**2. Instability of fluids under small temperature gradient: Rayleigh Bênard convection**

The convective motions occur in a fluid layer heated underside in which a small temperature gradient is maintained across its boundaries. The maintained temperature across the boundaries must surpass a certain value before the instability can manifest itself. This Phenomenon was discovered by Bénard [26] in 1900. In most of his experiments, he found that if a fluid layer is heated underside, the layer at the bottom expands due to higher temperature. This makes the fluid density lighter at the bottom than that on the top making the system top heavy. Here viscosity and thermal diffusivity tend to oppose the convective motions but with the application of higher temperature gradient across the fluid layer, the thermal convection process gets initiated showing the pattern of cellular motions (called Bénard convection). Bénard [27] performed an experiment with metallic plate with a thin non-volatile liquid layer of 1 mm depth maintained under constant temperature.

Keeping the upper layer of fluid exposed to free air, he observed that the fluid layer was decomposed into number of cells (showing cellular motion) called Bénard cells. Thus in the standard Bénard problem, density difference due to variation in

temperature of nanofluids. Tzou [35, 36] investigated the onset of convective instability of nanofluids using Buongiorno's model analytically and established that nanofluids exhibit much lower stability than regular fluids. The results depict the inverse relationship of density and heat capacity of nanoparticles with their thermal conductivity and the shape factor. The results also include that the heat transfer coefficient of nanofluid is enhanced relative to volume fraction of nanoparticles. Nield and Kuznetsov [37] reconsidered the instability problem for nanofluids to get the expression for Rayleigh number and found conditions for the existence of oscillatory convection. It was established that the buoyancy coupled with the conservation of nanoparticles lead to higher instability of nanofluids. Alloui et al. [38] considered the shallow cavity to study Rayleigh Bénard instability for nanofluids. They concluded that rate of heat transfer in nanofluids depends on the strength of

*Convection Currents in Nanofluids under Small Temperature Gradient*

*DOI: http://dx.doi.org/10.5772/intechopen.88887*

convection and volume fraction of nanoparticles while the presence of

**3. Conservation equations for a nanofluid layer**

motion except the term representing the external body force.

pared to that of regular fluids.

nanoscale effects as follows.

**117**

coefficient due to Brownian motion is given by

nanoparticles increase the stability of the system. Thermal instability for a horizontal nanofluid layer was considered by Yadav et al. [39]. They found an expression of thermal Rayleigh number and observed that the temperature gradient delays the convective motions while volumetric fraction of nanoparticles and the ratio of the density of nanoparticles to that of base fluid have destabilizing impact on the layer. The joint behaviour of nano-effects (Brownian motion and thermophoresis) creates destabilizing effect and can reduce the values of critical Rayleigh number as com-

We start this section with the description of Boussinesq approximation which is

As is the case of regular fluid [32], equations of nanofluids are difficult to solve because of their non-linear character. Therefore some mathematical approximations are to be used to simplify the basic equations without violating the physical laws. The contribution of Boussinesq [40] in the solution of thermal instability problems is in the form of approximations which is after his name. This approximation has been used by a many researchers for solving different problems of fluids. Boussinesq suggested that inertial effects of density variations can be neglected as compared to its gravitational effects as such situations exist in the domain of meteorology and oceanography. So, density is assumed to be constant everywhere in the equations of motion except in the term with external force. Therefore, we change *ρ*0½ � 1 þ *α*ð Þ *T*<sup>0</sup> � *T* by *ρ*<sup>0</sup> everywhere in the equations of

Anoop et al. [41] explained various experimental techniques using which nanoparticles can be suspended in the base fluid and that suspension remain stable for several weeks. Buongiorno [6] adopted the formalism of Bird et al. [42] and Chandrasekhar [32] to write conservation equations for nanofluids by considering nanoscale effects; Brownian diffusion and thermophoresis. A model for convective transport in regular fluids was reformulated for nanofluids to accommodate these

The random motion of nanoparticles is called Brownian motion and results into the continuous collisions with the base fluid molecules. The Brownian diffusion

, (1)

*DB* <sup>¼</sup> *kBT* 3*πμdp*

used to write the conservation equations of nanofluids in simplified form.

**Figure 2.** *A view of Rayleigh-Bénard convection.*

temperature across the upper and lower boundaries of the fluid becomes the main reason for the occurrence of instability. **Figure 2** shows the schematic representation of Rayleigh-Bénard convection. Rayleigh [28] was the first person who gave an analytical treatment of the problem related to identifying the conditions responsible for breakdown of basic state. As a subsequent work carried out by Rayleigh and Bénard, thermal instability of fluids is known as Rayleigh Bénard convection. The condition for convective motions (depends on temperature gradient) can be represented in dimensionless form by the critical Rayleigh number. He figured out the condition for the instability of free surfaces by showing that the instability would occur on a large temperature gradient *β* ¼ �*dT=dz* in such a way that the Rayleigh number; *RA* <sup>¼</sup> *αβgd*<sup>4</sup> *=κν*, exceeds a certain critical value; where acceleration due to gravity is represented by g, coefficient of thermal expansion by *α*, the depth of the layer by d, thermal diffusivity by *κ* and kinematic viscosity is given by *ν*. For the stabilizing viscous force, *RA* parameter gives the force of destabilizing buoyancy. Chandra [29] found discrepancy between the theoretical and experimental work for the convective motions in fluids when heated underside. He explained it by conducting an experiment on the layer in air and observed that instability of the fluid layer was dependent on its depth. A simplification in the partial differential equations describing the flow of compressible fluid is done by Spiegel and Veronis [30] by assuming very small depth of the layer as compared to the height. The basic equations of a fluid layer in porous medium (when heated underside) were formulated and derived by Joseph [31] by using Boussinesq approximation. The problem of thermal convection of a fluid layer has been put forward by Chandrasekhar [32] by considering the implications of various aspects of hydrodynamics and hydromagnetics. He depicted the result that addition of rotation and magnetic field increases the stability of the system. Kim et al. [33] considered the same problem of thermal convection for nanofluids. They showed that convective motion directly depends on the two physical properties (heat capacity and density) of nanoparticles and adversely depends on the conductivity of nanoparticles. Buongiorno [6] was the first scientist who formulated the conservation equations of nanofluids by assimilating the effects of diffusion due to Brownian motion and thermophoresis of nanoparticles. During his analysis, he concluded that Brownian and thermophoretic diffusion play a significant role in the absence of turbulent effects as compared to other seven mechanisms. Hwang et al. [34] treated this problem analytically and put forth the result of thermal instability of water based nanofluid with alumina nanoparticles in a rectangular container which is heated from below. They found that stability of the base fluid is enhanced by adding alumina nanoparticles and further it is enhanced by increasing the volume fraction of nanoparticles, the average temperature of the nanofluids and by decreasing the size of nanoparticles. They observed the decrease in heat transfer coefficient of nanofluids with the increase of the size of nanoparticles and decrease in the

*Convection Currents in Nanofluids under Small Temperature Gradient DOI: http://dx.doi.org/10.5772/intechopen.88887*

temperature of nanofluids. Tzou [35, 36] investigated the onset of convective instability of nanofluids using Buongiorno's model analytically and established that nanofluids exhibit much lower stability than regular fluids. The results depict the inverse relationship of density and heat capacity of nanoparticles with their thermal conductivity and the shape factor. The results also include that the heat transfer coefficient of nanofluid is enhanced relative to volume fraction of nanoparticles. Nield and Kuznetsov [37] reconsidered the instability problem for nanofluids to get the expression for Rayleigh number and found conditions for the existence of oscillatory convection. It was established that the buoyancy coupled with the conservation of nanoparticles lead to higher instability of nanofluids. Alloui et al. [38] considered the shallow cavity to study Rayleigh Bénard instability for nanofluids. They concluded that rate of heat transfer in nanofluids depends on the strength of convection and volume fraction of nanoparticles while the presence of nanoparticles increase the stability of the system. Thermal instability for a horizontal nanofluid layer was considered by Yadav et al. [39]. They found an expression of thermal Rayleigh number and observed that the temperature gradient delays the convective motions while volumetric fraction of nanoparticles and the ratio of the density of nanoparticles to that of base fluid have destabilizing impact on the layer. The joint behaviour of nano-effects (Brownian motion and thermophoresis) creates destabilizing effect and can reduce the values of critical Rayleigh number as compared to that of regular fluids.
