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.     

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.     

Nanofluid bioconvection: interaction of microorganisms oxytactic upswimming, nanoparticle distribution, and heating/cooling from below

The aim of this paper is to develop a theory describing the onset of convection instability (called here nanofluid bioconvecion) that is induced by simultaneous effects produced by oxytactic microorganisms, nanoparticles, and vertical temperature variation. The theory is developed for the situation when the nanofluid occupies a shallow horizontal layer of finite depth. The layer is defined as shallow as long as oxygen concentration at the bottom of the layer is above the minimum concentration required for the bacteria to be active (to actively swim up the oxygen gradient). The lower boundary of the layer is assumed rigid, while at the upper boundary both situations when the boundary is rigid or stress free are considered. Physical mechanisms responsible for the slip velocity between the nanoparticles and the base fluid, such as Brownian motion and thermophoresis, are accounted for in the model. A linear instability analysis is performed, and the resulting eigenvalue problem is solved analytically using the Galerkin method.     

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.     

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.     

The onset of double-diffusive convection in a nanofluid layer

The onset of double-diffusive convection in a horizontal layer of a nanofluid is studied analytically. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. In addition the thermal energy equations include regular diffusion and cross-diffusion terms. The stability boundaries for both non-oscillatory and oscillatory cases have been approximated by simple analytical expressions. Physical significance of the obtained results is 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.     

Radiative heat transfer in a hydromagnetic nanofluid past a non-linear stretching surface with convective boundary condition

Heat transfer characteristics of a two-dimensional steady hydromagnetic natural convection flow of nanofluids over a non-linear stretching sheet taking into account the effects of radiation and convective boundary condition has been investigated numerically. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. The local similarity solutions are obtained by using very robust computer algebra software Maple 13. The results corresponding to the dimensionless temperature profiles and the reduced Nusselt number, Sherwood number and skin friction coefficient are displayed graphically for various pertinent parameters. The results show that temperature within the boundary layer is enhanced with the increase of the Biot number, buoyancy due to nanoparticle concentration, strength of the applied magnetic field, Brownian motion parameter, and thermophoresis parameter. An opposite trend is observed for the increase of the buoyancy due to temperature, stretching index, and the radiation parameter. The results also show that the local rate of heat transfer strongly depends on the nonlinear stretching index, radiation parameter, Biot number, Brownian motion parameter, and thermophoresis parameter.     

Estimating the Hydraulic Conductivity of Two-Dimensional Fracture Networks Using Network Geometric Properties

Based on a modified Darcy–Brinkman–Maxwell model, a linear stability analysis of a Maxwell fluid in a horizontal porous layer heated from below by a constant flux is carried out. The non-oscillatory instability and oscillatory instability with different hydrodynamic boundaries such as rigid and free surfaces at the bottom are studied. Compared with the rigid surface cases, onset of fluid motion due to non-oscillatory instability and oscillatory instability is found to occur both more easily for the system with a free bottom surface. The critical Rayleigh number for onset of fluid motion due to non-oscillatory instability is lower with a constant flux heating bottom than with an isothermal heating bottom, but the result due to oscillatory instability is in contrast. The effects of the Darcy number, the relaxation time, and the Prandtl number on the critical Rayleigh number are also discussed.     

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.     

Analysis of periodic and aperiodic convective stability of double diffusive nanofluid convection in rotating porous layer

The onset of periodic and aperiodic convection in a binary nanofluid saturated rotating porous layer is studied considering constant flux boundary conditions. The porous medium obeys Darcy’s law, while the nanofluid envisages the effects of the Brownian motion and thermophoresis. The Rayleigh numbers for stationary and oscillatory convection are obtained in terms of various non-dimensional parameters. The effect of the involved physical parameters on the aperiodic convection is studied graphically. The results are validated in comparison with the published literature in limiting cases of the present study.     

Natural Convection in a Nanofluid Saturated Rotating Porous Layer: A Nonlinear Study

The present paper deals with linear and nonlinear analysis of thermal instability in a rotating porous layer saturated by a nanofluid. Momentum equation with Brinkman term, involving the Coriolis term and incorporating the effect of Brownian motion along with thermophoresis has been considered. Linear stability analysis is done using normal mode technique, while for nonlinear analysis, a minimal representation of the truncated Fourier series, involving only two terms, has been used. Stationary and oscillatory modes of convection have been studied. A weak nonlinear analysis is used to obtain the concentration and thermal Nusselt numbers. The behavior of the concentration and thermal Nusselt numbers is investigated by solving the finite amplitude equations using a numerical method. Obtained results have been presented graphically and discussed in details.     

Instability of binary nanofluids with magnetic field

The present paper investigates the effects of a vertical magnetic field on the double diffusive nanofluid convection. The effects of the Brownian motion and thermophoresis due to the presence of nanoparticles and the effects of the Dufour and Soret parameters due to the presence of solute are included in the investigated model. The normal mode technique is used to solve the conservation equations. For the analytical study, valid approximations are made in the complex expression for the Rayleigh number to get useful and interesting results. The bottom heavy binary nanofluids are more stable than the regular binary fluids, while the top heavy binary nanofluids are less stable than the regular binary fluids. The critical wave number and the critical Rayleigh number increase whereas the frequency of oscillation (for the bottom heavy configuration) decreases when the Chandrasekhar number increases. The numerical results for the alumina-water nanofluid are studied by use of the MATHEMATICA software.     

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.     

Flow of Oldroyd-B fluid with nanoparticles and thermal radiation

The two-dimensional boundary layer flow of an Oldroyd-B fluid in the presence of nanoparticles is investigated. Convective heat and mass conditions are considered in the presence of thermal radiation and heat generation. The Brownian motion and thermophoresis effects are retained. The nonlinear partial differential equations are reduced into the ordinary differential equation (ODE) systems. The resulting ODE systems are solved for the series solutions. The results are analyzed for various physical parameters of interest. Numerical values of the local Nusselt and Sherwood numbers are also computed and analyzed.     

Unsteady heat and mass transfer in a rotating nanofluid layer

The present paper studies the effect of rotation on the thermal instability in a horizontal layer of a Newtonian nanofluid which incorporates the effect of Brownian motion along with thermophoresis. In order to find the concentration and the thermal Nusselt numbers for unsteady state, a nonlinear analysis, using a minimal representation of the truncated Fourier series of two terms, has been performed. The results obtained are then presented graphically. It is observed that rotation delays the rate of heat and mass transferred, representing a delay in the onset on convection. This shows a stabilizing effect for a rotating system against a nonrotating system.     

Convective Transport in a Nanofluid Saturated Porous Layer With Thermal Non Equilibrium Model

The effect of local thermal non-equilibrium on linear and non-linear thermal instability in a horizontal porous medium saturated by a nanofluid has been investigated analytically. The Brinkman Model has been used for porous medium, while nanofluid incorporates the effect of Brownian motion along with thermophoresis. A three-temperature model has been used for the effect of local thermal non-equilibrium among the particle, fluid, and solid-matrix phases. The linear stability 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. The critical conditions for the onset of convection and the heat and mass transfer across the porous layer have been obtained numerically.     

Effects of a.c. Electric Field and Rotation on Bénard–Marangoni Convection

The coupled buoyancy and thermocapillary instability, the Bénard–Marangoniproblem, in an electrically conducting fluid layer whose upper surface is deformed and subject to a temperature gradient is studied. Both influences of an a.c. electric field and rotation are investigated. Special attention is directed at the occurrence of convection both in the form of stationary motion and oscillatory convection. The linear stability problem is solved for different values of the relevant dimensionless numbers, namely the a.c. electric Rayleigh number, the Taylor, Rayleigh, Biot, Crispation and Prandtl numbers. For steady convection, it is found that by increasing the angular velocity, one reinforces the stability of the fluid layer whatever the values of the surface deformation and the applied a.c. electric field. We have also determined the regions of oscillatory instability and discussed the competition between both stationary and oscillatory convections. This revised version was published online in July 2006 with corrections to the Cover Date.     

Magnetohydrodynamic and heat transfer effects on the stagnation-point flow of an electrically conducting nanofluid past a porous vertical shrinking/stretching sheet in the presence of variable stream conditions

A steady two-dimensional magnetohydrodynamic stagnation-point flow of an electrically conducting fluid and heat transfer with thermal radiation of a nanofluid past a shrinking and stretching sheet is investigated numerically. The model used for the nanofluid incorporates the effects of the Brownian motion and thermophoresis. A similarity transformation is used to convert the governing nonlinear boundary-layer equations into coupled higher-order nonlinear ordinary differential equations. The result shows that the velocity, temperature, and concentration profiles are significantly influenced by the Brownian motion, heat radiation, and thermophoresis particle deposition.