heat transfer rate whilst the thermophoresis decreases the heat transfer rate. It is also found that Lewis number and the convective heat transfer parameter enhance the nanoparticle volume fraction rate whilst the thermophoresis parameter decreases nanoparticle volume fraction rate. A very good agreement is found between numerical results of the present article for special case and published results. This close agreement supports the validity of our analysis and the accuracy of the numerical computations.     

Natural Convection in a Nanofluid-Saturated Rotating Porous Layer: A More Realistic Approach

The present work aims at studying the thermal instability in a rotating porous layer saturated by a nanofluid based on a new boundary condition for the nanoparticle fraction, which is physically more realistic. The model used for nanofluid combines the effect of Brownian motion along with thermophoresis, while for a porous medium Brinkman model has been used. A more realistic set of boundary conditions where the nanoparticle volume fraction adjusts itself including the contributions of the effect of thermophoresis so that the nanoparticle flux is zero at the boundaries has been considered. Using linear stability analysis, the expression for critical Rayleigh number has been obtained in terms of various non-dimensional parameters. The effect of various parameters on the onset of instability has been presented graphically and discussed in detail.     

Thermal Instability in a Porous Medium Layer Saturated by a Nanofluid: Brinkman Model

The onset of convection in a horizontal layer of a porous medium saturated by a nanofluid is studied analytically. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. For the porous medium, the Brinkman model is employed. Three cases of free–free, rigid–rigid, and rigid–free boundaries are considered. The analysis reveals that for a typical nanofluid (with large Lewis number), the prime effect of the nanofluids is via a buoyancy effect coupled with the conservation of nanoparticles, whereas the contribution of nanoparticles to the thermal energy equation is a second-order effect. It is found that the critical thermal Rayleigh number can be reduced or increased by a substantial amount, depending on whether the basic nanoparticle distribution is top-heavy or bottom-heavy, by the presence of the nanoparticles. Oscillatory instability is possible in the case of a bottom-heavy nanoparticle distribution.     

Natural Convection in a Nanofluid Saturated Rotating Porous Layer with Thermal Non-Equilibrium Model

In the present article, we study the effect of local thermal non-equilibrium on the linear and non-linear thermal instability in a nanofluid saturated rotating porous layer. The Darcy Model has been used for the porous medium, while the nanofluid layer incorporates the effect of Brownian motion along with thermophoresis. A three-temperature model is been used for the effect of local thermal non-equilibrium among the particle, fluid, and solid–matrix phases. The linear stability analysis is based on normal mode technique, while for nonlinear analysis a minimal representation of the truncated Fourier series analysis involving only two terms has been used.     

Laminar Free Convection Over a Vertical Wavy Surface Embedded in a Porous Medium Saturated with a Nanofluid

Numerical analysis is performed to examine laminar free convective of a nanofluid along a vertical wavy surface saturated porous medium. In this pioneering study, we have considered the simplest possible boundary conditions, namely those in which both the temperature and the nanoparticle fraction are constant along the wall. Non-similar transformations are presented for the governing equations and the obtained PDE are then solved numerically employing a fourth order Runge–Kutta method with shooting technique. A detailed parametric study (nanofluid parameters) is performed to access the influence of the various physical parameters on the local Nusselt number and the local Sherwood number. The results of the problem are presented in graphical forms and discussed.     

Effect of variable thermal conductivity models on the combined convection heat transfer in a square enclosure filled with a water–alumina nanofluid

In this numerical study, the effects of variable thermal conductivity models on the combined convection heat transfer in a two-dimensional lid-driven square enclosure are investigated. The fluid in the square enclosure is a water-based nanofluid containing alumina nanoparticles. The top and bottom horizontal walls are insulated, while the vertical walls are kept at different constant temperatures. Five different thermal conductivity models are used to evaluate the effects of various parameters, such as the nanofluid bulk temperature, nanoparticle size, nanoparticle volume fraction, Brownian motion, interfacial layer thickness, etc. The governing stream–vorticity equations are solved by using a second-order central finite difference scheme coupled with the conservation of mass and energy. It is found that higher heat transfer is predicted when the effects of the nanoparticle size and bulk temperature of the nanofluid are taken into account.     

Radiation Effects on Mixed Convection over a Wedge Embedded in a Porous Medium Filled with a Nanofluid

The problem of steady, laminar, mixed convection boundary-layer flow over an isothermal vertical wedge embedded in a porous medium saturated with a nanofluid is studied, in the presence of thermal radiation. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis with Rosseland diffusion approximation. The wedge surface is maintained at a constant temperature and a constant nanoparticle volume fraction. The resulting governing equations are non-dimensionalized and transformed into a non-similar form and then solved by Keller box method. A comparison is made with the available results in the literature, and our results are in very good agreement with the known results. A parametric study of the physical parameters is made, and a representative set of numerical results for the velocity, temperature, and volume fraction, the local Nusselt and Sherwood numbers are presented graphically. The salient features of the results are analyzed and discussed.     

Radiation effects on mixed convection about a cone embedded in a porous medium filled with a nanofluid

The problem of steady, laminar, mixed convection boundary-layer flow over a vertical cone embedded in a porous medium saturated with a nanofluid is studied, in the presence of thermal radiation. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis with Rosseland diffusion approximation. The cone surface is maintained at a constant temperature and a constant nanoparticle volume fraction. The resulting governing equations are non-dimensionalized and transformed into a non-similar form and then solved by Keller box method. A comparison is made with the available results in the literature, and our results are in very good agreement with the known results. A parametric study of the physical parameters is made and a representative set of numerical results for the local Nusselt and Sherwood numbers are presented graphically. Also, the salient features of the results are analyzed and discussed.     

Thermal Instability in a Porous Medium Layer Saturated with a Viscoelastic Nanofluid

The onset of convection in a horizontal layer of a porous medium saturated with a viscoelastic nanofluid was studied in this article. The modified Darcy model was applied to simulate the momentum equation in porous media. An Oldroyd-B type constitutive equation was used to describe the rheological behavior of viscoelastic nanofluids. The model used for the viscoelastic nanofluid incorporates the effects of Brownian motion and thermophoresis. The onset criterion for stationary and oscillatory convection was analytically derived. The effects of the concentration Rayleigh number, Prandtl number, Lewis number, capacity ratio, relaxation, and retardation parameters on the stability of the system were investigated. Oscillatory instability is possible in both bottom- and top-heavy nanoparticle distributions. Results indicated that there is competition among the processes of thermophoresis, Brownian diffusion, and viscoelasticity that causes the convection to set in through oscillatory rather than stationary modes. Regimes of stationary and oscillatory convection for various parameters were derived and are discussed in detail.     

Onset of Convection with Internal Heating in a Porous Medium Saturated by a Nanofluid

Linear stability analysis was applied to the onset of convection due to internal heating in a porous medium saturated by a nanofluid. A model in which the effects of thermophoresis and Brownian motion are taken into account is employed. We utilized more realistic boundary conditions than in the previous work on this subject; now the nanofluid particle fraction is allowed to adapt to the temperature profile induced by the internal heating, subject to the requirement that there is zero perturbation flux across a boundary. The results show that the presence of the nanofluid particles leads to increased instability of the system. We identified two combinations of dimensionless parameters that are the major controllers of convection instability in the layer.     

The Effect of Vertical Throughflow on Thermal Instability in a Porous Medium Layer Saturated by a Nanofluid

The effect of vertical throughflow on the onset of convection in a horizontal layer of a porous medium saturated by a nanofluid is studied analytically. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. The dependences of the critical Rayleigh number for the non-oscillatory and oscillatory modes of instability on the thermophoresis and Brownian motion parameters for the cases with and without throughflow are investigated.     

Effects of walls temperature variation on double diffusive natural convection of Al2O3–water nanofluid in an enclosure

The effect of wall temperature variations on double diffusive natural convection of Al₂O₃–water nanofluid in a differentially heated square enclosure with constant temperature hot and cold vertical walls is studied numerically. Transport mechanisms of nanoparticles including Brownian diffusion and thermophoresis that cause heterogeneity are considered in non-homogeneous model. The hot and cold wall temperatures are varied, but the temperature difference between them is always maintained 5 °C. The thermophysical properties such as thermal conductivity, viscosity and density and thermophoresis diffusion and Brownian motion coefficients are considered variable with temperature and volume fraction of nanoparticles. The governing equations are discretized using the control volume method. The results show that nanoparticle transport mechanisms affect buoyancy force and cause formation of small vortexes near the top and bottom walls of the cavity and reduce the heat transfer. By increasing the temperature of the walls the effect of transport mechanisms decreases and due to enhanced convection the heat transfer rate increases.     

Effect of Local Thermal Non-equilibrium on the Onset of Convection in a Porous Medium Layer Saturated by a Nanofluid

The onset of convection in a horizontal layer of a porous medium saturated by a nanofluid is analytically studied. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. For the porous medium, the Darcy model is employed. The effect of local thermal non-equilibrium among the particle, fluid, and solid-matrix phases is investigated using a three-temperature model. The analysis reveals that in some circumstances the effect of LTNE can be significant, but for a typical dilute nanofluid (with large Lewis number and with small particle-to-fluid heat capacity ratio) the effect is small.     

Analytical and numerical solutions for axisymmetric flow of nanofluid due to non-linearly stretching sheet

This article reports the laminar axisymmetric flow of nanofluid over a non-linearly stretching sheet. The model used for nanofluid contains the simultaneous effects of Brownian motion and thermophoretic diffusion of nanoparticles. The recently proposed boundary condition is considered which requires the mass flux of nanoparticles at the wall to be zero. Analytic solutions of the arising boundary value problem are obtained by optimal homotopy analysis method. Moreover the numerical solutions are computed by Keller–Box method. Both the solutions are found in excellent agreement. The behavior of Brownian motion on the fluid temperature and wall heat transfer rate is insignificant. Further the nanoparticle volume fraction distribution is found to be negative near the vicinity of the stretching sheet.     

Analytical considerations of flow/thermal coupling of nanofluids in foam metals with local thermal non-equilibrium (LTNE) phenomena and inhomogeneous nanoparticle distribution

Foam metals with micro pores own excellent thermal performance, however, poor heat conductive ability of most heat-transfer fluids restricts further heat transfer improvement. Combination of foam metal and nanofluid with highly conductive nanoparticles is a promising solution. Convective thermal characteristics of nanofluids in porous foams are theoretically investigated in this work. Effects of Brownian motion and thermophoretic diffusion of nanoparticles in the base fluid on thermal performance are considered. The nanoparticle and the base-fluid are considered to be in thermal equilibrium and the temperature difference between the nanofluid and foam ligaments is especially considered. Compared with the base-fluid flow in a duct, the velocity distribution for the nanofluid flow in a porous foam is more uniform with a decreased dimensionless temperature. The pressure drop of the nanofluid increases with an increase in the concentration of the nanoparticles. By employing foam metals and nanofluid, the cross-sectional temperature becomes closer to the wall temperature. Simultaneously, notable difference between solid and fluid temperatures can be observed, revealing the LTNE effect of the nanofluid on the porous foam. It is found that the Nusselt number first increases and then decreases with an increase in nanoparticle concentration. Furthermore, the Nusselt number decreases with an increase in the foam porosity. It is found that the thermal performance of a nanofluid in a plain tube is different from that in the foam metals.     

Mathematical modelling of thermal conductivity for nanofluid considering interfacial nano-layer

Maxwell’s classical model for predicting effective thermal conductivity of colloidal solution predicts the thermal conductivity of nanofluids quite satisfactorily. However, Maxwell’s model does not consider the effect of interfacial layer, Brownian motion of nano-particle and nanoparticle aggregation. In this paper, the effect of interfacial layer on thermal conductivity is considered. A simple expression has been derived to determine thermal conductivity of nanofluid considering interfacial layer formed on the nano particles. The thermal conductivity of the interfacial layer has been precisely determined and results are found to be closer to the experimental values, hence, further improving the results of classical Maxwell model.     

Non-linear Convective Transport in a Binary Nanofluid Saturated Porous Layer

In this article, we study double-diffusive convection in a horizontal porous medium saturated by a nanofluid, for the case when the base fluid of the nanofluid is itself a binary fluid such as salty water. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis, while the Darcy model is used for the porous medium. The thermal energy equations include the diffusion and cross-diffusion terms. The linear stability is studied using normal mode technique and for non-linear analysis, a minimal representation of the truncated Fourier series analysis involving only two terms has been used. For linear theory analysis, critical Rayleigh number has been obtained, while non-linear analysis has been done in terms of the Nusselt numbers.